Juuso Välimäki
Lecture Notes on Mechanism Design
Spring 2016
Mechanism Design: Review of Basic Concepts
Single Agent
We start with a review of incentive compatibility in the simplest possible
setting: A Principal is contracting with a single Agent. At the point of
contracting, the agent possesses private information θ ∈ Θ that is valuable
for deciding an allocation. In general,we will let allocations to be denoted by
a ∈ A, and we shall specialize to settings with money etc. as the need arises.
The two parties have preferences over the allocations parametrized by the
information of the agent;
uP (a, θ), uA (a, θ).
Often, we will assume that a = (x, t), where x is a physical allocation
(e.g. the amount of trade between the two parties) and t is a transfer from
the agent to the principal. Often we also assume that the payoff functions
are quasi-linear:
uP (a, θ) = v P (x, θ) + t, uA (a, θ) = v A (x, θ) − t.
In Principal-Agent models, we assume that the Principal has all the bargaining power and the agent has autonomy over the use of her private information. Let F (θ) denote the prior distribution (of the Principal over the
types of the agent). We interpret the meaning of full bargaining to the principal as follows: The Principal asks the agent to send a message (or report)
m ∈ M from some set M of allowed messages. She also commits to an
allocation ϕ(m) conditional on the report:
ϕ : M → ∆(A)).
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A pair (M, ϕ) is called a mechanism. A reporting strategy for the agent in is
a function
σ : Θ → ∆(M ).
Since the agent has autonomy over her reports, she will maximize her own
utility over her reports: For an optimal reporting strategy σ ∗ ,
σ ∗ (θ) ∈ arg max uA (ϕ(m), θ)
m
for deterministic mechanisms, and
σ ∗ (θ) ∈ arg max Eϕ(m) uA (a, θ)
m
when ϕ(m) ∈ ∆(A).
In first year micro course, you saw that when deciding which allocation
rules, i.e. functions
ψ : Θ → ∆(A),
can be implemented (i.e. that can arise as equilibra for general mechanisms),
there is no loss of generality in restricting attention to the case where M = Θ,
and the agent reports her type truthfully: For all θ,
σ ∗ (θ) = θ.
We say that the mechanism is incentive compatible if truthful reporting is
optimal for the agent.
This is called the Revelation Principle. Hence from this point onwards,
we will discuss settings where messages are reported types, i.e. M = Θ. Such
games are called direct mechanisms. The main question is when the agent
will have the right incentives to report truthfully. In other words, how can
we characterize allocation rules ϕ(m) such that for all θ,
θ ∈ arg max uA (ϕ(m), θ),
m
and similarly for random mechanisms.
2
Revenue Equivalence
For the remainder in this part, we concentrate on the incentives of the
agent, and therefore we drop the superscripts in the payoff functions. The
key result for characterizing the (equilibrium) maximized payoff to the agent
is the envelope theorem. We follow here closely the treatment in Milgrom &
Segal (2002).
We consider a family of parametrized optimization problems where the
objective functions are given by {f (x, θ)}θ∈Θ . The choice variable x ∈ X,
where X can be left quite general. The parameters set is assumed to be an
interval of the real line: Θ = [θ, θ]. It is crucial for the results that we obtain
to have connected parameter sets.
Let
V (θ) = sup f (x, θ) ,
x∈X
∗
and X (θ) := arg maxx∈X f (x, θ) . A function x∗ (θ) is a selection from
X ∗ (θ) if for all θ, x∗ (θ) ∈ X ∗ (θ) .
Theorem 1 (Milgrom and Segal, 2002) Assume that
• f (x, ·) is absolutely continuous for all x ∈ X (and hence differentiable
for almost every θ).
• There exists an integrable b (θ) such that for all x ∈ X and almost every
θ,
|fθ (x, θ)| ≤ b (θ) .
• X ∗ (θ) ̸= ∅ for all θ.
Then V (θ) is absolutely continuous, and for any selection x∗ (θ) from
X ∗ (θ) , V ′ (θ) = fθ (x∗ (θ), θ) for almost every θ and therefore
∫
V (θ) = V (θ) +
θ
fθ (x∗ (θ), s) ds.
θ
Proof. i) V (θ) is absolutely continuous.
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′ |V (θ) − V (θ )| = sup f (x, θ) − sup f (x, θ )
x
x
′
≤ sup |f (x, θ) − f (x, θ′ )|
x
∫ θ
= sup fθ (x, s) ds
x
θ
∫ θ
≤
sup |fθ (x, s)| ds
∫
x
θ
θ
≤
b (θ) ds.
θ
Hence V (θ) is absolutely continuous and
∫
V (θ) = V (θ) +
θ
V ′ (θ) ds.
θ
ii) V ′ (θ) = fθ (x∗ (θ) , θ) for any selection x∗ (θ) from X ∗ (θ) for almost
all θ.
Fix any selection x∗ (θ) . Then by definition of V (θ) :
V (θ) = f (x∗ (θ) , θ) ≥ f (x∗ (θ′ ) , θ) ,
V (θ′ ) = f (x∗ (θ′ ) , θ′ ) ≥ f (x∗ (θ) , θ′ ) ,
and therefore:
V (θ) − V (θ′ ) ≤ f (x∗ (θ) , θ) − f (x∗ (θ) , θ′ ) .
Consider the dividing this inequality by (θ − θ′ ) and taking the limits as
θ′ ↑ θ and θ′ ↓ θ respectively, we get
V ′ (θ−) ≤ fθ (x∗ (θ) , θ) ,
V ′ (θ+) ≥ fθ (x∗ (θ) , θ) .
Hence if V ′ (θ) exists, we have
V ′ (θ) = fθ (x∗ (θ) , θ)
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whenever fθ (x∗ (θ) , θ) exists. Hence
∫
θ
V (θ) = V (θ) +
fθ (x∗ (θ), s) ds.
θ
Notice that a simple sufficient condition for the theorem is that f (x, θ)
be Lipschitz continuous in θ uniformly over x, i.e. there is a B < ∞ such
that for all x ∈ X,
|f (x, θ) − f (x, θ′ )| ≤ B |θ − θ′ | .
To see that some uniform bound on the slope of the f (x, ·) is needed, note
that even if each f (x, θ) is arbitrarily smooth in θ, supx f (x, θ) may fail to
be continuous and hence the theorem fails.
To see how this connects to the mechanism design problem, consider the
family of functions {u (ϕ (m) , θ)}θ∈Θ . Let
V (θ) = sup u (ϕ (m) , θ) .
m∈Θ
Incentive compatibility requires that θ ∈ m∗ (θ) for all θ. Hence we can
compute
∫ θ
V (θ) = V (θ) +
uθ (ϕ (s) , s) ds.
θ
If we have quasi-linear preferences and u (x, t, θ) = v (x, θ) − t, this simplifies
to
∫
θ
V (θ) = V (θ) +
vθ (x (s) , s) ds.
θ
By noting that V (θ) = v (x (θ) , θ) − t (θ) , we see that
∫
t (θ) = v (x (θ) , θ) − V (θ) −
θ
vθ (x (s) , s) ds.
θ
Hence apart from the additive constant V (θ) , the transfer from the agent in
any incentive compatible mechanism is uniquely pinned down by the physical
allocation function x (θ) . This is the Revenue Equivalence Theorem.
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To see that no such result is possible when the type sets are discrete, it
is sufficient to consider the problem of selling an indivisible good (i.e. x ∈
{0, 1}) to an agent with θ ∈ {0, 1}.Let v (x, θ) = xθ. Consider mechanisms
where the agent gets the good if and only if θ = 1. Then any (x (θ) , t (θ))
with x (θ) = θ, t (θ) = θp for p ∈ [0, 1] is incentive compatible and has the
same allocation function. The utility of θ = 0 is zero for all p, but the transfer
(and expected payoff) of θ = 1 depends on p. Hence Revenue Equivalence
Theorem fails.
Remark 1 This result generalizes to smoothly connected parameter sets Θ ⊂
Rk by considering parametrized single dimensional paths in the parameter set.
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Aside: Monotone Comparative Statics
You may recall from the first year courses some ideas of when parametrized
optimization problems have solutions that are monotonic in the parameter.
Given the prominence of these ideas in Mechanism Design problems, we
review some of the most basic concepts briefly here.
The simplest setting for such considerations is with a family of functions
{f (x, θ)}θ∈Θ , where x ∈ X is the choice variable for some X ⊂ R, and θ ∈ Θ
is a real-valued parameter for the problem: Θ ⊂ R.
The basic question here is to determine which properties of f ensure that
any selection from the set of optimal choices x(θ) ∈ X(θ) is a monotone
function. We focus on the case where x(θ) is increasing (or non-decreasing).
Definition 1 (Single Crossing) A function g : Θ → R is single crossing
′′
′
if for all θ > θ ,
′
′′
g(θ ) ≥ (>)0 =⇒ g(θ ) ≥ (>)0.
Definition 2 (Strictly Single Crossing) A function g (s) is strictly single crossing if for all θ′′ > θ′ ,
′′
g(θ′ ) ≥ 0 =⇒ g(θ ) > 0.
Definition 3 (Single Crossing Differences) A family of functions
{f (·, θ)}θ∈Θ has (strictly) single crossing differences if for all x′′ > x′ , the
function
δ (θ) := f (x′′ , θ) − f (x′ , θ)
is (strictly) single crossing.
Notice that these are ordinal properties. It is a good exercise to show
that if {f (·, θ)}θ∈Θ has single crossing difference and h (x, θ) is increasing in
x, then {g(·, θ)}θ∈Θ where g (x, θ) := f (h (x, θ) , θ) has also single crossing
differences.
Definition 4 (Strictly Increasing Differences) A family of functions
{f (·, θ)}θ∈Θ has strictly increasing differences (SID) if δ (θ) is strictly increasing. Alternatively, we say that f (x, θ) has SID.
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Remark 2 It is easy to see that if f (x, θ) has SID, then it also satisfies the
strict single crossing property, and if f is differentiable in θ, then fθ (x, θ) is
increasing in x.
Consider a family of sets Y (θ) parametrized by θ.
Definition 5 Y (θ) is increasing (in the strong set order) if for θ′ > θ,
x′ ∈ Y (θ′ ) and x ∈ Y (θ) ⇒ max{x′ , x} ∈ Y (θ′ ) and min{x′ , x} ∈ Y (θ) .
The main tool in monotone comparative statics analysis is the following
theorem due to Milgrom and Shannon in Milgrom & Shannon (1994).
Theorem 2 (Milgrom and Shannon) The family {f (·, θ)}θ∈Θ has single
crossing differences if and only if arg maxx∈Y f (x, θ) is increasing in θ for
all Y ⊆ X.
Hence if f (x, θ) has SID, the set of maximizers is increasing (in the strong
set order). The proof of this Theorem is left as an exercise.
I have found the lecture notes written by John Quah (available at
http://users.ox.ac.uk/∼ssfc0023/new teaching.html) to be extremely useful
on this topic (and the related topic of comparative statics in games).
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Characterizing Incentive Compatibility
In this subsection, we assume that the conditions for the Milgrom-Segal Envelope Theorem are satisfied. For example, we could have a B < ∞ such
that for all m,
|u (ϕ (m) , θ) − u (ϕ (m) , θ′ )| ≤ B |θ − θ′ | .
We start with the quasi-linear case where a direct mechanism can be
identified with (x (θ) , t (θ)) .
Lemma 1 If (x (θ) , t (θ)) is an incentive compatible mechanism, then the
equilibrium payoff to the agent satisfies condition (LIC):
∫ θ
vθ (x (s) , s) ds.
V (θ) = V (θ) +
θ
Proof. Milgrom-Segal Envelope Theorem.
Lemma 2 If (x (θ) , t (θ)) is an incentive compatible mechanism, and v (x, θ)
has strictly increasing differences, then the optimal allocation x(θ) satisfies
condition (M):
x (θ) is non-decreasing in θ.
Proof. Milgrom and Shannon.
Lemma 3 If conditions (M) and (LIC) hold and if v (x, θ) has strictly increasing differences, then (x (θ) , t (θ)) is incentive compatible.
Proof. Take any m, θ ∈ Θ. Then
v (x (θ) , θ) − t (θ) − (v (x (m) , θ) − t (m))
(
∫ θ
∫
=
vθ (x (s) , s) ds − v (x (m) , θ) − v (x (m) , m) +
θ
∫
θ
θ
(vθ (x (s) , s) − vθ (x (m) , s)) ds
=
m
≥ 0.
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)
m
vθ (x (s) , s) ds
The first equality uses (LIC), the second uses absolute continuity of
v (x, s) , and the last inequality uses monotonicity of x (θ) and SID of v.
Theorem 3 If v (x, θ) satisfies SID, then incentive compatibility is equivalent to (LIC) and (M).
Notice that the above does not really use the full power of SID. A slight
generalization of the sufficient conditions follows directly from he above proof.
Theorem 4 Suppose that (LIC) holds, and that for all θ, m ∈ Θ, we have
(vθ (x (θ) , θ) − vθ (x (m) , θ)) (θ − m) ≥ 0.
Then (x (θ) , t (θ)) is incentive compatible.
Remark 3 This Theorem is equally valid for cases where X is not a subset of
the real line. In a regulation problem, it could represent e.g. various outputs
of a multi-product monopolist. Each individual output xi (θ) for i ∈ {1, ..., k}
need not be monotonic in θ, but on average, monotonicity must hold. If
v (x, θ) is twice differentiable in this case, a sufficient condition for incentive
compatibility is
Σki=1 vθxi (x (θ) , θ) x′i (θ) ≥ 0.
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General Case: Rochet’s Theorem and Cyclical Monotonicity
A much more general result on the implementability of allocation rules in
quasilinear settings was proved in Rochet (1987). It is a result of implementability in dominant strategies and hence there is no real loss of generality considering the single agent case. An added bonus of the construction
by Rochet is that it handles general type sets (including finite and multidimensional cases, convex and discrete, etc.) all at the same time.
An allocation rule x is weakly monotone if for all θ and θ′ ,
v(θ, x(θ)) − v(θ′ , x(θ)) ≥ v(θ, x(θ′ )) − v(θ′ , x(θ′ )).
It is an easy exercise to show that the allocation rule in any incentive compatible mechanism is weakly monotone. Rochet’s contribution was to strengthen
weak monotonicity to what is known as cyclical monotonicity to get necessary and sufficient conditions for dominant strategy implementability. To
understand better the condition that follows, write the definition for weak
monotonicity as follows:
(v(θ′ , x(θ)) − v(θ, x(θ))) + (v(θ, x(θ′ )) − v(θ′ , x(θ′ ))) ≤ 0.
Notice that here we consider the changes in payoffs as first we change
the type of the agent keeping the allocation fixed and then again change the
type back, keeping the allocation fixed. Weak monotonicity implies that the
sums of payoff changes be nonpositive. Cyclical monotonicity extends this
idea to longer cycles of types. For a fixed k, consider θ1 , θ2 , ..., θk+1 with the
requirement that θ1 = θk+1 . An allocation rule x is cyclically monotone if
for all k and all θ1 , θ2 , ..., θk+1 such that θ1 = θk+1 , we have:
k
∑
v(θi+1 , xi ) − v(θi , xi ) ≤ 0,
i=1
i
where we let x = x(θi ).
Theorem 5 (Rochet’s Theorem) There is an incentive compatible mechanism (x, t) if and only in x is cyclically monotone.
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Proof. i) Necessity is again easy and based on summing together the incentive compatibility inequalities.
ii) Sufficiency is somewhat more delicate. Fix an arbitrary type θ0 ∈ Θ,
and denote by S(θ) the set of finite sequences in Θ with θ1 = θ0 and θk = θ.
Define
V (θ) =
sup
k
∑
v(θi+1 , xi ) − v(θi , xi ).
(θ1 ,...,θ k )∈S(θ) i=1
To show that V is a well-defined function, observe first that V (θ0 ) = 0.
By construction of V , we have
V (θ0 ) ≥ V (θ) + v(θ0 , x(θ)) − v(θ, x(θ)),
which implies that V is finite. Similarly, we have for all θ, θ′ :
V (θ) ≥ V (θ′ ) + v(θ, x(θ′ )) − v(θ′ , x(θ′ )).
Thus the proof is completed by setting:
t(θ) := v(θ, x(θ)) − V (θ).
Finally, a connection between the monotonicity results using SID and
cyclical monotonicity can be found as follows. Suppose that Θ ⊂ R and
X ⊂ R, and that vxθ (x, θ) ≥ 0 for all θ, x. Then it is a good exercise to
show that weak monotonicity is equivalent to incentive compatibility directly
from the definitions. Next, one shows that in the single-dimensional case,
weak monotonicity implies cyclical monotonicity. This can be done based on
induction on the length of the cycle.
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Many Agents
Set-up
A Mechanism Designer or Principal has the bargaining power and N agents
i ∈ {1, 2, ..., N } have private information. For each agent i , her type is
θi ∈ Θi and is known only to agent i. All agents and the principal share the
same common prior on the types given by the distribution function F (θ) ,
where θ = (θ1 , ..., θN ) ∈ Θ = ×N
i=1 Θi . An agent i of type θi has interim beliefs
F (θ−i |θi ) on the type vector θ−i of the other players.
There is a set of potential allocation A with a typical element a ∈ A. The
agents and the principal have payoff function (von Neumann - Morgenstern)
over the allocations and type vectors. Agent i has a payoff function ui (a, θ)
and the principal has a payoff function u0 (a, θ) .
The principal chooses message spaces Mi for each of the agents, and
commits to an allocation rule (possibly random) ϕ : M → ∆ (A) , where
M = ×N
i=1 Mi with a typical element m ∈ M. The agents announce their
messages simultaneously, in other words they engage in a Bayesian game
with (pure) strategies
σi : Θi → Mi ,
and payoffs (for pure allocation rules) given by ui (ϕ (mi , m−i ) , θi , θ−i ) . Again
we call the pair (M, ϕ) a mechanism.
We have now different possible solution concepts for the message game
between the agents: Dominant strategy equilibrium, Bayes-Nash equilibrium,
Ex-post equilibrium.
1. A vector σ = (σ1 , ..., σN ) is a dominant strategy equilibrium in pure
strategies if for all i, θi and for all θ−i , m−i ,
ui (ϕ (σi (θi ) , m−i ) , θi , θ−i ) ≥ ui (ϕ (mi , m−i ) , θi , θ−i ) for all mi .
2. A vector σ = (σ1 , ..., σN ) is an ex-post equilibrium in pure strategies if
for all i, θi and for all θ−i ,
ui (ϕ (σi (θi ) , σ−i (θ−i )) , θi , θ−i ) ≥ ui (ϕ (mi , σ−i (θ−i )) , θi , θ−i ) for all mi .
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3. A vector σ = (σ1 , ..., σN ) is an Bayes-Nash equilibrium in pure strategies if for all i, θi and for all mi ,
Eθi ui (ϕ (σi (θi ) , σ−i (θ−i )) , θi , θ−i ) ≥ Eθi ui (ϕ (mi , σ−i (θ−i )) , θi , θ−i )
Clearly a dominant strategy equilibrium is an ex-post equilibrium and
an ex-post equilibrium is a Bayes-Nash equilibrium. For all three equilibrium concepts, one can prove that the revelation principle holds: For any
mechanism (M, ϕ) and an equilibrium σ in the Bayesian game induced by
(M, ϕ) , there is another mechanism (M ′ , ϕ′ ) such that truthful reporting, i.e.
σi (θi ) = θi for all i, θi is an equilibrium in the game induced by (M ′ , ϕ′ ) and
the outcomes in the two games coincide, i.e. ϕ (σ (θ)) = ϕ′ (θ) .The formal
proof for the three equilibrium concepts is left as an exercise.
Notions of Incentive Compatibility
We define here the corresponding three notions for incentive compatibility.
A direct mechanism is said to be incentive compatible if truthful reporting
is optimal for every type of every agent. Since what follows in these lectures
is in the quasilinear setting, I write direct mechanism as (x (θ) , t (θ)) where
x denotes again a physical allocation and t denotes a vector of transfers. We
write ui (x, t, θ) = vi (x, θ) − ti for i ∈ {1, ..., N } and u0 (x, t, θ) = v0 (x, θ) + t0 .
1. A direct mechanism (x (m) , t (m)) is dominant strategy incentive compatible if for all i, θi , θ−i , m−i , we have:
θi ∈ arg max vi (x(mi , m−i ), θi , θ−i ) − ti (mi , m−i ),
mi ∈Θi
i.e. truthtelling is optimal for each type of each player regardless of
what the type and announcement of the other players.
2. The mechanism (x(m), t (m)) is ex post incentive compatible if for all
i, θi , θ−i , we have
θi ∈ arg max vi (x(mi , θ−i ), θi , θ−i ) − ti (mi , θ−i ),
mi ∈Θi
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i.e. truthtelling is optimal for all θi given that m−i = θ−i . In other
words, as long as the other players are truthful, their type is not important for a players’ incentives for truthful reporting. Notice also that
for private values environments, the two notions coincide in the static
case in the sense that for each ex post incentive comaptible mechanism,
thee esists a dominant strategy incentive compatible mechanism that
implements the same allocation. This claim is not true in dynamic
settings even with independent private information.
3. The mechanism (x(m), t (m)) is Bayes-Nash incentive compatible if for
all i, θi , we have
θi ∈ arg max Eθ−i [vi (θi , θ−i , q(mi , θ−i )) − ti (mi , θ−i ) |θi ] ,
mi ∈Θi
i.e. truthtelling is optimal given that m−i = θ−i and given the distribution F (θ−i |θi ) for the types of other players.
Clearly, a dominant strategy incentive compatible mechanism is also ex
post incentive compatible and an ex post IC mechanism is Bayes-Nash IC.
Individual Rationality
Sometimes it makes sense to consider settings where the agents have an option of option gout from the mechanism. In general, the payoff from this
option could depend on the type, but for these notes, I will assume that this
outside option value is constant on type and normalized to 0. There are
different notions of participation constraints depending on when the participation decision is taken. We consider here incentive compatible mechanisms
and assume truthful reports by the other players.
1. A mechanism (x (·) , t (·)) is ex-post individually rational if for all i and
all θ,
vi (x (θ) , θ) − t (θ) ≥ 0.
