Selfishness Level of Strategic Games Krzysztof R. Apt CWI, - PowerPoint PPT Presentation

Selfishness Level of Strategic Games Krzysztof R. Apt CWI, Amsterdam, the Netherlands , University of Amsterdam based on joint work with Guido Sch¨ afer CWI, Amsterdam, the Netherlands , VU University Amsterdam Selfishness Level of Strategic Games – p. 1/23

Strategic Games: Review Strategic game for | N | ≥ 2 players: G : = ( N , { S i } i ∈ N , { p i } i ∈ N ) . For each player i (possibly infinite) set S i of strategies, payoff function p i : S 1 × ... × S n → R . Selfishness Level of Strategic Games – p. 2/23

Main Concepts Notation: s i , s ′ i ∈ S i , s , s ′ , ( s i , s − i ) ∈ S 1 × ... × S n . s is a Nash equilibrium if ∀ i ∈ { 1 ,..., n } ∀ s ′ i ∈ S i p i ( s i , s − i ) ≥ p i ( s ′ i , s − i ) . Social welfare of s : n ∑ SW ( s ) : = p j ( s ) . j = 1 s is a social optimum if SW ( s ) is maximal. Selfishness Level of Strategic Games – p. 3/23

Altruistic Games Given G : = ( N , { S i } i ∈ N , { p i } i ∈ N ) and α ≥ 0 . G ( α ) : = ( N , { S i } i ∈ N , { r i } i ∈ N ) , where r i ( s ) : = p i ( s )+ α SW ( s ) . When α > 0 the payoff of each player in G ( α ) depends on the social welfare of the players. G ( α ) is an altruistic version of G . Selfishness Level of Strategic Games – p. 4/23

Selfishness Level (1) G is α -selfish if a Nash equilibrium of G ( α ) is a social optimum of G ( α ) . If for no α ≥ 0 , G is α -selfish, then its selfishness level is ∞ . Suppose G is finite. If for some α ≥ 0 , G is α -selfish, then α ∈ R + ( G is α -selfish ) min is the selfishness level of G . Selfishness Level of Strategic Games – p. 5/23

Selfishness Level (2) Suppose G is infinite. If for some α ≥ 0 , G is α -selfish and α ∈ R + ( G is α -selfish ) min exists, then it is the selfishness level of G . Otherwise the selfishness level of G is undefined. Note SW ( s ) in G ( α ) equals ( 1 + α n ) SW ( s ) in G , so the social optima of G and G ( α ) coincide. Selfishness Level of Strategic Games – p. 6/23

Three Examples (1) Prisoner’s Dilemma C D C 2 , 2 0 , 3 D 3 , 0 1 , 1 The Battle of the Sexes F B F 2 , 1 0 , 0 B 0 , 0 1 , 2 Matching Pennies H T H 1 , − 1 − 1 , 1 T − 1 , 1 , − 1 1 Selfishness Level of Strategic Games – p. 7/23

Three Examples (2) Prisoner’s Dilemma: selfishness level is 1. C D C D C C 2 , 2 0 , 3 6 , 6 3 , 6 D D 3 , 0 1 , 1 6 , 3 3 , 3 The Battle of the Sexes: selfishness level is 0. F B F 2 , 1 0 , 0 B 0 , 0 1 , 2 selfishness level is ∞ . Matching Pennies: H T H 1 , − 1 − 1 , 1 T − 1 , 1 , − 1 1 Selfishness Level of Strategic Games – p. 8/23

Selfishness Level vs Price of Stability Recall Price of stability = SW ( s ) / SW ( s ′ ) , where s is a social optimum and s ′ a Nash equilibrium with the highest social welfare. Note Selfishness level of a finite game is 0 iff price of stability is 1. Theorem For every finite α > 0 and β > 1 there is a finite game with selfishness level α and price of stability β . Theorem There exists a game that is α -selfish for every α > 0 , but is not 0-selfish. Selfishness Level of Strategic Games – p. 9/23

Stable Social Optima Social optimum s stable if no player is better off by unilaterally deviating to another social optimum. That is, s is stable if for all i ∈ N and s ′ i ∈ S i if ( s ′ i , s − i ) is a social optimum, then p i ( s i , s − i ) ≥ p i ( s ′ i , s − i ) . Notes If s is a unique social optimum, then it is stable. Stable social optima don’t need to exist: take the Matching Penny game. Selfishness Level of Strategic Games – p. 10/23

Characterization Result Theorem If G is finite, then its selfishness level is finite iff it has a stable social optimum. If G is infinite, then its selfishness level is finite iff a stable social optimum s exists for which p i ( s ′ i , s − i ) − p i ( s i , s − i ) α ( s ) : = i , s − i ) , max SW ( s i , s − i ) − SW ( s ′ i ∈ N , s ′ i ∈ R ( i , s ) is finite, where R ( i , s ) : = { s ′ i ∈ S i | p i ( s ′ i , s − i ) > p i ( s i , s − i ) and SW ( s i , s − i ) > SW ( s ′ i , s − i ) }, We can compute the selfishness level by minimizing α ( s ) . Selfishness Level of Strategic Games – p. 11/23

