Mechanism design introduction

1. Mechanism designintroduction EitanYanovsky

2. Outline • Election • Mechanisms with money • Incentive compatible mechanism • Incomplete information • Characterizations of incentive compatible mechanisms

3. Election • Two candidates election • Intuitive solution fits, take majority vote. • Three or more candidates election • Condorcet’s paradox: • The notation means that voter i prefers candidate a to candidate b. The joint majority choice is not consistent

4. Election • Voting methods • Plurality • Borda count These methods encourage strategic voting

5. Election • Formally: • A - the set of candidates (alternatives). • I - the set of n voters. • L - the set of linear orders on A ( : is a total order on A). • Preferences of voter iare given by • Definition 1.1: • is called social welfare function (s.w.f). • is called a social choice function (s.c.f).

6. Election • Definition 1.2: • s.w.f F satisfies unanimity if • Voter iis a dictator in F if F is not a dictatorship if no iis a dictator in it. • s.w.f F satisfies independence of irrelevant alternatives if the social preferences between any two alternatives a and b only depends on the voters preferences between a and b, formally if we denote then

7. Election – Arrow’s theorem • Theorem1.13 • Every s.w.f over a set A s.t |A|>2 that satisfies unanimity and independence of irrelevant alternatives is a dictatorship. • Arrow’s theorem has devastating strategic implications. • Escape route, mechanism with money.

8. Mechanisms with money • Replace order with valuation • A player wishes to maximize its utility which is derived from its valuation and some price paid or gained at each chosen alternative • Single item auction: • . • where is the price player iis “willing to pay” for a. • social welfare achieved with second price auction. is commonly known set of possible valuation functions for player i

9. Incentive compatible mechanisms • Notations: • Similarly etc’ • Definition 1.14 • A mechanism consist of: • a social choice function • Payment functions

10. Incentive compatible mechanisms • Definition 1.15 • A mechanism is called incentive compatible if if we denote then • Intuitively this means that player iwhose valuation is would prefer “telling the truth” rather than any possible “lie” since this gives him higher utility.

11. Incentive compatible mechanisms • Definition 1.16 • A mechanism is VCG mechanism if (f maximizes social welfare) • For some functions ( does not depend on ) The main idea is that each player is paid an amount equal to the sum of the values of all other players, thus this mechanism aligns all players incentives with the social goal of maximizing social welfare.

12. Incentive compatible mechanisms • Theorem 1.17 • Every VCG mechanism is incentive compatible. Proof: Let be player i truthful valuation and some different valuation. We need to show that the utility for declaring is higher than declaring . the utility when declaring is respectivelyand since f maximizes social welfare over all alternatives then and the same holds when subtracting

13. Incentive compatible mechanisms • Definition 1.18 • A mechanism is individual rational if • A mechanism has no positive transfer if • Definition 1.19 (Clark pivot rule) • Each player pays his damage to the others.

14. Incentive compatible mechanisms • Lemma 1.20 • A VCG mechanism with Clarke Pivot payments makes no positive transfers and if then it is also individually rational. Proof: Let be the alternative maximizing and b the alternative maximizing . Player’s i utility . No positive transfers since • This down side can be addressed by a modified rule that chooses b as to maximize the social welfare “when idoes not participate”.

15. Incentive compatible mechanisms • Examples • A single item auction • Bilateral trade • Seller Buyer outcome • Public project • Government does a project at price C, each citizen ihas a value for the project (may be negative) • Reverse Auction • Tender

16. Incomplete information • Definition 1.21 • A game with strict incomplete information: • For every player i, a set of actions • For every player i, a set of types , a value is the private information that player ihas. • For every player i, a utility function each player choose action knowing only its own information, i.e.

