note 3a mechanism design - peter cramton

Lecture Note 3: Mechanism Design

• Games with Incomplete Information – Imperfect Information vs. Incomplete Information

– Bayesian Games

– The Revelation Principle

1

Imperfect Information vs. Incomplete

Information

Definitions

• Game of imperfect information: one or more players do not know the full history of the game. (Porter’s model on cartel maintenance).

• Game of incomplete information: the players have different private information about their preferences and abilities.

2

Imperfect Information vs. Incomplete

Information

The key to analyzing games of incomplete information is to transform them into games of imperfect information by letting nature move first, randomly selecting each player's payoff function.

3

Example 1: Symmetric oligopoly model with unknown costs

• Firm i's marginal cost ci may be either low or high {l,h}.

• Firm i observes its cost (nature's choice of l or h) but not the costs of the other firms.

• In the game of imperfect information, j does not observe natures choice of ci, but j holds probabilistic beliefs about the likelihoods of nature's choice, summarized by a probability p that nature chose l.

4

Example 2: First-Price, Sealed-Bid Auction

• Suppose a seller decides to use a first-price, sealed-bid auction to allocate a good to one of two buyers.

• Let nature's choice of the buyers' valuations for the good, v1 and v2, be independently and uniformly distributed on [0,1].

• Based on vi, player i submits a bid bi(vi). The player with the highest bid gets the good and pays her bid.

5

Bayesian Games (Harsanyi, Management Science 1967-8)

• normal form game G = {A1,...,An; u1,...,un}

• Bayesian game = {A1,...,An; T1,...,Tn; p1,...,pn; u1,...,un}

• Ai = strategy set for i, actions: a = (a1,...,an) A = A1...An.

• Ti = type space for i, types: t = (t1,...,tn) T = T1...Tn

• pi = beliefs for i, pi(t-i | ti) = i's belief about types t-i given type ti.

• ui = utility function for i, ui(a,t) depends on both actions a and types t.

6

Bayesian Games (Harsanyi, Management Science 1967-8)

• Beliefs {p1,...,pn} are consistent if they can be derived from Bayes' rule from a common joint distribution p(t) on T; i.e., there exists p(t) such that

where

for all i and ti.

• Beliefs are consistent if nature moves first and types are determined according to p(t) and each i is informed only of ti.

7

p t |tp(t)

p(ti -i i

i

( ))

p(t p(t ti -i i

t T-i -i

) , )

• Types ti are the valuations vi ; Ti = [0,1].

• Actions ai are the bids bi ; Ai = [0,).

• pi(tj|ti) = 1 for all ti and tj.

•

b(vi) = vi/2 is the unique symmetric equilibrium bidding strategy 8

Example: First-Price Auction

u a, t

t a if a a

(t a if a a

if a a

i

i i i j

i i i j

i j

( ) ) /

.

RS|T|

2

0

• A strategy for i is a plan of action for each of i's possible types i:TiAi. That is, what to do in every possible contingency (each of the possible types).

• A strategy profile = (1,...,n) is a Bayesian equilibrium of if

9

Definitions

p (t t u t), t]

p (t t u t a t] i, a A

i -i i i

t T

i -i i i -i -i i

t T

i i

-i -i

-i -i

| ) [ (

| ) [( ( ), ), .

• Existence of a Bayesian equilibrium when the type sets and pure-strategy spaces are finite follows from the standard existence theorem for finite games.

• Given consistent beliefs, a Bayesian equilibrium of is simply a Nash equilibrium of the game with imperfect information in which nature moves first.

• Any game of incomplete information with consistent beliefs can be transformed into a standard normal form game.

10

Bayesian Equilibrium

Revelation Principle (Myerson, Econometrica 1979 and others)

An equilibrium of a Bayesian game can be represented by a simple equilibrium of a modified Bayesian game ' as follows:

• = {A1,...,An; T1,...,Tn; p1,...,pn; u1,...,un}

• ' = {A1',...,An'; T1,...,Tn; p1,...,pn; u1',...,un'}

• Ai' = Ti (each player reports her private information (possibly dishonestly))

• ui'(a',t) = ui[(a'),t] (by reporting type ti you get the payoff that ti gets by playing the equilibrium strategy i(ti) in )

11

• For any Bayesian equilibrium of , reporting your true type is a Bayesian equilibrium of '.

• In the game , types are mapped into actions via strategies and then these actions are mapped into outcomes via the utility functions.

• In the game ', types are mapped directly into outcomes, by the composition of the utility and strategy functions.

12

Revelation Principle

Direct Revelation Game

13

Original Game ()

Strategy

Payoffu

TypesT

Actions A

Outcomes

n


14

Original Game ()

Strategy

Payoffu

TypesT

Actions A

Outcomes

n

Direct Revelation Game (’)

IdentityI

Payoffu'=u o

TypesT

Actions A

Outcomes

n


• In the direct revelation game, the players' equilibrium strategy profile is simply the identity map I(t) = t.

