Auction Theory
Class 5 – single-parameter
implementation and risk aversion
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Outline
•
What objective function can be implemented in
equilibrium?
–
•
Characterization result for single-parameter environments.
Revenue effect of risk aversion.
–
Comparison of 1st and 2nd price auctions.
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Implementation
•
Many possible objective functions:
–
–
Maximizing efficiency; minimizing gaps in the society;
maximizing revenue; fairness; etc.
Many exogenous constraints imply non-standard objectives.
•
Problem: private information
•
Which objectives can be implemented in equilibrium?
–
•
We saw that one can maximize efficiency in equilibrium.
What about other objectives?
We will show an exact characterization of
implementable objectives.
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Reminder: our setting
•
Let v1,…,vn be the private values (“types”) of the
players (drawn from the interval [a,b])
•
Each player can eventually win or lose.
–
–
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An allocation function: Q:[a,b]nq1,…,qn
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Winning gains the player a value of vi, losing gains her 0.
(More general than single-item auction.)
qi = the probability that player i wins.
Given an allocation function Q, let Qi(vi) be the
probability that player i wins.
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In average, over all other values.
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Characterization
Recall that an auction consists of an allocation function
Q and a payment function p.
Theorem: An
An auction
auction (Q,p)
(Q,p) isis truthful
truthful ifif and
and only
only if
if
Theorem:
1. (Monotonicity.)
(Monotonicity) QQi()
is non-decreasing for every i.
1.
i() is non-decreasing for every i.
vivi
2. (Unique
(Unique payments.)
payments) PPi(v
)= v ·Q (v ) –  Q (x)dx
2.
i(vi i)= vi i·Qi i(vi i) – aa Qi i(x)dx
Conclusion: only monotone objective functions are
implementable.
–
Indeed, the efficient allocation is monotone (check!).
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Reminder: our setting
Proof:
We actually already proved:
truthfulness  (monotonicity) + (unique payments)
Let’s see where we proved monotonicity:
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Proof
• Consider some auction protocol A, and a bidder i.
• Notations: in the auction A,
– Qi(v) = the probability that bidder i wins when he bids v.
– pi(v) = the expected payment of bidder i when he bids v.
– ui(v) = the expected surplus (utility) of player i when he
bids v and his true value is v.
ui(v) = Qi(v) v - pi(v)
• In a truthful equilibrium: i gains higher surplus when
bidding his true value v than some value v’.
–
Qi(v) v - pi(v) ≥ Qi(v’) v - pi(v’)
=ui(v)
=ui(v’)+ ( v – v’) Qi(v’)
We get: truthfulness  ui(v) ≥ ui(v’)+ ( v – v’) Qi(v’)
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Proof
• We get: truthfulness 
ui(v) ≥ ui(v’)+ ( v – v’) Qi(v’) or
u i (v) - u i (v’ )
 Qi (v' )
v – v’
• Similarly, since a bidder with true value v’ will not prefer
bidding v and thus
u i (v) - u i (v’ )
ui(v’) ≥ ui(v)+ ( v’ – v) Qi(v) or
 Q (v )
v – v’
Let dv = v-v’
u i (v' dv) - u i (v’ )
Qi (v' ) 
 Qi (v' dv)
dv
Taking dv  0 we get:
du i (v' )
 Qi (v' )
dv'
i
Given
that
v>v’
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Rest of the proof
We will now prove the other direction:
if a mechanism satisfies
(monotonicity)+(unique payments)
then it is truthful.
In other words: if the allocation is monotone, there is a
payment scheme that defines a mechanism that
implements this allocation function in equilibrium.
Let’s see graphically what happens when a bidder with
value v’ bids v>v’.
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Proof: monotonicity truthfulness
Proof: We saw that truthfulness is equivalent to:
for every v,v’ : ui(v) - ui(v’) ≤ ( v – v’) Qi(v)
We will show that (monotonicity)+(unique payments) implies the
above inequality for all v,v’. (Assume w.l.o.g. that v>v’)
We first show:
ui (v' )  v' Qi (v' )  pi (v' )
v'

