1. 6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 22

2. Characterizations of Incentive Compatible Mechanisms

3. 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?

4. Direct Characterization

5. A mechanism is incentive compatible iff it satisfies the following conditions for every i and every : i.e., for fixed, there is an advertised price per alternative ; the bidder is free to affect the chosen alternative and through that the corresponding price that she’ll pay; (ii) i.e., for every, we have alternative where the quantification is over all alternatives in the range of (i) p i depends on only through the alternative

6. Direct Characterization (cont’d) Proof (if part): Denote, where respectively is the bidder’s true value and is a potential lie. Since the mechanism optimizes for i, the utility he receives when telling the truth is not less than the utility he receives when lying.

7. Direct Characterization (cont’d) Proof (cont): (only if part; (i)) Suppose that for some, but. WLOG, assume that. Then a player with type will increase his utility by declaring. (only if part; (ii)): Suppose and let instead s.t. Now a player with type will increase his utility by declaring.

8. Weak Monotonicity

9. ●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.

10. 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.

11. 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.

12. 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

13. 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.

14. 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.

15. Weak Monotonicity WMON is a good characterization of implementable social choice functions, but is a local one (i.e. a collection of local conditions). Is there a global characterization of what functions can be implemented, e.g. maximization of social welfare, etc.?

16. Weighted VCG

17. 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 i.e. maximization only over A’ weighted social welfare + bonus per alternative

18. 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. the appropriate generalization of the VCG payment rule

19. 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.

20. 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 exist incentive compatible mechanisms beyond VCG. |A|=2  single-parameter domains the valuation function of each bidder is described by 1 parameter; i.e, for every bidder i there exists subset W i  A (winning set) and a parameter r i such that the bidder receives value r i for any outcome in W i and 0 otherwise; e.g. single-item/multi-unit auctions, etc.

21. Bayesian Mechanisms

22. 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 x 1,…,x n is :

23. 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 s i, 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 x i ’; still the expectation is computed with respect to the actions of the other bidders assuming they are using their strategies

24. 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 s 1 (t 1 )=t 1 /2 and s 2 (t 2 )=t 2 /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: On the board. 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 !

25. Revenue Equivalence Theorem 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. 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 t 0 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 t i. In particular, if for each player i there exists a type t 0 i 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.

26. 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).