Motivation. Game Theory 24. Mechanism Design. Setting. Preference relations contain no information about by how much one candidate is preferred.

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1 Motivation Game Theory 24. Mechanism Design Preference relations contain no information about by how much one candidate is preferred. Idea: Use money to measure this. Albert-Ludwigs-Universität Freiburg Use money also for transfers between players for compensation. Bernhard Nebel Robert Mattmüller Research Group Foundations of Artificial Intelligence July 20, 2015 July 20, 2015 BN, RM Game Theory 2 / 32 Setting Formalization: Set of alternatives A. Second price auctions: There are n players bidding for a single item. Set of n players I. Valuation functions v i : A R such that v i (a) denotes the value player i assigns to alternative a. Player i s private valuations of item: w i. Desired outcome: Player with highest private valuation wins bid. Payment functions specifying amount p i R that player i pays. Utility of player i: u i (a) = v i (a) p i. Players should reveal their valuations truthfully. Winner i pays price p and has utility w i p. Non-winners pay nothing and have utility 0. July 20, 2015 BN, RM Game Theory 3 / 32 July 20, 2015 BN, RM Game Theory 5 / 32

2 Vickrey Auction Definition (Vickrey Auction) Formally: A = N v i (a) = { w i if a = i 0 else What about payments? Say player i wins: p = 0 (winner pays nothing): bad idea, players would manipulate and publicly declare values w i w i. p = w i (winner pays his valuation): bad idea, players would manipulate and publicly declare values w i = w i ε. better: p = max w j (winner pays second highest bid). The winner of the Vickrey Auction (aka second price auction) is the player i with the highest declared value w i. He has to pay the second highest declared bid p = max w j. Proposition (Vickrey) Let i be one of the players and w i his valuation for the item, u i his utility if he truthfully declares w i as his valuation of the item, and u i his utility if he falsely declares w i as his valuation of the item. Then u i u i. Proof See July 20, 2015 BN, RM Game Theory 6 / 32 July 20, 2015 BN, RM Game Theory 7 / 32 Idea: Generalization of Vickrey auctions. Preferences modeled as functions v i : A R. Let V i be the space of all such functions for player i. Definition (Mechanism) A mechanism f,p 1,...,p n consists of a social choice function f : V 1 V n A and for each player i, a payment function p i : V 1 V n R. Unlike for social choice functions: Not only decide about chosen alternative, but also about payments. Definition ( Compatibility) A mechanism f,p 1,...,p n is called incentive compatible if for each player i = 1,...,n, for all preferences v 1 V 1,...,v n V n and for each preference v i V i, v i (f (v i,v i )) p i (v i,v i ) v i (f (v i,v i )) p i (v i,v i ). July 20, 2015 BN, RM Game Theory 9 / 32 July 20, 2015 BN, RM Game Theory 10 / 32

3 If f,p 1,...,p n is incentive compatible, truthfully declaring ones preference is dominant strategy. The Vickrey-Clarke-Groves mechanism is an incentive compatible mechanism that maximizes social welfare, i.e., the sum over all individual utilities n i=1 v i(a). Idea: Reflect other players utilities in payment functions, align all players incentives with goal of maximizing social welfare. Definition (Vickrey-Clarke-Groves mechanism) A mechanism f,p 1,...,p n is called a Vickrey-Clarke-Groves mechanism ( mechanism) if 1 f (v 1,...,v n ) argmax a A n i=1 v i(a) for all v 1,...,v n and 2 there are functions h 1,...,h n with h i : V i R such that p i (v 1,...,v n ) = h i (v i ) v j (f (v 1,...,v n )) for all i = 1,...,n and v 1,...,v n. Note: h i (v i ) independent of player i s declared preference h i (v i ) = c constant from player i s perspective. Utility of player i = v i (f (v 1,...,v n ))+ v j (f (v 1,...,v n )) c = n j=1 v j(f (v 1,...,v n )) c = social welfare c. July 20, 2015 BN, RM Game Theory 12 / 32 July 20, 2015 BN, RM Game Theory 13 / 32 Theorem (Vickrey-Clarke-Groves) Every mechanism is incentive compatible. Proof Let i, v i, v i and v i be given. Show: Declaring true preference v i dominates declaring false preference v i. Let a = f (v i,v i ) { and a = f (v i,v i). v i (a) + Utility player i v j (a) h i (v i ) = v i (a ) + v j (a ) h i (v i ) Alternative a = f (v i,v i ) maximizes social welfare v i (a) + v j (a) v i (a ) + v j (a ). if declaring v i if declaring v i So far: payment functions p i and functions h i unspecified. One possibility: h i (v i ) = 0 for all h i and v i. Drawback: Too much money distributed among players (more that necessary). Further requirements: Players should pay at most as much as they value the outcome. Players should only pay, never receive money. v i (f (v i,v i )) p i (v i,v i ) v i (f (v i,v i)) p i (v i,v i). July 20, 2015 BN, RM Game Theory 14 / 32 July 20, 2015 BN, RM Game Theory 15 / 32