Some Examples Prisoner’s dilemma for n players Each S i = { 0 , 1 } , p i ( s ) : = 1 − s i + 2 ∑ s j . j � = i 1 Proposition Selfishness level is 2 n − 3 . Two players, S i = { 2 ,..., 100 } , Traveler’s dilemma  s i if s i = s − i   p i ( s ) : = s i + 2 if s i < s − i s − i − 2 otherwise .   Proposition Selfishness level is 1 2 . Selfishness Level of Strategic Games – p. 12/23

Cournot Competition One infinitely divisible product (oil), n companies decide simultaneously how much to produce, price is decreasing in total output. Each S i = R + , n ∑ � � p i ( s ) : = s i a − b s j − cs i j = 1 for some a , b , c , where a > c and b > 0 . Price of the product: a − b ∑ n j = 1 s j . Production cost: cs i . Proposition For each n > 1 the selfishness level is ∞ . Selfishness Level of Strategic Games – p. 13/23

Tragedy of the Commons Contiguous common resource (shared bandwidth), Each S i = [ 0 , 1 ] , s i : chosen fraction of the common resource payoff function: s i ( 1 − ∑ n if ∑ n j = 1 s j ) j = 1 s j ≤ 1 � p i ( s ) : = otherwise 0 Intuition: the payoff degrades when the resource is overused. Proposition For each n > 1 the selfishness level is ∞ . Selfishness Level of Strategic Games – p. 14/23

Potential Games G : = ( N , { S i } i ∈ N , { p i } i ∈ N ) is an ordinal potential game if for some P : S 1 × ... × S n → R for all i ∈ N , s − i ∈ S − i and s i , s ′ i ∈ S i p i ( s i , s − i ) > p i ( s ′ i , s − i ) iff P ( s i , s − i ) > P ( s ′ i , s − i ) . Theorem Every finite ordinal potential game has a finite selfishness level. Proof Each social optimum with the largest potential is a stable social optimum. Selfishness Level of Strategic Games – p. 15/23

Congestion Games Congestion game: G = ( N , E , { S i } i ∈ N , { d e } e ∈ E ) , where E is a finite set of facilities, S i ⊆ 2 E is the set of facility subsets available to player i , d e ∈ N is the delay function for facility e ∈ E . Let x e ( s ) be the number of players using facility e in s . The goal of a player is to minimize his individual cost c i ( s ) : = ∑ e ∈ s i d e ( x e ( s )) . Social cost: SC ( s ) = ∑ n i = 1 c i ( s ) . Selfishness Level of Strategic Games – p. 16/23

Linear Congestion Games Linear congestion game: each delay function is of the form d e ( x ) = a e x + b e , where a e , b e ∈ N . Let L be the maximum number of facilities that any player can choose: L : = max i ∈ N , s i ∈ S i | s i | . ∆ max : = max e ∈ E ( a e + b e ) , ∆ min : = min e ∈ E ( a e + b e ) . Proposition Selfishness level of a linear congestion game is 2 ( L · ∆ max − ∆ min − 1 ) . ≤ 1 Note This bound does not depend on the number of players. Selfishness Level of Strategic Games – p. 17/23

Fair Cost Sharing Games (1) Fair cost sharing game: G = ( N , E , { S i } i ∈ N , { c e } e ∈ E ) , where E is the set of facilities, S i ⊆ 2 E is the set of facility subsets available to player i , c e ∈ Q + is the cost of facility e ∈ E . Let x e ( s ) be the number of players using facility e in s . The cost of facility e ∈ E is evenly shared. So c e c i ( s ) : = ∑ e ∈ s i x e ( s ) . Social cost: SC ( s ) = ∑ n i = 1 c i ( s ) . Selfishness Level of Strategic Games – p. 18/23

Fair Cost Sharing Games (2) Let L : = max i ∈ N , s i ∈ S i | s i | . c max : = max e ∈ E c e . 2 L · c max − 1 . Proposition Selfishness level is ≤ 1 Note This bound does not depend on the number of players. Selfishness Level of Strategic Games – p. 19/23

Concluding Remark Other games and equilibria notions can be studied. Example Centipede game and subgame perfect equilibrium. C C C C C C 1 2 1 2 1 2 ( 6 , 5 ) S S S S S S ( 1 , 0 ) ( 0 , 2 ) ( 3 , 1 ) ( 2 , 4 ) ( 5 , 3 ) ( 4 , 6 ) In its unique subgame perfect equilibrium the resulting payoffs are ( 1 , 0 ) . We have 5 +( 6 + 5 ) α ≥ 6 +( 4 + 6 ) α iff α ≥ 1 . So the (redefined) selfishness level is 1. Selfishness Level of Strategic Games – p. 20/23

Some Quotations Dalai Lama: The intelligent way to be selfish is to work for the welfare of others. Microeconomics: Behavior, Institutions, and Evolution , S. Bowles ’04. An excellent way to promote cooperation in a society is to teach people to care about the welfare of others. The Evolution of Cooperation , R. Axelrod, ’84. Selfishness Level of Strategic Games – p. 21/23

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