2. A mechanism (x (·) , t (·)) is interim individually rational if for all i and
all θi ,
Eθ−i [vi (x (θ) , θ) − t (θ) |θi |] ≥ 0.
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3. A mechanism (x (·) , t (·)) is ex-ante individually rational if for all i and
all θi ,
Eθ [vi (x (θ) , θ) − t (θ)] ≥ 0.
Clearly an ex-post IR mechanism is interim IR and ex-ante IR. For the
ex-post IR, it is sometimes useful to consider also the case where the other
players are not truthful and then one would have the requirement that for
all i and all m−i , θ,
vi (x (θi , m−i ) , θ) − t (θi , m−i ) ≥ 0.
Bayesian and Dominant Strategy Mechanisms
How much more restrictive is the requirement of dominant strategy incentive
compatibility in comparison to Bayes-Nash incentive compatibility? The
answer here turns out to depend on the class of models as well as the notion
of similarity between mechanisms.
The issue is basically the following: For dominant strategy incentive compatibility, the appropriate monotonicity requirement (recall last lecture) on
x (m) is that for all i, and θi , vi (x (mi , m−i ) , θi ) be monotonic in mi . For
Bayes-Nash incentive compatibility, the requirement is that
Eθ−i vi (x (mi , θ−i ) , θi )
be monotonic in mi for all i and all θi .
It is easy to come up with examples of allocation rules x (mi , m−i ) that
are monotonic in mi in expectation over θ−i under truthful announcements
but that are not monotonic for all m−i . If vi (x, θi ) = θi x, then a Bayesian IC
mechanism (x (·) , t (·)) that implements x(θ) exists even though no dominant
strategy mechanisms implementing this allocation rule x (·) exist. Hence the
set of DIC mechanisms is in this sense a strict subset of the set of BIC
mechanisms.
It is therefore quite remarkable that for the linear case where vi (x, θi ) =
θi vi (x) , there is another sense in which the two classes are equivalent. Suppose one changes attention to expected equilibrium payoffs Vi (θi ) . Say that
a mechanism γ = (x (·) , t (·)) implements a payoff assignment V γ (θ) :=
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(V1γ (θ1 ) , ..., VNγ (θN )) if Viγ (θi ) is the expected payoff of player i with type θi
in γ. Let ΓD denote the set of dominant strategy IC mechanisms and let ΓB
denote the Bayesian IC mechanisms. Define
V D = {V γ (θ) γ ∈ ΓD }, and V B = {V γ (θ) γ ∈ ΓB }.
Since ΓD ⊂ ΓB , we have V D ⊂ V B . For the case of linear valuations,
Gershkov, Goeree, Kushnir, Moldovanu & Shi (2013) show that V D = V B .
The paper is written for the case of finite X and Θ, but it is clear that the
construction generalized to infinite X and Θ. Hence for linear environments,
any Bayesian incentive compatible equilibrium payoff vectors is also a payoff vector for some dominant strategy mechanism. The geometric approach
proposed in Goeree and Kushnir: ”A Geometric Approach to Mechanism
Design” also touches this issue and the approach is remarkably simple and
yields many of the existing results in mechanism design in a straightforward
manner.
Efficient Mechanisms
In this subsection, we address the incentive compatibility of efficient decision
rules in models with private values and quasi-linear payoffs.
Definition 6 The model has private values if for all i ∈ {0, 1, ..., N },
′
′
vi (x, θi , θ−i ) = vi (x, θi , θ−i
) for all θ−i , θ−i
∈ Θ−i .
∑
We start with the simplest setting where v0 = 0, and t0 = N
i=1 ti . In
other words, the mechanism designer collects the payments made by the
agents. With this specification, Pareto-efficiency reduces to surplus maximization, and efficiency is unambiguous.
Let
N
N
∑
∑
x∗ (θ) ∈ arg max
vi (x, θ) = arg max
vi (x, θi ),
x∈X
x∈X
i=1
i=1
where the last equality follows from the private values assumption.
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Consider the direct mechanism with choice rule x∗ (m) and with transfer
function
∑
t∗i (m) = −
vj (x∗ (m), mj ) + τi (m−i ).
(1)
j̸=i
It is easy to see that for an arbitrary announcement m−i by the other players,
mi = θi is the optimal report for an arbitrary function τi . The mechanism
(x∗ (θ) , t∗ (θ)) as above is called a Vickrey-Clarke-Groves mechanism (VCG
mechanism). Green and Laffont show in Green & Laffont (1979) that if there
are no restrictions on the domain of preferences for the players, then VCG
mechanisms are the only mechanisms that make truthful revelation a dominant strategy for the efficient transfer rule x∗ (θ) . Basically this observation
is a direct consequence of the Revenue Equivalence Theorem (separately for
each announcement m−i by the other agents), and therefore the set of parameters Θ must be sufficiently regular for the uniqueness result (e,g, a convex
subset of Rk will do). As equation (1) makes clear, the transfers are pinned
down by the efficient allocation rule x∗ (·) up to the constant τi (m−i ) .
A particularly useful VCG mechanism is the pivot mechanism or the
externality mechanism, where
∑
vj (x, mj ).
τi (m−i ) = max
x∈X
j̸=i
If we denote the maximized surplus for a coalition S of players by
∑
WS (θ) = max
vi (x, θi ),
x∈X
i∈S
then we see that the utility in the pivotal mechanism going to player i is
given by
max
x∈X
N
∑
i=1
vi (x, θ)−max
x∈X
∑
vj (θj , q) = W{1,...,N } (θ)−W{1,...,N }\{i} (θ) := Mi (θ) ,
j̸=i
i.e. in the pivotal mechanism, each player gets her marginal contribution to
the social welfare as her payoff.
Notice that since (x∗ (·) , t∗ (·)) is dominant strategy incentive compatible, correlation in the agents’ types makes no difference for the incentive
compatibility of the mechanism. It works for all private values settings with
quasi-linear payoffs.
18
Budget Balance
An unfortunate feature of the VCG mechanism is that it is not budget balanced (BB) between the agents. In some problems this is not problematic:
the mechanism designer may be a real seller that does not know the valuations of potential buyers. In other economic situations, positive sums
of transfers from the agents represent wasted money, and negative sums of
transfers feature financing originating outside the model.
We say that the mechanism (x (·) , t (·)) satisfies ex-post budget balance if
for all θ,
N
∑
ti (θ) = 0.
i=1
It satisfies ex-ante budget balance if
Eθ
N
∑
ti (θ) = 0.
i=1
A less demanding requirement for mechanisms would be that they have no
deficits (again either in the ex-post and ex-ante senses), i.e. Σi ti ≥ 0.
For an environment with independent types, we define the AGV or expected externality mechanism by x∗ (m), and transfers
∑
vj (x∗ (mi , m−i ) , mj ) + τi (m−i ) .
ti (m) = −Eθ−i
j̸=i
It is again easy to see that mi = θi is an optimal announcement in a Bayesian
equilibrium (not a dominant strategy though, why?).
Let
∑
Ei (mi ) = Eθ−i
vj (x∗ (mi , θ−i ) , θj ),
j̸=i
∑
and set
Ej (mj )
.
N −1
Then the mechanism is by construction ex-post BB.
Notice also that this mechanism does not work for correlated private
values. The reason is essentially the same as for not having the dominant
j̸=i
τi (m−i ) =
19
strategy property. It is a useful exercise to show that the AGV mechanism
is not a VCG mechanism. It is also a good exercise to think about the
interim expected utilities of the agents in this mechanism. Suppose that each
agent has the option of not participating in the mechanism (after learning
her type) and getting zero payoff. Will all types of agents participate in the
AGV mechanism? More on this in the next subsection. You may also wonder
about ex ante participation decisions. If the agents choose their participation
before learning their type, will they participate?
It is possible to design variants with similarities to the AGV mechanism
that work also in the correlated values setting. The main objection to VCG
mechanisms is that they often run a deficit, i.e. the sum of transfers is
negative. It is a useful exercise to examine the properties of the following
(
)
mechanism x∗ (·) , b
t (·) , where
[
]
∑
∑
b
vj (x∗ (m), mj ) + Eθi
ti (·) = −
vj (x∗ (θi , θ−i ), θj ) |θ−i .
j
j̸=i
Is this a VCG -mechanism? What can be said about the sum of the transfers.
What can be said about incentives to participate interim (i.e. after learning
θi ) and ex ante (prior to learning θi )?
BB in BNE
The previous chapter raises the question of what more can be done with BNE
as the solution concept since this is less restrictive that dominant strategy.
As long as the types are independent, we know from the characterization
of incentive compatibility that all BNE implementing the efficient decision
rule have the same expected payoffs apart from the utility of the lowest type
player. Hence the expected payments etc. can be calculated using VCG
mechanisms.
IC, BB and Participation
Assume now that the type sets of the agents are intervals on the real line
Θi = [θi , θi ]. Assume also that vi (x, θi ) satisfies strictly increasing differences.
20
Then we know from the revenue equivalence theorem that in any VCG mechanism, type θi of player i gets the lowest expected equilibrium payoff. If the
outside option for each i is to get payoff 0, then interim Individual Rationality
is satisfied for all players in an incentive if and only if Vi (θi ) ≥ 0.
Amongst interim individually rational VCG mechanisms, the following
transfer payment rule collects the largest expected payment from the agents:
tPi (θ) = W (θi , θ−i ) − W−i (θ).
We call this the revenue pivotal mechanism. It is immediately clear that with
these transfers, the lowest type of each agent has a zero expected payoff.
Hence a very useful observation is that if the VCG mechanism with payment
function tP runs a deficit, then there is no efficient IR mechanism that satisfies
budget balance.
It is slightly less trivial (and not so important) to show that if (x∗ (·) , tP (·))
results in a surplus and the types are independent, then there exists another
efficient, IR mechanism that satisfies budget balance. The construction uses
both the pivotal mechanism and the AGV mechanism and is left as an exercise.
We discuss next the issue of individual rationality or participation in a
particular mechanism. To fix ideas, we consider a game of public goods
provision. A project can be completed at cost c > 0 or alternatively it
may remain uncompleted. at zero cost. Therefore x ∈ {0, 1}. Assume that
vi (0, θ) = 0 for all i and vi (1, θ) = θi for all i, and θi . Then the efficient
decision rule is to have
{
1 if Σi mi ≥ c.
∗
x (m) =
0 otherwise.
Consider the pivot mechanism for this problem:
(
)
ti (m) = θi x∗ θi , m−i + (x∗ (m) − x∗ (θi , m−i ))(c − Σj̸=i mj )
As a very easy exercise, verify that all players have a dominant strategy
to report mi = θi . Suppose next that all player have the opportunity to opt
out from the mechanism and get a utility of 0. Then it is immediate from
the revenue equivalence and the transfer function that the largest transfers
21
consistent with participation by all types are given by the pivot mechanism
transfers. If we understand budget balance to mean that the sum of transfers
covers the cost c, then we can simply evaluate the expected payments in the
pivot mechanism and ask whether the cost c is covered. It is easy to show
that cost is covered only in cases where N θ > c or c < N θ. In all other cases,
there is a deficit. The computation of this is left as an exercise.
The same line of argument can be used in e.g. the bilateral trade model.
Here a buyer has a valuation v ∈ [v, v], and a seller has a cost of producing the
good given by c ∈ [c, c]. The cost and the valuation are private information
to the seller and the buyer, respectively. Assume that v > c. The buyer’s
expected utility from the good is xv − t, where x is the probability of trade,
The seller’s gross utility is t − xc. Construct the pivot mechanism. Argue
that it has the largest transfers, and show that the pivot mechanism transfers
result in a deficit.
Linear Environments
Assume next the simple linear form of preferences:
ui (x, t, θ) = θi vi (x) − ti ,
where vi is an increasing function of the physical allocation x. We also as[
]
sume that for all i, θi ∈ θi , θi and that the distribution admits a well defined
density function with full support. Furthermore, we assume that for i ̸= j,
θi ⊥ θj . We look for a characterization of the truthful equilibria of direct
mechanisms. Denote an arbitrary announcement by player i by mi , and denote by Ti (mi , θi ) the expected transfer given true type θi and announcement
mi . Notice that by our assumption of independence of types, we can write
Ti (mi , θi ) = Ti (mi ) .
Similarly, let
Xi (mi ) = Eθ−i [vi (x (mi , θ−i ))] .
The key here is that agent i′ s expected payment conditional on report mi is
independent of θi by the independence of the valuations. This is clearly not
22
the case when types are correlated and this is the main reason why revenue
equivalence fails in that setting.
Because of the linearity of the utilities, we may now write the expected
payoff from announcing mi while of type θi .
Eθ−i [ui (x (mi , θ−i ) , θ)] = θi Xi (mi ) − Ti (mi ) ,
Define also the following functions for i = 1, ..., N :
Vbi (θi , mi ) = θi Xi (mi ) − Ti (mi ) .
Now we are in a position to characterize the outcomes that are achievable in
Bayesian Nash equilibria of mechanism design problems. Let
Vi := Vbi (θi , θi ) .
Since the payoff function θi Xi satisfies strictly increasing differences for all
i, we have from the results on single agent implementability the following
theorem.
Proposition 1 (Bayesian incentive compatibility of outcomes)
The mechanism (x (·) , t (·)) is Bayesian incentive compatible if and only
if for all i ∈ {1, ..., N },
1. Xi (mi ) is nondecreasing.
∫θ
2. Vi (θi ) = Vi (θi ) + θ i Xi (s) ds for all θi .
i
Optimal Myerson Auctions
We continue with the IPV case of auctions.This case was covered in Myerson (1981). There are N bidders, each with a type θi drawn according to
the distribution function Fi (θi ). We assume that each Fi has a continuous
density function fi and that the types are statistically independent. For each
allocation decision x = (x1 , ..., xN ) , we have the payoff of bidder i as
ui (θ, q, t) = θi xi − ti .
23
We also require that x ∈ RN
+ and
∑
xi ≤ 1. Let
∫
∏
Xi (θi ) = Eθ−i xi (θi , θ−i ) =
xi (θi , θ−i )
fj (θj ) dθ−i ,
i
Θ−i
j̸=i
∫
and
Ti (θi ) = Eθ−i ti (θi , θ−i ) =
ti (θi , θ−i )
∏
Θ−i
fj (θj ) dθ−i .
j̸=i
The expected payoff for bidder i of type θi from announcing type mi is
Vbi (θi , mi ) = θi Xi (mi ) − Ti (mi ) .
By the previous lecture, an allocation Xi (θi ) is incentive compatible if and
only if we have:
1. Vi (θi ) = Vbi (θi , θi ) = Vi (θi ) +
∫ θi
θi
Xi (s) ds.
2. Xi′ (θi ) non-decreasing.
The expected payment going to the auctioneer is
∫ ∑
∑
∏
E
Ti (θi ) =
ti (θi , θ−i )
fi (θi ) dθ.
Θ
i
i
i
Using the envelope result, we have
∫
∑
∑∫
E
Ti (θi ) =
[θi Xi (θi ) − Vi (θi ) −
i
i
θi
Xi (si ) ds]fi (θi ) dθi .
θi
Θi
If the outside option for the bidders is fixed at 0, it is optimal to set Vi (θi ) = 0.
Integrating by parts or interchanging the order of integration we get:
∫
∫ ∫ θi
(1 − Fi (si ))
Xi (si ) dsi fi (θi ) dθi =
Xi (si )
fi (si ) dsi .
fi (si )
Θi
Θi θi
Therefore using the definition of Xi (θi ) , we have:
∫ ∑
∑
(1 − Fi (θi ))
E
Ti (θi ) =
]f (θ) dθ.
[θi xi (θ) − xi (θ)
f
(θ
)
i
i
Θ
i
i
24
The seller’s revenue is maximized if x (θ) is chosen to maximize the above
integral. A good strategy is to first ignore the constraint that Xi (θi ) must
be non-decreasing in θi. Since the expected revenue is linear in x, x∗ (θ) maximizes the integral for each θ if:
{
i (θi ))
i (θi ))
i (θi ))
1 if θi − (1−F
≥ θj − (1−F
and θi − (1−F
≥ 0.
∗
fi (θi )
fj (θj )
fi (θi )
xi (θ) =
0 otherwise.
We call
θi −
1 − Fi (θi )
fi (θi )
the virtual type of player i. If the virtual surplus is nondecreasing in θi , then
we have arrived at a full solution of the problem. Otherwise we must take the
constraint Xi′ (θi ) ≥ 0 into account and use optimal control theory to find the
optimal feasible solution. A sufficient condition for having a nondecreasing
virtual surplus is that
fi (θi )
1 − Fi (θi )
be nondecreasing in θi .In this case, the optimal auction thus allocates the
good to the buyer with the maximal virtual type if this type is non-negative
for some bidder and leaves the good unallocated otherwise.
Comments on Optimal Auction
1. Symmetric Fi : optimal reserve prices that are independent of N.
2. Asymmetric Fi : optimal auction favors the disadvantaged bidder.
3. Notice the trade-off between efficiency and information rent
4. A good exercise is to use the revenue equivalence theorem to conclude
the following: the expected revenue from an optimal auction with n
symmetric bidders is dominated by the expected revenue from a second price auction without reserve prices with n + 1 bidders from the
same distribution. Hence the amount that can be gained by optimally
tailoring the auction rules (reservation prices) is quite limited.
25
Aside: Symmetric Equilibria Through Revenue Equivalence
FPA
From revenue equivalence theorem, we have:
∫
θi
V (θi ) = Vi (θi ) +
Xi (s) ds.
θi
If the first price auction has an equilibrium with strictly increasing strategies
bi (θi ), then we have:
Xi (θi ) = F (θ)N −1 ,
and
Vi (θi ) = 0.
On the other hand, in first price auction, the payoff to bidder i is:
(θi − bi (θi ))Xi (θi ),
or
∫
θi
(θi − bi (θi ))Xi (θi ) =
Xi (s) ds,
θi
or
(θi − bi (θi ))F (θ)
N −1
∫
θi
=
F (θ)N −1 ds,
θi
or
∫ θi
bi (θi ) = θi −
θi
F (θ)N −1 ds
.
F (θ)N −1
Since the model satisfies strictly increasing differences, the payoffs are
calculated using the Revenue Equivalence Theorem and since the allocation
rule is monotonic, we know tha the direct mechanism associated with the
first-price auction rules is globally incentive compatible. Hence the only
remaining issue that must be checked is that there are no profitable bids
outside the range of equilibrium bids in the mechanism. This is left as a
simple exercise.
War of attrition
In this auction format the highest bidder wins the object, all losers bid
their own bid and the winner pays the highest losing bid. This can be seen
26
as a model where the bidders compete for a price by expending resources (as
expressed in their bids). The contest ends when all but one bidders give up.
At that point, the winner collect the prize.
We look for a symmetric equilibrium in increasing strategies. Any such
equilibrium allocates the prize efficiently (i.e. the highest bidder is the bidder
with the highest valuation. Hence the allocation is the same as in the second
price auction. By revenue equivalence theorem, the expected payment must
also coincide with the expected payment SPA. Denote the bid of bidder i in
the symmetric equilibrium of the war of attrition by bW (θi ) and the expected
payment by ΠW . We have:
Π
W
= (1−F
N −1
∫
θi
W
W
(θi ))b (θi )+
b (z)dF
N −1
∫
(z) =
0
θi
zdF N −1 (z) = ΠSP A .
0
Since this holds for all θi we may differentiate to get:
dbW (θi )
(1 − F N −1 (θi )) = θi dF N −1 (θi ),
dθi
and therefore
∫
W
b (θi ) =
0
θi
θi dF N −1 (θi )
.
1 − F N −1 (θi )
Again, the final step is to prove that no bids outside the range of equilibrium bids yields a profitable deviation. This is again a imple matter to
check.
We turn next to the issue of efficient mechanisms. We shall see how the
revenue equivalence theorem turns out to be useful for deducing properties
of social surplus maximizing allocation schemes.
27
Mechanism Design with Correlated Types
Until now, types θi have been assumed
1. Independent across i,
2. Private values:
′
vi (x, θi , θ−i ) = vi (x, θi , θ−i
)
′
.
for all i, θi , θ−i , θ−i
In the next two sections, we shall relax these assumptions. For more
details on correlated values, see Cremer & McLean (1985) and Cremer &
McLean (1988).
First example
A single indivisible good is auctioned between two bidders with three possible
types (valuations) θi ∈ {1, 2, 3} . We assume private values so that the payoff
in a direct mechanism (x (·) , t (·)) for bidder i is simply
θi xi (mi , mj ) − ti (mi , mj ) ,
where xi (m) is the probability of getting the good if the reported type vector
is m. Since there is only a single good, we require that for all m,
x1 (m) + x2 (m) ≤ 1.
Assume that the types are distributed according to the following table:
θ1 \θ2
1
2
3
1
2
3
1
6
1
12
1
12
1
12
1
6
1
12
1
12
1
12
1
6
The types are positively correlated, but not perfectly correlated. Notice
that the type spaces are discrete, and therefore we could not use revenue
equivalence theorem even in the case with independent valuations. Consider
28
a second price auction with no reserve price. This auction is efficient in the
sense that it allocates the good to a bidder with the highest value. It realizes
total expected surplus of 14
and an expected revenue of 10
to the seller. By
6
6
setting a reserve price of 2, the revenue would increase to 11
at the cost of
6
13
social surplus going down to 6 .
With correlated values, the seller can improve on this by quite a bit.
Write the conditional probability vectors p1 (θ2 |θ1 ) that player 1 of type θ1
assigns to the types of player 2 in matrix form:

P1 (θ2 |θ1 ) = 
1
2
1
4
1
4
1
4
1
2
1
4
1
4
1
4
1
2

.
So for example p23 is the probability that player 1 of type θ1 = 2 assigns to
the event {θ2 = 3}.
Consider now the second price auction without reserve prices. The bidders’ interim expected payoffs Vi (θi ) are as follows: Vi (1) = 0, Vi (2) = 41 ,
and Vi (3) = 34 .Notice that the matrix Pi has full rank. Therefore there exists
a vector t = (t1 , t2 , t3 ) such that


1
2
1
4
1
4
1
4
1
2
1
4
1
4
1
4
1
2

  
t1
0
  t2  =  1  .
