Mechanism Design with Interdependent Values

Mechanism Design with Interdependent Values

Fuhito Kojima1

April 21, 2013

1

Interdependent Values

Interdependent values: your utility depends not only on your own private information but also on others’.

Interdependent values seem important in many economic situations.

1 _{A painting may be real or a fake, and someone might know} the answer.

2 Oil drilling companies bidding for a tract of land, when the amount of oil is unknown.

3 Hiring an assistant professor.

4 Elections: The “jury theorem”; selecting the “right” president. 5 _{Other examples?}

Interdependent Values

Consider the case of allocating one good, withN bidders i = 1, . . . ,N.

Each bidder receives a one-dimensional signalsi.

Valuation ofi is

vi(s1,s2, . . . ,sN), ∂vi/∂si ≥0.

Private Values

Private values is a simple case wherevi(·) only depends onsi.

We know that the followingsecond price auctionworks: 1 Each agent reports her valuation.

2 Allocate the object to the highest-value bidder i. 3 i pays max_j6=iv_j(s) while no one else pays. Theorem

Under the second price auction, reporting valuations truthfully is a weakly dominant strategy for each agent.The mechanism is

efficient, that is, the object is assigned to the highest-value bidder.

We know that the idea of the second price auction can be

Second Price Auction Fails with Interdependent Values

Based on Example 3 of Dasgupta and Maskin (2000 QJE).

Three bidders bid for one object.

Valuations are

v1(s1,s2,s3) =s1+ 1 2s2+

1 4s3,

v2(s1,s2,s3) =s2+ 1 4s1+

1 2s3,

v1(s1,s2,s3) =s3.

Note that

(s1,s2,s3) = (1,1,1 +ε)→v2>v1,v3,

(s1,s2,s3) = (1,1,1−ε)→v1>v2,v3.

But in the 2nd price auction, bidders can bid based only on their signals.

A Single Crossing Condition

We’ll consider direct revelation mechanisms.

We will need thesingle crossing condition: For anyi 6=j and signals = (s1, . . . ,sN) where vi =vj = max{v1,v2, . . . ,vN},

∂vi(s) ∂si

> ∂vj(s) ∂si

.

The Single Crossing Condition is “Necessary”

Example 4 of Dasgupta and Maskin.

There are two wildcatters.

Oil price is 4.

1’s fixed cost is 1, and marginal cost is 2.

2’s fixed cost is 2, and marginal cost is 1.

1 privately discovers the amounts1of oil, whiles2is uninformative.

v1(s1,s2) = (4−2)s1−1 = 2s1−1,

v2(s1,s2) = (4−1)s1−2 = 3s1−2,

Then

2 = ∂v1 ∂s1

<∂v2 ∂s1

= 3,

Keeping in mind that

v1(s1,s2) = 2s1−1,

v2(s1,s2) = 3s1−2, Efficiency demands

1 obtains the right if

2s1−1>3s1−2 and 2s1−1>0⇔1/2<s1 <1 2 obtains the right if 1<s1

Let transfer function byt(·). Fors1 >1>s10 >1/2,

t(s1)≥2s1−1 +t(s10), (incentive compatibility at s1)

2s₁0 −1 +t(s₁0)≥t(s1), (incentive compatibility at s10) resulting in

Generalized VCG

Assume the single crossing condition.

Consider the following mechanism, ageneralized VCG

mechanism(Cremer and McLean 1985; see Krishna’s textbook): Letσi(s−i) := min{si :vi(si,s−i) = maxjvj(si,s−i)}.

That is,σi(s−i) is the lowest of the signals that makes i be the

highest-value bidder given signalss−i of others. Define the

generalized VCG by

1 Direct Revelation Mechanism: each agent reports her signal. 2 Allocate the object to the highest-value bidder i.

3 i paysv_i(σ_i(s_−i),s_−i) while no one else pays.

vi(σi(s−i),s−i) isi’s valuation when i has the lowest signal among

those that makei the highest-valuation agent.

Efficiency of Generalized VCG

Theorem

Suppose that utility functions of bidders satisfy the single crossing condition. Then, in the generalized VCG, it is an ex post

equilibrium for all bidders to report their signals truthfully.

So in this case, there is an efficient and incentive compatible mechanism!

Ex post equilibrium means that each agent does not want to change their reports even after they learn signals of others (i.e., no regret).

Remark: Truthtelling is not a dominant strategy.

Proof Sketch

Suppsei is the highest-value bidder unders. Then

1 As long as i reports a signal such thati wins, the payment is unchanged from truthtelling (obvious from the rule).

2 Ifi reports a signal such thati does not win, then he loses the object while not paying anything, being (weakly) worse off.

Suppsei is not the highest-value bidder unders. Then

1 As long as i reports a signal such thati does not win, the allocation and payment are unchanged from when she reports the true signal.