• A direct mechanism ' in which truthful reporting is a Bayesian equilibrium is call incentive compatible.

• The revelation principle states that without loss of generality, the analysis of Bayesian equilibria can be restricted to incentive compatible direct mechanisms.

15

Provision of a Public Good

(Groves, Econometrica 1973)

Example: Paving a road

• Three households

• Cost of paving the road: c

• Independent private valuation: vi Fi

• If the road is built, the cost c must be paid from some combination of funds from the three households; no subsidies from outside sources are available.

16

Procedure 1

• The three households simultaneously announce their values: b1, b2, and b3.

• Decision rule: Pave the road if b1+b2+b3 c,

and i pays

• Problem: bi = vi is not an equilibrium.

• Best response: bi < vi, the more honest the others are the more you should lie. This misrepresentation leads to inefficiency.

17

b

b b bci

1 2 3

Procedure 2: Groves Mechanism

• Each household simultaneously reports its valuation by making the bid bi(vi).

• If b = b1 + b2 + b3 > c, then the road is paved, and each contributes the amount the other players' bids fall short of c. That is,

18

i 1 2 3 i

i j k j k

i j k

u (b ,b ,b ,v )

0 if b c

v (c-b b ) if b c and b b c,

v if b c and b b c.

Groves Mechanism

Claim : Truth-telling is a dominant strategy.

• Your bid does not influence how much you pay, only whether the road is paved.

• If b-i c, then ui = vi and bi=vi is a BR.

• If b-i < c, then

• Your payoff is maximized by bidding so that the road is paved whenever vi - (c - b-i ) 0, which is accomplished by bidding bi = vi.

19

uv -(c -b ) if b c -b

0 otherwisei

i -i i -i

RST

Groves Mechanism

Problems:

• Does not satisfy budget balance

The sum of the payments 3c - 2b c, since bc, so having a benefactor to make up the deficit is essential.

• Incentive to collude

The more the households value the road, the more the benefactor must contribute. Hence, the players have a strong incentive to collude and overstate their valuations, so that the benefactor pays a larger share of the road.

20

Second-Price Auction

• n bidders

• Each has a valuation vi for the good being auctioned, where each bidders' valuation is private information.

• Suppose the seller uses a second-price auction to allocate the good: each bidder simultaneously submits a bid and the good goes to the highest bidder, who pays the seller a price equal to the second highest bid.

21

Second-Price Auction

Claim: Truth telling is a dominant strategy.

• As in Grooves mechanism, player i's bid does not influence the price paid if i wins, but does affect whether the player wins.

• The optimal bid is such that player i wins whenever p < vi. This is accomplished by bidding vi: by bidding bi < vi, i stands to lose some profitable opportunities, and by bidding more than vi, i may lose by winning.

22


• Bilateral Trading Mechanisms

– War of Attrition

– Simultaneous Offers

– The Public Choice Problem

23

Games of Timing: War of Attrition

• Two animals are fighting for a prize

• Each knows the value of the prize to himself, but not to the other

• The valuations vi (i=1, 2), are i.i.d. with distribution F and density f on [0,1]

• Each incurs a cost of c for each minute the fight continues

How long should an animal i with valuation vi wait before conceding?

24

War of Attrition: Symmetric Bayesian equilibrium

• Let ti(vi) be the stopping time for animal i with valuation vi

• Suppose ti’ > 0

• Let xi(ti) = ti-1(ti) be the valuation of animal i if it

concedes at time ti

• Animal i's payoff is

25

u v ,v , t , tv ct if t t

ct if t ti 2

i j j i

i j i

( ).1 2 1

RST


• i seeks to maximize her expected utility given animal j's strategy tj(); that is for each vi, ti is chosen to

• F.O.C.

26

max ] ) [ ))].t

i j j j

x

i j ii

j(ti )

[v ct f(v dv ct F(x (tXZY

0

1

x (t v f(x (t c F(x (tj i i j i j i) )) [ ))] .1 0


• Imposing symmetry:

• In equilibrium, x(t) = v, and x'(t) = 1/t'(v), so

27

x (t) =c F(x(t

vf(x(t

[ ))]

))

1

t (v) =vf(v

c F(v

)

[ )]1


• Equilibrium Strategy

28

t(v) =zf(z

c F(zdz

0

v)

[ )]1

XZY

Simultaneous Offers (Chatterjee & Samuelson, Operations Research 1983)

• A seller and a buyer are engaged in the trade of a single object worth s to the seller and b to the buyer.

• Valuations are known privately, as summarized below

29

TradersValue

DistributedPayoff

PrivateInfo

CommonKnowledge

Strategy(Offer)

Seller s sF on [s, s] u =P – s s F, G p(s)

Buyer b bG on [ , ]b b v =b – P b F, G q(b)

Simultaneous Offers

• Independent private value model: s and b are independent random variables.

• Ex post efficiency: trade if and only if s < b.

• Game: Each player simultaneously names a price; if p q then trade occurs at the price P = (p + q)/2; if p > q then no trade (each player gets zero).

