Bayes Nash Implementation

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Complete information games (you know the type of every other agent, type = payoff)◦ Nash equilibria: each players strategy is best response to the

other players strategies

Incomplete information game (you don’t know the type of the other agents)◦ Game G, common prior F, a strategy profile actions – how to play game (what to bid, how to answer…) ◦ Bayes Nash equilibrium for a game G and common prior F is a

strategy profile s such that for all i and is a best response when other agents play where

Complete vs. Incomplete

Bayes Nash Implementation

There is a distribution Di on the types Ti of Player i

It is known to everyone The actual type of agent i, ti 2Di

Ti is the

private information i knows A profile of strategis si is a Bayes Nash

Equilibrium if for i all ti and all t’i

Ed-i[ui(ti, si(ti), s-i(t-i) )] ¸ Ed-i

[ui(t’i, s-i(t-i)) ]

Bayes Nash: First Price Auction

First price auction for a single item with two players.

Private values (types) t1 and t2 in T1=T2=[0,1] Does not make sense to bid true value –

utility 0. There are distributions D1 and D2 Looking for s1(t1) and s2(t2) that are best

replies to each other Suppose both D1 and D2 are uniform.

Claim: The strategies s1(t1) = ti/2 are in Bayes Nash Equilibrium

t1Cannot winWin half the time

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If ◦ Other agent bids half her value (uniform [0,1])◦ I bid b and my value is v

No point in bidding over max(1/2,v) The probability of my winning is 2b My Utility is he derivative is set to zero to

get This means that maximizes my utility

First Price, 2 agents, Uniform [0,1]

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Bayes Nash equilibria (assumes priors)◦ Today: characterization

Special case: Dominant strategy equilibria (VCG), problem: over “full domain” with 3 options in range (Arrow? GS? New: Roberts) – only affine maximizers (generalization of VCG) possible.

Implementation in undominated strategies: Not Bayes Nash, not dominant strategy, but assumes that agents are not totally stupid

Solution concepts for mechanisms and auctions (speical case of mechanisms) (?)

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Characterization of Equilibria

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What happens to the Bayes Nash equilibria characterization when one deals with arbitrary service conditions: ◦ A set is the set of allowable characteristic vectors◦ The auction can choose to service any subset of

bidders for whom there exists a characteristic vector

Prove the characterization of dominant truthful equilibria.

Homework #1

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Characterization of Equilibria

Claim1: If (¯1;¯2; : : : ;¯n) is a Bayes-Nash equilibrium, (agent i bids ¯ i (vi )when vi is her value), then, for all i:

1. The probability of allocation ai (vi ) is monotone increasing in vi .

2. The expected utility ui (vi ) (expected utility of agent i when agents withvalue vj bid bj (vj )) is a convex function of vi ,

ui (vi ) =Z vi

0ai (z)dz:

3. The expected payment

pi (vi ) = vi ai (vi ) ¡Z vi

0ai (z)dz =

Z vi

0za0

i (z)dz:

Claim2: If (¯1;¯2; : : : ;¯n) are such that either (1) and (2) hold or (1) and (3)hold then (¯1;¯2; : : :;¯n) are a Bayes-Nash equilibria.

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Let ui (w;v) bethe(expected) utility of agent i when shebids ¯ i (w) and hervalue is v. Let vi be the truevalue of agent i.

Choose some i, ¯x all ¯ j , j 6= i, u = ui , v = vi , a = ai .If ¯1; : : : ;¯n is a Bayes Nash Equilibrium then

u(v;v) = va(v) ¡ p(v) ¸ va(w) ¡ p(w) = u(w;v):

But, if the true value of agent i was w we also get that

u(w;w) = wa(w) ¡ p(w) ¸ wa(v) ¡ p(v) = u(v;w):

Adding these two(v ¡ w) (a(v) ¡ a(w)) ¸ 0:

If v ¸ w then a(v) ¸ a(w). I.e., ai is monotonic for all i.