4 Individual Rationality, Positive Transfers Clarke Pivot Function Definition (Clarke pivot function) Definition (individual rationality) The Clarke pivot function is the function A mechanism is individually rational if all players always get a nonnegative utility, i.e., if for all i = 1,...,n and all v 1,...,v n, v i (f (v 1,...,v n )) p i (v 1,...,v n ) 0. h i (v i ) = max v j (b). This leads to payment functions Definition (positive transfers) A mechanism has no positive transfers if no player is ever paid money, i.e., for all preferences v 1,...,v n, p i (v 1,...,v n ) 0. p i (v 1,...,v n ) = max v j (b) v j (a) for a = f (v 1,...,v n ). Player i pays the difference between what the other players could achieve without him and what they achieve with him. Each player internalizes the externalities he causes. July 20, 2015 BN, RM Game Theory 16 / 32 July 20, 2015 BN, RM Game Theory 17 / 32 Clarke Pivot Function Example Lemma (Clarke pivot rule) Players I = {1,2}, alternatives A = {a,b}. Values: v 1 (a) = 10, v 1 (b) = 2, v 2 (a) = 9 and v 2 (b) = 15. Without player 1: b best, since v 2 (b) = 15 > 9 = v 2 (a). With player 1: a best, since v 1 (a) + v 2 (a) = = 19 > 17 = = v 1 (b) + v 2 (b). With player 1, other players (i.e., player 2) lose v 2 (b) v 2 (a) = 6 units of utility. Clarke pivot function h 1 (v 2 ) = 15 payment function p 1 (v 1,...,v n ) = max v j (b) v j (a) = 15 9 = 6. j 1 j 1 A mechanism with Clarke pivot functions has no positive transfers. If v i (a) 0 for all i = 1,...,n, v i V i and a A, then the mechanism is also individually rational. Proof Let a = f (v 1,...,v n ) be the alternative maximizing n j=1 v j(a), and b the alternative maximizing v j (b). Utility of player i: u i = v i (a) + v j (a) v j (b). Payment function for i: p i (v 1,...,v n ) = v j (b) v j (a). Since b maximizes v j (b): p i (v 1,...,v n ) 0 (no positive transfers).... July 20, 2015 BN, RM Game Theory 18 / 32 July 20, 2015 BN, RM Game Theory 19 / 32

5 Vickrey Auction as a Mechanism Proof (ctd.) Individual rationality: Since v i (b) 0, u i = v i (a) +v j (a) v j (b) Since a maximizes n j=1 v j(a), and hence u i 0. n j=1 v j (a) n j=1 n j=1 v j (b) v j (a) n j=1 Therefore, the mechanism is also individually rational. v j (b). July 20, 2015 BN, RM Game Theory 20 / 32 A = N. Valuations: w i. v a (a) = w a, v i (a) = 0 (i a). a maximizes social welfare n i=1 v i(a) iff a maximizes w a. Let a = f (v 1,...,v n ) = argmax j A w j be the highest bidder. Payments: p i (v 1,...,v n ) = max v j (b) v j (a). But max v j (b) = max \{i} w b. Winner pays value of second highest bid: p a (v 1,...,v n ) = max v j (b) v j (a) j a j a Non-winners pay nothing: For i a, = max \{a} w b 0 = max \{a} w b. p i (v 1,...,v n ) = max v j (b) v j (a) = max \{i} w b w a = w a w a = 0. July 20, 2015 BN, RM Game Theory 21 / 32 Example: Bilateral Trade Example: Bilateral Trade (ctd.) Seller s offers item he values with 0 w s 1. Potential buyer b values item with 0 w b 1. Alternatives A = {trade, no-trade}. Valuations: v s (no-trade) = 0, v s (trade) = w s, v b (no-trade) = 0, v b (trade) = w b. mechanism maximizes v s (a) + v b (a). We have v s (trade) + v b (trade) = w b w s, v s (no-trade) + v b (no-trade) = 0 Requirement: if no-trade is chosen, neither player pays anything: p s (v s,v b ) = p b (v s,v b ) = 0. To that end, choose Clarke pivot function for buyer: h b (v s ) = max a A v s(a). For seller: Modify Clarke pivot function by an additive constant and set h s (v b ) = max a A v b(a) w b. i.e., trade maximizes social welfare iff w b w s. July 20, 2015 BN, RM Game Theory 22 / 32 July 20, 2015 BN, RM Game Theory 23 / 32