30

Simultaneous Offers

• Payoffs:

– Seller

– Buyer

where the trading price is P = (p + q)/2 31

u(p,q,s,b) =P - s if p q

0 if p > q

RST

v(p,q,s,b) =b - P if p q

0 if p > q

RST

• Let F and G be independent uniform distributions on [0,1].

• Equilibrium conditions:

32

Example

(1) s [s,s],p(s) argmaxE {u(p,q,s,b)|s,q( )}

(2) b [b,b],q(b) argmaxE {v(p,q,s,b)|b,p( )

pb

qs

}

• Assume p and q are strictly increasing.

• Let x() = p-1() and y() = q-1().

• Optimization in (1) can be stated as

• First-order condition

-y'(p)[p - s] + [1 - y(p)]/2 = 0,

since q(y(p)) = p 33

Seller’s Problem

max [(p q(b)) / 2 s]dbp

y(p)

1

XZY

• Optimization in (2) can be stated as


x'(q)[b - q] - x(q)/2 = 0,

since p(x(q)) = q.

34

Buyer’s Problem

max [b - (p(s) q) / 2]dsq

x(q)

XZY0

Equilibrium

• Equilibrium condition:

s=x(p) and b=y(q)

• Equilibrium first-order conditions:

(1') -2y'(p)[p - x(p)] + [1 - y(p)] = 0,

(2') 2x'(q)[y(q) - q] - x(q) = 0.

35

• Solving (2') for y(q) and replacing q with p yields

• Substituting into (1') then yields

36

Solution

(2 ) y(p) p1

2

x(p)

x (p), so y (p)=

3

2

1

2

x(p)x (p)

[x (p)]2

(1 ) [x(p) -p] 3 -x(p)x (p)

[x (p)]1 p

1

2

x(p)

x (p)=0

2

LNM

OQP

LNM

OQP

• Linear Solution:

x(p) = p + .

with = 3/2 and = -3/8.

• Using (2") yields

y(q) = 3/2 q - 1/8.

• Inverting these functions results in

p(s) = 2/3 s + 1/4 and q(b) = 2/3 b + 1/12 37

Analytical Solution

Figure 1

38

0.0

1.0

Valuations s, b

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0

Offers p, q

p(s)

q(b)

• Trade occurs if and only if

p(s) q(b), or b - s 1/4

• The gains from trade must be at least 1/4 or no trade takes place.

The outcome is inefficient 39

Outcome

The Public Choice Problem

• There are two members of society i {1,2}

• Public project: do it/not do it, d {0,1}

• Individual benefits: vi (-,), where vi is known privately to i

• Ex post efficiency requires:

Find mechanism to implement efficient choice rule

40

d (v ,v )1 if v +v 0,

0 otherwise.*

1 21 2

RST

Mechanism Design

By the Revelation Principle, restrict attention to incentive-compatible direct mechanisms {d(v),t(v)}

– v = {v1,v2}, d:2{0,1} determines the decision as a function of the reports

– t:22 determines the transfers between the players where t(v) = {t1(v),t2(v)} and ti(v) is the transfer that player i receives.

41

Mechanism Design

• Find a mechanism that satisfies:

(1) efficient social choice: d(v) d*(v), and

(2) dominant-strategy incentive compatibility:

for all,

42

jv̂

v argmaxv d(v ,v ) t (v ,v )iv

i i j i i ji

.


If the reported types are , then

(D)

and

for some function hi:.

43

v (v ,v )1 2

d(v)1 if v v 0,

0 otherwise,

1 2

RST

t (v) d(v)v h (v )i j i j


Dominant strategy:

vi solves

if and only if vi solves

But this is the case, since reporting

makes equal to 1 if and only if

Truth is a dominant strategy 44

maxv d(v ,v ) + t (v ,v )v

i i j i i ji

maxd(v ,v )[v + v ].v

i j i ji

v vi i

d(v ,v )i j v + v 0i j


Problem: Transfers do not satisfy budget balance.

• Budget balance requires:

• But this cannot happen if hi is independent of vi 45

t (v) t (v) (v v )d(v ,v ) h (v ) h (v ) 0,

or

h (v ) h (v )=-(v v ) if v v 0,

0 otherwise.

1 2 1 2 1 2 1 2 2 1

1 2 2 11 2 1 2

RST

Procedure 2: Bayesian Game

Solution: weaken incentive compatibility criterion, so that truth is merely a Bayesian equilibrium rather than a dominant strategy.

(2') Bayesian incentive compatibility:

replacing (2) with (2') allows us to satisfy

(3) Budget balance: t1(v) + t2(v) = 0 for all v 46

j

i

i v i i j i i j iv̂

ˆ ˆv argmax E [v d(v ,v )+t (v ,v ) | v ]


• = {A1,A2; V1,V2; p1,p2; u1,u2}

• Ai = Vi = , ui(a,v) = vid(a) + ti(a)

• pi(vj | vi) = fj(vj), so types are independent

We wish to construct a mechanism satisfying (1), (2'), and (3). Our decision rule must be as in (D) to satisfy (1)

47


Budget Balance

• Consider the transfers ti(v) = gi(vi) - gj(vj), where

Clearly, the transfers balance.

