Computational Game TheoryAmos Fiat

Spring 2012

More on Social choice and implementations

Using slides by

Uri Feige Robi Krauthgamer Moni Naor

Costas Daskalakis

Examples: Second Price Auction

Second Price auction: hi(t-i) = maxj(w1, w2,…, wi-1, wi+1,…, wn)

= maxb 2 A j i tj(b)

Multiunit auction: if k identical items are to be sold to k individuals. A={S wins |S ½ I, |S|=k} and

vi(S) = 0 if i2S and vi(i)=wi if i 2 S Allocate units to top k bidders. They pay the

k+1th priceClaim: this is

maxS’ ½ I\{i} |S’| =k j i vj(S’)-j i vj(S)

Generalized Second Price Auctions

Multiunit auction: if k identical items are to be sold to k individuals. A={S wins |S ½ I, |S|=k} and

vi(S) = 0 if iS and vi(i)=wi if i 2 S

VCG with Clarke Pivot Payments: Allocate units to top k bidders. Each pays bid j+1.GSP: The k items are not identical (ad slots)vi(S) = 0 if iS and vi(j)=wij if i is given item j

Agents bid one valuei’ th top bidder gets slot i at price of bid i+1

Common in web advertisingClaim: this is not incentive compatible

VCG with Public Project

Want to build a bridge:◦ Cost is C (if built) ◦ Value to each individual vi

◦ Want to built iff i vj ¸ C

Player with vj ¸ 0 pays only if pivotal

j i vj < C but j vj ¸ Cin which case pays pj = C- j i vj

In general: i pj < C

Payments do not cover project cost’s Subsidy necessary!

A={build, not build}

Equality only when

i vj = C

Set A of alternatives: all s-t paths

Buying a (Short) Path in a Graph

A Directed graph G=(V,E) where each edge e is “owned” by a different player and has cost ce.

Want to construct a path from source s to destination t.

How do we solicit the real cost ce?◦ Set of alternatives: all paths from s to t◦ Player e has cost: 0 if e not on chosen path and –ce

if on◦ Maximizing social welfare: finding shortest s-t

path: minpaths e 2 path ce

A VCG mechanism pays 0 to those not on path p: pay each e0 2 p: e 2 p’ ce - e 2 p\{e0} ce

where p’ is shortest path without eo

If e0 would not have woken up in the morning, what would otheredges earn? If he does wake up, what would other edges earn?

VCG (+CPP) is not perfect• Requires payments & quasilinear utility functions• In general money needs to flow away from the system

– Strong budget balance = payments sum to 0– Impossible in general [Green & Laffont 77]

• Vulnerable to collusions• Maximizes sum of players’ valuations (social welfare)

– (not counting payments, but does include “COST” of alternative) But: sometimes [usually, often??] the mechanism is not

interested in maximizing social welfare:– E.g. the center may want to maximize revenue– Minimize time– Maximize fairness– Etc., Etc.

Black slides from Costas Daskalakis, MIT6.853: Topics in Algorithmic Game Theory Fall 2011

Vickrey’s auction and VCG are both single round and direct-revelation mechanisms.

We now consider a general model of mechanisms. It can model multi-round and indirect-revelation mechanisms.

Games with Strict Incomplete Information

Def: A game with (independent private values and) strict incomplete information for a set of n players is given by the following ingredients:

(ii)

(i)

(iii)

Strategy and Equilibrium

Def: A strategy of a player i is a function

Def: Equilibrium (ex-post Nash and dominant strategy)

A profile of strategies is an ex-post Nash equilibrium if for all i, all , and all we have that

A profile of strategies is a dominant strategy equilibrium if for all i, all , and all we have that

Equilibrium (cont’d)

Proposition: Let be an ex-post Nash equilibrium of a game

Define , then is a dominant strategy equilibrium in the game

Mechanism

Def: A (general-non direct revelation) mechanism for n players is given by

The game with strict incomplete information induced by the mechanism has the same type spaces and action spaces, and utilities

Implementing a social choice function

Given a social choice function

Ex: Vickrey’s auction implements the maximum social welfare function in dominant strategies, because is a dominant strategy equilibrium, and maximum social welfare is achieved at this equilibrium.

Similarly we can define ex-post Nash implementation.

Remark: We only requires that for some equilibrium and allows other equilibria to exist.