6 Example: Bilateral Trade (ctd.) Example: Bilateral Trade (ctd.) For alternative no-trade, p s (v s,v b ) = max a A v b(a) w b v b (no-trade) = w b w b 0 = 0 and p b (v s,v b ) = max a A v s(a) v s (no-trade) For alternative trade, = 0 0 = 0. p s (v s,v b ) = max a A v b(a) w b v b (trade) = w b w b w b = w b and p b (v s,v b ) = max a A v s(a) v s (trade) = 0 + w s = w s. Because w b w s, the seller gets at least as much as the buyer pays, i.e., the mechanism subsidizes the trade. Without subsidies, no incentive compatible bilateral trade possible. Note: Buyer and seller can exploit the system by colluding. July 20, 2015 BN, RM Game Theory 24 / 32 July 20, 2015 BN, RM Game Theory 25 / 32 Example: Public Project Example: Public Project (ctd.) Project costs C units. Each citizen i privately values the project at w i units. Government will undertake project if i w i > C. Alternatives: A = {project, no-project}. Valuations: v G (project) = C, v G (no-project) = 0, v i (project) = w i, v i (no-project) = 0. mechanism with Clarke pivot rule: for each citizen i, ) h i (v i ) = max a A = ( v j (a) + v G (a) { w j C, if w j > C 0, otherwise. Citizen i pivotal if j w j > C and w j C. Payment function for citizen i: ( ) p i (v 1..n,v G ) = h i (v i ) v j (f (v 1..n,v G )) + v G (f (v 1..n,v G )) Case 1: Project undertaken, i pivotal: ( ) p i (v 1..n,v G ) = 0 w j C = C w j Case 2: Project undertaken, i not pivotal: ( ) ( ) p i (v 1..n,v G ) = w j C w j C Case 3: Project not undertaken: p i (v 1..n,v G ) = 0 = 0 July 20, 2015 BN, RM Game Theory 26 / 32 July 20, 2015 BN, RM Game Theory 27 / 32

7 Example: Public Project (ctd.) Example: Buying a Path in a Network I.e., citizen i pays nonzero amount only if he is pivotal. C w j He pays difference between value of project to fellow citizens and cost C, in general less than w i. Generally, p i (project) C i i.e., project has to be subsidized. Communication network modeled as G = (V, E). Each link e E owned by different player e. Each link e E has cost c e if used. Objective: procure communication path from s to t. Alternatives: A = {p p path from s to t}. Valuations: v e (p) = c e, if e p, and v e (p) = 0, if e / p. Maximizing social welfare: minimize e p c e over all paths p from s to t. Example: c a = 4 c d = 12 s i t c b = 3 c e = 5 July 20, 2015 BN, RM Game Theory 28 / 32 July 20, 2015 BN, RM Game Theory 29 / 32 Example: Buying a Path in a Network (ctd.) Example: Buying a Path in a Network (ctd.) For G = (V,E) and e E let G \ e = (V,E \ {e}). VGC mechanism with Clarke pivot function: h e (v e ) = max p G\e e p c e i.e., the cost of the cheapest path from s to t in G \ e. (Assume that G is 2-connected, s.t. such p exists.) Payment functions: for chosen path p = f (v 1,...,v n ), p e (v 1,...,v n ) = h e (v e ) e e p Case 1: e / p. Then p e (v 1,...,v n ) = 0. Case 2: e p. Then p e (v 1,...,v n ) = max c e p G\e e p c e. e e p c e. Example: c a = 4 s i t c b = 3 c d = 12 c e = 5 Cost along b and e: 8 Cost without e: 3 Cost of cheapest path without e: 15 (along b and d) Difference is payment: 15 ( 3) = 12 I.e., owner of arc e gets payed 12 for using his arc. Note: Alternative path after deletion of e does not necessarily differ from original path at only one position. Could be totally different. July 20, 2015 BN, RM Game Theory 30 / 32 July 20, 2015 BN, RM Game Theory 31 / 32

8 Summary New preference model: with money. mechanisms generalize Vickrey auctions. mechanisms are incentive compatible mechanisms maximizing social welfare. With Clarke pivot rule: even no positive transfers and individually rational (if nonnegative valuations). Various application areas. July 20, 2015 BN, RM Game Theory 32 / 32