Sequential Groves mechanisms for public project problems Krzysztof R. Apt CWI, Amsterdam, the Netherlands , University of Amsterdam joint work with Arantza Est´ evez-Fern´ andez Free University, Amsterdam Sequential Groves mechanisms for public project problems – p.1/24
Executive Summary Groves mechanisms allow us to implement desired decisions by imposing on the players taxes. In Groves mechanisms truth-telling is a dominant strategy. In Groves mechanisms for the public project problems taxes � = 0 are unavoidable. So we study the sequential Groves mechanisms. We show that other dominant strategies than truth-telling may then exist. They can be used to maximize social welfare in public projects problems. Sequential Groves mechanisms for public project problems – p.2/24
Reminder Sequential Groves mechanisms for public project problems – p.3/24
Decision problems Assume players 1 , . . ., n , set of decisions D , for each player a set of types Θ i and a utility function R v i : D × Θ i → that he wants to maximize. Decision rule: a function f : Θ 1 × · · · × Θ n → D . We call ( D, Θ 1 , . . ., Θ n , v 1 , . . ., v n , f ) a decision problem. Sequential Groves mechanisms for public project problems – p.4/24
Decisions, Decisions, . . . One studies the following sequence of events: 1. each player i receives type θ i , 2. each player i announces to the central planner a type θ ′ i , 3. the central planner takes the decision d := f ( θ ′ 1 , . . ., θ ′ n ) , and communicates it to each player, 4. the resulting utility for player i is then v i ( d, θ i ) . Problem to solve: Each player i wants to manipulate the choice of d ∈ D so that his utility v i ( d, θ i ) is maximized. Sequential Groves mechanisms for public project problems – p.5/24
Decision rules A decision rule f is called strategy-proof if for all θ ∈ Θ , i ∈ { 1 , . . ., n } and θ ′ i ∈ Θ v i ( f ( θ i , θ − i ) , θ i ) ≥ v i ( f ( θ ′ i , θ − i ) , θ i ) . efficient if for all θ ∈ Θ and d ′ ∈ D n n � � v i ( d ′ , θ i ) . v i ( f ( θ ) , θ i ) ≥ i =1 i =1 Sequential Groves mechanisms for public project problems – p.6/24
Groves Mechanisms Central authority takes the decision d := f ( θ ′ ) and computes the sequence of taxes t 1 ( θ ′ ) , . . ., t n ( θ ′ ) , where � t i ( θ ′ ) := v j ( f ( θ ′ ) , θ ′ j ) + h i ( θ ′ − i ) , j � = i R is an arbitrary function. h i : Θ − i → player’s i ex post utility is u i (( f, t )( θ ′ i , θ − i ) , θ i ) := v i ( f ( θ ′ i , θ − i ) , θ i ) + t i ( θ ′ i , θ − i ) . Intuition: j � = i v j ( f ( θ ′ ) , θ ′ � j ) is the social welfare with i excluded from decision f ( θ ′ ) . Sequential Groves mechanisms for public project problems – p.7/24
Groves Mechanisms, ctd Groves Theorem Suppose f is efficient. Then in each Groves mechanism ( f, t ) is strategy-proof. Groves mechanism with the tax function t := ( t 1 , . . ., t n ) is feasible if � n i =1 t i ( θ ) ≤ 0 for all θ , pay only if t i ( θ ) ≤ 0 for all θ and i . Sequential Groves mechanisms for public project problems – p.8/24
Special Case: VCG Mechanism j � = i v j ( d, θ ′ � One uses h i ( θ − i ) := − max d ∈ D j ) . So t i ( θ ′ ) := � j � = i v j ( f ( θ ′ ) , θ ′ j � = i v j ( d, θ ′ � j ) − max d ∈ D j ) . Intuition: j � = i v j ( f ( θ ′ ) , θ ′ � j ) is the social welfare from f ( θ ′ ) with i excluded, j � = i v j ( d, θ ′ � max d ∈ D j ) is the maximal social welfare from f ( θ ′ ) with i excluded. Note If with player i excluded better decision can be taken, i is pivotal. His tax is then � = 0 . Note VCG mechanism is pay only (so feasible). Sequential Groves mechanisms for public project problems – p.9/24
Example Public project problem: D = { 0 , 1 } , R + , [0 , c ] ⊆ Θ i ⊆ cost of building the bridge: c , v i ( d, θ i ) := d ( θ i − c n ) , � 1 if � n i =1 θ i ≥ c f ( θ ) := 0 otherwise Note f is efficient. Sequential Groves mechanisms for public project problems – p.10/24
Example, ctd Suppose c = 300 and n = 3 . R + player value set of types ( Θ i ) tax u i R + A 60 -20 -20 R + B 70 -10 -10 C 150 0 0 The project does not get through ( d = 0 ) since 60 + 70 + 150 < 300 . Yet both A and B have to pay a tax. Sequential Groves mechanisms for public project problems – p.11/24
Optimality of VCG Mechanism Groves mechanism h ′ is superior to h if for all θ and i h ′ i is better for player i than h i : h i ( θ − i ) ≤ h ′ i ( θ − i ) , for some θ and i h ′ i is strictly better for player i than h i : h i ( θ − i ) < h ′ i ( θ − i ) . Theorem In the public project problem for no c > 0 and n ≥ 2 a pay only Groves mechanism exists that is superior to VCG mechanism. Sequential Groves mechanisms for public project problems – p.12/24