48

g (v ) v d(v ,v )f (v )dvi i j i j j j jXZY

.


Incentive Compatibility

• Player i chooses the report to solve

• Substituting the definition of ti() yields

49

max [v d(v ,v ) + t (v ,v )]f (v )dvv

i i j i i j j j ji

.

XZY

max (v + v )d(v ,v )f (v )dvv

i j i j j j ji

,

XZY


Incentive Compatibility (cont.)

• Previous expression is equivalent to

when the decision rule in (D) is used.


so

50

max (v + v )f (v )dvv

i j j j j

vii

,

XZY

(v - v f ( vi i j i ) ) 0 v vi i


Problem: It may not satisfy individual rationality

(4) Interim individual rationality:

51

U (v ) = [v d(v ,v ) + t (v ,v )]f (v )dv

for all v V

i i i i j i i j j j j

i i

XZY

0 .


Substituting for d() and t() yields:

52

i

i

i i i j j j jv

i i i j i i

i j i j j j jv̂

i i i j i i

U (v ) (v +v )f (v )dv

v f (v )[1-F (-v )]dv

v [1-F (-v )]+ v f (v )dv

v f (v )[1-F (-v )]dv .


• Note that Ui'(vi) = 1 - Fj(-vi) 0, so if interim individual rationality fails, it fails for the lowest values of vi.

• Assume that the means and variances of vi and vj are finite. Then as vi - the first and second terms approach zero.

• If the integral in the third term is positive then for sufficiently low values of vi,

Ui(vi) < 0. 53


• Bilateral Trading Mechanisms

– A General Model

– Efficiency in Games with Incomplete Information

– Durability

54

A General Model (Myerson and Satterthwaite, JET 1983)

Direct Revelation Game:

– Bilateral Exchange with independent private value.

– s F with positive pdf f on

– b G with positive pdf g on

– F and G are common knowledge

In the DRG, the traders report their valuations and then an outcome is selected. Given the reports (s,b), an outcome specifies a probability of trade (p)

and the terms of trade (x). 55

[s, s]

[b,b]

Definition

Direct Mechanism

A direct mechanism is a pair of outcome functions p,x, where:

– p(s,b) is the probability of trade given the reports (s,b), and

– x(s,b) is the expected payment from the buyer to the seller.

56

Payoffs

Ex post utilities:

• Seller's ex post utility:

u(s,b) = x(s,b) - sp(s,b)

• Buyer's ex post utility:

v(s,b) = bp(s,b) - x(s,b)

Both traders are risk neutral and there are no income effects

57

Payoffs

Define:

X(s) is the seller's expected revenue given s

Y(b) is the buyer's expected payment given b

P(s) is the seller's probability of trade

Q(b) is the buyer's probability of trade 58

X(s) x(s,b)g(b)db Y(b) x(s,b)f(s)ds

P(s) p(s,b)g(b)db Q(b) p(s,b)f(s)ds.

b

b

s

s

b

b

s

s

XZY

XZY

XZY

XZY

Payoffs • Interim Utilities:

U(s) = X(s) - sP(s) V(b) = bQ(b) - Y(b)

• The mechanism p,x is incentive compatible if for all s, b, s’, and b’:

(IC) U(s) X(s') - sP(s') V(b) bQ(b') - Y(b')

• The mechanism p,x is individually rational if for all and

(IR) U(s) 0 V(b) 0.

59

s [s, s] b [b, b]

Lemma 1 (Mirrlees, Myerson)

The mechanism p,x is IC if and only if P() is decreasing, Q() is increasing, and

(IC’)

60

U(s) U(s) P(t)dt

V(b) V(b) Q(t)dt

s

s

b

b

XZY

XZY

Lemma 1: Proof

Only if:

• By definition, U(s) = X(s) - sP(s) and U(s') = X(s') - s'P(s'). This and (IC) imply

U(s) X(s') - sP(s') = U(s') + (s' - s)P(s'), and

U(s') X(s) - s'P(s) = U(s) + (s - s')P(s)

• Putting these inequalities together yields

(s' - s)P(s) U(s) - U(s') (s' - s)P(s')

61

Lemma 1: Proof

• Taking s' > s implies that P() is decreasing

• Dividing by (s' - s) and letting s' s, then yields dU(s)/ds = -P(s)

• Integrating produces (IC')

• The same is true for the buyer

62

Lemma 1: Proof

If:

• To prove (IC) for the seller, note that it suffices to show that

s[P(s) - P(s')] + [X(s') - X(s)] 0 for all s, s’

• Substituting for X(s') and X(s) using (IC') and the definition of U(s) yields

63

[s, s]

X(s) sP(s) U(s) P(t)dts

s

XZY .