4
3
t3
4
Consider next the second price auction with the modified payment scheme:
t′i (mi , mj ) = tSP
i (mi , mj ) − tmj .
Since the second part in the payment is independent of mi , it is a dominant
strategy to announce truthfully in (x∗ (m) , t′ (m)). Hence the outcome is
efficient. By construction, all bidders earn a zero interim expected payoff.
Hence the seller extracts full surplus in revenues. The next subsection shows
that the insight in the above result generalizes to most mechanism design
settings.
29
Cremer-McLean result
We analyze the case where all players’ types are drawn from finite sets. Let
Ni be the number of types for player i. Let π(θ) denote the prior distribution
on the vectors of types. The following restriction on the conditional beliefs
of the players on each others’ types turns out to be important.
Denote by π(θ−i | θi ) the posterior probability that type θi of player i
assigns to the types of the other players. These probability vectors satisfy
the Cremer-McLean condition if there does not exist a type θi (for any player)
i −1
such that :
and a vector λ ∈ RN
+
∑
λ(θi′ )π(θ−i | θi′ ) = π (θ−i |θi ) .
θi′ ̸=θi
Hence the C-M condition is a weakening of linear independence of the
conditional probability vectors. It is violated if two or more of these are
identical. Thus independent types case where all the vectors are identical is
ruled out. In fact, the insight contained in the example is very general and
takes the form of the following Theorem.
Theorem 6 Consider a mechanism where the allocation function is x(θ)
and the transfer function is given by t(θ), and suppose that Cremer-McLean
condition above is satisfied. Then there is another incentive compatible mechanism given by x(θ) and τ (θ) where the interim expected payment for each
type θi of each player i is the same under the two mechanisms.
How to prove this? Recall Separating hyperplane theorem:
Theorem 7 Let A and B be two closed and convex subsets of Rn such that
A ∩ B = ∅. Then there is a vector b and a constant c such that for all
x ∈ A, b · x > c,
and for all
y ∈ B, b · y ≤ c.
30
To apply this theorem, consider
i −1
A = {λ(θi′ ) · π(θ−i | θi′ ) | λ(θi′ ) ∈ RN
}
+
∏
In other words, A is the positive cone generated in R+ j̸=i
π(θ−i | θi′ )
Let
Nj
by the vectors
B = π(θ−i | θi ).
Cremer-McLean condition states that A∩B = ∅. Therefore, we can apply
∏ the
Nj
separating hyperplane theorem to conclude the existence of b(θi ) ∈ R+ j̸=i
and c(θi ) ∈ R such that .
b(θi ) · π(θ−i | θi′ ) > c(θi )
for all π(θ−i | θi′ ), and
b(θi ) · π(θ−i | θi ) = c(θi ).
Interpret b(θi ) − c(θi ) as the vector of transfers (one for each type vector
of other bidders) that player i has to pay. Consider the game where players
are only asked to report their types and then the transfers above are enacted.
The above construction shows that for all i and all θi , the players have strict
incentives to report their true types. Multiplying the transfers by a large
enough positive constant overwhelms any other incentives that might arise
from the original incentive incompatible mechanism. Since this reporting
game involves no interim expected transfers under truthful reporting, we
conclude the proof of the theorem.
Similar ideas can be used to construct efficient mechanisms that satisfy
additional constraints such as balanced budget. Recent work by David Rahman available at https://sites.google.com/site/davidrahmanwork/ is at the
cutting edge of this literature.
Perhaps it is best to note some more or less obvious problems with the
proposed mechanisms. If types are almost independent, then the required
transfers are very large stretching the limits of the assumed quasi-linearity
and hence risk-neutrality. These mechanisms are prone to collusion by the
31
agents. They rely on the common knowledge of fine details about the players’
probability assessments of each others’ types.
Finally,the result depends on the assumption that for each belief that
player i has on the other players’ types, there is a single payoff type. From
a Harsanyi type space point of view, this seems peculiar. There is a lively
debate on whether this assumption should be thought to hold in general or
not. In the literature, you will find papers discussing whether BDP (Beliefs
Determine Payoffs) should be regarded a generic property in the universal
type set or not. Additional reading on this can be found in Neeman (2004)
and Heifetz & Neeman (2006)
32
Mechanism Design with Common Values
In this lecture, we drop the assumption of private values. The simplest
example of such a case is the Wallet Game. An indivisible object is auctioned
to one of two potential bidders. Bidder i has type θi . The bidders have pure
common values:
vi = θ1 + θ2
Assume for simplicity that the θi are i.i.d. random variables with a continuous density on [0, 1].We start wit the analysis of the first price auction
by finding a symmetric equilibrium in strictly increasing strategies. In such
equilibria, player i wins if θi > θj . Given the continuous distribution of types,
ties can be ignored. Therefore the expected payoff to player i with type θi
from bidding b(θi′ ) is:
U (θi , θi′ ) = (θi + E[θj | θi′ > θj ] − b(θi′ ))F (θi′ ).
In equilibrium, we must have:
θi′ = θi .
Let
U (θi , θi ) ≡ V (θi ).
It is easy to see that the conditions for the Milgrom& Segal envelope theorem
are satisfied, and we have:
V ′ (θi ) = F (θi ).
Since V (0) = 0, we also have
∫
θi
F (s)ds = (θi + E[θj | θi > θj ] − b(θi ))F (θi ).
0
Therefore we can solve:
∫ θi
b(θi ) = (θi + E[θj | θi > θj ]) −
33
0
F (s)ds
.
F (θi )
Consider for example a uniform distribution. Then we have:
1
1
b(θi ) = θi + θi − θi = θi .
2
2
We could perform the same steps for second price auction to get to the
stage:
∫
θi
F (s)ds = (θi + E[θj − b(θj ) | θi > θj ])F (θi ).
0
Since both sides are equal for all values of θi , the derivatives of the two sides
of the equality must coincide. Therefore:
F (θi ) = F (θi ) + θi f (θi ) + θi f (θi ) − b(θi )f (θi ).
Solving for b(θi ) gives:
b(θi ) = 2θi .
Is there a direct argument for the result? You want to win if and only if
the payoff exceeds your payment. If you bid twice your value and win, you
pay 2 min[θi , θj ] whereas your value is
θi + θj > 2 min[θi , θj ].
Therefore you win if and only if your value exceeds your payment.
Notice that this equilibrium is not in dominant strategies (why?) as opposed to the private values case. Nevertheless the equilibrium seems to have
nice properties. First of all, beliefs about the distribution of your opponent’s
value do not appear at all in the calculation of the optimal strategy. Furthermore, no bidder has an incentive to change their strategy even if they learn
the other bidders’ types. In other words bidding according to b(θi ) = 2θi .is
an ex-post equilibrium.
In an ex post equilibrium, players do not care about each others’ types
at all conditional that they report truthfully. It turns out that the property
of having a BNE that is independent of the prior beliefs is closely connected
to the equilibrium being an ex post equilibrium.
34
Theorem 8 A mechanism (x (·) , t (·)) is ex-post incentive compatible if and
only if it is Bayes-Nash incentive compatible for all prior beliefs on the type
sets.
The literature on robust mechanism design revisits this theorem in the
context of Harsanyi types and higher order beliefs.
Implementability of Efficient Decision Rules
In the quasi-linear setting with private values, VCG-mechanisms implement
the efficient decision rules in dominant strategies. Is this possible with common values. In the Wallet Game example, we saw that dominant strategy
equilibria will fail to exist in general for common values settings. What about
Bayesian or ex post equilibria? The following example shows that it is not
possible to implement the efficient decision rule in common values settings.
Recall that in the case of independent types, a necessary condition for implementability is that the allocation rule be monotonic in each player’s type.
Efficient rules are clearly monotonic in this sense in the private values case
since an increase in the type of one agent leaves the values of other agents
unchanged.
Failure of Monotonicity
How to allocate a single indivisible object between two players? Only player
1 has private information θ1 ∈ [0, 2]. Both players’ payoffs depend on this
information:
v1 = θ1 , v2 = 2θ1 − 1.
Efficient allocation of the object (letting x(θ1 ) be the probability of assigning the good to 1):
θ1 > 1 ⇒ x(θ1 ) = 0,
θ1 ≤ 1 ⇒ x(θ1 ) = 1.
Is there a transfer function that could make this allocation incentive compatible? At most two transfers can be used (transfers must be constant
35
conditional on the allocation). (Why?) Let t(0) be the transfer payment
when 1 gets the object, and t(1) the payment that 1 makes when 2 gets the
object. Then we have from incentive compatibility:
θ1 − t(0) ≥ −t(1) for all θ1 ≤ 1,
and
−t(1) ≥ θ1 − t(0) for all θ1 > 1.
These inequalities are seen to be incompatible by e.g. summing them up.
The efficient decision rule fails to be implementable in this example because
monotonicity of the decision rule fails. The reason for this failure can be
traced to the fact that player 1’s type has a larger effect on the valuation of
player 2 than on the valuation of player 1 herself.
Single Crossing
Consider next that case where all players i ∈ 1, ..., N have private information
represented by a single dimensional type θi . Assume for now that the types
are independent across players, and they are distributed on an interval [θ, θ].
Write the payoff functions of the players as:
b θ) − t(θ).
b
vi (x(θ),
We say that the efficient decision rule x∗ (θ) is monotonic if for all θ−i and
all x
b, the set Si (b
x) = {θi | x∗ (θi , θ−i ) = x
b} is connected.
Notice that the notion of monotonicity we have here is quite general. Since
the sets Si (b
x) are connected for all x
b, we can define an order (for each θ−i )
on X by the requirement that for all θi ≤ θi′ we have x∗ (θi , θ−i ) ≤ x∗ (θi′ , θ−i ).
Recall that the model satisfies strictly increasing differences if for all i
and all θ−i , and x ≤ x′ in the order defined above, we have:
∂vi (x′ , θ)
∂vi (x, θ)
≥
∂θi
∂θi
Consider for example the case of a single object auction. Suppose the bidders
get zero utility whenever they are not allocated the good and utility ui (θ)
36
when they are allocated the good. Then monotonicity and SID are satisfied
if
∂vi (θ)
∂vj (θ)
≥
for all j.
∂θi
∂θi
Monotonicity and single crossing are sufficient for ex post implementation of
the efficient decision rule.
Theorem 9 Suppose that the efficient decision rule x∗ (θ) is monotonic and
the payoffs have SID. Then there exists a schedule of transfers t(θ) such that
x∗ (θ) is an ex post incentive compatible mechanism. We call such mechanisms
generalized VCG- mechanisms.
So how exactly did our simple example violate the conditions of this theorem? Write the valuation of bidder i as θi x∗ (θ), where x∗ is the probability
of giving the object to i. Then the order given by the efficient allocation says
that x∗ and θi are inversely ordered. But with this ordering, single crossing
of ui fails. (It holds for the ’natural ordering’ on the real line for x, θi ).
What does the generalized VCG mechanism look like in simple applications?
• Is this the second price auction? Ascending auction?
• Why different transfers?
• Is it possible to construct pivotal mechanisms where the equilibrium
payoff of each player coincides with her contribution to the societal
welfare?
For more on efficient interdependent value auctions with single-dimensional
types, see Dasgupta & Maskin (2000).
Common Values and Multi-Dimensional Signals
We conclude this section with the simplest example of multidimensional signals. There are two agents, only agent 1 has private information. Her type
is two-dimensional: (θ, s) ∈ [0, 1] × [0, 1] . Payoffs are given by:
v1 (x, θ, s) = (3θ + 2s) x,
v2 (x, θ, s) = (1 + 2θ + s) (1 − x) ,
37
where 0 ≤ x ≤ 1 is the probability with which agent 1 gets the object.
In any efficient allocation, x∗ (θ, s) = 1 if θ + s > 1 and x∗ (θ, s) = 0 if
θ+s < 1. Notice that x∗ (θ, s) is increasing in each component and v1 satisfies
strictly increasing differences. Furthermore the single crossing condition from
the previous subsection is satisfied. It is still easy to show that there are no
incentive compatible mechanisms that implement the efficient allocation rule.
This is left as an exercise. For more on this topic, see Jehiel & Moldovanu
(2001).
38
Aside: Existence of Monotone Equilibria in
Pure Strategies in Bayesian Games
The following exposition follows essentially Athey (2001). A game of incomplete information is given by (I, T, A, u, f ) , where I = {1, ..., I} is the set
of players, T = ×Ii=1 Ti is the set of types, A = ×Ii=1 Ai is the set of actions,
u = ×Ii=1 ui is the set of Bernoulli utilities where ui : A × T → R, and f is
a prior probability on T. we assume throughout that actions and types are
real numbers with the natural order on them.
Definition 7 The Single Crossing Condition (SCC) is satisfied if for each
i ∈ I and a−i (t−i ) where each aj (tj ) for j ̸= i is nondecreasing, we have
∫
Ui (ai , ti ; a−i (t−i )) := ui (ai , a−i (t−i ) ; ti , t−i ) f (t−i |ti ) dt−i
satisfies the single crossing property in (ai , ti ) .
For a discussion of the sufficient conditions on the primitives (in terms of
the ui (·) and f (·)) to guarantee (SCC), see Quah & Strulovici (2012).
Consider first the case of games where A is a finite set with cardinality
M on the real line. and Ti is an interval of the real line [ti , ti ]. Then nondecreasing strategies may be represented by a vector x of cutoff points. We say
that xi is consistent with ai (ti ) if xi gives the cutoffs in the pure strategy
ai (ti ) . Notice that each vector has at most M elements. If there are fewer
cutoffs, we can take the remaining cutoffs to be at ti . Then each xi ∈ TiM .
Let x = (x1 , ..., xI ).
We denote by aBR
(ti , x) the best responses of player i to a−i (t−i ) that
i
are consistent with x−i . Define for each i,
{y ∈
TiM
Γi (x) =
∃ai (ti ) consistent with y such that for all ti , ai (ti ) ∈ aBR
(ti , x) }.
i
Let Γ = (Γ1 , ..., ΓI ) . If we can find a fixed point for the correspondence Γ,
then we’re done.
(ti , x)
The key claim is that Γi (x) (and therefore also Γ (x)) is convex if aBR
i
is increasing in the strong set order in ti . Showing this is a good exercise. We
39
can apply Kakutani’s fixed point theorem if we show that Γ(x) has a closed
graph. But this is a standard continuity/compactness argument (actually
even easier since A is finite) so it is also left as an exercise.
Theorem 10 Under SCC, the game of incomplete information has a monotone (nondecreasing) pure strategy equilibrium.
So for finite games, all the work is in establishing SCC. Consider next
games with a continuous action space.
Theorem 11 Assume that Ai = [ai , ai ] and that ui (a, t) is continuous in a
and for all finite subsets A′ ⊂ A, the finite game has a pure strategy Nash
equilibrium in nondecreasing strategies. Then the continuous action game
also has a monotone PSNE
The proof is an application of Helly’s selection theorem that shows that
the every sequence of nondecreasing bounded functions has a (pointwise)
convergent subsequence. For this you start with an approximating sequence
of finite games with An and let An → A in the sense that for all ε > 0 and
for all a ∈ A, there is a n (ε, a) such that for all n > n (ε, a) , the distance
d (An , a) < ε, where
d (An , a) = min
d (a′ , a) .
′
a ∈An
Next you consider a sequence of monotone PSNE a∗n (t) of {An } which exist
by assumption. Use Helly’s selection theorem to conclude the existence of
a convergent subsequence, and the limit along this subsequence must be a
monotone PSNE of the continuous game. The details are left as an exercise.
In the paper by Athey, you will find all kinds of sufficient conditions for
SCC. These cover, in particular, the case of affiliated auctions that we discuss
next.
40
Comparing Fixed Mechanisms: Auctions with
Affiliated Values
A General Model of Affiliation
This section is based on Milgrom & Weber (1982) and the book ’Auction
Theory’ by Vijay Krishna. Consider a vector of random variables X =
(X1 , ..., XN ) on [0, 1]. Assume that these variables have a strictly positive
joint density function f (x) that is symmetric in its xi components. Symmetry
means simply that the joint distribution is tha same for all permutatios of
the indices of the random variables. For any two vectors x′ , x in RN , let
x′ ∧ x denote their componentwise minimum and x′ ∨ x their componentwise
maximum. We say that the random variables are affiliated if for all (x′ , x)
we have:
f (x′ ∧ x) f (x′ ∨ x) ≥ f (x′ ) f (x) .
It is easy to see that if f is twice differentiable, then affiliation is equivalent
to
∂ 2 ln f (x)
≥ 0 for all i, j.
∂xi ∂xj
Furthermore, it is easy to see that for two non-negative affiliated functions
h (x) and g (x) , the function given by
f (x) = g (x) h (x)
is also affiliated.
Let Yk denote the k th order statistic amongst (X2 , ...XN ) for k ∈ {1, ..., N −
1}. Then by the symmetry of f in x, we have
f (x1 , y1 , ..., yN −1 ) = (n − 1)!f (x1 , y1 , ..., yN −1 ) 1y1 ≥y2 ≥...≥yN −1 ,
where 1A is the indicator function on A. By the observation on the products of affiliated functions, we see that (x1 , y1 , ..., yN −1 ) are affiliated if the
components of x are affiliated.
Finally, for any non-decreasing function u (x) , The conditional expectation
E[u (X) |x1 , ..., xN ]
41
is increasing in xi for all i.
Conditionally I.I.D. Model (Mineral Rights)
The most commonly used variant of the affiliated model is one where the
joint density of the signals and an underlying true value V is written as
f (x, v) = f (v)
N
∏
f (xi |v ),
i=1
where f (v) is the marginal distribution of V, and the conditionals f (xi |v )
across the players’ signals and satisfy the monotone likelihood ratio property.
Definition 8 f (xi |v ) satisfies the monotone likelihood ratio property if for
all x′i ≥ xi and v ′ ≥ v,
f (x′i |v ′ )
f (x′i |v )
≥
.
f (xi |v ′ )
f (xi |v )
It is an easy exercise to verify that the model is then affiliated according
to the above definition. Therefore
E[V |x1 , ..., xN ] and E[V |x1 , y1 , ..., yN −1 ]
are increasing in all arguments.
For the next section, it is useful to define the following function.
v (x, y) = E[V |X1 = x, Y1 = y ].
Make also the following normalization:
v (0, 0) = 0.
By the properties of affiliated random variables, v is increasing in both arguments.
Second-Price Auctions
Proposition 2 Symmetric equilibrium strategies in a second-price auction
are given by:
β II (x) = v (x, x) .
42
Proof. Suppose that players other than 1 bid according to the proposed
equilibrium strategy. Bidder 1’s payoff from bidding b with signal x is
∫ β −1 (b)
Π (b, x) =
(v (x, y) − β (y)) g (y |x) dy
0
∫
β −1 (b)
(v (x, y) − v (y, y)) g (y |x ) dy,
=
0
where g (y |x ) is the density of Y1 conditional on X1 = x.
Since v is increasing in its first argument,
v (x, y) − v (y, y) ≥ 0
if and only if
x ≥ y.
Thus Π (b, x) is maximized by choosing b so that β −1 (b) = x or b = β(x).
Notice that the payment is constant in own bid conditional on the allocation (determined by other players’ bids).Therefore own bid only changes
the allocation in the auction. It is clearly optimal to bid in such a manner
that there is an expected surplus if and only if the auction is won. Notice
also that this equilibrium is an ex post equilibrium.
First-Price Auction
We give here only a heuristic derivation of the equilibrium bid function.
Let β I (x) be the proposed equilibrium bid function. Clearly, it does not pay
to bid more that β (1) or less than β (0) . Let Π (z, x) denote the expected
payoff from bidding β (z) when signal is x. Then
∫ z
Π (z, x) =
(v (x, y) − β (z)) g (y |x) dy
0
∫ z
=
v (x, y) g (y |x ) dy − β (z) G (z |x ) .
0
The first order condition for optimal choice of z is
(v (x, z) − β (z)) g (z |x ) − β ′ (z) G (z |x ) = 0.
If β (x) is an equilibrium, then it must be optimal to bid β (x) so that z = x
and we have:
g (x |x )
β ′ (x) = (v (x, x) − β (x))
.
G (x |x )
43
With boundary condition β (0) = v (0, 0) = 0, the above differential equation
characterizes the optimal bidding function. The explicit solution can be
written as:
∫ x
I
β (x) =
v (y, y) dL (y |x ) ,
0
where
( ∫
L (y |x ) = exp −
y
x
)
g (t |t )
dt .
G (t |t )
Revenue in English versus Second-Price Auction
To see that English auction always raises at least as much revenue as
second-price auction, recall the definition of v (x, y) from above. Write also
u (X1 , Y1 , ..., YN −1 ) = E[V |X1 , Y1 , ..., YN −1 ]
We have
v (y, y) = E[u (X1 , Y1 , ..., YN −1 ) |X1 = y, Y1 = y ]
= E[u (Y1 , Y1 , ..., YN −1 ) |X1 = y, Y1 = y ]
≤ E[u (Y1 , Y1 , ..., YN −1 ) |X1 = x, Y1 = y ],
where the last inequality follows from affiliation. Expected revenue in second
price auction is
E[β II (Y1 ) |X1 > Y1 ]
= E[v (Y1 , Y1 ) |X1 > Y1 ]
≤ E[E[u (Y1 , Y1 , ..., YN −1 ) |X1 = x, Y1 = y ] |X1 > Y1 ]
= E[u (Y1 , Y1 , ..., YN −1 ) |X1 > Y1 ].
In an English auction, Y2 , ..., YN −1 are known to the last remaining two bidders. Therefore their value for the object is:
E[u (Y1 , Y1 , ..., YN −1 ) |X1 > Y1 , Y2 = y2 , ..., YN −1 = yN −1 ],
and therefore the equilibrium bidding strategy for the last two remaining
bidders is given by
β 2 (x; y2 , ..., yN −1 ) = E[u (Y1 , Y1 , ..., YN −1 ) |X1 > Y1 , Y2 = y2 , ..., YN −1 = yN −1 ],
44
and thus
E[β 2 (Y1 , ..., YN −1 )] = E[u (Y1 , Y1 , ..., YN −1 ) |X1 > Y1 ],
and we have shown that
E[β II (Y1 ) |X1 > Y1 ] ≤ E[β 2 (Y1 , ..., YN −1 ) |X1 > Y1 ]
as claimed.
