Sequential mechanism design Krzysztof R. Apt (so not Krzystof and definitely not Krystof) CWI, Amsterdam, the Netherlands , University of Amsterdam based on joint works with A. Est´ evez-Fern´ andez E. Markakis Sequential mechanism design – p. 1/3
Executive Summary Mechanism design: how to arrange our economic interactions so that, when everyone behaves in a self-interested manner, the result is something we all like. Important question: how to avoid manipulations? This can be done, but is costly. Our objective: minimize these costs. We study the problem in sequential setting for public project problem, single unit auctions. Sequential mechanism design – p. 2/3
Recap: Direct Mechanisms (1) Given: set of decisions D , for each player i a set of types Θ i , initial utility function v i : D × Θ i → R . Sequential mechanism design – p. 3/3
Recap: Direct Mechanisms (2) We consider the following sequence of events: each player i has an initial utility v i ( d, θ i ) , and a type (e.g., valuation of an item) θ i , each player i announces to the central authority a type (e.g., a bid) θ ′ i , the central authority computes decision and taxes d := f ( θ ′ 1 , . . ., θ ′ n ) and ( t 1 , . . ., t n ) := t ( θ ′ 1 , . . ., θ ′ n ) , and communicates to each player i the pair ( d, t i ) . Player’s i final utility: u i (( f, t )( θ ) , θ i ) := v i ( f ( θ ) , θ i ) + t i ( θ ) . Social welfare: � n i =1 u i (( f, t )( θ ) , θ i ) . Sequential mechanism design – p. 4/3
Recap: Direct Mechanisms (3) A direct mechanism ( f, t ) is feasible if always � n i =1 t i ( θ ) ≤ 0 . (External funding not needed.) incentive compatible if no player is better off when submitting a false type ( θ ′ i � = θ i ). (Manipulations do not pay off or truth-telling is a dominant strategy.) Sequential mechanism design – p. 5/3
Public Project Problem Each person is asked to report his or her willingness to pay for the project, and the project is undertaken if and only if the aggregate reported willingness to pay exceeds the cost of the project. (15 October 2007, The Royal Swedish Academy of Sciences, Press Release, Scientific Background) Sequential mechanism design – p. 6/3
Public Project Problem Formally D = { 0 , 1 } , for each player i Θ i = [0 , c ] , where c > 0 , v i ( d, θ i ) := d ( θ i − c n ) , � 1 if � n i =1 θ i ≥ c f ( θ ) := 0 otherwise Sequential mechanism design – p. 7/3
Incentive Compatibility Theorem (Clarke ’71): � min(0 , n − 1 k � = i θ k + θ ′ n c − � k � = i θ k ) if � i < c t i ( θ ′ i , θ − i ) := k � = i θ k − n − 1 min(0 , � n c ) otherwise yields an incentive compatible mechanism. Example c = 300 . player type submitted type tax u i A 110 110 − 10 0 B 80 80 0 − 20 C 110 110 − 10 0 Sequential mechanism design – p. 8/3
Optimality Result (1) Theorem [Apt, Conitzer, Guo, Markakis, WINE’08] Consider the public project problem. No direct mechanism exists that is feasible, incentive compatible, ‘better’ than Clarke’s tax. Sequential mechanism design – p. 9/3
However . . . Clarke’s tax is not optimal in the public project problem when the payments per player can differ. Note: Pivotal mechanism then ceases to be anonymous. Sequential mechanism design – p. 10/3
(Single Item) Sealed Bid Auction argsmax θ := µi ( θ i = max j ∈{ 1 ,...,n } θ j ) . D = { 1 , . . ., n } , for each player i Θ i = R + , � θ i if d = i v i ( d, θ i ) := 0 otherwise f ( θ ) := argsmax θ. Sequential mechanism design – p. 11/3
Vickrey Auction as a Direct Mechanism θ ∗ : the reordering of θ in descending order. � − θ ∗ 2 if i = argsmax θ t V i ( θ ) := 0 otherwise Example: player bid tax to authority u i A 18 0 0 B 24 − 21 3 C 21 0 0 Theorem : Vickrey auction is incentive compatible. Sequential mechanism design – p. 12/3
Bailey-Cavallo Mechanism i ( θ ) + ( θ − i ) ∗ t i ( θ ) := t V 2 n Example: player bid tax to authority why? u i A (= 1 / 3 of 21 ) 18 0 7 B (= 24 − 2 − 7 − 6 ) 24 − 2 9 C (= 1 / 3 of 18 ) 21 0 6 Theorem: Bailey-Cavallo mechanism is feasible and incentive compatible. Warning: Bailey-Cavallo mechanism does not satisfy the participation constraint. Sequential mechanism design – p. 13/3
Optimality Result (2) Theorem [Apt, Conitzer, Guo, Markakis, WINE’08] Consider the sealed bid auction. No tax-based mechanism exists that is feasible, incentive compatible, ‘better’ than Bailey-Cavallo mechanism. Sequential mechanism design – p. 14/3