A mechanism implements in dominant strategies if for some dominant strategy equilibrium of the induced game, we have that for all , .

outcome of the mechanism at the equilibriumoutcome of the social choice function

The Revelation Principle

Revelation Principle

We have defined direct revelation mechanisms in previous lectures. Clearly, the general definition of mechanisms is a superset of the direct revelation mechanisms.

But is it strictly more powerful? Can it implement some social choice functions in dominant strategy that the incentive compatible (direct revelation dominant strategy implementation) mechanism can not?

Revelation Principle

Proposition: (Revelation principle) If there exists an arbitrary mechanism that implements in dominant strategies, then there exists an incentive compatible [direct revelation] mechanism that implements . The payments of the players in the incentive compatible mechanism are identical to those, obtained at equilibrium, of the original mechanism.

Incentive Compatible

utility of i if he says the truth

utility of i if he lies

i.e. no incentive to lie!

Def: A mechanism is called incentive compatible, or truthful , or strategy-proof iff for all i, for all and for all

Revelation Principle

Proposition: (Revelation principle) If there exists an arbitrary mechanism that implements in dominant strategies, then there exists an incentive compatible mechanism that implements . The payments of the players in the incentive compatible mechanism are identical to those, obtained at equilibrium, of the original mechanism.

Proof idea: Simulation

Revelation Principle (cont’d)

original mechanism

new mechanism

Proof of Revelation PrincipleProof: Let be a dominant strategy equilibrium of the original mechanism such that , we define a new direct revelation mechanism:

Since each is a dominant strategy for player i, for every , we have that

Thus in particular this is true for all and any we have that

which gives the definition of the incentive compatibility of the mechanism.

Revelation Principle (cont’d)

Corollary: If there exists an arbitrary mechanism that ex-post Nash equilibrium implements , then there exists an incentive compatible mechanism that implements . Moreover, the payments of the players in the incentive compatible mechanism are identical to those, obtained in equilibrium, of the original mechanism.

Proof sketch: Restrict the action spaces of each player. By the previous proposition, we know in the restricted action spaces, the mechanism implements the social choice function in dominant strategies. Now we can invoke the revelation principle to get an incentive compatible mechanism.

Characterizations of Incentive Compatible Mechanisms

Characterizations

Only look at incentive compatible mechanisms (revelation principle)

When is a mechanism incentive compatible? Characterizations of incentive compatible mechanisms.

Maximization of social welfare can be implemented (VCG). Any others? Basic characterization of implementable social choice functions.

What social choice functions can be implemented?

Direct Characterization

Direct Characterization

A mechanism is incentive compatible iff it satisfies the following conditions for every i and every :

(i)

i.e., for every , there exists a price , when the chosen alternative is , the price is

(ii)

i.e., for every , we have alternative where the quantification is over all alternatives in the range of

Direct Characterization (cont’d)

Proof: (if part) Denote and . Since the mechanism optimizes for i,

the utility of i when telling the truth is not less than the utility when lying.

Direct Characterization (cont’d)

Proof (cont): (only if part; (i)) If for some , but . WLOG, we assume . Then a player with type will increase his utility by declaring .

(only if part; (ii)) If , we fix

and Now a player with type

will increase his utility by declaring .

Weak Monotonicity

Weak Monotonicity

●The direct characterization involves both the social choice function and the payment functions.

● Weak Monotonicity provides a partial characterization that only involves the social choice function.

Weak Monotonicity (WMON)

Def: A social choice function satisfies Weak Monotonicity (WMON) if for all i, all we have that

i.e. WMON means that if the social choice changes when a single player changes his valuation, then it must be because the player increased his value of the new choice relative to his value of the old choice.

Weak Monotonicity

Theorem: If a mechanism is incentive compatible then satisfies WMON. If all domains of preferences are convex sets (as subsets of an Euclidean space) then for every social choice function that satisfies WMON there exists payment function such that is incentive compatible.

Remarks: (i) We will prove the first part of the theorem. The second part is quite involved, and will not be given here.

(ii) It is known that WMON is not a sufficient condition for incentive compatibility in general non-convex domains.

Weak Monotonicity (cont’d)

Proof: (First part) Assume first that is incentive compatible, and fix i and in an arbitrary manner. The direct characterization implies the existence of fixed prices for all (that do not depend on ) such that whenever the outcome is then i pays exactly .