Claim 1 proof: Monotonic

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Claim 1 proof: Convex

u(v) = u(v;v) = supw

u(w;v) = supw

f va(w) ¡ p(w)g:

The supremum of a family of convex functions is convex

f convex:f (®x + (1¡ ®)z) · ®f (x) + (1¡ ®)f (z):

Ergo, is convexui (v)

If f : [a;b] 7! < is convex then it is the integral of it's (right) derivative

f (t) = f (a) +Z t

af+(x)dx:

where f+(x) is the right derivative at x

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Claim 1 proof: u’(v)=a(v), u(v) = int(a(z), z=0..v)For every v and w:

u(v) ¸ u(w;v) ¸ va(w)¡ p(w) = (wa(w) ¡ p(w))+(v¡ w)a(w) = u(w)+(v¡ w)a(w)

Or,u(v) ¡ u(w)

v ¡ w¸ a(w):

If v approaches w from above, the left derivative u0(w) ¸ a(w). If v ap-proaches w from below the right derivative u0(w) · a(w). If u is di®erentiableat w then

u0(w) = a(w):

Since a convex function is the integral of it's right derivativewe have that

u(v) ¡ u(0) =Z v

0a(z)dz:

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Since

Claim 1, end

u(v) = va(v) ¡ p(v)

p(v) = va(v) ¡ u(v)

pi (vi ) = vi ai (vi ) ¡Z vi

0ai (z)dz =

Z vi

0za0

i (z)dz:

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From (condition 3)

pi (vi ) = vi ai (vi ) ¡Z vi

0ai (z)dz =

Z vi

0za0

i (z)dz:

it follows that

u(v) =Z v

0a(z)dz:

u(w;v) = va(w) ¡ p(w) = (v ¡ w)a(w) +Z w

0a(z)dz:

As ai is monotonic (condition 1) this implies that

u(v) ¸ u(w;v):

Characterization: Claim 2 proof

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If bidding truthfully ( for all i) is a Bayes Nash equilibrium for auction A then A is said to be Bayes Nash incentive compatible

Bayes Nash Incentive Compatible Auctions

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For every auction A with a Bayes Nash equilibrium, there is another “equivalent” auction A’ which is Bayes-Nash incentive compatible (in which bidding truthfully is a Bayes-Nash equilibrium )

A’ simply simulates A with inputs ◦ A’ for first price auctions when all agents are

U[0,1] runs a first price auction with inputs The Big? Lie: not all “auctions” have a

single input.

The Revelation Principle

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Dominant strategy truthful: Bidding truthfully maximizes utility irrespective of what other bids are. Special case of Bayes Nash incentive compatible.

Dominant strategy truthful equilibria

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The probability of ) is weakly increasing in - must hold for any distribution including the distribution that gives all mass on

The expected payment of bidder i is

Dominant strategy truthful auctions

pi (vi ) = vi ai (vi ) ¡Z vi

0ai (z)dz =

Z vi

0za0

i (z)dz:

over internal randomization

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The probability of ) is weakly increasing in - must take values 0,1 only

The expected payment of bidder i is ◦ There is a threshold value such that the item is

allocated to bidder i if but not if ◦ If i gets item then payment is

Deterministic dominant truthful auctions

Expected RevenuesExpected Revenue:

◦ For first price auction: max(T1/2, T2/2) where T1 and T2

uniform in [0,1]◦ For second price auction min(T1, T2)

◦ Which is better? ◦ Both are 1/3.◦ Coincidence?

Theorem [Revenue Equivalence]: under very general conditions, every two Bayesian Nash implementations of the same social choice function if for some player and some type they have the same

expected payment then◦ All types have the same expected payment to the player◦ If all player have the same expected payment: the expected

revenues are the same

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If A and A’ are two auctions with the same allocation rule in Bayes Nash equilibrium then for all bidders i and values we have that

Revenue Equivalence

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F strictly increasing If is a symmetric Bayes-Nash equilibrium

and strictly increasing in [0,h] then

◦ | This is the revenue from the 2nd price auction

IID distributions highest bidder wins

pi (vi ) = vi ai (vi ) ¡Z vi

0ai (z)dz =

Z vi

0za0

i (z)dz:

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w

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First price auctions

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n bidders U[0,1]