Lemma 1: Proof

If:

• Then it suffices to show for every s,s’ that

which holds because P() is decreasing.

• The proof for the buyer is similar.

64

[s, s]

0 s[P(s) P(s )] + s P(s ) + P(t)dt sP(s) P(t)dt

(s s)P(s ) P(t)dt [P(t) P(s )]dt ,

s

s

s

s

s

s

s

s

XZY

XZY

XZY

XZY

Lemma 2

An incentive compatible mechanism p,x is individually rational if and only if

(IR') and V( ) 0.

65

U(s) 0 b_

Proof

– Clearly, (IR') is necessary for p,x to be IR

– By Lemma 1, U() is decreasing; hence, (IR') is sufficient as well

66

Lemma 2

• An incentive-compatible, individually rational mechanism p,x satisfies

67

Corollary

(*) U(s) + V(b)

b1 G(b)

g(b)s

F(s)

f(s)p(s,b)f(s)g(b)dsdb 0

b s

sb

LNM

OQP

XZY z .

Proof

• Using (IC') and the definition of U(s) yields

68

Lemma 2: Corollary

X(s) sP(s) U(s) P(t)dt.s

s

z

Lemma 2: Corollary

• Taking the expectation with respect to s (and substituting in the definitions of X(s) and P(s)) shows that

69

b

b

s

s

b

b

s

s

b

b

s

s

x(s,b)f(s)g(b)dsdb

U(s) sp(s,b)f(s)g(b)dsdb

p(s,b)F(s)g(b)dsdb.

XZYXZY

XZYXZY

XZYXZY

• The third term in the right hand side follows, since

70

Lemma 2: Corollary

s

s

s

s

s

s

s

t

s

s

p(t,b)f(s)dtds

p(t,b)f(s)dsdt p(s,b)F(s)ds .

XZYXZY

XZYXZY

XZY

• Preceding analogously for the buyer yields

• Equating the right-hand sides of the last two equations and applying (IR') completes the proof

71

Lemma 2: Corollary

b

b

s

s

b

b

s

s

b

b

s

s

x(s,b)f(s)g(b)dsdb

= -V(b) bp(s,b)f(s)g(b)dsdb

- p(s,b)f(s)[1- G(b)]dsdb.

XZYXZY

XZYXZY

XZYXZY

Theorem

72

If it is not common knowledge that gains

exist (the supports of the traders'

valuations have non-empty intersection),

then no incentive-compatible, individually

rational trading mechanism can be ex-

post efficient.

• A mechanism is ex-post efficient if and only if trade occurs whenever s b:

73

Proof

p(s,b)1 if s b

0 if s b.

RST

Proof

• To prove that ex-post efficiency cannot be attained, it suffices to show that the inequality (*) in the Corollary fails when evaluated at this p(s,b). Hence,

74

b s

min{b,s}b

b1 G(b)

g(b)s

F(s)

f(s)f(s)g(b)dsdbz

LNM

OQP

XZY

Proof

75

min{b,s} min{b,s}b b

b bs s

b b

b b

b b

b s

b

[bg(b)+G(b) 1]f(s)dsdb [sf(s)+F(s)]dsg(b)db

[bg(b)+G(b) 1]F(b)db min{bF(b),s}g(b)db (by parts)

[1 G(b)]F(b)db (b s)g(b)db

[1 G(b)]F(b)db

b b

s

s

b

[1 G(b)]db (by parts)

[1 G(t)]F(t)dt 0, since b s.

Proof

• The second term in the second line follows, since by integrating by parts

Since ex-post efficiency is unattainable, we need a weaker efficiency criterion with which to measure a mechanism's performance

76

[sf(s) F(s)]ds xF(x)s

x

XZY .

Definition

Pareto optimality:

• An allocation is Pareto optimal if there does not exist an alternative allocation that makes no parties worse off and at least one party strictly better off

77

Efficiency in Games with Incomplete Information

“A decision rule is efficient if and only if no other feasible decision rule can be found that may make some individuals better off without ever making any other individual worse off.”

78


Problems:

• What is meant by a feasible decision rule? Are we to recognize incentive constraints?

• What is meant by better off or worse off? On what information should the expectation be conditioned? Three alternatives are:

1. Ex ante information: a planner's information at the beginning of the game (no knowledge of types).

2. Interim information: a player's private information at the beginning of the game.

3. Ex post information: all the private information.

79


Problems:

• Who is to "find" the potentially better decision rule, and at what information stage? If a player proposes a particular decision rule after learning her private information, the other players may infer something about the player's type from the information.

80

Bayesian Game

• = {A1,...,An;T1,...,Tn;p1,...,pn;u1,...un}

• Each action set Ai and type set Ti is finite

• Beliefs pi are consistent.

• Let D be the set of probability distributions over A = A1...An.

• A decision rule (or direct mechanism) :TD maps reports into a randomization over feasible actions.

• The utility function ui(d,t):DT maps the decision and types into payoffs.

