1. 6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 21

2. Review: Direct revelation Mechanisms, VCG

3. Direct Revelation Mechanisms Setup: – Set of alternatives A – n bidders; bidder i has a (private) valuation function ; – bidder i’s value for alternative a is ; – if alternative a is chosen and bidder i pays price p i the utility of the bidder is (quasi-linear utility). Def: A direct revelation mechanism is a collection of functions (f, p 1, p 2,…, p n ) where: – chooses an alternative – chooses the payment of bidder i “Direct Revelation” because it asks bidders to reveal their whole valuation function (i.e. doesn’t involve rounds of communication). Def: A direct revelation mechanism (f, p 1, p 2,…, p n ) is called Incentive Compatible iff for all i, and : ))))

4. Vickrey-Clarke-Groves Mechanisms Def: A mechanism (f, p 1, p 2,…, p n ) is a VCG mechanism if – (ie chooses SW maximizing alternative) – the payment of bidder i has the form: Theorem(Vickrey-Clarke-Groves): Any VCG mechanism is IC. Def: A payment function p i is called Clarke pivot payment if I.e. bidder pays the harm he causes to the other bidders. Theorem: VCG with Clarke pivot payments makes no positive transfers (i.e. sum of prices charged is always positive). Also if the valuation functions are non-negative, it is individually rational (bidders never have never negative utility, i.e. value-price is always non-negative). best social welfare without bidder i )

5. VCG Examples Auctioning a single item (Vickrey auction) Multi-unit Auctions Reverse auction Public Project

6. Power of Non-Direct revelation Mechanisms?

7. 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)

8. 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

9. Formal Definition of Mechanisms

10. General Mechanisms Vickrey’s auction and VCG are both single round and direct-revelation mechanisms. We will give a general model of mechanisms. It can model multi-round and indirect-revelation mechanisms.

11. Mechanism Def: A (general-non direct revelation) mechanism for n players is defined by The game with strict incomplete information induced by the mechanism has the same type spaces and action spaces, and utilities : setup mech

12. 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 require that for some equilibrium and allow 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 equilibrium outcome of the social choice function

13. The Revelation Principle

14. 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?

15. 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.

16. [Incentive Compatibility (restated) utility of i if he says the truth utility of i if he lies i.e. no incentive to lie about your type! ] Def: A direct revelation mechanism mechanism is called incentive compatible, or truthful, or strategy-proof iff for all i, for all and for all

17. 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

18. Revelation Principle (cont’d) original mechanism new mechanism

19. Proof of Revelation Principle Proof: 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. So we get that which gives the definition of the incentive compatibility of the direct revelation mechanism that we obtained above.

20. 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. Technical Lemma (gedanken experiment: from ex-post to dominant strategy equilibria): Let be an ex-post Nash equilibrium of a game. Define new action spaces. Then is a dominant strategy equilibrium of the game. Proof sketch: Restrict the action spaces in the original mechanism to the sets. Our technical lemma implies that now is a dominant strategy equilibrium. Now invoke the revelation principle on dominant strategies.