 v'
 v' Qi (v' )   v' Qi (v' )   Qi (v)dv    Qi (v)dv
a

 a
Due to the
Due to
unique-payment
monotonicity
assumption
Now,
v
v'
v
a
a
v'
ui (v)  ui (v' )   Qi ( x)dx   Qi ( x)dx   Qi (v)dv  v  v'Qi (v)
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Single vs. multi parameter
Theorem: An auction (Q,p) is truthful if and only if
1. (Monotonicity.) Qi() is non-decreasing for every i.
(Unique payments.) Pi(vi)= vi·Qi(vi) – a Qi(x)dx
vi
2.
A comment: this characterization holds for general
single-parameter domains
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Single parameter domains: a private value is one
number.
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Not only for auction settings.
Or alternatively, an ordered set.
Multi-dimensional setting are less well understood.
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Goal for extensive recent research.
We will discuss it soon.
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Outline
•
What objective function can be implemented in
equilibrium?
–
•
Characterization result for single-parameter environments.
Revenue effect of risk aversion.
–
Comparison of 1st and 2nd price auctions.
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Risk Aversion
We assumed so far that the bidders are risk-neutral.
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Utility is separable (quasi linear), vi-pi
Now: bidders are risk averse )‫(שונאי סיכון‬.
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All other assumptions still hold.
We assume each bidder has a (von-NeumannMorgenstern) utility function u(∙).
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u is an increasing function (u’>0): u($10)>u(5$)
Risk aversion: u’’ < 0
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Risk Aversion – reminder.
u($10)
u( ½* $10 + ½*$5 )
½*u($10) + ½*u($5)
u($5)
A risk averse bidder
prefers the expected
value over a lottery
with the same
expected value.
$5
7.5
$10
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Auctions with Risk Averse Bidders
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The revenue equivalence theorem does not hold
when bidders are risk averse.
•
We would like to check:
with risk-averse bidders, should a profit maximizing
seller use 1st-price or 2nd-price auction?
•
Observation: 2nd-price auctions achieve the same
revenue for risk-neutral and risk-averse bidders.
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Bids are dominant-strategy, no uncertainty.
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Auctions with Risk Averse Bidders
Theorem:
Assume that
1. Private values, distributed i.i.d.
2. All bidders have the same risk-averse utility u(∙)
Then, E[1st price revenue] ≥ E[2nd price revenue]
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Intuition:
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risk-averse bidders hate losing.
Increasing the bid slightly increases their potential
payment, but reduces uncertainty.
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Gain ε when you win, but risk losing vi-bi
 The equilibrium bid is higher than in the risk-neutral case.
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1st price + risk aversion: proof
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Let β(v) be a symmetric and monotone equilibrium
strategy in a 1st-price auction.
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Notation: let the probability that n-1 bidders have
values of at most z be G(z), and G’(z)=g(z).
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Bidder i has value vi and needs to decide what bid
to make (denoted by β(z) ).
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That is, G(z)=F(z)n-1
Will then win with probability G(z).
Maximization problem: max z G( z)  ux   ( z)
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1st price + risk aversion: proof
Proof:
FOC:
Or:
max z G( z)  uv1   ( z)
g ( z)  uv1   ( z)  G( z)  u' v1   ( z)  ' ( z)  0
 ' ( z) 
u v1   ( z )  g ( z )

u ' v1   ( z )  G ( z )
But, since β(v) is best response of bidder 1, he must
choose z=x:
u v   (v )  g (v )
 ' ( z) 
1
1

1
u ' v1   (v1 )  G (v1 )
We didn’t use risk aversion yet…
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1st price + risk aversion: proof
Fact: if u is concave, for all x we have (when u(0)=0) :
u x 
x
u ' x 






or
u
x

x

u
'
x




u(x)
xu’)x(
x
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1st price + risk aversion: proof
For risk-averse bidders:
u x 
x
u ' x 
u v1   (v1 )  g (v1 )
g (v1 )
 ' ( z) 

 v1   (v1 )  
u ' v1   (v1 )  G (v1 )
G (v1 )
With risk-neutral bidders (u(x)=x, u’(x)=1 for all x).
b' ( z )  v1   (v1 )  
Therefore, for every v1
g (v1 )
G (v1 )
 ' (v1 )  b' (v1 )
Since β(0) = b(0) =0, we have that for all v1  (v1 )  b(v1 )
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1st price + risk aversion: proof
Summary of proof:
in 1st-price auctions, risk-averse bidders bid higher
than risk-neutral bidders.
 Revenue with risk-averse bidders is greater.
Another conclusion: with risk averse bidders,
Revenue in
1st-price
auctions
>
Revenue in
2nd-price
auctions
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Summary
•
We saw today:
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Monotone objectives can be implemented (and only them)
Risk aversion makes sellers prefer 1st-price auctions to 2ndprice auctions.
So far we discusses single-item auction in private value
settings.
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Next: common-value auctions, interdependent values,
affiliated values.
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