81

Bayesian Game

• A decision rule {:TD} is incentive compatible if for all i and tiTi

(IC)

for all

• Let * = {:TD} be the set of all incentive-compatible decision rules. By the revelation principle, we can restrict attention to *.

82

i i i i

i i i i i i i i i i

t T t T

ˆp (t |t )u ( (t),t) p (t |t )u ( (t ,t ),t)

t Ti i

Expected Utility

For a decision rule (), the expected utility at each of the three information stages are

• (1) Ex Ante Utility:

• (2) Interim Utility:

• (3) Ex Post Utility:

83

U ( ) p(t)u ( (t), t)i i

t T

U ( |t ) p (t |t )u ( (t), t)i i i i i i

t Ti i

U ( |t) u ( (t), t)i i

Domination

• ex ante dominates iff Ui() Ui() i with at least one strict inequality;

• interim dominates iff Ui(|ti) Ui(|ti) i and tiTi with at least one strict inequality;

• ex post dominates iff Ui(|t) Ui(|t) i and tT with at least one strict inequality.

84

Efficiency

85

• is ex post (classically) efficient iff there does not exist that ex post dominates

• is ex ante (incentive) efficient iff there does not exist * that ex ante dominates

• is interim (incentive) efficient iff there does not exist * that interim dominates

Trading Game (Myerson and Satterthwaite)

• Ex ante efficient mechanism that maximizes the expected gains from trade:

86

U(s)f(s)ds V(b)g(b)dbs

s

b

bXZY

XZY

Trading Game (Myerson and Satterthwaite)

• Myerson and Satterthwaite show that the ex ante efficient decision rule (probability of trade) is:

where

and is chosen so that U( ) = V( ) = 0. 87

p s,bif c(s, ) d(b, )

if c(s, ) d(b, )

( )

RST1

0

c( ,s) = s +F(s)

f(s)d( ,b) = b

1 G b

g b

( )

( )

s b

Remarks

• If = 0, then p is ex post efficient (all the weight on the objective function).

• If = 1, p maximizes the expression in (*); a constrained maximization.

• The ex ante efficient trading rule has the property that, given the reports, trade either occurs with probability one or not at all.

88

Example

Valuations are uniformly distributed on [0,1]

• Ex ante efficient mechanism: linear equilibrium in which trade occurs if and only if the gains from trade are at least 1/4 (Chatterjee & Samuelson)

• If the traders cannot commit to walking away from gains from trade, then they would be unable to implement this mechanism

• So long as it is not common knowledge that gains exist, the traders will, with positive probability, make incompatible demands in situations where gains from trade exist 89

Accomplishments

• Characterization of the set of all BE of all bargaining games in which the players' strategies map their private valuations into a probability of trade and a payment from buyer to seller

• Proof that ex post efficiency is unattainable if it is uncertain that gains from trade exist

• Determination of the set of ex ante efficient mechanisms

• Proof that ex ante efficiency is incompatible with sequential rationality

90

Durability

Are there further restrictions on the feasible set of decision rules that should be made?

• If the players have limited abilities to make binding commitments, then this limitation poses a further restriction on the set of feasible decision rules

• A second restriction on the set of feasible decision rules can come from the process of deciding on which decision rule to implement

91

Durability

Is it ever the case that a player, knowing her type, could suggest an alternative decision rule that the others would surely prefer?

92

Example

• Each of two players, 1 and 2, is of one of two types, a or b

• The players' utilities as a function of their types and which of three possible decisions {A,B,C} are:

93

2

1

0

1a 1b 2a 2b

0 4 9

2 1 0

2 1

-8

d=A d=B d=C

Decision Rule

• The ex ante efficient decision rule that maximizes the sum of the players' payoffs is:

(1a,2a) = A (1a,2b) = B

(1b,2a) = C (1b,2b) = B.

• No outsider could suggest an alternative decision rule that would make some type better without making another type worse off

94

Decision Rule

Problem:

• If player 1's type is 1a, then player 1 can suggest to 2 that decision A be adopted, and 2 would surely accept such a proposal

• In the words of Holmstrom and Myerson, the decision rule is not durable

A decision rule is durable iff the players would

never unanimously approve a change to any

other decision rule

95


• Multilateral Trading Mechanisms

– Dissolving a Partnership

– Optimal Auctions

96

Dissolving a Partnership (Cramton, Gibbons, and Klemperer, 1987)

• n traders. Each trader i {1,...,n} owns a share ri 0 of the asset, where r1 + ... + rn = 1

• As in MS, player i's valuation for the entire good is vi

• The utility from owning a share ri is rivi

• Private values, vi’s are iid F() on

• A partnership (r,F) is fully described by the vector of ownership rights r = {r1,...,rn} and the traders' beliefs F about valuations

97

[v,v]

Dissolving a Partnership

MS Case:

• n = 2 and r = {1,0}

• There does not exist a BE of the trading game such that:

(1) is (interim) individually rational and

(2) is ex post efficient

CGK Case:

• If the ownership shares are not too unequally distributed, then it is possible to satisfy both (1) and (2), (satisfying IC, IR, EE and BB) 98

Dissolving a Partnership

A partnership (r,F) can be dissolved efficiently if there exists a Bayesian Equilibrium of a Bayesian trading game such that is interim individually rational and ex post efficient

99

Theorem

• The partnership (r,F) can be dissolved efficiently if and only if

(*)

where vi* solves F(vi)

n-1 = ri and G(u) = F(u)n-1

100

[1 F(u)]udG(u) F(u)udG(u)v

v

v

v

i 1

n

i*

i*

LNM

OQP z z

0

Example

• n=3, F(vi) = vi.