Groves Auctions A sealed bid auction with redistribution: t i ( θ ) := t V i ( θ ) + r i ( θ − i ) . Theorem [Groves ’73] Each Groves auction is incentive compatible. Sequential mechanism design – p. 15/3
Sequential Mechanisms Players move sequentially. Player i submits his/her type after he has seen the types of players 1 , . . ., i − 1 . The decisions and taxes are computed using a given direct based mechanism. Sequential mechanism design – p. 16/3
Strategies Assume a sequential mechanism Seq . A strategy of player i in Seq : s i : Θ 1 × . . . × Θ i → Θ i . Strategy s i ( · ) of player i is optimal in Seq if for all θ ∈ Θ and θ ′ i ∈ Θ i u i (( f, t )( s i ( θ 1 , . . ., θ i ) , θ − i ) , θ i ) ≥ u i (( f, t )( θ ′ i , θ − i ) , θ i ) . Sequential mechanism design – p. 17/3
Intuitions Strategy of player j is memoryless if it does not depend on the types of players 1 , . . ., j − 1 . Then s i ( · ) is optimal iff for all θ ∈ Θ it yields a best response to all joint strategies of players j � = i assuming players i + 1 , . . ., n use memoryless strategies (or move jointly with player i ). In particular, an optimal strategy is a best response to truth-telling by players j � = i . Sequential mechanism design – p. 18/3
Optimality Result (3) Theorem [Apt, Estévez-Fernández, SAGT’09] Consider public project problem and Clarke’s tax. Strategy if � i θ i j =1 θ j < c and i < n , 0 (!) if � i s i ( θ 1 , . . ., θ i ) := j =1 θ j < c and i = n , � i c (!) if j =1 θ j ≥ c is optimal for player i in the sequential pivotal mechanism. Under certain natural circumstances s i simultaneously maximizes the final utility of the other players. Sequential mechanism design – p. 19/3
Example 1 c = 300 . Pivotal mechanism: player type submitted type tax u i A 110 110 − 10 0 B 80 80 0 − 20 C 110 110 − 10 0 Now: player type submitted type tax u i A 110 110 0 10 B 80 80 0 − 20 C 110 300 − 10 0 Sequential mechanism design – p. 20/3
Example 2 c = 300 . Pivotal mechanism: player type submitted type tax u i A 110 110 0 0 B 80 80 − 10 − 10 C 100 100 0 0 Now: player type submitted type tax u i A 110 110 0 0 B 80 80 0 0 C 100 0 0 0 Sequential mechanism design – p. 21/3
Optimality Result (4) Theorem [Apt, Estévez-Fernández, SAGT’09] Consider public project problem and Clarke’s tax. Strategy if � i j =1 θ j < c and i < n , θ i if � i 0 (!) j =1 θ j < c and i = n , s i ( θ 1 , . . ., θ i ) := 0 (!!) if � i j =1 θ j = c, θ i > c n and i = n, otherwise c (!) is optimal for player i in the sequential pivotal mechanism. When all players follow s i ( · ) , maximal social welfare is generated in the universe of optimal strategies. Sequential mechanism design – p. 22/3
Example 3 c = 300 . Before: player type submitted type tax u i A 110 110 0 10 B 80 80 0 − 20 C 110 300 − 10 0 Now: player type submitted type tax u i A 110 110 0 0 B 80 80 0 0 C 110 0 0 0 Sequential mechanism design – p. 23/3
Proof Idea (1) Lemma 1 Let s ′ i ( · ) be an optimal strategy for player i . Suppose � i j =1 θ j < c and i < n . Then s ′ i ( θ 1 , . . ., θ i ) = θ i . Suppose � i j =1 θ j < c and i = n . Then � n − 1 j =1 θ j + s ′ i ( θ 1 , . . ., θ n ) < c . Suppose � i j =1 θ j = c and i < n . Then s ′ i ( θ 1 , . . ., θ i ) ≥ θ i . j =1 θ j > c . Then � i − 1 Suppose � i j =1 θ j + s ′ i ( θ 1 , . . ., θ i ) ≥ c . Proof In each case by case analysis. Sequential mechanism design – p. 24/3
Proof Idea (2) Lemma 2 s i ( · ) maximizes social welfare in the universe of optimal strategies, assuming that players who follow i are truthful. Proof By Lemma 1 and case analysis: Case 1 � i j =1 θ j < c and i < n . Case 2 � i j =1 θ j < c and i = n Case 3 � i j =1 θ j = c, θ i > c n and i = n . Case 4 � i j =1 θ j = c, θ i ≤ c n and i = n . Case 5 ( � i j =1 θ j = c and i < n ) or � i j =1 θ j > c . Sequential mechanism design – p. 25/3
Nash Implementation Suppose players submit their strategies simultaneously, for each vector of initial types their final utilities are determined using the pivotal mechanism. Game-theoretic interpretation: sequential pre-Bayesian games. Theorem Vectors of strategies from Theorems 1 and 2 form a Nash equilibrium in the universe of optimal strategies. The result does not hold if deviations to non-optimal strategies are allowed. Sequential mechanism design – p. 26/3
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