Econ 805 – Advanced Micro Theory I
Dan Quint
Fall 2007
Lecture 7 – Sept 27 2007
Tuesday: Amit Gandhi on empirical auction stuff
Up till now, we’ve mostly been analyzing auctions under the following assumptions:
1. Bidders (and seller) are risk-neutral
2. Bidders are ex-ante symmetric
3. Bidders’ types are independent
4. Bidders have private values
The next several lectures, we’ll be relaxing each of these assumptions. Today, we relax riskneutrality. Next week, we relax symmetry. The following week, we relax both independence
and private values.
Specifically, today, we compare the first- and second-price auctions when bidders are riskaverse (PATW 4.3.1), and discuss Maskin and Riley, “Optimal Auctions with Risk-Averse
Bidders.”
• We’ve talked some about optimal mechanisms; but in some cases, implementing a
complicated direct-revelation mechanism is unrealistic
• Empirically, most auctions for a single good are either a first- or second-price auction
with a reserve price (or something that’s strategically equivalent to one of these)
• These require minimal information on the part of the seller, generally give “pretty
good” performance
• (We saw with symmetric IPV, with the right reserve price, either is optimal)
• So a reasonable question when we come to risk-averse bidders: when bidders are
risk-averse, which auction performs better, a first- or a second-price auction?
• Maskin and Riley give a very general formulation of risk-averse bidder preferences
with private values; but here’s the simplest/most natural one:
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• Under risk-neutrality with private values, we’ve been assuming a bidder’s payoff is
(
0
if lose
u=
t − b if win
Instead, have each bidder maximize the expected value of
(
U (0)
if lose
u=
U (t − b) if win
where U is some increasing, concave von Neumann-Morgenstern utility function
• In this setting, we get a nice sharp revenue-ranking result:
Theorem 1. Suppose U is strictly concave and differentiable. Then
• In a second-price auction with a reserve price r, bidders bid the same as they would
if they were risk-neutral: bidders with valuations below r do not submit serious bids,
and bidders with valuations above r bid their value
• In a first-price auction with a reserve price r, every bidder with types t > r bids higher
than in the equilibrium when bidders are risk-neutral
With risk-neutral bidders, the two auctions are revenue-equivalent; so with risk-averse bidders, the first-price auction yields strictly higher expected revenue.
• The proof that b(ti ) = ti is a dominant strategy is exactly the same as before. So
risk aversion doesn’t change equilibrium bids, and therefore revenue, in a second-price
auction. (It does change bidder payoffs, of course.)
• For the first-price auction, the intuition is this: fixing the opponents’ bid distribution,
my optimal bid in a first-price auction is higher when I’m risk-averse. This is because,
starting at the bid that maximizes my expected (risk-neutral) gain, raising my bid
a little bit more is the same as buying partial insurance that’s priced very close to
actuarily fair. To put it another way, I’m better off when I win than when I lose,
which means my marginal utility of wealth is lower when I win; so I’m happy to give
up more in those cases (by bidding higher) to improve my outcome in some of the
cases where I was losing.
• To see this, let G be the probability distribution of the highest of everyone elses’ bids.
If I’m risk-neutral and have private value t, I maximize
(t − b)G(b)
with first-order condition
(t − b)g(b) − G(b)
so if bRN is my risk-neutral best-response, g(bRN ) = G(bRN )/(t − bRN )
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• Now suppose I’m risk-averse, with some Bernoulli utility function U , and have the
same type t and am facing the same opponent distribution G. If we normalize U (0) =
0, then I maximize
U (t − b)G(b)
with first-order condition
U (t − b)g(b) − U 0 (t − b)G(b)
Let’s plug in bRN and check the sign:
U (t − bRN )g(bRN ) − U 0 (t − bRN )G(bRN ) =
U (t − bRN )
G(bRN ) − U 0 (t − bRN )G(bRN )
t − bRN
• By the intermediate-value theorem,
U (t − b) − U (0)
= U 0 (a)
t−b
for some a ∈ (0, t − b), so this is
¡
¢
G(bRN ) U 0 (a) − U 0 (t − bRN )
but since a < t − bRN and U is concave, this is positive. So at the same type and the
same opponent bid distribution, I bid higher when I’m risk-averse.