Linkage Principle
Denote by W I (z, x) the expected payment by the winner in first-price
auction if signal is x and the bid is z. Similarly, let the winning payment in
a second-price auction be W II (z, x) . Let Wik denote the partial derivative
of W k (z, x) with respect to the ith argument. Then
W I (z, x) = β I (z) and W II (z, x) = E[β II (Y1 ) |X1 = x, Y1 < z ].
Proposition 3 Consider two auctions A and B in which only the highest
bidder pays. Suppose that each has a symmetric increasing equilibrium such
that i) W2A (x, x) ≥ W2B (x, x) for all x and ii) W A (0, 0) = 0 = W B (0, 0) .
Then the expected revenue in A is at least as high as the expected revenue in
B.
Proof. In auction k ∈ {A, B}, the bidder chooses to bid β k (z) maximizes
∫ z
v (x, y) g (y |x ) dy − W k (z, x) G (z |x ) .
0
FOC and equilibrium require that
(
) g (x |x)
,
W1k (x, x) = v (x, x) − W k (x, x)
G (x |x)
where Wjk (x, x) denotes the partial derivative with respect to j th argument
in the expected winning price of auction k. Therefore,
(
) g (x |x )
W1A (x, x) − W1B (x, x) = − W A (x, x) − W B (x, x)
G (x |x )
45
Let
∆ (x) = W A (x, x) − W B (x, x) .
Then
∆′ (x) = W1A (x, x) − W1B (x, x) + W2A (x, x) − W2B (x, x)
g (x |x )
= −∆ (x)
+ W2A (x, x) − W2B (x, x) .
G (x |x )
If
∆ (x) ≤ 0, then ∆′ (x) ≥ 0.
Since
∆ (0) = 0,
by assumption, we have
∆ (x) ≥ 0 for all x.
Notice that this Proposition immediately implies that second-price auctions collect a higher expected revenue than first-price auctions (why?).
46
Information Aggregation and Large Auctions
Motivation
The theory of competitive markets rests on the idea of a price that serves
a dual role. On the one hand, it serves as a sufficient signal to equilibrate
demand and supply across various separate markets. On the other hand,
price is supposed to aggregate the total information in the economy to ensure efficient allocation of resources. Yet there is no generally accepted theory
explaining the emergence of prices. If prices are to reflect information in the
economy, they should be influenced by individual actions (maybe communication, maybe trades). The requirement of price taking, rules this possibility
out by assumption. Hence one has to wonder where the prices really come
from.
In these lectures, we adopt the point of view that maybe the Walrasian
outcome can be seen as the equilibrium of a large auction market where
participants engage in strategic interactions. Maybe if all traders are small
enough, we can safely ignore the distortions that arise from attempts to
manipulate prices. At the same time, it is clear that Myerson-Satterhwaite
theorem is working against us?
The first part of these lectures looks at markets with private values. The
informational role of prices is limited in this setting. The main question
is whether we can find an auction whose equilibrium price and allocation
converges to the Walrasian outcome as the number of participants grows.
The second part of these lectures addresses the informaitonal role of
prices. Can we find an auction environment that provides the foundations
for a fully revealing rational expectations equilibrium where the price aggregates all separately held information about the value of the objetcs for
saleThe proper setting for studying this latter question is within large common values auctions.
Private values auctions
We start with the simplest possible model. There are kn objects for sale
and n potential buyers. The valuations vi are drawn independently from a
47
common distribution F (vi ) for all potential bidders. We assume that F has
a continuous density function f .
Suppose now that the objects are allocated according to the following
rule: kn buyers with the highest bids receive the object and pay the price of
the (kn + 1)st highest bid. In case of a tie between the knth and the (kn + 1)st
bidder the object is randomly allocated between them.
It is an easy exercise to show that this is a VCG mechanism for the
allocation problem. Therefore there exists a dominant strategy equilibrium
where the bidders bid according to their valuation. I.e.
bi (vi ) = vi
∀i, vi .
Recall the concept of order statistic. The random variable Yn (kn ) denotes
the knth highest draw amongst n independent draws from the distribution F .
(We omit the dependence of Y on F for notational simplicity). The questions
about the convergence of the bids and prices in the auction can then be cast
simply in terms of the order statistics. We have
Pn = Yn (kn + 1)
in the auction with n bidders and k objects. Assume that F is concentrated on [0, 1]. Let
kn
κ = lim
.
n→∞ n
Then
Pn → F −1 (κ).
All limits of random variables in this lecture are understood in the sense
of convergence in probability.
I.e.
Xn → Yn ⇔
∀δ, ϵ > 0, ∃N such that n > N ⇒ P (| Xn − Yn |> ϵ) < δ.
The result follows as a simple application of the (weak) law of large numbers.
48
Hence we have for our very first simple auction the result that as the
number of bidders grows, the auction converges to a fixed price mechanism
with the Walrasian price.
Private value double auctions
The next step is to consider a model where the supply as well as demand is
determined endogenously. Therefore we start in a setting where all the kn
identical objects are initially held by potential sellers. We let the (private)
values of the sellers cj be i.i.d. according to a continuously differentiable
distribution G. Recall that in the case where k1 = n = 1, the MyersonSatterthwaite theorem tells us that efficient allocation is not possible under
balanced budgets. In a double auction, sellers receive all payments made
by the buyers and all players can secure their no-trade interim utility. The
relevant question is then whether this inefficiency persists and whether the
market price converges to the Walrasian price. For the remainder in this
section, we set kn = n and have an equal number of buyers and sellers in the
market.
Let’s start by analyzing the ’limit case’. Assume that there is a continuum
of buyers and sellers, and assume that the law of large numbers holds for this
case. Then the demand curve in the economy is given by:
D(p) = 1 − F (p).
Similarly, the supply curve is given by
S(p) = G(p).
A competitive equilibrium price p∗ satisfies:
1 − F (p∗ ) = G(p∗ ).
Clearly a unique equlibrium exists for our model. The aim in this subsection is to show that this equilibrium is the unique limit of large double
auctions as well. What is the issue here? In contrast to the Walrasian market
where the price is exogenously given, the price in a double auction market
49
must be formed based on the bids (or in direct mechanism terms types) of
the bidders. There will always be an incentive to manipulate the bid or the
report so as to make a gain in the market. The key for the efficiency limit
result that we get is an argument that shows that the probability that the
individual bid changes the equilibrium price is small. Therefore bidders behave almost as if they were faced with a fixed price and therefore the result
follows.
The simplest method for guaranteeing convergence to the walrasian outcome would be through the following approximately efficient fixed price mechanism. Calculate p∗ as above and ask the buyers and seller that are willing
to trade at that price to report at a market place. Buyers and sellers are
matched randomly so that all traders on the short side are matched, but the
traders on the long side of the market are matched with equal probability.
Within the matches, the trades are executed at price p∗ . Unmatched players
that reported at the market place leave with their initial endowments.
Claim 1 In this fixed price mechanism, sellers have a dominant strategy to
report at the market place if and only of their valuation is below p∗ and buyers
have a dominant strategy to report at the market place if and only if their
valuation exceeds p∗ . Furthermore, this mechanism is efficient in the limit
in the sense that for large markets, the expected share of realized gains from
trade to total gains from trade converges to 1 as n grows.
Proof. The payoff from not reporting to a seller is cj and for a buyer, it is
0. Reporting at the market place gives a higher payoff if matched and the
same if unmatched. This proves the first claim.
Let Bn denote the random number of buyers that report and Sn the
random number of sellers that report when the total number of sellers is n.
An upper bound for the efficiency loss is given by
| Bn − Sn | .
Efficiency loss per trader is then bounded from above by
| Bn − Sn |
,
n
50
and weak law of large numbers implies that
| B n − Sn |
→0
n
as n → 0.
A problem with the above mechanism is that it does not use information
well at all. The rationing stage is assumed to be independent of the true
valuations of the traders. This is essential to get the mechanism to have the
dominant strategy solution. It is clear that efficient rationing would do a
much better job, but this particular mechanism does not allow that. Hence
we move to discuss briefly a better mechanism for the private values double
auction setting.
We discuss next (n + 1)st price auctions. My exposition follows closely
Mark Satterthwaite and Steven Williams: ”The Rate of Convergence to Efficiency in the Buyers Bid Double Auction as the Market Becomes Large’,
Review of Economic Studies, (1989).
In these auctions, buyers submit bids b(vi ) and sellers submit offers s(cj ).
Put all bids and offers in an decreasing order: s1 ≤ s2 ≤ ... ≤ s2n . The price
that each buyer receiving the object pays is equal to the pn = sn+1 highest
bid. The allocation is organized as follows: all sellers that have submitted a
bid strictly below pn trade with a buyer whose bid is at least pn If there are
no ties at the lowest winning bid, the market clears, if there are ties and the
market does not clear, then market is cleared according to efficient rationing.
The payoffs are simply vi − pn for the buyers that trade, 0 for the buyers
that do not trade, pn for the sellers that trade, and cj for the sellers that do
not trade.
It is clear that bidding one’s valuation is not a dominant strategy for the
buyers. (Remember that the (n + 1)st bid might come from a buyer, and in
this case the seller’s bid would affect the transfer even for a fixed allocation.
For the sellers, bidding the value is still a dominant strategy.
We shall look for an equilibrium where the sellers bid according to their
dominant strategies, and the buyers use symmetric strategies. In other words,
we are looking for a pair of functions S(c), B(v) such that whenever other
players in the double auction follow these strategies, it is optimal for an
51
individual buyer to bid according to B(v) and for the seller to offer according
to S(c).
Proposition 4 If (S, B) is an equilibrium in the double auction, then the
function B has the following properties: (i) 0 < B(v) for all v ∈ (0, 1], (ii)
B(v) ≤ v for all v ∈ [0, 1), (iii) B(v) is strictly increasing on [0, 1] and
differentiable almost everywhere.
Proof. Note that buyer i trades with a strictly positive probability at price
b if she submits any bid b ∈ (0, 1). This implies i) and ii) in the theorem.
Write P (b; S, B) for the probability that an individual buyer trades if she
submits the bid b and other players follow their equilibrium strategies. By
standard arguments, incentive compatibility implies that:
P (B(v); S, B) ≥ P (B(v ′ ); S, B)
whenever v > v ′ . Since P (·; S, B) is increasing, we conclude that B(v) ≥
B(v ′ ).
We now show by contradiction that B cannot be constant over any interval with non-empty interior. Suppose that B(v) = b′ for all v in such an
interval I. By i) and ii), 0 < b′ < 1. We claim: P (b; S, B) is discontinuous at
b = b′ . This is true because the following events occur simultaneously with
positive probability: (i) each buyer’s reservation value is in I and therefore
all buyers bid b′ , (ii) at least one seller’s offer is less than b′ , and (iii) at least
one seller’s offer is greater than b′ . Note that (ii) and (iii) require that m ≥ 2.
Then price is b′ , the market fails to clear at this price. Therefore there is an
ϵ > 0 such that P (b′′ ; S, B) > P (b′ ; S, B) + ϵ for all b′′ > b′ .
The rise in the buyer’s expected payment ∆C(b′′ , b′ ) is bounded from
above by
b′′ (P (b′′ ; S, B) − P (b′ ; S, B)) + (b′′ − b′ ).
By ii), there is a v ′ such that B(v ′ ) = b′ < v ′ . We claim that this type v ′
can gain by an upwards deviation. Raising the bid to b′′ ∈ (b′ , v ′ ) induces a
change in payoff calculated as:
v ′ (P (b′′ ; S, B) − P (b′ ; S, B)) − ∆C(b′′ , b′ )
52
≥ (v ′ − b′′ )ϵ + (b′ − b′′ )
by the earlier results. Since b′′ can be chosen arbitrarily close to b′ , this
establishes the desired contradiction. All increasing functions are almost
everywhere differentiable.
We consider next what the equilibria look like and how close they are to
efficient. We start by defining probabilities of some events that turn out to
be important for the incentives. Let
)(
)
n−1 (
∑
n−1 n−1
Kn (b) =
G(b)n−1−i (1−G(b))i F (B −1 (b))i (1−F (B −1 (b)))n−1−i ,
i
n
−
i
i=0
where B −1 (b) is the type of the buyer whose equilibrium bid is b. The inverse
function exists by Proposition 1.
)
n−1 ( )(
∑
n n−2
Ln (b) =
G(b)n−i (1−G(b))i F (B −1 (b))i−1 (1−F (B −1 (b)))n−i−1 ,
i
i
−
1
i=1
)( )
n−1 (
∑
n−1 n
Mn (b) =
G(b)n−i (1−G(b))i F (B −1 (b))i (1−F (B −1 (b)))n−1−i .
i
i
i=0
Kn (b) is the probability that sn−1 < b < sn in a sample of n − 1 buyers
and sellers using equilibrium strategies. Ln (b) is the probability that sn−1 <
b < sn in a sample of n − 2 buyers and n sellers using equilibrium strategies.
Mn (b) is the probability that sn < b < sn+1 in a sample of n − 1 buyers and
n sellers using equilibrium strategies.
Consider a buyer with valuation v that raises her bid from b to b + ∆b.
The change in her expected utility is
[ng(b)Kn (b)∆b + (n − 1)
f (v)
Ln (b)∆b](v − b − ∆b) − Mn (b)∆b,
B ′ (v)
where v solves B(v) = b.
Taking ∆b → 0 and requiring no profitable deviations from the equilibrium bid gives:
1
B ′ (v)
=
Mn − (v − B(v))ng(B(v))Kn
.
(v − B(v))(n − 1)f (v)Ln
53
Since
1
B ′ (v)
> 0, we get
(v − B(v)) ≤
Mn
.
ng(B(v))Kn
Hence we have a strong convergence result if it can be shown that
Mn
= γ ∈ R.
n→∞ Kn
This intuitive result can be shown, but it is a semi-involved exercise in
combinatorics and therefore omitted here.
lim
54
Information Aggregation in Common Values Auctions
Fixed Supply
Here we follow Ilan Kremer: ’Information Aggregation in Common Value
Auctions’, Ecnometrica (2006) and Wolfgang Pesendorfer and Jeroen Swinkels:
’The Loser’s Curse and Information Aggregation in Common Value Auctions,Econometrica,1997.
Motivation
In private values markets, price reflect the aggregate demand and supply
conditions, and measures the equilibrium level of scarcity. With common
values, price also reflects aggregate information on the true value of goods
for sale. Financial markets are a prime example. In general equilibrium
theory with private information, the most common solution concept is fully
revealing rational expectations equilibrium (REE). The price reflects all the
private information regardless of how individual traders believe. Hence my
information is reflected in the price even if I trade the same quantity at all
my information sets. (This is off the equilibrium path in those models, but
as always events off the equilibrium path give incentives for the equilibrium
path.)
Grossman & Stiglitz have an early paper demonstrating the problematic
nature of REE. They construct an example trade reflects private information
if price is uninformative and trade is independent of information if price is
revealing. Could large auctions and in particular large double auctions be
used as models of almost competitive trade? These game-theoretic models
account well for the informational effects of actions. We shall address the
following questions: i) Does the price converge the Walrasian price in common values models? ii) Does the price aggregate all the information that the
bidders have?
We shall see that many properties of the equilibrium outcome can be
inferred from certain order statistics of the vector of bidders’ signals.
55
Setup
• True value of the object: random variable V n with a realization vn ∈
[0, 1] in an auction with n bidders.
• Bidder i ∈ {1, ..., n} receives a private signal Si with a realization
si ∈ [0, 1] correlated with Vn .
• Tthe vector of signals S = (S1 , ..., Sn )
• The lth order statistic of S is denoted by Yn (l).
• In the sample of all bidders but i, it is denoted by Yni (l).
• Price (random) in the auction with n bidders is Pn .
• Bidding strategy in the auction with n bidders is bn (si )
Assumption 1 Prior distribution on Vn is independent of n and denoted by
F (vn ). We assume that the densities f (vn ) of the value and the conditionals
density on signals f (si | vn ) are well defined. Let V := E(Vn ).
Assumption 2 Monotone likelihood ratio property (MLRP) holds : For all
i, s′i > si , and vn′ > v n ,
f (s′i | vn′ )
f (s′i | vn )
.
>
f (si | vn′ )
f (si | vn )
Furthermore we make the assumption that the information content in
individual signals is bounded.
Assumption 3 There is a constant κ > 0 such that
∀si , vn ,
1
> f (si | vn ) > κ.
κ
We concentrate on the kn unit auction where kn identical units of a good
are sold. The kn highest bidders receive the object and each pays the (kn +1)st
highest bid.
56
Milgrom & Weber (1982) show that this auction has a symmetric equilibrium in strictly increasing strategies where an individual bid is calculated
as:
bn (si ) = E(V | Yn (k) = Yn (k + 1) = si ).
The proof that this is actually an equilibrium is left as an exercise. (Recall
the reasoning for the wallet game).
Analysis
Definition 9 An auction is competitive if E(Pn ) → V .
Definition 10 A competitive auction aggregates information if Pn −Vn → 0
in probability.
Definition 11 An auction is informative if E(Vn | Pn ) − Vn → 0 in probability.
Call the bidder whose bid is taken as the price a pivotal bidder. Denote
the information available to the pivotal bidder by Fn . The following result
is an immediate consequence of the characterization of the equilibrium bid
function.
Lemma 4
Pn ≤ E(Vn | Fn ).
The following two theorems establish the informational properties of competitive auctions in general.
Theorem 12 If the auction is competitive, then Pn − E(Vn | Fn ) → 0.
Theorem 13 If the auction is competitive, then Pn − E(Vn | Pn ) → 0.
Hence a competitive auction aggregates information if and only if it is
informative.
57
In order to get concrete results for the kn unit auction, we assume that
the si are i.i.d. conditional on the true v and consider the case where kn = αn
for some α ∈ (0, 1). Then weak law of large numbers implies that
−1
Yn (αn + 1) → Fs|v
(1 − α)
−1
in probability. Under MLRP, Fs|v
(1 − α) is strictly increasing when viewed
as a function of v, we conclude that
E(Vn | Yn (kn + 1)) → Vn
in probability.
Therefore we know that if the auction is competitive, then it aggregates
information. This final step is in the following theorem.
Theorem 14 The kn unit auction is competitive.
Proof. Let
b∗n (s) ≡ E(Vn | Yn (kn + 1) = s),
let
hn (s) = b∗n (s) − bn (s),
and
b∗∗
n (s) ≡ E(Vn | Yn (kn ) = s).
By MLRP,
∗
b∗∗
n (s) < bn (s) < bn (s).
∗
∗
By weak law of large numbers, b∗n (s) → v ∗ and b∗∗
n (s) → v , where v solves
F (s | v) = 1 − α.
Therefore hn (s) → 0 and the proof follows from the fact that
∫ 1
V − E(Pn ) =
hn (s)fYn (kn ) (s)ds
0
and that the limiting distribution of Yn (kn ) is given by F −1 (1 − α | v) and
therefore fYn (kn ) (s) converges to a finite limit.
58
Notes on Further Literature
’Convergence to Efficiency in a Simple Market with Incomplete Information’
by Aldo Rustichini, Mark A. Satterthwaite, Steven R. Williams in Econometrica, Vol. 62, No. 5 (Sep., 1994), pp. 1041-1063 obtains a slightly more
efficient mechanism than the one discussed above, but the setup is essentially
the same, and the gain is marginal in terms of economic understanding.
’Efficiency of Large Double Auctions’, Econometrica, vol. 74(1) Jan 2006,
pages 47-92 by Martin Cripps and Jeroen Swinkels gives a very thorough
investigation of equilibria in large double auctions with private values as
the number of participants increases. The paper relaxes the assumptions
of symmetry, independence and unit demands and supplies from the earlier
literature. Main conclusions are:
• Equilibrium exists. (For non-trivial equilibria, this result is based on
earlier work by Jackson and Swinkels).
• Equilibria either involve no trade (trivial equilibria) or are almost efficient (non-trivial equilibria).
• Efficiency losses are asymptotically proportional to
1
n2
as n → ∞.
’Existence of equilibrium in large double auctions’ by Drew Fudenberg ,
Markus Mobius , Adam Szeidl in JET, 2007 introduces a nice perturbation
technique for proving equilibrium existence in setting similar to Cripps and
Swinkels (2007).
Lecture Notes on Dynamic Mechanism Design
The Dynamic Pivot Mechanism
In this lecture, we generalize the idea of the pivot mechanism to dynamic
environments with private information. We design an intertemporal sequence
of transfer payments which allows each agent to receive her flow marginal
contribution in every period. In other words, after each history, the expected
transfer that each agent must pay coincides with the dynamic externality
cost that she imposes on the other agents. In consequence, each agent is
willing to truthfully report her information in every period.
59
Model
Uncertainty We consider an environment with private and independent
values in a discrete-time, infinite-horizon model. The flow utility of agent
i ∈ {1, 2, ..., I} in period t ∈ N is determined by the current allocation
xt ∈ x, the current monetary transfer pi,t ∈ R and a state variable θi,t ∈ Θi .
The von Neumann Morgenstern utility function ui of agent i is quasi-linear
in the monetary transfer:
ui (xt , pi,t , θi,t ) ≜ vi (xt , θi,t ) − pi,t .
The current allocation xt ∈ X is an element of a finite set x of possible
allocations. The state of the world θi,t for agent i is a general Markov process
on the state space Θi . The aggregate state of the system is given by the vector
θt = (θ1,t , ..., θI,t ) with Θ = ×Ii=1 Θi .
There is a common prior Fi (θi,0 ) regarding the initial type θi,0 of each
agent i. The current state θi,t and the current action xt define a probability
distribution for next period state variables θi,t+1 on Θi . We assume that this
distribution can be represented by a stochastic kernel
Fi (θi,t+1 |θi,t , xt ) .
The utility functions ui (·) and the probability transition functions Fi are
common knowledge at t = 0. The common prior Fi (θi,0 ) and the stochastic
kernels Fi (θi,t+1 |θi,t , xt ) are assumed to be independent across agents. At
the beginning of each period t, each agent i observes θi,t privately. At the
end of each period, an action xt ∈ X is chosen and payoffs for period t are
realized. The asymmetric information is therefore generated by the private
observation of θi,t in each period t. We observe that by the independence
of the priors and the stochastic kernels across i, the information of agent i,
θi,t+1 , does not depend on θj,t for j ̸= i. The expected absolute value of the
flow payoff of every agent is assumed to be bounded for every allocation plan
x′ : Θ → x :
∫
|vi (x′ (θ′ ) , θi′ )| dFi (θi |θi , xt ) < K,
for some K < ∞ for all i, x and θi . We could accommodate even more
general payoff structures where the periodic utility of i depends on the entire
sequence of allocations xt := (x0 , ..., xt ) .