Multi-Dimensional Signals (Example 5 of DM)

Example: Two wildcatters. Oil price is 1.

There are two regionsi = 1,2 within the tract of land for auctioning.

Wildcatteri has private fixed cost of ci and no marginal cost, and

discovers that amountqi of oil is in regioni.

So the payoff ofi is

Remember the payoff ofi is

vi =q1+q2−ci.

Say that an equilibrium isregular if a player’s decision is the same for different signals for her as long as her decision problem is the same.

Considerc1<c2 <c1+ ∆. Efficiency demands

1 obtains the right forc1 and not forc1+ ∆.

But (q1,c1) and (q1+ ∆,c1+ ∆) gives the same decision problem for 1.

Impossibility of Efficient Auctions in Regular Equilibria

Theorem (Proposition 3 of Dasgupta and Maskin)

Suppose that there is an agent i with multidimensional signals. Assume si is distributed independently of s−i. If there exist si0 and s_i00 such that

1 v_i(s0

i,·) =vi(si00,·), but 2 there exists s0

−i such that

[arg maxjvj(s_i0,s−0 i)]∩[arg maxjvj(s

00

i ,s

0

Multi-Dimensional Signals

Based on Jehiel and Moldovanu (2001 EMA).

There areK social alternatives,k = 1, . . . ,K andN agents, i= 1, . . . ,N.

Each agenti receives a private signalsi = (s_ki)k∈K →Roughly,s_ki is i’s information about alternativek.

Utility ofi when alternativek is chosen and signals are s= (s1, . . . ,sN) is

v_ki(s) := N

X

j=1

aj_kis_kj

=a1_kis_k1+a2_kis_k2+. . .aN_kis_kN

aj_ki represents the impact ofs_kj (j’s signal aboutk) oni’s valuation.

Impossibility

Bayesian incentive compatibility turns out to be a difficult requirement.

Theorem (Theorem 4.3 of Jehiel and Moldovanu)

Assume that efficient and Bayesian incentive compatible

mechanism exists. Then, under some regularity conditions, it must satisfy

ai_ki ai_k0_i

= PN

j=1aikj PN

j=1aik0_j

for all i , k,k0.

The condition is a congruence requirement: the rate of

substitution is the same between an individuali and the society.

Note that this is a very demanding condition: generically violated (equality requirement).

Special cases in which the condition is satisfied:

1 _{Private values: a}i

kj= 0 for allj6=i. 2 “Common Values” case: ai

Intuition (Example 4.4 of JM)

Two agents 1, 2, and 3 alternatives, A, B, C.

Only agent 1 receives a signal,s = (sA,sB,sC).

Mechanism can be written byx = (xA,xB,xC), wherexk is transfer

to agent 1 when social alternativek is chosen.

Consider two signalss∗ ands∗∗ where A and B are both efficient.

Incentive compatibility and efficiency (and continuity) imply

v_A1(s∗) +xA=vB1(s∗) +xB,

v_A1(s∗∗) +xA=vB1(s∗∗) +xB.

Rearranging terms,

Keep in mind

v_A1(s∗)−v_B1(s∗) =v_A1(s∗∗)−v_B1(s∗∗).

Since both A and B are efficient,

v_A1(s∗) +v_A2(s∗) =v_B1(s∗) +v_B2(s∗),

v_A1(s∗∗) +v_A2(s∗∗) =v_B1(s∗∗) +v_B2(s∗∗).

In this particular case, we obtain

One-Dimensional Signals Again

Now assume that each agenti has a one-dimensional signal, si _in

an interval.

Utility ofi when alternativek is chosen and signals are s= (s1, . . . ,sN) is

v_ki(s) :=

N

X

j=1

a_kij sj

=a1_kis1+a_ki2s2+. . .aN_kisN

The “weak congruence condition”,

ai_ki ai_k0_i

>1→

PN

j=1aikj

PN

j=1aki0_j

>1.

The “weak congruence condition” ai_ki

ai_k0_i

>1→

PN

j=1aikj

PN

j=1aik0_j >1,

turns out to be key.

Theorem (Theorem 5.1 of Jehiel and Moldovanu)

Assume that the weak congruence condition is satisfied. Then there exists an efficient and Bayesian incentive compatible mechanism.

Wilson Doctrine

In some cases (mostly one-dimensional cases), we could find efficient and incentive compatible mechanisms.

But they are direct revelation mechanisms, where agents submit their signals.

It may be a strong assumption: the mechanism designer (and agents) needs to know the physical signal spaces of all agents as well as their valuation functions.

What about Ex Post?

The general impossibility of Jehiel and Moldovanu (2001) is based on Bayesian incentive compatibility.

A related question: What about ex post incentive compatibility? Of course, ex post incentive compatibility implies Bayesian incentive compatibility.

→So full efficiency is impossible.

Jehiel et al. (2006 EMA) show a stronger impossibility theorem for ex post incentive compatibility: The only ex post incentive