Sequential pivotal mechanisms for public project problems Krzysztof R. Apt (so not Krzystof and definitely not Krystof) CWI, Amsterdam, the Netherlands , University of Amsterdam joint work with A. Est´ evez-Fern´ andez Vrije Universiteit, Amsterdam Sequential pivotal mechanisms for public project problems – p. 1/2
Executive Summary We study the public project problem. Our objective: to maximize social welfare. We study strategies in sequential setting. They can yield a higher social welfare. Sequential pivotal mechanisms for public project problems – p. 2/2
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 pivotal mechanisms for public project problems – p. 3/2
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 pivotal mechanisms for public project problems – p. 4/2
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 pivotal mechanisms for public project problems – p. 5/2
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 pivotal mechanisms for public project problems – p. 6/2
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 pivotal mechanisms for public project problems – p. 7/2
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 pivotal mechanisms for public project problems – p. 8/2
Background: an Optimality Result Theorem [Apt, Conitzer, Guo, Markakis, ’08] Consider the public project problem. No direct mechanism exists that is feasible, incentive compatible, ‘better’ than Clarke’s tax. Sequential pivotal mechanisms for public project problems – p. 9/2
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 pivotal mechanisms for public project problems – p. 10/2
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 pivotal mechanisms for public project problems – p. 11/2
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 pivotal mechanisms for public project problems – p. 12/2
Optimality Result (1) Theorem 1 Consider public project problem and Clarke’s tax. Strategy if � i j =1 θ j < c and i < n , θ i 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 pivotal mechanisms for public project problems – p. 13/2
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 pivotal mechanisms for public project problems – p. 14/2
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 pivotal mechanisms for public project problems – p. 15/2
Optimality Result (2) Theorem 2 Consider public project problem and Clarke’s tax. Strategy if � i j =1 θ j < c and i < n , θ i if � i j =1 θ j < c and i = n , 0 (!) s i ( θ 1 , . . ., θ i ) := 0 (!!) if � i j =1 θ j = c, θ i > c n and i = n, c (!) otherwise 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 pivotal mechanisms for public project problems – p. 16/2
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 pivotal mechanisms for public project problems – p. 17/2
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 pivotal mechanisms for public project problems – p. 18/2
Conclusions Social welfare can be increased if the players move sequentially. Dalai Lama: The intelligent way to be selfish is to work for the welfare of others. Microeconomics: Behavior, Institutions, and Evolution , S. Bowles ’04. Recent work: similar analysis for sequential Bailey-Cavallo mechanism for single item auctions ([Apt, Markakis, WINE’09]). Sequential pivotal mechanisms for public project problems – p. 19/2
THANK YOU Sequential pivotal mechanisms for public project problems – p. 20/2
More on Optimal Strategies Consider the sequential pivotal mechanism. Lemma s i ( · ) is an optimal strategy for player i iff the following holds: 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 . Suppose � i j =1 θ j > c . Then � i − 1 j =1 θ j + s i ( θ 1 , . . ., θ i ) ≥ c . Sequential pivotal mechanisms for public project problems – p. 21/2
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