17. Incomplete information • Definition 1.22 • A strategy of player iis a function • A profile is an ex-post-Nash equilibrium if we have that • A strategy is a dominant strategy if we have that • A profile is a dominant strategy equilibrium if each is a dominant strategy

18. Incomplete information • Definition 1.24 • A mechanism for n players is given by: • Private information spaces • Action spaces • Alternative set A • Valuations • Outcome function • Payments • The game induced by the mechanism is by using the private information spaces ,action spaces and the utilities

19. Incomplete information • Definition 1.24 cont’ • A mechanism implements a s.w.f in dominant strategies if for some dominant strategy equilibrium of the induced game we have that for • Similarly a mechanism implements a s.w.f in ex-post-equilibrium.

20. Incomplete information • Proposition 1.25 • (Revelation principle) If there exists an arbitrary mechanism that implements f in dominant strategies, then there exists an incentive compatible mechanism that implements f. The payment of the players in the incentive compatible mechanism is identical to those, obtained at equilibrium, of the original mechanism. Proof: The new mechanism will simulate the equilibrium strategies.Let be a dominant strategy equilibrium, we define a new direct revelation mechanism: and . Since each is dominant for player i, then for every we have that this in particular this is true for all and any which gives the definition of incentive compatibility of the mechanism

21. Characterizations of incentive compatible mechanisms • Proposition 1.27 • A mechanism is incentive compatible if and only if it satisfies the following condition for every iand every : • The payment does not depend on , but only on the alternative chosen • The mechanism optimizes for each player. I.e. for every we have that

22. Characterizations of incentive compatible mechanisms • Proof: • <= Denote The utility of iwhen telling the truth is which is not less than the utility when declaring ( ) since the mechanism optimize for i( ) • => (i) If for some but then player with this valuation will increase his gain by declaring • => (ii) If , fix in the range of and thus for some . Now a player with this valuation will increase his gain by declaring

23. Characterizations – Weak Monoticity • Definition 1.28 • A s.c.f f satisfies weak monoticityif we have that in simply words, 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. • Theorem 1.29 • If a mechanism is incentive compatible then f satisfies WMON. If all preferences are convex sets, then for every s.c.ff that satisfies WMON, there exists such that is incentive compatible.

24. Characterizations – Weak Monoticity • Proof (one way) • Assume is incentive compatible, fix iand . Proposition 1.27 implies the existence of prices for all (that do not depend on ) s.t whenever the outcome is a the bidder ipays it. Assume since a player with valuation does not prefer declaring we have that similarly a since player with does not prefer declaring we have that . By subtracting the second from the first we get as required

25. Characterizations – Weighted VCG • Definition 1.30 • A s.c.ff is called an affine maximizer if for some , for some player weights and for some outcome weights for every we have that

26. Characterizations – Single parameter domain • Restrict to be one dimensional, a player has a scalar value for “winning” and 0 value for “losing”. • Definition 1.33 • a single parameter domain is defined by a publicly known and a range of values . is the set of such that for some for all and otherwise. • Definition 1.34 • A s.c.ff is called monotone in if and every we have that

27. Characterizations – Single parameter domain • Definition 1.35 • The critical value of a monotone s.c.ff is the critical value is undefined if iwins for every • Theorem 1.36 • A normalized mechanism is incentive compatible if and only if the following conditions hold: • f is monotone in every • Every winning bid pays the critical value. Formally, such that we have that (If is undefined the winner pays some constant value)

28. Characterizations – Uniqueness of prices The payment function is essentially uniquely determined by the social choice function, means that we can modify a payment by adding some function that do not depend on the player valuation. Conclusion: • The only incentive compatible mechanisms that maximizes social welfare are those with VCG payments. • In the example (bilateral) the only i.c.m that maximizes social welfare, must subsidize the trade.

29. Characterizations – Randomized mechanisms • Definition 1.38 • A randomized mechanism is a distribution over deterministic mechanism(all with the same players, types spaces and outcome) • A randomized mechanism is incentive compatible in the universal sense if every deterministic mechanism in the support is incentive compatible • A randomized mechanism is incentive compatible in expectation if truth is dominant strategy in the game induced by expectation. I.e where are random variables denoting the outcome and payment of iwhen he bids respectively