• Then (*) becomes

101

r 3 4i3 2

i 1

3

1/3,1/3,1/3

(0, .3, .7)

(0, .3, .7) (.3, 0, .7)

(1, 0, 0)

(.7, 0, .3)

(0, 0, 1)

(0, 1, 0)

(.3, .7, 0)

(.7, .3, 0)

Proposition

For any distribution F, the one-owner partnership r = {1,0,0,...,0} cannot be dissolved efficiently.

– The one-owner partnership can be interpreted as an auction

– Ex post efficiency is unattainable because the seller's value v1 is private information: the seller finds it in her best interest to set a reserve price above her value v1

– An optimal auction maximizes the seller's expected revenue over the set of feasible (ex post inefficient) mechanisms

102

Theorem

• If a partnership (r,F) can be dissolved efficiently, then the unique symmetric equilibrium of the following bidding game is interim individually rational and achieves ex-post efficiency: given an arbitrary minimum bid b,

– the players choose bids bi [b,)

– the good goes to the highest bidder

– each bidder i pays

103

p (b , ,b ) b1

n 1bi 1 n i j

j i

n

Theorem

– each player receives a side-payment, independent of the bidding,

104

c (r , , r ) udG(u) udG(u).i 1 n

v

v

1n

v

v

j 1

ni*

j*

XZY

XZY

Optimal Auctions (Myerson, 1981)

• Let the n buyers be indexed by i {1,...,n}

• Let each buyer i's willingness to pay for the object be

• ti independently drawn from fi()

• The seller's type, t0, is common knowledge

105

t [a ,a ]i i i

Bayesian Auction

• A Bayesian auction consists of bids spaces {B1,...,Bn} and outcome functions

and

where B = B1...,Bn, is the probability that player i gets the object when the bids are b = {b1,...,bn} B, and is the payment from i to the seller when the bids are bB. For each b,

which allows for the possibility that the seller may keep the object.

106

~p :B [0,1]i ~x :Bi ~pi

~xi

~pi

i 1

n

1

Utility Function

• The utility functions are

,

where t = {t1,...,tn}.

107

~u (b, t)i

~ ~ ~u (b,t) t p (b) - x (b)i i i i

Bayesian Game

• = {B1,...,Bn; T1,...,Tn; f1,...,fn; ,..., }

• A strategy for bidder i in this game is bi:TiBi

• A strategy profile b={b1,...,bn} is a Bayesian equilibrium if for each ti Ti, the prescribed bid bi(ti) is a best response to the n - 1 other strategies b-i.

108

~u1~un

Maximization Problem

109

• Choose {Bi, , } to maximize the expected

revenue

• subject to

~pi~xi

E 1 p (b) t x (b)b i

i 1

n

0 i

i 1

n

LNM

OQP

RSTUVW

~ ~

E t p (b ,b ) - x (b ,b )|b = b (t )b i i i -i i i -i i i i-i

~ ~ .l q 0

Seller’s Problem

Choose outcome functions pi(t) and xi(t) to

(ER)

subject to

• i, t, pi(t) 0 and p1(t) +...+ pn(t) 1

• (IC) Vi(ti) vi(ti,ti) vi(i,ti) i, i, ti Ti

• (IR) Vi(ti) 0 for all i.

110

max t 1 p (t) x (t) f(t)dt0 i

i 1

n

i

i 1

n

T

FHG

IKJ

RSTUVW

XZY

Definition

Probability of Trade

is the conditional probability that player i gets the object when i's type is ti.

111

P (t ) p (t)f (t )dti i i i i iT i

z

Lemma 1

{pi(),xi()} satisfies (IC) and (IR) iff i

(i) Pi(ti) is weakly increasing

(ii) for all ti Ti

(iii) Vi(ai) 0.