Assume . Since the mechanism is incentive compatible, we have

Thus, we have

Minimization of Social Welfare

We know maximization of social welfare function can be implemented.

How about minimization of social welfare function?

No! Because of WMON.

Minimization of Social Welfare

Assume there is a single good. WLOG, let . In this case, player 1 wins the good.

If we change to , such that . Then player 2 wins the good. Now we can apply the WMON.

The outcome changes when we change player 1’s value. But according to WMON, it should be the case that . But . Contradiction.

Weak Monotonicity

WMON is a good characterization of implementable social choice functions, but is a local one.

Is there a global characterization?

Weighted VCG

Affine Maximizer

Def: A social choice function is called an affine maximizer if for some subrange , for some weights and for some outcome weights , for every , we have that

Payments for Affine Maximizer

Proposition: Let be an affine maximizer. Define for every i,

where is an arbitrary function that does not depend on . Then, is incentive compatible.

Payments for Affine Maximizer

Proof: First, we can assume wlog . The utility of player i if alternative is chosen is . By multiplying by this expression is maximized when is maximized which is what happens when i reports truthfully.

Roberts Theorem

Theorem [Roberts 79]: If , is onto , for every i, and is incentive compatible then is an affine maximizer.

Remark: The restriction is crucial (as in Arrow’s theorem), for the case , there do exists incentive compatible mechanisms beyond VCG.

Bayesian Mechanism Design

Def: A Bayesian mechanism design environment consists of the following:

setup

mech

+ for every bidder a distribution is known; bidder i’s type is sampled from it

Def: A Bayesian mechanism consists of the following:

The utility that bidder i receives if the players’ actions are x1,…,xn is :

Strategy and Equilibrium

Def: A strategy of a player i is a function .

Def: A profile of strategies is a Bayes Nash equilibrium if for all i, all , and all we have that

expected utility of bidder i for using si, where the expectation is computed with respect to the actions of the other bidders assuming they are using their strategies

expected utility if bidder i uses a different action xi’; still the expectation is computed with respect to the actions of the other bidders assuming they are using their strategies

First Price Auction

Theorem: Suppose that we have a single item to auction to two bidders whose values are sampled independently from [0,1]. Then the strategies s1(t1)=t1/2 and s2(t2)=t2/2 are a Bayes Nash equilibrium.

Remark: In the above Bayes Nash equilibrium, social welfare is optimized, since the highest winner gets the item.

Proof: ???.

Expected Payoff?

E[ max(X/2, Y/2)], where X, Y are independent U[0,1] random variables.

=1/3

Expected Payoff of second price auction?E[ min(X, Y)], where X, Y are independent U[0,1] random variables.

=1/3 !

Revenue Equivalence TheoremAll single item auctions that allocate (in Bayes Nash equilibrium) the item to the bidder with the highest value and in which losers pay 0 have identical expected revenue.

Given a social choice function we say that a Bayesian mechanism implements if for some Bayes Nash equilibrium we have that for all types :

Generally:

Revenue Equivalence Theorem: For two Bayesian-Nash implementations of the same social choice function f, * we have the following: if for some type t0

i of player i, the expected (over the types of the other players) payment of player i is the same in the two mechanisms, then it is the same for every value of ti.

In particular, if for each player i there exists a type t0i where the two mechanisms

have the same expected payment for player i, then the two mechanisms have the same expected payments from each player and their expected revenues are the same.

Revenue Equivalence Theorem (cont.)All single item auctions that allocate (in Bayes Nash equilibrium) the item to the bidder with the highest value and in which losers pay 0 have identical expected revenue.

So unless we modify the social choice function we won’t increase revenue.

E.g. increasing revenue in single-item auctions:

- two uniform [0,1] bidders

- run second price auction with reservation price ½ (i.e. give item to highest bidder above the reserve price (if any) and charge him the second highest bid or the reserve, whichever is higher.

Claim: Reporting the truth is a dominant strategy equilibrium. The expected revenue of the mechanism is 5/12 (i.e. larger than 1/3).

Bayesian Nash Implementation

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

It is known to everyone The value ti 2Di

Ti is the private information i

knows A profile of strategis si is a Bayesian Nash

Equilibrium if for i all ti and all x’i

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

[ui(ti, s-i(t-i)) ]

Bayesian Nash: First Price Auction

First price auction for a single item with two players.

Each has a private value 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 Bayesian Nash Equilibrium

t1Cannot winWin half the time

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