60
The nature of the state space Θ will depend on the application at hand.
At this point, we stress that the formulation accommodates the possibility of
random arrival or departure of the agents. The arrival or departure of agent
i can be represented by an inactive state θi , where vi (xt , θi ) = 0 for all xt ∈ x
and a random time τ at which agent i privately observes her transition in or
out of the inactive state.
Social Efficiency All agents discount the future with a common discount
factor δ, 0 < δ < 1. The socially efficient policy is obtained by maximizing
the expected discounted sum of valuations. Given the Markovian structure,
the socially optimal program starting in period t at state θt can be written
as:
{∞
}
I
∑
∑
W (θt ) ≜ max
E
δ s−t
vi (xs , θi,s ) .
∞
{xs }s=t
s=t
i=1
Alternatively, we can represent the social program in its recursive form:
{ I
}
∑
W (θt ) = max E
vi (xt , θi,t ) + δEW (θt+1 ) .
xt
i=1
The socially efficient policy is denoted by x∗ = {x∗t }∞
t=0 . The social externality cost of agent i is determined by the social value in the absence of agent
i:
}
{∞
∑
∑
vj (xs , θj,s ) .
W−i (θt ) ≜ max
E
δ s−t
∞
{xs }s=t
s=t
j̸=i
{
}∞
The efficient policy when agent i is excluded is denoted by x∗−i = x∗−i,t t=0 .
The marginal contribution Mi (θt ) of agent i at signal θt is defined by:
Mi (θt ) ≜ W (θt ) − W−i (θt ) .
(2)
The marginal contribution of agent i is the change in the social value due to
the addition of agent i.
Mechanism and Equilibrium We focus attention on direct mechanisms
which truthfully implement the socially efficient policy x∗ . A dynamic direct mechanism asks every agent i to report her state θi,t in every period
61
t. The report ri,t ∈ Θi may or may not be truthful. The public history
in period t is a sequence of reports and allocations until period t − 1, or
ht = (r0 , x0 , r1 , x1 , ...rt−1 , xt−1 ), where each rs = (r1,s , ..., rI,s ) is a report
profile of the I agents. The set of possible public histories in period t is
denoted by Ht . The sequence of reports by the agents is part of the public
history and we assume that the past reports of each agent are observable
to all the agents. The private history of agent i in period t consists of the
public history and the sequence of private observations until period t, or
hi,t = (θi,0 , r0 , x0 , θi,1 , r1 , x1 , ..., θi,t−1 , rt−1 , xt−1 , θi,t ) . The set of possible private histories in period t is denoted by Hi,t . An (efficient) dynamic direct
mechanism is represented by a family of allocations and monetary transfers,
∗
I
{x∗t , pt }∞
t=0 : xt : Θ → ∆ (x) , and pt : Ht × Θ → R . With the focus on efficient mechanisms, the allocation x∗t depends only on the current (reported)
state rt ∈ Θ. In contrast, the transfer pt may depend on the entire history
of reports and actions.
A (pure) reporting strategy for agent i in period t is a mapping from the
private history into the state space: ri,t : Hi,t → Θi . For a given mechanism,
the expected payoff of agent i from reporting strategy ri = {ri,t }∞
t=0 given
∞
the strategies r−i = {r−i,t }t=0 is:
E
∞
∑
δ t [vi (x∗ (rt ) , θi,t ) − pi (ht , rt )] .
t=0
Given the mechanism {x∗t , pt }∞
t=0 and the reporting strategies r−i , the optimal
strategy of bidder i can be stated recursively:
Vi (hi,t ) = max E {vi (x∗t (ri,t , r−i,t ) , θi,t ) − pi (ht , ri,t , r−i,t ) + δVi (hi,t+1 )} .
ri,t ∈Θi
The value function Vi (hi,t ) represents the continuation value of agent i given
the current private history hi,t . We say that a dynamic direct mechanism is
interim incentive compatible, if for every agent and every history, truthtelling
is a best response given that all other agents report truthfully. We say
that the dynamic direct mechanism is periodic ex-post incentive compatible
if truthtelling is a best response regardless of the history and the current
state of the other agents.
62
In the dynamic context, the notion of ex-post incentive compatibility
is qualified by periodic as it is ex-post with respect to all signals received
in period t, but not ex-post with respect to signals arriving after period t.
The periodic qualification arises in the dynamic environment as agent i may
receive information at some later time s > t such that in retrospect she would
wish to change the allocation choice in t and hence her report in t.
Finally we define the interim participation constraints of each agent. After each history ht , each agent i may opt out (permanently) from the mechanism. The value of the outside option is denoted Oi (hi,t ) and it is defined by
the payoffs that agent i receives if the planner pursues the efficient policy x∗−i
for the remaining agents. The periodic participation constraint requires that
each agent’s equilibrium payoff after each history weakly exceeds Oi (hi,t ).
For the remainder of the text we shall say that a mechanism is ex-post incentive compatible and individually rational if it satisfies the periodic incentive
and participation constraints.
Scheduling: An Example
We consider the problem of allocating time to use a central facility among
competing agents. Each agent has a private valuation for the completion of a
task which requires the use of the central facility. The facility has a capacity
constraint and can only complete one task per period. The cost of delaying
any task is given by the discount rate δ < 1. The agents are competing for
the right to use the facility at the earliest available time. The objective of
the social planner is to sequence the tasks over time so as to maximize the
sum of the discounted utilities.
An allocation policy in this setting is a sequence of choices xt ∈ {0, 1, ..., I},
where xt denotes the bidder chosen in period t. We allow for xt = 0 and hence
the possibility that no bidder is selected in t. Each agent has only one task
to complete and the value θi,0 ∈ R+ of the task is constant over time and
independent of the realization time (except for discounting). The transition
function is then given by:
{
0, if xt = i;
θi,t+1 =
θi,t if xt ̸= i.
63
The utility function vi (xt , θi,t ) for bidder i from the efficient policy x∗ is:
{
θi,t , if
xt = i;
vi (xt , θi,t ) =
0, if otherwise.
For this scheduling model, we find the marginal contribution of each
agent and derive the associated dynamic pivot mechanism. We determine
the marginal contribution of bidder i by comparing the value of the social
program with and without i. With the constant valuations over time for all
i, the optimal policy is clearly given by assigning in every period the alternative j with the highest remaining valuation. To simplify notation, we define
vi ≜ θi,0 . We may assume without loss of generality (after relabelling) that
the valuations vi are ordered with respect to the index i: v1 ≥ · · · ≥ vI ≥ 0.
The descending order of valuations allows us to identify each task i with the
period i + 1 in which it is completed along the efficient path and so:
W (θ0 ) =
I
∑
δ t−1 vt .
(3)
t=1
Similarly, the efficient program in the absence of task i assigns the tasks in
ascending order, but necessarily skips task i in the assignment process:
W−i (θ0 ) =
i−1
∑
δ
t−1
vt +
t=1
I−1
∑
δ t−1 vt+1 .
(4)
t=i
By comparing the social program with and without i, (3) and (4), respectively, we find that the assignments for agents j < i remain unchanged after
i is removed, but that each agent j > i is allocated the slot one period earlier
than in the presence of i. The marginal contribution of i from the point of
view of period 0 is:
Mi (θ0 ) = W (θ0 ) − W−i (θ0 ) =
I
∑
δ t−1 (vt − vt+1 ) .
t=i
The social externality cost of agent i is established in a straightforward manner. At time t = i − 1, agent i completes her task and hence realizes a gross
value of vi . The immediate opportunity cost is the next highest valuation
64
vi+1 . But this overstates the externality, because in the presence of i all less
valuable tasks are realized one period later. The externality cost of agent i is
hence equal to the next valuable task vi+1 minus the improvement in future
allocations due to the delay of all tasks by one period:
pi (θt ) = vi+1 −
I
∑
δ
t−i
(vt − vt+1 ) = (1 − δ)
t=i+1
I
∑
δ t−i vt+1 .
(5)
t=i
Since we have by construction vt − vt+1 ≥ 0, the externality cost of agent
i in the intertemporal framework is less than in the corresponding single
allocation problem where it would be vi+1 . Consequently, the final expression
states that the externality of agent i is the cost of delay imposed on the
remaining and less valuable tasks.
Scheduling: Further Considerations
We show next that the efficient allocation can be realized through a bidding mechanism rather than a direct revelation mechanism. We consider a
dynamic version of the ascending price auction where the contemporaneous
use of the facility is auctioned. As a given task is completed, the number of
effective bidders decreases by one. We can then use a backwards induction
algorithm to determine the values for the bidders starting from a final period
in which only a single bidder is left without effective competition.
Consider then an ascending auction in which all tasks except that of
bidder I have been completed. Along the efficient path, the final ascending
auction will occur at time t = I − 1. Since all other bidders have vanished
along the efficient path at this point, bidder I wins the final auction at a price
equal to zero. By backwards induction, we consider the penultimate auction
in which the only bidders left are I − 1 and I. As agent I can anticipate to
win the auction tomorrow even if she were to loose it today, she is willing to
bid at most
bI (vI ) = vI − δ (vI − 0) ,
(6)
namely the net value gained by winning the auction today rather than tomorrow. Naturally, a similar argument applies to bidder I − 1, by dropping
65
out of the competition today bidder I − 1 would get a net present discounted
value of δωI−1 and hence her maximal willingness to pay is given by
bI−1 (vI−1 ) = vI−1 − δ (vI−1 − 0) .
Since bI−1 (vI−1 ) ≥ bI (vI ), given vI−1 ≥ vI , it follows that bidder I − 1 wins
the ascending price auction in t = I − 2 and receives a net payoff:
vI−1 − (1 − δ) vI .
We proceed inductively and find that the maximal bid of bidder I − k in
period t = I − k − 1 is given by:
(
(
))
bI−k (vI−k ) = vI−k − δ vI−k − bI−(k−1) vI−(k−1)
(7)
In other words, bidder I − k is willing to bid as much as to be indifferent
between being selected today and being selected tomorrow, when she would
be able to realize a net valuation of vI−k − bI−(k−1) , but only tomorrow, and
so the net gain from being selected today rather than tomorrow is:
(
)
vI−k − δ vI−k − bI−(k−1)
(8)
The maximal bid of bidder I − (k − 1) generates the transfer price of bidder
I − k and by solving (7) recursively with the initial condition given by (6),
we find that the price in the ascending auction equals the externality cost
in the direct mechanism. In this class of scheduling problems, the efficient
allocation can therefore be implemented by a bidding mechanism.1
We end this section with a minor modification of the scheduling model
to allow for multiple tasks. For this purpose it will suffice to consider an
example with two bidders. The first bidder has an infinite series of single
1
The nature of the recursive bidding strategies bears some similarity to the construction
of the bidding strategies for multiple advertising slots in the keyword auction of Edelman,
Schwartz and Ostrovski, AER, 2007. In the auction for search keywords, the multiple
slots are differentiated by their probability of receiving a hit and hence generating a value.
In the scheduling model here, the multiple slots are differentiated by the time discount
associated with different access times.
66
period tasks, each delivering a value of v1 . The second bidder has only a
single task with a value v2 . The utility function of bidder 1 is thus given by
{
v1 if xt = 1 for all t,
v1 (xt , θ1,t ) =
0 if
otherwise.
whereas the utility function of bidder 2 is as described earlier.
The socially efficient allocation in this setting either has xt = 1 for all t
if v1 ≥ v2 or x0 = 2, xt = 1 for all t ≥ 1 if v1 < v2 . For the remainder of this
example, we will assume that v1 > v2 . Under this assumption the efficient
policy will never complete the task of bidder 2. The marginal contribution
of each bidder is:
δ
M1 (θ0 ) = (v1 − v2 ) +
v1 ,
1−δ
and
M2 (θ0 ) = 0.
Along any efficient allocation path, we have Mi (θ0 ) = Mi (θt ) for all i and
the social externality cost of agent 1, p∗1 (θt ) for all t, is p∗1 (θt ) = − (1 − δ) v2 .
The externality cost is again the cost of delay imposed on the competing
bidder, namely (1 − δ) times the valuation of the competing bidder. This
accurately represent the social externality cost of agent 1 in every period
even though agent 2 will never receive access to the facility.
We contrast the efficient allocation and transfer with the allocation resulting in the dynamic ascending price auction. For this purpose, suppose that
the equilibrium path generated by the dynamic bidding mechanism would
be efficient. In this case bidder 2 would never be chosen and hence would
receive a net payoff of 0 along the equilibrium path. But this means that
bidder 2 would be willing to bid up to v2 in every period. In consequence
the first bidder would receive a net payoff of v1 − v2 in every period and her
discounted sum of payoff would then be:
1
(v1 − v2 ) < M1 (θ0 ) .
1−δ
(9)
But more important than the failure of the marginal contribution is the fact
that the equilibrium will not support the efficient assignment policy. To see
67
this, notice that if bidder 1 looses to bidder 2 in any single period, then the
task of bidder 2 is completed and bidder 2 will drop out of the auction in all
future stages. Hence the continuation payoff for bidder 1 from dropping out
in a given period and allowing bidder 2 to complete his task is given by:
δ
v1 .
1−δ
(10)
If we compare the continuation payoffs (9) and (10) respectively, then we see
that it is beneficial for bidder 1 to win the auction in all periods if and only
if
v2
v1 ≥
,
1−δ
but the efficiency condition is simply v1 ≥ v2 . It follows that for a large
range of valuations, the outcome in the ascending auction is inefficient and
will assign the object to bidder 2 despite the inefficiency of this assignment.
The reason for the inefficiency is easy to detect in this simple setting. The
forward looking bidders consider only their individual net payoffs in future
periods. The planner on the other hand is interested in the level of gross
payoffs in the future periods. As a result, bidder 1 is strategically willing
and able to depress the future value of bidder 2 by letting bidder 2 win today
to increase the future difference in the valuations between the two bidders.
But from the point of view of the planner, the differential gains for bidder 1
is immaterial and the assignment to bidder 2 represents an inefficiency. The
rule of the ascending price auction, namely that the highest bidder wins, only
internalizes the individual equilibrium payoffs but not the social payoffs.
This small extension to multiple tasks shows that the logic of the marginal
contribution mechanism can account for subtle intertemporal changes in the
payoffs. On the other hand, common bidding mechanisms may not resolve
the dynamic allocation problem in an efficient manner. Indirectly, it suggests
that suitable indirect mechanisms have yet to be devised for scheduling and
other sequential allocation problems.
The Dynamic Pivot Mechanism
We now construct the dynamic pivot mechanism for the general model described in Section . The marginal contribution of agent i is her contribution
68
to the social value. In the dynamic pivot mechanism, the marginal contribution will also be the information rent that agent i can secure for herself if the
planner wishes to implement the socially efficient allocation. In a dynamic
setting if agent i can secure her marginal contribution in every continuation
game of the mechanism, then she should be able to receive the flow marginal
contribution mi (θt ) in every period. The flow marginal contribution accrues
incrementally over time and is defined recursively:
Mi (θt ) = mi (θt ) + δEMi (θt+1 ) .
The flow marginal contribution can be expressed directly in terms of the
social value functions, using the definition of the marginal contribution given
in (2), as:
mi (θt ) ≜ W (θt ) − W−i (θt ) − δE [W (θt+1 ) − W−i (θt+1 )] .
(11)
The continuation payoffs of the social programs with and without i, respectively, may be governed by different transition probabilities as the respective
social decisions in period t, x∗t ≜ x∗ (θt ) and x∗−i,t ≜ x∗−i (θ−i,t ), may differ.
The expected continuation value of the socially optimal program, conditional
on current allocation xt and current state θt is:
W (θt+1 |xt , θt ) ≜ EF (θt+1 ;xt ,θt ) W (θt+1 ) ,
where the transition from state θt to state θt+1 is controlled by the allocation xt . For notational ease we omit the expectations operator E from the
conditional expectation. We adopt the same notation for the marginal contributions Mi (·) and the individual value functions Vi (·). The flow marginal
contribution mi (θt ) is expressed as:
mi (θt ) =
−
∑
vj
(
x∗−i,t , θj,t
)
I
∑
vj (x∗t , θj,t )
(12)
j=1
)]
(
[
+ δ W−i (θt+1 |x∗t , θt ) − W−i θt+1 x∗−i,t , θt .
(13)
j̸=i
A monetary transfer p∗i (θt ) such that the resulting flow net utility matches
the flow marginal contribution leads agent i to internalize her social externalities:
(14)
p∗i (θt ) ≜ vi (x∗t , θi,t ) − mi (θt ) .
69
We refer to p∗i (θt ) as the transfer of the dynamic pivot mechanism. The
transfer p∗i (θt ) depends only on the current report θt and not on the entire
public history ht . We can express p∗i (θt ) in terms of the flow utilities and the
social continuation values:
∑[ (
)
]
p∗i (θt ) =
vj x∗−i,t , θj,t −vj (x∗t , θj,t )
(15)
j̸=i
[
]
(
)
+δ W−i θt+1 x∗−i,t , θt −W−i (θt+1 |x∗t , θt ) .
(16)
The transfer p∗i (θt ) for agent i depends on the report of agent i only through
the determination of the social allocation which is a prominent feature of the
static Vickrey-Clarke-Groves mechanisms. The monetary transfers p∗i (θt ) are
always non-negative as the policy x∗−i,t is by definition an optimal policy to
maximize the social value of all agents exclusive of i. It follows that in every
period t the sum of the monetary transfers across all agents generates a weak
budget surplus.
Theorem 15 (Dynamic Pivot Mechanism)
The dynamic pivot mechanism {x∗t , p∗t }∞
t=0 is ex-post incentive compatible and
individually rational.
Proof. By the unimprovability principle, it suffices to prove that if agent i
receives as her continuation value her marginal contribution, then truthtelling
is incentive compatible for agent i in period t, or:
vi (x∗ (θt ) , θi,t ) − p∗i (θt ) + δMi (θt+1 |x∗t , θt ) ≥
(17)
vi (x∗ (ri,t , θ−i,t ) , θi,t ) − p∗i (ri,t , θ−i,t ) + δMi (θt+1 |x∗ (ri,t , θ−i,t ) , θt ) ,
for all ri,t ∈ Θi and all θ−i,t ∈ Θ−i and we recall that we denote the socially
efficient allocation at the true state profile θt by x∗t ≜ x∗ (θt ). By construction
of p∗i in (15), the lhs of (17) represents the marginal contribution of agent
i. We can express the marginal contributions Mi (·) in terms of the different
social values to get:
W (θt ) − W−i (θt ) ≥ vi (x∗ (ri,t , θ−i,t ) , θi,t ) − p∗i (ri,t , θ−i,t )
∗
∗
+δ (W (θt+1 |x (ri,t , θ−i,t ) , θt ) − W−i (θt+1 |x (ri,t , θ−i,t ) , θt )) .
70
(18)
We then insert the transfer price p∗i (ri,t , θ−i,t ), see (15), into (18) to obtain:
W (θt ) − W−i (θt ) ≥ vi (x∗ (ri,t , θ−i,t ) , θi,t )
∑ (
)
(
)
−
vj x∗−i,t , θj,t − δW−i θt+1 x∗−i,t , θt
j̸=i
+
∑
vj (x∗ (ri,t , θ−i,t ) , θj,t ) + δW (θt+1 |x∗ (ri,t , θ−i,t ) , θt ) .
j̸=i
But now we reconstitute the entire inequality in terms of the respective social
values:
W (θt ) − W−i (θt ) ≥
I
∑
vj (x∗ (ri,t , θ−i,t ) , θj,t )
j=1
+δW (θt+1 |x∗ (ri,t , θ−i,t ) , θt ) − W−i (θt ) .
(19)
The above inequality holds for all ri,t by the social optimality of x∗ (θt ) in
state θt .
In “The Dynamic Pivot Mechanism”, Econometrica (2010), some conditions for the uniqueness of the above payment rule are given. Needless
to say, these are really strong conditions. This scheme has properties that
other VCG schemes do not necessarily have. All payments are online in the
sense that once an agent is irrelevant for future allocations, she is not asked
to make any payments. Furthermore the property of equating equilibrium
payoffs with marginal contributions gives the individual agents the socially
correct incentives to engage in productive investments in θi .
Learning and Licensing
In this section, we show how our general model can be interpreted as one
where the bidders learn gradually about their preferences for an object that is
auctioned repeatedly over time. We use the insights from the general pivot
mechanism to deduce properties of the efficient allocation mechanism. A
primary example of an economic setting that fits this model is the leasing of
a resource or license over time.
In every period t, a single indivisible object can be allocated to a bidder
i ∈ {1, ..., I}, and the allocation decision xt ∈ {1, 2, ..., I} simply determines
71
which bidder gets the object in period t. In order to describe the uncertainty
explicitly, we assume that the true valuation of bidder i is given by ωi ∈ Ωi =
[0, 1]. Information in the model represents therefore the bidder’s prior and
posterior beliefs on ωi . In period 0, bidder i does not know the realization of
ωi , but she has a prior distribution θi,0 (ωi ) on Ωi . The prior and posterior
distributions on Ωi are assumed to be independent across bidders. In each
subsequent period t, only the winning bidder in period t−1 receives additional
information leading to an updated posterior distribution θi,t on Ωi according
to Bayes’ rule. If bidder i does not win in period t, we assume that she gets
no information, and consequently the posterior is equal to the prior. In the
dynamic direct mechanism, the bidders simply report their posteriors at each
stage.
The socially optimal assignment over time is a standard multi-armed
bandit problem and the optimal policy is characterized by an index policy .