112

V (t ) V (a ) + P ( )di i i i i i i

a

t

i

i

XZY

Lemma 2

If {pi(),xi()} satisfies (IC) and (IR), then (ER) becomes

(ER')

113

t V (a ) t1 F (t )

f (t )t p (t) f(t)dt.0 i i

i 1

n

ii i

i i

0

i 1

n

i

T

RST

UVWLNMM

OQPP

XZY

Lemma 2

Proof: By definition,

and by (ii) and the definition of Pi(ti),

114

V (t ) [t p(t , t ) x (t , t )]f (t )dti i i i i i i i i i i

T i

XZY

,

i

i

i

i

t

ii i i i i ia

t

ii i i -i -i -i -i i

a

V (t ) V (a )+ P ( )d

V (a )+ ( ,t )f (t )dt d .

iT

p

Lemma 2

Proof (cont.): Rearranging terms yields

115

x (t)f (t )dt =

- V (a ) + t p (t) p ( , t )d f (t )dt

i i i i

T

i i

T

i i i i -i i

a

t

-i -i -i

-i

-i i

i

XZY

XZY

XZY

LNMM

OQPP

Lemma 2

Proof (cont.): Integrating with respect to fi(ti) produces

after changing the order of integration.

Substituting into (ER) then completes the proof.

116

x (t)f(t)dt = -V (a ) + t p (t) p ( , t )d f(t)dt

= -V (a ) + t1 F (t )

f (t )p (t)f(t)dt

i

T

i i

T

i i i i -i i

a

t

i i

T

ii i

i i

i

i

iXZY

XZY

XZY

LNMM

OQPP

XZY

LNM

OQP

Lemma 2

Solving the problem of optimal auction design

• For fixed {pi()} choose

(EP)

• Suppose the seller sets Vi(ai) = 0 for each i. Then the problem has simplified to choosing {pi()} to maximize (ER') subject to (i) and the feasibility constraint.

117

x (t) = t p (t) p (t , )di i i i -i i i

a

t

i

i

XZY ,

Lemma 2 Optimal Auction Design

Solving the problem of optimal auction design

• Consider choosing {pi()} to maximize (ER') subject only to the feasibility constraint.

• Define:

and for fixed t let j maximize ci(ti) over i{1,...,n} such that cj(tj) ci(ti) for all i.

118

c (t ) t1 F (t )

f (t )i i i

i i

i i

Optimal Auction Design

Point-wise Optimization

For each fixed t,

• if cj(tj) - t0 > 0 then set pj(t) = 1 and pi(t) = 0 for all ij, and

• if cj(tj) - t0 0 then set pi(t) = 0 for all i.

This defines an optimal auction if (i) holds.

119


Regular Case:

For each i, ci(ti) is weakly increasing in ti

Consider i < ti

• Then ci(i) ci(ti), and so pi(i,t-i) pi(ti,t-i) for any t-i

• Therefore Pi(ti) is weakly increasing, completing the optimization

120


Irregular Case:

Define new functions that are constructed from the functions ci(ti) and are guaranteed to be increasing.

121

_ ci

ci

_ ci, ci

c (t )i i

Efficiency

Is the optimal auction ex post efficient ?

To maximize expected revenue when the buyers know their types but the seller does not, the seller may need to design an auction that sometimes fails to award the object to the player with the highest willingness to pay.

122

Inefficiency 1: Seller withholds

Example 1:

• Let t0 = 0, and Fi(ti) = ti for ti on [0,1]

• Then c(ti) = ti - (1 - ti) = 2ti - 1 and ci(ti) - t0 > 0 if and only if ti > 1/2

The seller sets a reserve price that deters half the types from collecting the object, even though this would be efficient since t0 = 0 because this reservation price increases the bids of the other half of the types.

123

Inefficiency 2: Seller misassigns

Example 2: Asymmetric Bidders

•

•

•

•

•

so that 1 gets the object even though 2 values it more 124

Let n = 2

f (t ) 1 (a a ) on [a ,a ]

f (t ) 1 (a a ) on [a ,a ]

Let t 0

Then c (t ) 2t a , and it could happen that

2t a 2t a 0 and t t

1 1 1 1 1 1

2 2 2 2 2 2

0

i i i i

1 1 2 2 2 1

,

Practical Results: Second-Price Auction

• For each i, Ti = T1, fi(ti) = f1(t1), and

• Suppose that c1(t1) is strictly increasing, so c-1() exists.

• For fixed t, let j denote the bidder with highest type:

tj > ti for all ij.

125

c (t ) t1 F (t )

f (t )1 1 1

1 1

1 1

Practical Results: Second-Price Auction

• In the optimal auction, bidder j gets the object if c(tj) - t0 > 0, or tj > c-1(t0), and pays

(EP)

which is simply

the second-highest type.

126

x (t) t p (t) p (t , )d ,j j j j j j j

a

t

j

j

XZY

max{c (t ),max t }10

i ji

,

Practical Results: Revenue Equivalence Theorem

One form of the Revenue Equivalence Theorem states that if

(1) V(ai) = 0 for all i, and

(2) for each t,

then the seller's expected revenue is the expected value of the second-highest type.

127

p (t) = 1 if t > maxtj jj i

i

,

Remarks

• In the absence of reservation prices, symmetric equilibria of the English, Dutch, first-price (sealed bids), and second-price (sealed bids) auctions satisfy these conditions

• Predominance of the English auction in practice:

– violation of the private values assumption

– noisy signals about a common value

128

note 3a mechanism design - peter cramton

Documents