• Of course, that isn’t a proof; we need to make sure that this holds up in equilibrium,
that is, when everyone else shifts their equilibrium bid functions as well. The proof is
from Putting Auction Theory to Work, pages 122-125.
Before we get to the proof, a word on notation.
• Thus far, we’ve been using the convention that at a type (or signal) ti , a bidder’s
value from winning the object is exactly ti
• Works well with private values; but as we get into common-value and other more
general auctions, we’ll need bidder’s valuation as some function of everyone’s types
• We’ll use the notation vi (t) for bidder i’s value from winning the object, given a vector
of types t = (t1 , . . . , tN )
• We can see IPV as the special case where vi depends only on ti , not other bidders’
types
• Once we’re assuming that the value of winning the object is vi (ti ), not ti , there’s no loss
of generality from assuming that signals ti are drawn from a particular distribution,
say, the uniform distribution on the interval [0, 1]
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• (This is the normalization/convention used in the Milgrom book)
• That is, in a general model, bidder i’s type is drawn from an arbitrary distribution
Fi , and leads him to value the object at vi (ti ); we’ve been using the normalization
that vi (ti ) = ti , Milgrom uses the normalization that Fi (ti ) = ti .
• The mapping between the two notations is very clean: vi in one world is simply Fi−1
in the other. That is, start in our world, where vi (ti ) = ti and ti is drawn from the
distribution Fi . If instead of observing ti , the bidder observes Fi (ti ), that is, where
his type lies in its distribution, then we’re in Milgrom’s world. So it’s just a notation
adjustment.
• For today, we’ll stay in our old notation; once we get into interdependent values,
though, we’ll move into the other notation. But if you look in PATW for the proof,
it’s done in the v notation.
Lemma 1. If log U is concave and differentiable, then the unique symmetric equilibrium
βU of the first-price auction with reserve price r is the solution to the differential equation
(N − 1)f (t)
U 0 (t − βU (t))
=
F (t)βU0 (t)
U (t − βU (t))
with boundary condition βU (r) = r.
The bidder’s problem is to maximize
Pr(win)U (t − b) + Pr(lose)U (0)
Normalize U (0) = 0, so the second term vanishes, and take the log, giving the bidder’s
problem as
max log U (t − b) + log H(b)
b
where H(b) is the probability of winning, conditional on bidding b. Differentiating with
respect to b gives
1
U 0 (t − b)
+
H 0 (b)
−
U (t − b)
H(b)
If everyone else is bidding according to the equilibrium bid function βU , then
¡
¢N −1
H(b) = Pr(βU (tj ) < b∀j 6= i) = Pr(tj < βU−1 (b)∀j 6= i) = F (βU−1 (b))
and so
¢N −2
¡
¢0
¡
¢N −2
¡
f (βU−1 (b))
f (βU−1 (b)) βU−1 (b) = (N −1) F (βU−1 (b))
H 0 (b) = (N −1) F (βU−1 (b))
1
βU0 (βU−1 (b))
We want the first-order condition to hold with equality at b = βU (t), or βU−1 (b) = t, so
plugging this in, the first-order condition becomes
−
U 0 (t − βU (t))
1
1
+ N −1 (N − 1)F N −2 (t)f (t) 0
=0
U (t − βU (t))
F
(t)
βU (t)
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or
(N − 1)f (t)
U 0 (t − βU (t))
=
0
F (t)βU (t)
U (t − βU (t))
Bidders with t < r must bid below r, and bidders with t > r must bid at least r
(since that gives them a positive probability of winning) but below t, so it’s easy to show
that βU (r) = r. Standard differential-equation stuff says there’s a unique solution to the
differential equation with the boundary condition βU (r) = r, and the equation itself makes
it clear any solution must be increasing.