In particular, we can compute for every bidder i the index based exclusively
on the information about bidder i. The index of bidder i after private history
hi,t is the solution to the following optimal stopping problem:
{ ∑τ i l
}
E l=0 δ vi (xt+l )
∑i l
γi (hi,t ) = max
,
τi
δ
E τl=0
where xt+l is the path in which alternative i is chosen l times following a given
past allocation (x0 , ..., xt ). An important property of the index policy is that
the index of alternative i can be computed independent of any information
about the other alternatives. In particular, the index of bidder i remains
constant if bidder i does not win the object. The socially efficient allocation
policy x∗ = {x∗t }∞
t=0 is to choose in every period a bidder i if γi (hi,t ) ≥ γj (hj,t )
for all j.
In the dynamic direct mechanism, we construct a transfer price such that
under the efficient allocation, each bidder’s net payoff coincides with her flow
marginal contribution mi (θt ). We consider first the payment of the bidder i
who has the highest index in state θt and who should therefore receive the
object in period t. In order to match her net payoff to her flow marginal
contribution, we must have:
mi (θt ) = vi (hi,t ) − pi (θt ) .
72
(20)
The remaining bidders, j ̸= i, should not receive the object in period t and
their transfer price must offset the flow marginal contribution: mj (θt ) =
−pj (θt ). We expand mi (θt ) by noting that i is the efficient assignment and
that another bidder, say k, would be the efficient assignment in the absence
of i:
mi (θt ) = vi (hi,t ) − vk (hk,t ) − δ (W−i (θt+1 |i, θt ) − W−i (θt+1 |k, θt )) .
The continuation value without i in t + 1, but conditional on having assigned
the object to i in period t is simply equal to the value conditional on θt ,
or W−i (θt+1 |i, θt ) = W−i (θt ). The additional information generated by the
assignment to agent i only pertains to agent i and hence has no value for the
allocation problem once i is removed. The flow marginal contribution of the
winning agent i is therefore:
mi (θt ) = vi (hi,t ) − (1 − δ) W−i (θt ) .
It follows that p∗i (θt ) = (1 − δ) W−i (θt ), which is the flow externality cost
of assigning the object to agent i. A similar analysis leads to the conclusion
that the losing bidders make zero payments: p∗j (θt ) = −mj (θt ) = 0.
Theorem 16 (Dynamic Second Price Auction)
The socially efficient allocation rule x∗ is ex-post incentive compatible in the
dynamic direct mechanism with the payment rule p∗ where:
{
(1 − δ) W−j (θt )
if x∗t = j,
∗
pj (θt ) =
0
if x∗t ̸= j.
The incentive compatible pricing rule has a few interesting implications.
First, we observe that in the case of two bidders, the formula for the dynamic
second price reduces to the static solution. If we remove one bidder, the social
program has no other choice but to always assign it to the remaining bidder.
But then, the expected value of that assignment policy is simply equal to
the expected value of the object for bidder j in period t by the martingale
property of the Bayesian posterior. In other words, the transfer is equal to
the current expected value of the next best competitor. It should be noted,
73
though, that the object is not necessarily assigned to the bidder with the
highest current flow payoff.
With more than two bidders, the flow value of the social program without
bidder i is different from the flow value of any remaining alternative. Since
there are at least two bidders left after excluding i, the planner has the option
to abandon any chosen alternative if its value happens to fall sufficiently.
This option value increases the social flow payoff and hence the transfer that
the efficient bidder must pay. In consequence the social opportunity cost is
higher than the highest expected valuation among the remaining bidders.
Second, we observe that the transfer price of the winning bidder is independent of her own information about the object. This means that for
all periods in which the ownership of the object does not change, the transfer price stays constant as well, even though the value of the object to the
winning bidder may change.
Efficient Mechanisms with Balanced Budget
It is possible to construct a dynamic efficent mechanism that balances the
budget, “An Efficient Dynamic Mechanism”, Econometrica (2013) by Athey
and Segal construct a Team Mechanism that gives each agent the entire
expected social surplus, and then based on this mechanism the dynamic
AGV mechanism.
The advantage of this is that the budget is balanced. The disadvantage
is that individual rationality is much harder to obtain (and sometimes requires posting bonds). The mechanism with budget balance do not have the
marginal contribution property either.
74
Sequential Screening
This part of the lectures draws heavily from the presentation of the topic
in Chapter 10 of the manuscript ”Lecture Notes on Mechanism Design”
by Tilman Borgers that is written by Daniel Krähmer and Roland Strausz.
Much of the original material is published in ”Sequential Screening” by Pascal Courty and Hao Li, Review of Economic Studies, 2000, and in ”Optimal
Information Disclosure in Auctions and the Handicap Auction” by Peter Esö
and Balazs Szentes, Review of Economic Studies, 2007. Also useful is ”Selling
Options: by Simon Board, Journal of Economic Theory, 2007.
Single Buyer Case: Preliminaries
Information Structure
Consider the case where a seller contracts with a single potential buyer of
an indivisible good for sale. The seller has no value for the good and is just
interested in maximizing her expected sales revenue. In the static case, we
know that the optimal incentive compatible sales mechanism takes the form
where the seller proposes a fixed price paid only in the event that trade takes
place and trade takes place if and only if the buyer’s valuation θ for the good
exceeds the fixed price. In the simplest dynamic screening problem, the
buyer has information that arrives over time. The simplest starting point is
one with two time periods t ∈ {0, 1}.
At the beginning of the game,i.e. in t = 0, the buyer is privately informed
about the value of the product, Let θ0 ∈ Θ0 = [θ0 , θ0 ] be the informational
type representing this information. Let G (θ0 ) denote the c.d.f. of the prior
information on θ0 and the corresponding density is denoted by g (θ0 ) . The
buyer observes additional information on her true valuation as time proceeds.
To capture this, we let θ1 ∈ Θ1 = [θ1 , θ1 ] denote the buyers’ true willingness
to pay for the object. This is without loss of generality since all information
that can be considered in the allocation decision arrives in periods Hence we
just denote by θ1 this total information available at the end of the game.
A key part of the model is the assumed dependence of θ1 on θ0 . If θ0 is
the initial best estimate of the final θ1 then it is reasonable to assume that θ1
75
is increasing (in some sense) in θ0 . We assume here that this dependence is
in the sense of First Order Stochastic Dominance. To formalize this, we let
F (θ1 |θ0 ) denote the conditional distribution of θ1 given θ0 . Thus we assume:
∂F (θ1 |θ0 )
≤ 0 for all θ0 .
∂θ0
∂F (θ 1 |θ0 )
∂F (θ |θ )
Observe that ∂θ10 0 =
= 0 since Θ1 is the maximal support for θ1 .
∂θ0
We assume also that the conditional density f (θ1 |θ0 ) is well defined and is
continuously
differentiable in both arguments. Furthermore, we require that
∂f (θ1 |θ0 ) ∂θ0 be uniformly bounded on Θ0 ×Θ1 . Finally, we assume that F (θ1 |θ0 )
has full support on Θ1 for all θ0 . Notice that G (θ0 ) and F (θ1 |θ0 ) induce a
distribution F (θ1 ) on Θ1 . We assume that θ1 is also private information to
the buyer. It is crucial to keep in mind that the moment of contracting in this
model is t = 0, and hence at this stage θ1 is not known to the buyer either.
It should be clear that the seller could always use static mechanisms based
on either G (θ0 ) or F (θ1 ) . It is a nice exercise for you to think about which
of the two would result in a higher expected revenue. Maybe in answering
this, you would like to insist on the martingale property:
E[θ1 |θ0 ] = θ0 for all θ0 .
The point of this model is to show how the seller can do better than either
of these alternatives in general. The idea is that the seller must leave the
information rent from the initial θ0 to the buyer. The allocation can be made
more efficient by using the information contained in θ0 . If this efficiency gain
can be realized without increasing the information rent to the buyer, then the
seller gains relative to the two static alternatives. The model of Sequential
Screening shows how this can be accomplished.
Payoffs
As usual in the literature on mechanism design, we assume that the players
have quasilinear preferences. Since we have defined θ1 to be the true value of
the object to the buyer, the simple model of sales induces preferences over the
76
probability of sales x and transfers t from the buyer to the seller as follows:
uS (θ1 , x, t) = t,
uB (θ1 , x, t) = θ1 x − t.
Dynamic Direct Mechanisms and Incentive Compatibility
For the purposes of this model, we can define Dynamic Direct Mechanisms
(DDM’s) as follows. Since we have assumed full support, we do not have to
condition the possible second period reports by the true first period type.
Definition 12 A dynamic direct mechanism is a pair of functions
x : Θ0 × Θ1 → [0, 1],
t : Θ0 × Θ1 → R+ .
Hence the definition of the mechanism is not taking any stance on the
issue of timing of the reports. We will nevertheless impose the restriction that
θ0 be reported before θ1 is realized. This will be reflected in the description of
the feasible reporting strategies for the DDM. We require that the reporting
strategy of the buyer in a DDM is a pair σ = (σ0 , σ1 ) with
σ 0 : Θ0 → Θ0 ,
σ1 : Θ0 × Θ1 × Θ0 → Θ1 .
Hence σ0 is the period t = 0 reporting strategy that can condition only on
θ0 . For the strategy in t = 1, we must allow for the possibility that the report
may depend on θ1 and also on the true and the reported type in t = 0. A
DDM induces a dynamic decision problem to the buyer and as usual, we are
interested in DDM’s where the reported types coincide with true types. The
following proposition shows that there is no loss of generality in restricting
attention to DDM’s where the buyer reports her type truthfully. We let Γ
denote any indirect mechanism that respects the same informational rules as
the DDM’s:
σ 0 : Θ0 → M0 ,
σ1 : M0 × Θ1 × Θ0 → M1 .
77
Proposition 5 (Dynamic Revelation Principle) For every Γ and every
optimal reporting strategy σΓ in Γ, there is a DDM Γ′ and an optimal strategy
σΓ′ in Γ′ such that
1. σ0,Γ′ (θ0 ) = θ0 for all θ0 ,
2. σ1,Γ′ (θ0 , θ1 , θ0 ) = θ1 for all θ0 , θ1 ,
and xΓ (m0 (θ0 ) , m1 (θ0 , θ1 , m0 (θ0 ))) = xΓ′ (θ0 , θ1 )
and tΓ (m0 (θ0 ) , m1 (θ0 , θ1 , m0 (θ0 ))) = tΓ′ (θ0 , θ1 ) for all θ0 , θ1 .
Proof. The proof is essentially the same as in the case of static mechanisms
and hence omitted.
Observe that we are claiming here that there is no loss of generality
in restricting attention to mechanisms where truthful reporting is optimal
on equilibrium path. Hence the requirement for truthful reporting off the
equilibrium path as well does not follow from the revelation principle. We
have to define the notion of incentive compatibility in such a manner that
this does not cause any problems. Denote an arbitrary reporting function
for the second period type θ1 as follows:
θ r : Θ1 → Θ 1 .
Given a DDM ((x (θ0 , θ1 ) , t (θ0 , θ1 )) , let u (θ0 , θ1 ) denote the ex post utility:
u (θ0 , θ1 ) = θ1 x (θ0 , θ1 ) − t (θ0 , θ1 ) .
Also as in the static case, let
b (θ′ |θ0 ) =
U
0
∫
u (θ0′ , θ1 ) f (θ1 |θ0 ) dθ1 ,
Θ1
and
b (θ0 |θ0 ) .
U (θ0 ) = U
We can now define incentive compatibility for DDM’s.
Definition 13 A DDM (x (θ0 , θ1 ) , t (θ0 , θ1 )) is incentive compatible (IC) if
78
1. u (θ0 , θ1 ) ≥ θ1 x (θ0 , θ1′ ) − t (θ0 , θ1′ ) for all θ0 , and all θ1 , θ1′ , and
∫
2. U (θ0 ) ≥ Θ1 θ1 x (θ0′ , θr (θ1 )) − t (θ0′ , θr (θ1 )) f (θ1 |θ0 ) dθ1 for all θ0 , θ0′
and all θr .
The first requirement can be thought of as ex post IC that applies at
the stage where all uncertainty has been resolved. The second refers to
interim incentive compatibility where uncertainty still remains over the final
realization of θ1 .Notice that we are allowing reporting strategies in the ex
post stage to depend on the past types and reports in an arbitrary manner
for the interim notion of IC.
Definition 14 A DDM (x (θ0 , θ1 ) , t (θ0 , θ1 )) is individually rational (IR) if
U (θ0 ) ≥ 0 for all θ0 .
Actually it is useful to note that incentive compatibility can be stated in
a slightly more user friendly manner as follows.
Proposition 6 A DDM is IC if and only if
1. u (θ0 , θ1 ) ≥ θ1 x (θ0 , θ1′ ) − t (θ0 , θ1′ ) for all θ0 , and all θ1 , θ1′ , and
b (θ′ |θ0 ) for all θ0 , θ′ .
2. U (θ0 ) ≥ U
0
0
Proof. Clearly IC implies 1. and 2. Assume then that 1 holds. Hence we
have
θ1 x (θ0′ , θ1 ) − t (θ0′ , θ1 ) ≥ θ1 x (θ0′ , θr (θ1 )) − t (θ0′ , θr (θ1 )) .
∫
Hence
[θ1 x (θ0′ , θ1 ) − t (θ0′ , θ1 )]f (θ1 |θ0 ) dθ1
Θ1
∫
[θ1 x (θ0′ , θr (θ1 )) − t (θ0′ , θr (θ1 ))]f (θ1 |θ0 ) dθ1 .
≥
Θ1
But then 2. implies that
b
U (θ0 ) ≥ U
(θ0′
∫
|θ0 ) ≥
[θ1 x (θ0′ , θr (θ1 )) − t (θ0′ , θr (θ1 ))]f (θ1 |θ0 ) dθ1 .
Θ1
Hence we see here that requiring incentives for telling the truth on equilibrium path also delivers optimality off path.
79
Single Buyer Case: Characterizing IC
In this section, we look for a useful characterization of IC mechanisms. The
best case scenario would obviously be a result that gives an if and only if characterization for IC DDM’s. In the static case, with single dimensional types,
such a characterization could be obtained under the additional assumption
of payoffs that satisfy Spence-Mirrlees single crossing condition. Recall that
in this case, global IC is equivalent to local IC (or envelope formula) together
with the monotonicity of the allocation rule in type.
In the current sales model, single crossing is clearly satisfied. We have
also assumed that the types across the two periods are linked in the FOSD
ordering. Hence there is a lot more structure in place than for the static
two-dimensional screening problem where little is known about the characterization of IC. One should also bear in mind the simplifications that come
from the sequential reporting structure.
We start with the easy part. For period t = 1, the first period report θ0
is fixed and hence the incentive compatibility for reporting θ1 reduces to the
static IC conditions.
Proposition 7 (Ex Post IC) A DDM is incentive compatible with respect
to the ex post type θ1 if and only if:
1. (Envelope Theorem)
∂u(θ0 , θ1
= x(θ0 , θ1 ).
∂θ1
2. (Implied Transfers)
∫
t(θ0 , θ1 ) = t(θ0 , θ1 ) + θ1 x(θ0 , θ1 ) − θ1 x(θ0 , θ1 ) −
θ1
x(θ0 , s)ds.
θ1
3. (Monotonicity) x(θ0 , θ1 ) is non-decreasing in θ1 for all θ0 , θ1 .
Proof. Since the proof is essentially the same as in the static case, it is
omitted.
80
One might hope that a similar proposition would hold for reporting the
type θ0 . Unfortunately this is not the case. Since the interim reports induce
lotteries of payoffs (depending on the realization of θ1 ), the static proof where
IC implies monotonicity fails (since now a similar monotonicity only needs
to hold in expectation). We will not get an if and only if statement for IC,
but we shall see that the envelope formula remains valid in a modified form
and that monotonicity implies IC. This is often helpful for determining the
optimal mechanism.
Proposition 8 (Interim Revenue Equivalence) In any incentive compatible DDM:
1. (Envelope Theorem) For almost all θ0 , we have:
∫
∂F (θ1 |θ0 )
′
U (θ0 ) = −
x(θ0 , θ1 )
dθ1 .
∂θ0
Θ1
2. (Implied Transfers)
∫
∫
t(θ0 , θ1 )f (θ1 |θ0 ) dθ1 =
Θ1
θ1 x(θ0 , θ1 )f (θ1 |θ0 ) dθ1
Θ1
∫
[t(θ0 , θ1 ) − θ1 x(θ0 , θ1 )]f (θ1 |θ0 ) dθ1
+
Θ1
∫
x(θb0 , θ1 )
+
Θ1
θ0
)
∂F θ1 θb0
(
θ0 ∫
∂θ0
dθ1 dθb0 .
Proof. 1. By IC,
∫
U (θ0 + δ) − U (θ0 ) ≥
(u(θ0 , θ1 )[f (θ1 |θ0 + δ ) − f (θ1 |θ0 )]dθ1 .
Θ1
Taking limits and dividing by δ gives:
∫
f (θ1 |θ0 + δ ) − f (θ1 |θ0 )
U (θ0 + δ) − U (θ0 )
≥ lim
u(θ0 , θ1 )
dθ1 .
lim
δ→0+ Θ
δ→0+
δ
δ
1
81
Since we have assumed that ∂f (θ∂θ10|θ0 ) is uniformly bounded, we can exchange the limits and integration by dominated convergence theorem. Hence
we have:
∫
U (θ0 + δ) − U (θ0 )
∂f (θ1 |θ0 )
lim
≥
u(θ0 , θ1 )
dθ1 .
δ→0+
δ
∂θ0
Θ1
A similar argument gives:
U (θ0 + δ) − U (θ0 )
lim
≤
δ→0−
δ
∫
(u(θ0 , θ1 )
Θ1
∂f (θ1 |θ0 )
dθ1 .
∂θ0
Hence if U ′ (θ0 ) exists, we have
∫
∂f (θ1 |θ0 )
′
U (θ0 ) =
u(θ0 , θ1 )
dθ1 .
∂θ0
Θ1
∂F (θ1 |θ0 )
∂F (θ |θ )
Integrating by parts and recalling that ∂θ10 0 =
= 0, we get:
∂θ0
∫
∂F (θ1 |θ0 )
′
U (θ0 ) = −
dθ1 .
x(θ0 , θ1 )
∂θ0
Θ1
2. This follows from
∫
U (θ0 ) = U (θ0 ) −
θ0
θ0
∫
x(θb0 , θ1 )
Θ1
( )
∂F θ1 θb0
∂θ0
dθ1 dθb0 ,
∫
[θ1 x(θ0 , θ1 ) − t(θ0 , θ1 )]f (θ1 |θ0 ) dθ1 .
U (θ0 ) =
Θ1
Notice that the second part in this proposition is just housekeeping. It is
a result of the fact that the interim utilities can be evaluated by the envelope
formula and directly through expected values from getting the object net
of expected payment. A similar (tedious) computation yields the following
proposition.
Proposition 9 (Full Revenue Equivalence) In any incentive compatible
DDM,
82
∫
t(θ0 , θ1 ) = t0 (θ0 ) + θ1 x(θ0 , θ1 ) −
θ1
x(θ0 , θb1 )dθb1 ,
θ1
where
∫
∫
t0 (θ0 ) = t(θ0 , θ1 ) − θ1 x(θ0 , θ1 )
( )
∫ θ0 ∫
∂F θ1 θb0
dθ1 dθb0
+
x(θb0 , θ1 )
∂θ0
θ0
Θ1
θ1
+
Θ1
[x(θ0 , ξ)f (θ1 |θ0 ) − x(θ0 , ξ)f (θ1 |θ0 ) dξdθ1 .
θ1
Proof. By manipulating the results in the last two propositions.
This proposition simply collects all the previous results to give a definitive
form to the payment rule as a function of the allocation. It uses the fact that
second period incentive compatibility pins down the payments as shown in
Proposition Ex Post IC. Then it uses the second part of the Interim Revenue
Equivalence result.
As you can see, the payment rule is pinned down except tor the payoff
going to the type (θ0 , θ1 ). This is really the ultimate form of revenue equivalence for our dynamic screening model. The final step in this section shows
that monotonicity of the allocation rule is a sufficient condition for incentive
compatibility.
Proposition 10 Suppose that x(θ0 , θ1 ) is increasing in both components.
Then there exists a transfer scheme t(θ0 , θ1 ) such that the DDM (x, t) is
incentive compatible.
Proof. Since x(θ0 , θ1 ) is assumed to be increasing in θ1 , Ex Post incentive
compatibility is guaranteed by any transfer schedule that results from the
Envelope Theorem. Hence we only need to show that
b (θ′ |θ0 ) = U
b (θ0 |θ0 ) − U
b (θ′ |θ′ ) + U
b (θ′ |θ′ ) − U
b (θ′ |θ0 )
∆U := U (θ0 ) − U
0
0 0
0 0
0
( )
∫ θ0
b θ0′ θb0
∂U
′ b
=
(U (θ0 ) −
)dθb0 .
∂θ0
θ0′
83
By an Envelope Argument, we get:
b (θ′ |θ0 )
∂U
0
=−
∂θ0
∫
x(θ0′ , θ1 )
Θ1
∂F (θ1 |θ0 )
dθ1 .
∂θ0
Hence
∫
θ0
[x(θ0′ , θ1 ) − x(θb0 , θ1 )]
∆U =
θ0′
)
∂F θ1 θb0
(
∫
Θ1
∂θ0
dθ1 dθb0 .
The proof is concluded by observing that monotonicity implies that (θ0 −
and [x(θ0′ , θ1 ) − x(θb0 , θ1 )] have the opposite sign and FOSD guarantees
1 |θ0 )
that ∂F (θ
is non-positive.
∂θ0
Finally, we should characterize individual rationality.
θ0′ )
Proposition 11 An incentive compatible DDM is individually rational if
and only if
U (θ0 ) ≥ 0.
Proof. only if is trivial. To see the if part, notice that property 1 in Interim Revenue Equivalence implies that interim expected utility is weakly
increasing in θ0 .
Determining the Optimal Selling Mechanism
We start by writing the seller’s expected revenue using the Revenue Equivalence results from the previous section.
∫
∫
Et(θ0 , θ1 ) =
[θ1 x(θ0 , θ1 ) − u(θ0 , θ1 )]f (θ1 |θ0 )g(θ0 )dθ1 dθ0
Θ0
∫
Θ1
∫
∫
θ1 x(θ0 , θ1 )f (θ1 |θ0 )g(θ0 )dθ1 dθ0 −
=
Θ0
Θ1
U (θ0 )g(θ0 )dθ0 .