Sequential Mechanism Design The players are randomly ordered. The following revised sequence of events then takes place: 1. each player i receives type θ i , 2. each player i in turn announces to the other players a type θ ′ i , 3. each player takes the decision d := f ( θ ′ 1 , . . ., θ ′ n ) . Crucial difference: Now each player announces his type after he saw the types of earlier players. Sequential Groves mechanisms for public project problems – p.13/24
Sequential Mechanisms: Dominant Strategies Set-up: sequential version of Example. Theorem θ i if � i j =1 θ j < c and i < n , if � i s i ( θ 1 , . . ., θ i ) := 0 j =1 θ j < c and i = n , � i if j =1 θ j ≥ c c is a dominant strategy for player i . Strategy s i ( · ) of player i minimizes the tax of every other player (subject to some conditions). Sequential Groves mechanisms for public project problems – p.14/24
Back to our Example A: 60 , B: 70 , C: 150 ; c = 300 . ordering t A t B t C A B C 0 0 0 A C B 0 − 10 0 B A C 0 0 0 B C A − 20 0 0 C A B 0 − 10 0 C B A − 20 0 0 In each ordering at least one player pays smaller taxes. In 2 orderings taxes are 0. Sequential Groves mechanisms for public project problems – p.15/24
Can we Reduce Taxes to 0? Theorem Assume each player follows s i ( · ) . For all c > 0 , n ≥ 2 and θ 1 , . . ., θ n for some permutation of players all taxes equal 0. Proof idea Not all players can be pivotal. Put a non-pivotal player at the end. However: If a pivotal player is the last one, then he has to pay a tax � = 0 . Sequential Groves mechanisms for public project problems – p.16/24
Theorem: Precise Version Theorem θ i if � i j =1 θ j < c and i < n , if � i s i ( θ 1 , . . ., θ i ) := 0 j =1 θ j < c and i = n , � i if j =1 θ j ≥ c c is a dominant strategy for player i . If s i ( · ) deviates from truth, then it simultaneously minimizes all taxes provided the decision is not changed. Formally: If s i ( θ 1 , . . ., θ i ) � = θ i , then t j ( s i ( θ 1 , . . ., θ i ) , θ − i ) ≥ t j ( θ ′ i , θ − i ) for all j � = i , θ ′ i , θ i +1 , . . ., θ n such that f ( θ ′ i , θ − i ) = f ( θ i , θ − i ) . Sequential Groves mechanisms for public project problems – p.17/24
Maximizing Social Welfare (1) R n , Θ 1 , . . ., Θ n , u 1 , . . ., u n , ( f, t )) . Fix a Groves mechanism ( D × Type θ ′ i is compatible with θ 1 , . . ., θ i if for all θ i +1 , . . ., θ n u i (( f, t )( θ ′ i , θ − i ) , θ i ) ≥ u i (( f, t )( θ i , θ − i ) , θ i ) . Intuition Given the announced types θ 1 , . . ., θ i − 1 and the received type θ i , player i may report θ ′ i without taking any ‘risk’. Sequential Groves mechanisms for public project problems – p.18/24
Example Suppose f is efficient, for all θ ∈ Θ , f ( s i ( θ 1 , . . ., θ i ) , θ − i ) = f ( θ i , θ − i ) . Then for all θ 1 , . . ., θ i s i ( θ 1 , . . ., θ i ) is compatible with θ 1 , . . ., θ i . Sequential Groves mechanisms for public project problems – p.19/24
Maximizing Social Welfare (2) Strategy s i ( · ) of player i is socially dominant if it is dominant, for all θ and all θ ′ i compatible with θ 1 , . . ., θ i n n � � u j (( f, t )( θ ′ u j (( f, t )( s i ( θ 1 , . . ., θ i ) , θ − i ) , θ j ) ≥ i , θ − i ) , θ j ) . j =1 j =1 Intuition s i ( · ) is socially dominant if it is dominant and among all types that player i might report without incurring any risk, s i ( · ) selects the one that yields the ex post maximal social welfare. Sequential Groves mechanisms for public project problems – p.20/24
Maximizing Social Welfare (3) Set-up: sequential version of Example. Theorem θ i if � i j =1 θ j < c and i < n , if � i 0 j =1 θ j < c and i = n , s i ( θ 1 , . . ., θ i ) := if � i j � = i θ j < n − 1 j =1 θ j = c, � 0 n c and i = n , otherwise . c is socially dominant strategy for player i . Sequential Groves mechanisms for public project problems – p.21/24
Example player value submitted value tax utility ( u i ) A 110 110 − 10 0 B 80 80 0 − 20 C 110 110 − 10 0 player value submitted value tax utility ( u i ) A 110 110 0 0 B 80 80 0 0 C 110 0 0 0 By submitting 0 instead of the true value, 110 , player C increased social welfare from − 20 to 0 . Sequential Groves mechanisms for public project problems – p.22/24
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