Since βU (t) < t, the right-hand side is positive, so βU is strictly increasing above r. We
derived β(t) from the first-order condition, so thinking of the objective function as f (x, t),
we showed by construction that fx (β(t), t) = 0. By the chain rule, then, since V = f (x(t), t),
V 0 (t) =
∂
f (β(t), t) = fx (β(t), t)β 0 (t) + ft (β(t), t) = 0 + ft (β(t), t)
∂t
which establishes the envelope theorem. We showed that for risk-neutral bidders, β increasing and satisfying the envelope theorem was sufficient for it to be a symmetric equilibrium;
we didn’t do the more general proof of sufficiency, but the result is more general, and so βU
is the unique symmetric equilibrium.
Next, we introduce a nice trick for comparing two functions.
Lemma 2. (Ranking Lemma.) Consider two continuous, differentiable functions g, h : < →
<. Suppose g(x∗ ) ≥ h(x∗ ), and for x ≥ x∗ ,
g(x) = h(x) → g 0 (x) > h0 (x)
Then for all x > x∗ , g(x) > h(x).
Proof. Suppose g(x) ≤ h(x) for some x > x∗ . Let x̂ ≡ inf {s > x∗ : g(s) ≤ h(s)}. Since
either g(x∗ ) > h(x∗ ) or g(x∗ ) = h(x∗ ) and g 0 (x∗ ) > h0 (x∗ ), x̂ > x∗ . By continuity, g(x̂) =
h(x̂), so by assumption, g 0 (x̂) > h0 (x̂), so g(s) < h(s) for s just below x̂, contradicting the
definition of x̂.
This leads us to a proof that equilibrium bids with risk aversion are higher than with
risk-neutrality.
Lemma 3. Let β be the symmetric equilibrium bidding strategy in a risk-neutral auction,
and βU the symmetric equilibrium bidding strategy in our risk-averse auction. For t > r,
βU (t) > β(t).
We know that β(r) = βU (r) = r. For t > r, we know
U 0 (t − βU (t))
(N − 1)f (t)
=
F (t)βU0 (t)
U (t − βU (t))
and, since the risk-neutral auction is the same but with U (s) = s,
(N − 1)f (t)
1
=
F (t)β 0 (t)
t − β(t)
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and so
β 0 (t)U 0 (t − βU (t))
β 0 (t)
(N − 1)f (t)
=
= U
t − β(t)
F (t)
U (t − βU (t))
We normalized U (0) = 0 and assumed U was strictly concave, so for x > 0,
Z x
Z x
U (x) =
U 0 (s)ds >
U 0 (x)ds = xU 0 (x)
0
so
so
0
1
U 0 (x)
<
U (x)
x
βU0 (t)
U 0 (t − βU (t))
β 0 (t)
= βU0 (t)
<
t − β(t)
U (t − βU (t))
t − βU (t)
So when βU (t) = β(t), βU0 (t) > β 0 (t); and we know that βU (r) = β(r) = r, so by the
ranking lemma, βU (t) > β(t) for all t > t∗ .
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Maskin and Riley, “Optimal Auctions with Risk-Averse Bidders”
So Maskin and Riley is a pretty long paper, and the math is pretty hard, so we’re not
going to go into all the details. As a general point, they mention that the introduction of
risk-averse bidders changes the seller’s problem in two ways:
• Since the seller is risk-neutral and bidders are risk-averse, the seller can profit by
“selling the bidders insurance.” That is, relative to a standard, say, second-price
auction, the seller can offer a deal to transfer some surplus from the “bidder i wins”
case to the “bidder i loses” case, at less than fair value, and the bidder will still accept
the deal
• In addition, risk-aversion gives the seller another way to “punish” high types who bid
low, allowing the seller to extract more of their surplus (or making them easier to
screen)
The first would lead toward the seller removing all risk from the bidders, but the second
makes it optimal to leave some.