Θ0
Integrating by parts allows us to write the second term as:
∫
∫
U (θ0 )g(θ0 )dθ0 =
(1 − G(θ0 ))U ′ (θ0 )dθ0 + U (θ0 ).
Θ0
Θ0
84
By Interim Revenue Equivalence, we can write this as:
∫ ∫
∂F (θ1 |θ0 )
−
(1 − G(θ0 ))x(θ0 , θ1 )
dθ0 + U (θ0 ).
∂θ0
Θ0 Θ1
Hence by collecting all the terms together, we get the expected revenue
as a function of the allocation rule and U (θ0 ) as follows:
Et(θ0 , θ1 )
∫
∫
=
Θ0
∂F (θ1 |θ0 )
1 − G(θ0 ) ∂θ0
]x(θ0 , θ1 )f (θ1 |θ0 )g(θ0 )dθ1 dθ0 − U (θ0 ).
[θ1 +
g(θ0 ) f (θ1 |θ0 )
Θ1
But now the solution of the relaxed problem where we ignore the global
incentive compatibility constraints is close. Since we are using the Envelope
formulas in computing the buyers’ equilibrium payoff, we are making sure
that the local incentive compatibility constraints hold. Clearly it is optimal
to make the individual rationality constraint binding and set
U (θ0 ) = 0.
Define
∂F (θ1 |θ0 )
1 − G(θ0 ) ∂θ0
.
ψ(θ0 , θ1 ) := θ1 +
g(θ0 ) f (θ1 |θ0 )
Since the value of the integral is linear in x, it is clearly optimal to set
x(θ0 , θ1 ) =1 whenever ψ(θ0 , θ1 ) ≥ 0 in the relaxed program. If ψ(θ0 , θ1 ) is
strictly increasing in both components, then this solution solves the revenue
maximization problem. Hence we assume from now on that ψ is increasing
in both arguments.
To complete the description of the optimal mechanism, define the following function
p(θ0 ) = min{θ1 ∈ Θ1 |ψ(θ0 , θ1 ) ≥ 0}.
Since ψ is increasing, p (·) is well defined. With the help of this function,
we can characterize the optimal selling mechanisms.
85
Theorem 17 (Optimal Screening Mechanism) If ψ(θ0 , θ1 ) is increasing in both arguments, then a DDM (x, t) maximizes the seller’s expected
profit in the class of incentive compatible mechanisms if and only if
x(θ0 , θ1 ) = 1{θ1 ≥p(θ0 )} ,
t(θ0 , θ1 ) = t0 (θ0 ) + p(θ0 )1{θ1 ≥p(θ0 )} ,
where 1A is the indicator function of the set A, and t0 (θ0 ) is as defined in
Proposition Full Revenue Equivalence with
∫
θ1
t(θ0 , θ1 ) =
θ1 f (θ1 |θ0 )dθ1 − p(θ0 )[1 − F (p(θ0 )|θ0 )] + θ1 x(θ0 , θ1 ).
p(θ0 )
Proof. For any θ1 < p(θ0 ), x(θ0 , θ1 ) = 0 and hence from Full Revenue
Equivalence, we get
t(θ0 , θ1 ) = t0 (θ0 ).
Similarly if θ1 > p(θ0 ), then x(θ0 , θ1 ) = 1, and again from Full Revenue
Equivalence, we have:
t(θ0 , θ1 ) = t0 (θ0 ) + p(θ0 ).
We also have that
∫
θ1 x(θ0 , θ1 ) − t(θ0 , θ1 )f (θ1 |θ0 )dθ1 .
U (θ0 ) =
Θ1
∫
θ1
=
θ1 f (θ1 |θ0 )dθ1 − p(θ0 )[1 − F (p(θ0 )|θ0 )] − t0 (θ0 ).
p(θ 0 )
Hence the result follows from setting U (θ0 ) = 0 and using the formula in
Proposition Full Revenue Equivalence for t0 (θ0 ).
Can we think of a nice indirect mechanism that implements this mechanism? We can consider option contracts for selling the good. In such a
contract, the buyer buys for an up-front fee t(θ0 ) the option of purchasing the good at strike price p(θ0 ). Hence the mechanism seems to bear
some relation to contracts that are actually observed in situations where
86
uncertainty is gradually revealed about the value of the alternatives. Let
{t (θ0 ) , p (θ0 )}θ0 ∈Θ0 denote the implementation of the optimal trading mechanism by these option contracts.
Single Buyer: How Much Information Rent Does the
Buyer Get?
Probability Transforms and Orthogonalizing New Information
Consider the following change to the information structure. Rather than
observing θ1 , the buyer now observes γ = F (θ1 |θ0 ). By our assumption of
FOSD, there is a bijection between γ and θ1 . Hence the buyer can perfectly deduce θ1 upon seeing γ. This means that from the point of view of
information to the buyer, the two models are equivalent.
It should be noted that γ and θ0 are statistically independent. To see
this, note that
Pr{γ < γ0 |θ0 } = Pr{θ1 < F −1 (γ0 |θ0 )} = F (F −1 (γ0 |θ0 ) = γ0 .
Hence γ is uniformly distributed on [0, 1] for all θ0 and thus γ is independent of θ0 . This allows us to think of γ as the new information relative to θ0
that is contained in θ1 .
Since the γ and θ1 are bijectively related, we can compute the implied
expected revenues to the seller easily from the previous analysis. Recall that
the expected revenue in the previous formulation is:
∫ ∫
ψ(θ0 , θ1 )x(θ0 , θ1 )f (θ1 |θ0 )g(θ0 )dθ1 dθ0 .
Θ0
Θ1
By change of variables γ = F (θ1 |θ0 ), we can rewrite this as:
∫
∫
Θ0
1
ψ(θ0 , F −1 (γ|θ0 ))x(θ0 , F −1 (γ|θ0 ))g(θ0 )dγdθ0 .
0
By defining ψ̃(θ0 , γ) = ψ(θ0 , F −1 (γ|θ0 )) and x̃(θ0 , γ) = x(θ0 , F −1 (γ|θ0 )),
we can write the expected revenue as:
87
∫
∫
1
ψ̃(θ0 , γ)x̃(θ0 , γ)g(θ0 )dγdθ0 .
Θ0
0
You should compute that
ψ̃(θ0 , γ) = F −1 (γ|θ0 ) −
1 − G(θ0 ) ∂F −1 (γ |θ0 )
.
g(θ0 )
∂θ0
Since the two integrals are really the same (since they are connected
to each other by change of variables), the maximization of the second is
equivalent to the maximization of the first integral. The second formulation,
however, allows us to make an easy comparison to a hypothetical case where
the seller also observes γ.
Publicly Observable New Information
We assume next that the seller also observes γ. Consider now a simple DM
where the buyer just reports her type θ0 but the allocation and the transfers
are also conditioned on (the random variable) γ. Write (x̃(θ0 , γ), t̃(θ0 , γ) for
a generic mechanism of this type. Then the expected payoff to the buyer
when participating in the mechanism is
∫
1
Ũ (θ0 ) =
[F −1 (γ|θ0 )x̃(θ0 , γ) − t̃(θ0 , γ)]dγ.
0
The usual definition of static incentive compatibility extends immediately to this case of random allocations and random payments. We say that
(x̃(θ0 , γ), t̃(θ0 , γ) is incentive compatible if for all θ0 , θ0′ ,
∫
1
Ũ (θ0 ) ≥
[F −1 (γ|θ0 )x̃(θ0′ , γ) − t̃(θ0′ , γ)]dγ.
0
We can once again use the envelope theorem to compute
∫
θ0
∫
Ũ (θ0 ) = Ũ (θ0 ) +
θ0
0
1
∂F −1 (γ|θb0 ) b
x̃(θ0 , γ)dγdθb0 .
∂θ0
The sellers expected revenue is the expected realized social surplus net of
the information rent going to the buyer:
88
∫
∫
1
F
Θ0
−1
∫
(γ|θ0 )x̃(θ0 , γ)g(θ0 )dγdθ0 −
0
Ũ (θ0 )g(θ0 )dθ0 .
Θ0
Finally, integrating by parts gives as the revenue:
∫
∫
1
ψ̃(θ0 , γ)x̃(θ0 , γ)g(θ0 )dγdθ0 .
Θ0
0
Hence the optimal expected revenue to the seller is exactly the same in
the model where she can observe the additional information γ and where γ
is private information to the buyer.
Combining this result with the previous section gives us a nice way of
seeing that in the optimal mechanism, the seller leaves no additional information rent to the buyer from the part of information that is independent of
the private information.
Word of Caution with Orhogonal Transformations
Suppose that the seller can reveal information that reveals less than full
information about θ1 . For example, the seller could commit to a releasing
a binary signal s (θ0 ) ∈ {sH (θ0 ) , sL (θ0 )}, where signal sH (θ0 ) reveals the
event that {θ1 ≥ pe (θ0 )} and sL (θ0 ) reveals the event that {θ1 < pe (θ0 )} for
some pe (θ0 ) . Notice that this signal is not independent of the first period
θ0 and thus it is not orthogonal information in that sense. Hence one is led
to ask whether it might ever be better for the seller to reveal {s (θ0 )}θ0 ∈Θ0
than full information on θ1 . Notice that we are here assuming a world where
the seller can commit to an information structure in the same sense as in the
literature on Bayesian Persuasion.
We make one more assumption on top of FOSD. We assume that a hazard
rate ordering between the types holds as follows. For all θ0 ≥ θ0′ ,
f ((θ1 |θ0 ))
f ((θ1 |θ0′ ))
≤
1 − F (θ1 |θ0 )
1 − F (θ1 |θ0′ .)
Fortunately enough, we have done enough preparatory work to answer this
question quite simply. A natural candidate is to start wit the optimal option
implementation from the first subsection. If the schedule {t (θ0 ) , p (θ0 )}θ0 ∈Θ0
89
solves the optimal sales problem in the case of full disclosure, consider the
modified scheme {e
t (θ0 ) , pe (θ0 )}θ0 ∈Θ0 , where
pe (θ0 ) = p (θ0 ) + δ,
∫ θ1
e
t (θ0 ) = − (1 − F (p (θ0 ) |θ0 )) δ +
(1 − F (θ1 |θ0 )) dθ1
p(θ0 )
)
∫ θ1 (
∫ θ1
∂F (p (θ0 ) |θ0 )
∂F (θ1 |θ0 )
−
δ+
dθ1 dθ0 .
−
∂θ0
∂θ0
θ1
p(θ0 )
The allocation rule is simply that the buyer buys the object if and only if
she sees signal sH (θ0 ) .
I leave it to the interested reader to complete the following steps: i) The
easiest step is to show that this scheme satisfies IC over reporting θ1 for small
enough δ when hazard rate dominance holds. ii) Show that the new scheme
e (θ0 ) satisfies:
is IR by verifying that the new interim expected payoff U
e (θ0 ) = 0, U
e ′ (θ0 ) ≥ 0 for all θ
U
for small enough δ. iii) Show that the seller’s expected profit is higher than
with the original contract for small enough δ by showing that the buyer’s
information rent is smaller and by noticing that the allocation rule is the
same across the two cases.
This example shows that some of the claims mad in Eso and Szentes
about the optimality of always disclosing all information has to be taken with
a pinch of salt. If the seller is able to manipulate the information structure
in a richer fashion, then other schemes will dominate full disclosure. This
analysis is taken from ”Discriminatory Information Disclosure” by Hao Li
and Xianwen Shi, mimeo, 2015.
Multiple Buyers: Handicap Auctions
In this subsection, we assume that there are N and the description of the
types and payoffs of all i ∈ {1, ..., N } is as in the first subsection. We
maintain the same structure of a Dynamic Direct Mechanism with obvious
modifications to reflect the fact that there are now N bidders that report
90
their types simultaneously. Obviously one has to now take into account
the fact that the types and reports of the other biders are not known to i
when she reports her θ0i and θ1i . The solution concept we use is Bayesian
Nash Equilibrium and hence incentive compatibility is to be interpreted in
this sense. Revelation Principle still holds so we can always take the other
agents’ announcements to be truthful
In order to maintain comparability with the IPV auction analysis in the
static case, we assume that the types of the bidders are statistically independent across i. Obviously we allow the types θ0i and θ1i to be correlated
((
) (
))
as in the previous subsections. Let (θ0 , θ1 ) = θ01 , ..., θ0N , θ11 , ..., θ1N and
θi = (θ0i , θ1i ) . We can then write a DDM (x, t) as:
x : Θ0 × Θ1 → ∆N ,
t : Θ0 × Θ1 → RN ,
(
)
i
i
where ∆N = { x1 , ..., xN ∈ RN
+ |Σi x ≤ 1 }. The interpretation is that x (θ0 , θ1 )
gives the probability that i is assigned the object if the reported types are
(θ0 , θ1 ). Let also
(
)
(
)
X i θ0i , θ1i = Eθ−i xi θi , θ−i
∫
∫
(
) ∏ j ( j j ) −i ∏ j ( j ) −i
=
f θ1 θ0 dθ1
g θ0 dθ0 ,
xi θ0i , θ0−i , θ1i , θ1−i
Θ−i
0
Θ−i
1
and
∫
=
Θ−i
0
∫
Θ−i
1
j̸=i
j̸=i
(
)
(
)
T i θ0i , θ1i = Eθ−i ti θi , θ−i
(
) ∏ j ( j j ) −i ∏ j ( j ) −i
ti θ0i , θ0−i , θ1i , θ1−i
f θ1 θ0 dθ1
g θ0 dθ0 .
j̸=i
j̸=i
The entire analysis of incentive compatibility and revenue equivalence
that we did in the case of a single buyer carries over to the case of many
bidders with functions (X i (θ0i , θ1i ) , T i (θ0i , θ1i )) . Hence we do not repeat the
steps but go directly to the formula for expected revenue from bidder i in
any incentive compatible mechanism:
∫ ∫
( i i)
(
)
(
) ( )
( )
( )
i
Eθi T θ0 , θ1 =
ψ i θ0i , θ1i X i θ0i , θ1i f i θ1i θ0i dθ1i g i θ0i dθ0i −U i θi0 ,
Θi0
Θi1
91
where ψ i (θ0i , θ1i ) is defined as before. Hence the total expected revenue is
∫
∫
Θ0
N
∑
ψ
i
(
θ0i , θ1i
)
i
x (θ0 , θ1 )
Θ1 i=1
∏
f
(
j
θ1j
N
∏ ( j)
∑
j)
( )
j
θ0 dθ1
g θ0 dθ0 −
U i θi0
j
j
i=1
To maximize this expected revenue without violating IR, it is clearly
( )
optimal to set U i θi0 = 0 for all i, and to use allocation rule
(
)
{
1 if ψ i (θ0i , θ1i ) ≥ max{0, ψ j θ0j , θ1j , for all j,
i
x (θ0 , θ1 ) =
0 otherwise.
The associated transfers with this mechanism are:
∫ θ1
( i i)
( i i)
i
i i
i
i
T θ0 , θ1 = T0 (θ0 ) + θ1 X θ0 , θ1 −
X i (θ0i , θb1i )dθb1i ,
θ1
∫
∫
T0i (θ0i ) = T i (θi0 , θi1 ) − θi1 X i (θi0 , θi1 )
( )
∫ θ0 ∫
∂F θ1i θb0i
dθ1i dθb0i
+
X i (θb0i , θ1i )
∂θ
0
Θ1
θ0
θ1
+
Θ1
( )
( )
[X i (θ0i , ξ)f θ1i θ0i − X i (θi0 , ξ)f θ1i θi0 dξdθ1i ,
θ1
( )
and T i (θi0 , θi1 ) is determined from U i θi0 = 0.
Observe that the transfer schemes here are in terms of the interim payments. It is possible to find the actual transfers too, but this is cumbersome.
In the special case where we can find for all i a function ϕi satisfying
( )
ϕi θ0i =
∂F i (θ1i |θ0i )
∂θ0i
,
f i (θ1i |θ0i )
the virtual valuation is simply
)
(
1 − G (θ0i ) i ( i )
ϕ θ0 .
ψ i θ0i , θ1i = θ1i +
g (θ0i )
Notice that the function ϕi (θ0i ) is negative. An example of such setting is
one where
θ1i = θ0i + εi ,
92
where ε is a random variable whose In this case, the optimal allocation mechanism a handicap auction. In the first period, each buyer selects a premium
pi (θ0i ) at a price ti0 (θ0i ). In the second round, a modified second price auction
without reserve prices is run. The payment is the second highest bid plus the
premium bought in the first round. This means that the buyer’s value from
winning the object is her valuation minus the premium. Since the second period auction is a second-price auction, each bidder has a dominant strategy
to bid θ1i − pi (θ0i ) . Hence if the seller sets
( )
1 − G (θ0i ) i ( i )
pi θ0i = −
ϕ θ0 .
g (θ0i )
If it is optimal for each bidder i with ex ante type θ0i to choose pi (θ0i ) amongst
the offered premia, then the resulting revenue is the same as in the optimal
mechanism by Revenue Equivalence Theorem. But this can be ensured by
setting the ti0 (θ0i ) so that they generate the same expected transfers as the
above optimal DDM.
Dynamic Mechanism Design: Single Agent
Setup
This subsection gives you an idea of how to extend the two-period case to an
infinite horizon. I am skipping a large number of details and for those I refer
you to Pavan, Segal and Toikka (2014). In the current setting, it is not hard
to check that the technical conditions for applying the Milgrom and Segal
(2002) envelope formula are satisfied. The relevant Revelation Principle for
the current model can be found in Myerson (1986): ”Multistage Games with
Communication”, Econometrica. This section uses heavily lecture notes that
Juuso Toikka has written on the topic.
We start again with the single agent case and consider a general discrete
time horizon, t ∈ {0, 1, ..., T } with T ≤ ∞. In each period, an allocation
]
[
xt ∈ Xt must be chosen. The buyer observes a new type θet ∈ Θt = θt θt
in each t, and again the initial type θe0 is her private information. The prior
distribution on Θ0 is given by the c.d.f. G (θ0 ) . Furthermore, we make the
93
Markov assumption that the distribution of θet+1 depends the past realizations
θt := (θ0 , ..., θt ) only through (θt , xt ) . We denote the conditional distribution
of θet+1 by Ft (θt+1 |θt , xt ). Hence we allow the type in future periods to
depend on the allocation as well as current state, but we make the Markov
assumption on the θt . We continue to assume that all distributions have
well defined conditional densities that are continuously differentiable with
uniformly ( in t, xt and θt ) bounded derivatives.
For example, we could have the following AR(1) process:
θet = γθt−1 + εet
where each εet is an i.i.d. draw from a single distribution H (·) . This process
has also a moving average representations as:
t
θt = γ θ0 +
t−1
∑
γ s εt−s .
s=0.
Since we allow for the case of infinite decision horizons, it is also natural to
discount future payoffs. In order not to bring in other intertemporal motives
for trading across periods, we assume that all buyer and the seller have the
same discount factor δ < 1. For obvious reasons, we will also denote transfers
with a letter different from t. Let p denote the transfers. Hence the payoff
to i from getting the object in t is
δ t (ut (θt , xt ) − pt ) .
We assume that ut (·) is continuously differentiable and bounded in both
arguments with uniformly (in t, θt , xt ) bounded derivatives.
We start by defining a Dynemic Direct Mechanism (DDM). Let θt =
(θ0 , ..., θt ) ∈ Θt , and ΘT := Θ, xt := (x0 , ..., xt ) ∈ X t , X T := X. In a DDM
the buyer reports her type θt at each t. Hence we define:
A DDM is a pair
x : Θ → X,
p : Θ → RT ,
94
where:
(
)
(
)
xt θt , θT −t = xt θt , θbT −t for all θT −t , θbT −t ∈ ×Ts=t+1 Θs ,
(
)
(
)
T −t
T −t
b
pt θt , θ
= pt θt , θ
for all θT −t , θbT −t ∈ ×Ts=t+1 Θs .
(21)
In other words, the allocations and payments in period t can only depend
on information available at t. Hence we can write the coordinates as xt (θ) =
xt (θt ) , pt (θ) = pt (θt ) . We say that the allocation rule is Markovian if
xt (θt ) = xt (θt , xt−1 ) .
Any Markovian allocation rule x induces a Markov process whose transitions are given by
( ( ))
θet+1 ∼ Ft+1 · θt , xt θt .
This process is denoted by λ[x].
By Revelation Principle, it is w.l.o.g. to consider DDM where the buyer
reports her type θi truthfully in each t. Such a mechanism is said to be
incentive compatible (IC).
Canonical State Representation
( )T
An alternative way of formulating the process θet
is by finding a sequence
t=0
of independent random variables or shocks (e
εt )Tt=1 where each εet has support
Et and a sequence of functions
Zt : Θt−1 × X t−1 × Et → Θt
for t = 1, ..., T such that for all t and all θt−1 , xt−1 ,
( )
Zt (θt−1 .xt−1 , εet ) ∼ Ft · θt−1 , xt−1 .
εt )Tt=1 .
In this sense, we can view the process as being generated by θ0 and (e
The advantage of this new formulation is that now the εet are independent of
θ0 by construction. Note that we are here extending the orthogonalization
process of the two-period model. To see this, let εet be i.i.d. draws from
UNIF[0, 1]. Define
)
( )
(
Zt θt−1 , xt−1 , ε′t = Ft−1 ε′t θt−1 , xt−1 .
95
If Ft (· |θt−1 , xt−1 ) is not one-to -one, define the generalized inverse as
{
( )
}
( )
Ft−1 ε′t θt−1 , xt−1 := inf θt : Ft θt θt−1 , xt−1 > ε′t .
Then
(
)
( )
Zt θt−1 , xt−1 , εet ∼ Ft · θt−1 , xt−1
as needed. Hence for any θs , θt with s < t, we can define
(
)
θt = Z(s),t θs , xt−1 , (ετ )tτ =s+1
( )T
We call this the canonical representation of the Markov Porcess θet
t=1
Example 1 Even though this example does not fit all the assumptions of the
model, it may be useful to shed light on the canonical representation. Consider
{
}
θt ∈ θH , θL . Let (e
εt )Tt=1 be uniform on [0, 1]. For the representations, set
{
}
{ H
θ if εet ≥ Pr θt = θH |θt−1 ,
{
}
Zt =
θL if εet < Pr θt = θH |θt−1 .
( )T
T
e
Then θ0 , (e
εt )t=1 , Zt form a canonical representation of θet
.
t=1
The reason for using this representation of the Markov Process, is that
by independence, we may now vary θ0 while keeping the sequence of future
shocks fixed. Notice also the similarity to the moving average representation
of the AR(1) process.
We define now the impulse response at history (θt , xt−1 ) as:
(
)
∂Zt (θ0 , xt−1 , εt )
,
It θt , xt−1 :=
∂θ0
where εt satisfies for all 0 ≤ s ≤ t:
(
)
θs = Zs θ0 , xs−1 , εt .
If Zs (θ0 , xs−1 , ·) is not one-to one, then an expectation over all εt satisfying
the above is taken. Similarly:
(
t
I(s),t θ , x
t−1
)
∂Z(s),t (θs , xt−1 , εt )
:=
∂θs
96
By chain rule,
∂Zt ∏ ∂Z(s−1),s
=
.
∂θ0
∂θs−1
s=1
t
Using the canonical representation, we have:


∂Ft (θτ |θτ −1 ,xτ −1 )
t
∏ 
(
)
∂θτi −1

I(s),t θt , xt−1 =
.
−
τ
−1
fτ (θτ |θτ −1 , x )
τ =s+1
To see this, note that
(
( )
)
∂Zs
∂Fs−1
=
and F Ft−1 εt θt−1 , xt−1 θt−1 , xt−1 ≡ εt ,
∂θs−1
∂θs−1
and totally differentiate the identity with respect to θs−1 . Notice that for this
to make sense, the canonical representation must be differentiable. This will
be assumed from now on and similarly, it is assumed that
∂Z(s),t (θs , xt−1 , εt ) ≤ Bt−s for all s < t
∂θs
for some sequence {Bt } such that
T
∑
δ t Bt < ∞.
t=0
The impulse response function isolates the effects of the true type at t
on the future evolution of θes for s > t. The other channel, i.e. through
the dependence on xt and therefore indirectly on θt takes place through the
announced types in the mechanism and not the true types. As always, we
proceed to show that information rents are determined by the direct effects
of payoffs. Effects through announcements vanish by the first order condition
for optimal announcements.
Payoff Equivalence
Consider the equilibrium payoff to agent with type history θs :
]
[ T
( ))
( )
∑ (
s
,
Vs (θs ) = Eλ[x]|θ
δ t ut (xt θet , θet ) − pt θet
t=s
97
where λ[x] |θs is the continuation process of λ[x] from history (θs , xs (θs ))
onwards.
Theorem 18 (ICFOC) If (x, p) is IC, then for all s, θs , Vs (θs ) is Lipschitz
continuous with derivative
( )


et , θet )
T
(
(
))
∂u
(x
θ
∑
t
t
s
.
Vs′ (θs ) = Eλ[x]|θ 
I(s),t θet , xt−1 θet−1 δ t
∂θ
t
t=s
Notice that this tells immediately that the allocation rule pins down the
IC payoff up to a constant as in the static case. To interpret the result, let
U (x, θ) :=
T
∑
δ t ut (xt , θt )
t=0
so that
∂U
∂ut
= δt
.
∂θt
∂θt
Then (ICFOC) becomes:
Vs′
λ[x]|θs
(θ ) = E
s
T
∑
t=s
I(s),t
∂U
.
∂θt
The impulse response function measures the effect of a small change in θs on
∂U
θt and ∂θ
measures the induced change in period t utility.
t
Proof. For t = 0. Assume (x, p) is IC. Let the agent report θ0 and εt in
t > 0. Since this is an equivalent representation of the problem, if (x, p) is
IC, then (b
x, pb) where x
b1 (θ0 , ε1 ) := x1 (θ0 , Z1 (θ0 , ε1 )) etc. is also IC. It is also
optimal for the agent to report truthfully in the problem where she is asked
only to report θ0 and where εt are observable. Truthtelling yields Vo (θ0 ) in
the modified mechanism by IC of the original mechanism. Hence
[
]
T
∑
(
)
t
t
V0 (θ0 ) = max E U (b
x (m0 , εe) , Z (θ0 , εe)) +
δ pbt m0 , εe ,
m0
t=0
98
where
( (
))T
Z (θ0 , εe) : = Zt θ0 , εet t+0 ,
εe : = (e
εt )Tt+0 ,
E is the expectation over εe.
So now we can apply Milgrom and Segal (2002) to see that V0 is Lipschitz
and
T
T
∑
∑
∂U ∂Zt
∂U
s
V0′ (θ0 ) = E
= Eλ[x]|θ
It
,
∂θ
∂θ
∂θ
t
t
t
t=0
t=0
where the second equality reinterprets the model wit the original process
( )T
.
θet
t=1
Transfers
This is just as in static case. Fix θ′ ∈ Θ and let
Dsλ[x]
λ[x]|θ s
(θ ) := E
s
T
∑
s=t
and
∫
Qsλ[x]
Put
θs
s
(θ ) :=
θs′
I(s),t
∂U
,
∂θt
(
)
Dsλ[x] θs−1 , λ dλ.
[
(
)]
( )
( )
s
λ[x] ( t )
λ[x]
pt θt = δ −t Qt
θ − Eλ[x]|θ δ −t Qt+1 θet+1 − ut (xt θt , θt )
Hence we can interpret pt (θt ) as the information rent of θt relative to type θ′
from period t perspective. It can be shown that this payment rule determines
the transfers in any IC mechanism (x, p) up to a constant.
Characterizing IC
Problem relative to static case: even though θt is unidimensional in each t,
the resulting allocation is multidimensional since reports affect the allocation
in all future periods. Consider first the following allocation rule
x ◦ ms (θ) = x (ms , θ−s ) .
99
This allocation just fixes the announcement of period t type at ms but otherwise follows exactly the true allocation. We have the following characterization for IC.
Theorem 19 (Integral Monotonicity) Allocation rule x is implementable
if and only if for all s, θs and ms ,
∫ θs
(
)
(
)
[Dsλ[x] θs−1 , λ − Dsλ[x◦ms ] θs−1 , λ ]dλ ≥ 0.
(22)
ms
The condition in (22) is known as the Integral Monotonicity condition.
It originates from the following simple Lemma that we already encountered
in the lectures on static mechanism design.
Lemma 5 Consider a function Φ : Θ × Θ → R such that for all θ′ ∈ Θ,
b (θ) := Φ (θ, θ) is a
Φ (θ, θ′ ) is a Lipschitz continuous function of θ and Φ
′
b (θ) ≥ Φ (θ, θ ) for all θ, θ′ ∈ Θ2 if and
Lipschitz continuous function. Then Φ
only if for all θ, θ′ ∈ Θ2
∫ θ
′
b ′ (q) − ∂Φ (q, θ ) ]dq ≥ 0.
[Φ
(23)
∂θ
θ′
b (θ) − Φ (θ, θ′ ) . For each fixed θ′ it is Lipschitz
Proof. Consider g (θ, θ′ ) := Φ
and therefore absolutely continuous and we have:
∫ θ
∫ θ
′
∂g (q, θ′ )
′
b ′ (q) − ∂Φ (q, θ ) ]dq.
g (θ, θ ) =
dq =
[Φ
∂q
∂θ
θ′
θ′
Hence for all θ, Φ (θ, θ′ ) is maximized at θ′ = θ if and only if equation (??)
holds.
In the static single dimensional case, this condition is equivalent to the
monotonicity of the allocation rule (under single-crossing). Here this is unλ[x]
fortunately not the case since the Ds (θs−1 , λ) are expectations themselves
and furthermore the expectations are taken over multi-dimensional allocations. Even though IM is not easy to work with in its general form, the
following two stronger conditions are easier to verify.
100
Definition 15 Allocation rule x satisfies average monotonicity if for all s
λ[x◦ms ]
and θs , Ds
(θs ) is nondecreasing in ms .
Definition 16 The allocation process satisfies strong monotonicity if the
( )T
does not depend on xt , if the transitions satisfy first order
process of θet
t=1
stochastic dominance and ut has strictly increasing differences in (xt , θt ) and
for all s, xs (θs ) is nondecreasing in θs (so that increasing ms increases xt for
all t ≥ s).
Example 2 Linear payoffs:
T
∑
δ t (θt xt − pt ) ,
t=0
autonomous process. Average monotonicity requires that
λ[x]|θs
E
T
∑
)
( ) (
t
I(s),t θet xt ms , θe−s
t=s
is nondecreasing in current message ms . Hence the expected discounted consumption must be nondecreasing where the discount refers both to δ and to
the impulse responses. To gain intuition, consider the fully persistent case.
The main use of Integral Monotonicity is as a verification theorem. It
is not easily handled as a constraint in optimization problems. Hence the
plan is to solve an easier relaxed problem that takes only local incentives into
account and then verify that the solution satisfies IM.
Optimal Dynamic Mechanisms
Consider next the optimization problem of a principal that has her own
dynamic payoffs given by
T
∑
δ t (pt − ct (xt ))
t=0
.
101
She designs a mechanism to maximize her own payoff. As always, we can
write the principal’s payoff as the difference between the social surplus and
the agent’s information rent. Hence the problem is to
max E
x
(x,p)
T
∑
δ
t
(
( ( )))
( )
t
e
e
ut (xt θ , θt ) − ct xt θet
( )
( )

T
(
)) ∂ut (xt θet , θet )
(
1 − G θe0
∑
 − V0 (θ0 ) .
( ) 
−Ex
δ t It θet , xt−1 θet−1
∂θ
e
t
g θ0
t=0
t=0
subject to (IC) and (IR):
V0 (θ0 ) ≥ 0.
( )
Denote the first line of the objective function by Ex S x, θe .
Note that we have built local incentive compatibility into the system by
using the envelope formula for buyer’s information rent. Clearly IR will bind
at optimum and thus V0 (θ0 ) = 0. If IR would bind at some future types, this
is not a problem since the buyer can be asked to post a bond at the beginning
of the contract that will be returned at the end of the contract at the same
discounted value.
In general solving
( )


T
( ) 1 − G θe0 ∑
∂e
ut 
( )
max Ex S x, θe −
δ t Iet
x
∂θt
g θe
t=0
0
involves dynamic programming and is not easy. However if the process of
( )T
θet
does not depend on xt and there are no intertemporal restrictions on
t=1
xt , then pointwise solution is possible. We turn next to special cases.
Applications: Progressive Screening
We return next to the single buyer case, and allow the buyer to have private information on the stochastic process itself. The model here is from
”Progressive Screening” by Boleslavsky and Said, REStud, 2013. Let
θ0 = 1, θt = αt θt−1 ,
102
where αt ∼ F (· |λ) on some positive interval A, and λ is private information
to the buyer and the prior distribution on λ is given by G (·). A direct
revelation mechanism asks the buyer to report λ in period t and θt in each
t > 0. Notice that formally we are outside of the Markov case since the firstperiod information affects the transition probabilities of all future periods.
At the same time, by taking logarithms, we see that the process can be
written as a sequence of additive shocks whose distrbutions depend only on
the initial private information about λ.
In this setting, we can define the impulse responses recursively:
I0 (λ) = 1,
∫
It (λ) =
−
A
∂F (αt |λ )
∂λ
f (αt |λ)
It−1 (λ) dF (αt |λ) .
If sales are made in all periods, then following usual steps, the seller maximizes the total surplus net of information rent (this starts accruing at t = 1
since θ0 is public information). The seller’s problem is then to
T
∑
t=1
∫ ∫
δ
t
(
t
)
t
xt α , λ [θ +
Λ
At
t
∑
1 − G (λ)
s=1
g (λ)
(
)
Is (λ)]dF αt |λ dG (λ) ,
where we let F (αt |λ ) denote the conditional distribution of (α1 , ..., αT ) .
Under suitable conditions on F (αt |λ) and G (λ) , the optimal pointwise
solution to the relaxed problem satisfies pointwise monotonicity in λ and αt
and hence the integral monotonicity also holds and the solution of the relaxed
problem is incentive compatible. For details, see the paper.
In the paper, you can find worked out solutions to the model when
∂F (αt |λ )
∂λ
f (αt |λ)
takes a simple form. Compute this expression for example in the case where
α = λe
z for some random variable ze defined on a positive support.
The paper also shows that since the only relevant information that the
seller needs from the buyer is the value of λ, the optimal contract can be
implemented as a menu of deterministic contracts. Hence this is quite similar
103
to the original Courty and Li paper. Another result that the paper gets
is that now the distortions do not die down as time progresses since in a
multiplicative model the impact of the initial information on future valuations
may be increasing in time. “Dynamic revenue Maximization: A ContinuousTime Approach” (2015) by Bergemann and Strack solves a number of related
problems within a continuous-time framework that yields nice closed-form
solutions.
This model is particularly tractable since the allocation decisions have
no intertemporal dependence. This is not true in cases where the process
governing the evolution of types depends on the allocation or if there is a
capacity constraint as in single unit auctions. The next two applications
address these issues in multi-agent contexts.
Many Agents
In the next two examples, we superscript the processes to denote the individual bidder i = 1, ..., N. There are a number of subtleties that one should
address when moving from the single buyer case to the N agent case. For example when discussing incentive compatibility, one must distinguish between
different types of mechanisms. The designer may choose a mechanism where
the past reports of the other bidders are not revealed in period t. Maybe
even the transfer payment is delayed to minimize information leakage. This
could be advantageous since this would imply fewer information sets at which
IC has to hold. The examples below are all simple enough to yield solutions
that satisfy IC even if past reports are made public.
Allocating in All Periods: Bandit Auctions
Pavan, Segal and Toikka (2014) also contains an application to a particular
type of a Bandit Auction. We covered the VCG mechanism resulting in
efficient allocations. The Bandit Model in PST satisfies the following: if
i
i
= θti + εi (ni (t)) where ni (t) is
= θti , if xit = 1, then θt+1
xit = 0, then θt+1
the number of periods up to t in which the good has been allocated to i. For
some stochastic processes, the number of observations from the process (here
ni (t)) and the current posterior mean (here θti ) form a sufficient statistic.
104
Consider for example the following class of models:
yti = ω i + εit ,
where ω i ∼ Gi (·) , εit ∼ H i (·) , and the εit are i.i.d. Let
[
]
θti = E ω i |y0 , ..., yt−1 .
Then if Gi (·) ,and H i (·) are Normal distributions, the learning model fits
these assumptions. Beta and Gamma learning models can also be accommodated.


i
i
∂Fti (θτ
+1 |θτ )
s
∏
− i ∂θτi i  = 1.
The nice thing about the Bandit Model is that
f (θτ +1 |θ )
τ =t
τ
τ
Hence the revenue maximization problem is now turned (again using the
usual steps) into a modified bandit problem where the seller maximizes
max E
x∈X
T ∑
N
∑
[
δ
t+0 i=1
t
θti
]
1 − Gi (θ0i ) i
−
xt (θ) ,
g i (θ0i )
(
)
i
where X = { x1 , ..., xN ∈ RN
+ |Σi x = 1 }. Notice that the difference to the
previous model is that now the good must be allocated to some bidder in
each period. Obviously we could add another bidder 0 with trivial valuation
θt0 = v for all t if we wanted to allow for the possibility that the good is
not sold in some periods. This problem can be obviously solved using the
usual Gittins as explained in the notes on the Dynamic Pivot Mechanism. It
can be verified that the solution satisfies average monotonicity and is hence
implementable.
Once and for All Auction: AR(1) Process
Consider next the example with
i
+ εit ,
θti = γθt−1
with θ0i ∼ Gi (θ0i ) , εit ∼ H i (·) , i.i.d. If we set γ = 1, we are essentially
dealing with the model of Board (2007).
105
We can now compute using the usual tricks
V0i
T
∑
( i)
1 − Gi (θ0i ) t i
θ0 = E
δt
γ xt (θ) .
g i (θ0i )
t+0
Hence the expected revenue to the seller is
E
T ∑
N
∑
t+0 i=1
[
δ
t
θti
]
1 − Gi (θ0i ) t i
−
γ xt (θ) ,
g i (θ0i )
Since the maximization is over policies that satisfy the informational requirement in (21), and the feasibility constraint x (θ) ∈ ∆T N , we can think of
the seller’s problem as being an optimal stopping problem. Her decision in
period t is whether to stop and collect
{
}
1 − Gi (θ0i ) t
i
max θt −
γ ,
i
g i (θ0i )
or to continue until t + 1 and draw a new valuation vector θt+1 = θt + εt
for the bidders. You can see that as t increases, the distortion relative to
the planner’s solution in the allocation diminishes. Stopping problem such
as the above are conceptually simple and have good existence properties
(see for example the references given in Board (2007)). You should try to
solve it with simple distributions to judge for yourself how easy this really
is if you move beyond the i.i.d. case for the θt . Nonetheless, it is easy to
see that the allocation policy maximizing equilibrium revenue satisfies the
integral monotonicity condition and is therefore incentive compatible. The
associated transfers are then calculated from the local incentive condition.
Stochastic Arrival of Bidders
We conclude this part of the course with a slightly different type of a question
in Dynamic Mechanism Design. We still assume that a seller has a single
good for sale and her problem is to find an incentive compatible mechanism
that maximizes her expected revenue. The twist is now that the bidders are
not necessarily present at the beginning of the game but arrive according to a
106
stochastic process. The process could be a Poisson arrival process of bidders
over time. The twist is now that the seller does not observe the arrivals of the
bidders and the bidders may delay reporting to the seller. A fully incentive
compatible mechanism makes the bidders report their true valuation to the
seller immediately as they arrive.
A good recent example of such papers is Gershkov, Moldovanu and Strack
(2014): ”Dynamic Allocation and Learning with Strategic Arrivals”, mimeo.
We shall not enter a full discussion of the paper here, but rather just give
some pointers to how the usual methods can be used to solve problems in
this class.
Setup
Time is continuous with a finite horizon, t ∈ [0, T ]. All bidders and the seller
discount future at rate r. A finite number K of goods are available for allocation at the beginning of the game. Bidders arrive randomly as the game
proceeds and n (t) denotes the number of bidders that have arrived. The
stochastic process controlling the arrivals is initially unknown, but (t, n (t))
forms a sufficient statistic for its future evolution. Bidders have private valuations θi ∼ F (·) with support [0, 1] . These valuations are independent across
bidders and also independent of the arrival time (is this a good assumption?).
Payoffs are quasilinear and ti denotes the payment made by i. If the good is
allocated to i at τ i and a payment pi is made at τbi , then the payoff to i is
given by
i
i
e−rτ θi − e−rbτ pi .
The twist to the usual model is now that the both the buyers’ valuation
θ , and her arrival time ai are private information. This means that the
mechanism designer must elicit information about n (t) as well in the direct
revelation mechanism. Consider for example the case of a Poisson arrival
process of buyers (up to T ) with an unknown intensity. The fewer buyers
have shown up, the more pessimistic the seller is about future arrivals and
hence the lower her reservation value of selling the good. Hence it might be a
good idea for a buyer to hide her arrival in order to induce a more favorable
deal in the future.
i
107
Assume ex post individual rationality. This is both reasonable and it also
rules out Cremer and McLean mechanisms that would otherwise be possible
since the arrival times are correlated.
Solving the Model: Observable Arrivals
Start with the hypothetical case where the arrivals are observable. Assume
that the mechanism designer makes all reported past arrivals known to the
buyer that arrives at ai (this can be shown to be wlog). Denote the history
at the arrival time ai by h (ai ). This includes the reported v i and all past
reported arrival times and valuations, and past assignments of goods. Let
[
( ( ))
( ( )) ( ) ]
i
i
U i h ai = E e−rτ (h(a )) θi − pi h ai h ai .
Hence the allocation policies τ i and the transfers pi must be adapted to
h (ai ) . The expectation covers future arrivals and future valuations of arriving buyers. By envelope theorem, we have in any incentive compatible
mechanism:
[
( i) ]
∂U i (h (ai ))
−rτ i (h(ai )) i =
E
e
θ
h a .
∂θi
The transfers can be backed from here. It is a nice exercise to show that the
Dynamic Pivot Mechanism solves the problem of efficient allocation in this
model.
By usual tricks (integration by parts or Fubini etc.) the revenue to the
seller in any mechanism that leaves zero rent to the bidders wit the lowest
type is given by:
[
(
)]
1 − F (θi )
−rτ i (a,θ)
i
E Σi e
θ −
.
f (θi )
Despite the apparent simplicity of this formula, the allocation decision for
the seller is really hard. Even in the case of a single good, it will in general
be a non-stationary optimal stopping problem where recall features as part
of the optimal solution (even for large T ).
Full incentive compatibility for the bidders boils down to monotonicity
of allocation times in θi . It can be shown that the solution to the relaxed
problem has this feature. Furthermore is a nice exercise to think of reasonable
indirect implementations of the revenue maximizing allocation.
108
Unobservable Arrivals
Assume next that the arrivals are not observable. Notice that there are
informational externalities amongst the buyers since the arrival of bidder i
conveys information about the other arrival times. To implement efficiency,
subsidies are then sometimes needed where early arrivals receive a subsidy
even if they are not allocated the good. Check this with Dynamic Pivot
Mechanism. This makes the efficient mechanism vulnerable to fake arrivals.
A stopping time τ is said to be monotone in arrival times if for all i, ai , b
ai ,
all a−i and all v,
((
) )
(( i −i ) )
τ i ai , a−i , v ≤ τ i b
a ,a ,v .
The main result in the paper states the following.
Theorem 20 If the arrival process is Markov and (τ, p) is such that
1. Under observable arrivals p implements τ,
2. τ is monotone with respect to arrivals,
3. (τ, p)leaves all bidders with θi = 0 with zero payoff,
then p implements τ under unobservable arrivals.
The proof is not hard. It consists of the following two steps. Misreporting
valuation is not profitable by 1. Delaying reporting is not optimal by 2. and
1. Combining the two, the result follows.
An unfortunate feature of the model is that the problem of deciding the
allocations is so hard even with observable allocations that it is not easy to
verify part 2. of the conditions.
References
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Strategy Equilibria in Games of Incomplete Information.” Econometrica
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URL: http://dx.doi.org/10.1111/1468-0262.00223
109
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URL: http://dx.doi.org/10.3982/ECTA10592
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