Maskin and Riley give a very general formulation of risk-averse preferences, then make a
bunch of assumptions, and then give several examples of more narrowly-defined formulations
that satisfy all their assumptions. Their general framework is that bidders are symmetric;
have independent types θi ; and have two utility functions, one, u, for when they win the
object, which is a function of wealth and θ; and one, w, for when they don’t win the object,
which is a function only of wealth. They normalize starting wealth to 0, so bidders maximize
E {Hi (s)u(−βi (s), θi ) + (1 − Hi (s))w(−αi (s))}
where
• θi is bidder i’s true type
• s is a vector of all the bidders’ reported types
• Hi is the probability that bidder i gets the object given reports s
• βi is what he pays if he gets it
• αi is what he pays if he doesn’t get it
• and the expectation is taken over everyone elses’ types, given their equilibrium strategies
Except that also, β and α are allowed to be stochastic, so the expectation is taken over
their realizations as well.
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They make two sets of assumptions on the utility functions u and w. The first set,
Assumption A, are very standard:
• u and w three times differentiable
• u and w increasing in wealth
• w(0) = 0 (normalization)
• u and w concave (risk aversion)
• u increasing in θ (higher types want the object more)
The second set (“Assumption B”) are harder to interpret, but they give some examples
where they hold:
• uxθ < 0
• uθθ < 0
• ux (−t1 , θ) < wx (−t2 ) → u(−t1 , θ) > w(−t2 )
• uxθθ ≥ 0
• uxxθ ≥ 0
The give four cases where assumptions A and B will hold, to make the case that the
assumptions aren’t too crazy.
Case 1 – “Certain Quality, Equivalent Monetary Value”
This is the case we looked at already – u(−t, θ) = U (θ − t) and w(−t) = U (−t), where U
is concave and increasing.
Case 2 – “Certain Quality, Additive Utility, No Equivalent Monetary Value”
This is a generalization of case 1, where u(−t, θ) = U (θ + Ψ(−t)), w(−t) = U (Ψ(−t)), with
U and Ψ concave and increasing.
In both these cases, assumptions A and B are satisfied if U is increasing, concave, and
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U ≥ 0.
Case 3 – same as Case 1, but with uncertain quality
This is a generalization of case 1 where the actual value of the object is stochastic, but
higher types value it more stochastically. u(−t, θ) = Ev|θ U (v − t) and w(−t) = U (−t),
where the distribution of v is increasing in θ, that is, for θ0 > θ, the distribution of v given
θ0 first-order stochastically dominates the distribution of v given θ.
In this case, A and B hold under some additional assumptions.
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Case 4 – “Intensification”
This is when higher θ also leads to higher marginal utility of income – u(−t, θ) = (θ +
1)U (θ − t), w(−t) = U (−t).
In this case, A and B hold if the coefficient of absolute risk aversion (−U 00 /U 0 ) is nonincreasing and greater than 2 everywhere.
Some Preliminary Results
Theorems 2-4 establish that at a given reserve price, the first-price auction outperforms the
second-price auction – this is what we already proved for Case 1, they show it generally
under Assumptions A and one more technical condition. Theorem 5 shows that if the
seller is also risk-averse, he still prefers the first-price auction. (This basically combines two
results we knew: one, risk-averse seller with risk-neutral buyers prefers first-price auction;
two, risk-neutral seller with risk-averse buyers prefers first-price; so it makes sense that
risk-averse seller with risk-averse buyers would also prefer first-price.)
Theorem 6 is that the seller does not gain by fully insuring the buyers. In general,
when you have a risk-neutral principal with a risk-averse agent, there’s profit to be made
by the principal from effectively selling insurance to the agent – in this case, insuring a
bidder of each type against the uncertainty created by the other bidders’ types. However,
in an auction, this insurance interferes with the seller’s ability to extract greater surplus
from higher types; Theorem 6 says that with case 1 preferences, a perfect insurance auction
generates the same expected revenue as a second-price auction; and we already know this
is lower than a first-price auction.
Back to the Optimal Auction Problem
As in the other papers we’ve been looking at recently, Maskin and Riley use direct revelation
mechanisms. They limit themselves to symmetric auctions. (No loss of generality, since if
an asymmetric auction was optimal, they could randomly permute the players ahead of
time and end up with a symmetric auction with the same revenue.)
First, they consider auctions where bidder i’s payment does not depend on θj , only on
θi and whether or not he is awarded the object. So let
• G(θi ) be probability of winning (expectation taken over other types, assuming truthful
revelation)
• b(θi ) be payment you make conditional on winning
• a(θi ) be payment you make conditional on not winning
The seller’s expected revenue, then, is
Z
N [G(θ)b(θ) + (1 − G(θ)a(θ)]dF (θ)
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So to calculate the “optimal deterministic auction,” this is what the seller maximizes,
subject to the usual constraints: individual rationality (everyone is willing to play the
game) and incentive compatibility (truthful revelation is an equilibrium).
Theorem 7 is basically that for a probability-of-winning function G to be feasible, it
must be that the probability of winning at type θ is less than or equal to the probability
that you have the highest type; and that if G is nondecreasing, this is sufficient.
Theorems 8 and 9 are where they solve for the optimal auction. Make Assumptions
A and B and one more technical condition. Theorems 8 and 9 state that if the solution
R
to maximizing [G(θ)b(θ) + (1 − G(θ))a(θ)]dF (θ) over the choice variables G, a, and b,
subject to the envelope equation (equivalent to incentive compatibility, as we’ve seen before),
individual rationality, and G increasing and bounded above (their feasibility condition)...
If this solution satisfies a technical condition, then it’s the optimal auction, not just among
deterministic auctions, but among all feasible auctions. (In the case of risk-neutrality,
the technical condition they require collapses to regularity; with risk-aversion, it’s hard to
interpret exactly, but it’s basically a limit on how fast F 0 can decline, or a limit on “how
concave” the type distribution F can be.)
Under these same conditions, then, the rest of the paper gives a partial characterization
of the optimal auction:
• the optimal auction is deterministic – bidder i’s payment depends only on his type
and whether he gets the object, not the other bidders’ types; and the seller doesn’t
use “unnecessary randomness” to punish low types in order to screen high types
• but on the other hand, marginal utility is lower when winning then when losing at all
but the highest type – so except for the highest possible type, bidders are not fully
insured
• the highest type, however, is perfectly insured
• bidders of all types strictly prefer when they win to when they lose
• for types with a positive probability of winning, the probability of winning, and the
payment when you win, are strictly increasing in type
• as for a(θ), the payment you make when you don’t get the object... in a neighborhood
of the lowest type that every wins, losers make a payment that is positive and increasing; but under an additional condition, in a neighborhood near the highest possible
type, losers are subsidized (get paid by the seller)
For the special case of Case 1 preferences with decreasing absolute risk aversion, they
give a further characterization:
• Bidders pay more when they win than when they lose
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• Bidders always pay something when they win
• Expected revenue from a given bidder is increasing in his type
• There are types who never win, so the object is not always sold
They also give an interesting interpretation of the case with only 1 buyer, so it’s just a
buyer-seller game. They point out that with a risk-averse buyer, the type space is divided
into three intervals: low types, who don’t get the object; medium types, who get the object
with positive probability less than 1; and high types, who get the object for sure. They
offer another interpretation, which is that G(θ) < 1 corresponds to selling an object of lower
quality, since G(θ) could correspond to getting something for sure, but that object falling
apart with positive probability. So they argue that with risk-averse buyers, it’s optimal for
a monopolist to sell less-than-the-highest-quality goods to some types, even if quality costs
nothing to improve, since this improves the ability to extract more surplus from the higher
types.
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