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 , Vrije Universiteit Amsterdam Selfishness Level of Strategic Games – p. 1/32

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. 2/32

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. 3/32

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. 4/32

Selfishness Level (1) G is α -selfish if a Nash equilibrium of G ( α ) is a social optimum of G ( α ) . Selfishness level of G : inf { α ∈ R + | G is α -selfish } . 0 ) = ∞ . Recall inf ( / Selfishness level of G is α + iff the selfishness level of G is α ∈ R + but G is not α -selfish. Selfishness Level of Strategic Games – p. 5/32

Selfishness Level (2) Intuition Selfishness level quantifies the minimal share of social welfare needed to induce the players to choose a social optimum. Selfishness Level of Strategic Games – p. 6/32

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. 7/32

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. 8/32

Another Example Game with a bad Nash equilibrium H T E H 1 , − 1 − 1 , − 1 , − 1 1 T − 1 , 1 , − 1 − 1 , − 1 1 E − 1 , − 1 − 1 , − 1 − 1 , − 1 The unique Nash equilibrium is ( E , E ) . The selfishness level of this game is ∞ . Selfishness Level of Strategic Games – p. 9/32

Invariance of Selfishness Level Lemma Consider a game G and α ≥ 0 . For every a , G is α -selfish iff G + a is α -selfish, For every a > 0 , G is α -selfish iff aG is α -selfish. Conclusion Selfishness level is invariant under positive linear transformations of the payoff functions. Selfishness Level of Strategic Games – p. 10/32

Selfishness Level vs Price of Stability (1) 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. Selfishness Level of Strategic Games – p. 11/32

Selfishness Level vs Price of Stability (2) Theorem For every finite α > 0 and β > 1 there is a finite game with selfishness level α and price of stability β . Proof Consider G : C D 0 , 2 α + 1 C 1 , 1 α + 1 2 α + 1 D β , 1 1 α + 1 , 0 β In each G ( γ ) with γ ≥ 0 , ( C , C ) is the unique social optimum. Consider G ( γ ) and stipulate that ( C , C ) is its Nash equilibrium. This leads to 1 + 2 γ ≥ ( γ + 1 ) 2 α + 1 α + 1 . This is equivalent to γ ≥ α . So the selfishness level is α . The price of stability is β . Selfishness Level of Strategic Games – p. 12/32

Selfishness Level can be α + Theorem There exists a game that is 0 + -selfish (so α -selfish for every α > 0 , but is not 0-selfish). Proof idea Plug the above games for each α > 0 and fixed β > 1 in: ... ... 0 0 0 0 ... ... 0 0 0 0 0 0 ... ... 0 0 ... ... 0 0 0 0 0 0 0 0 ... ... 0 0 0 0 ... ... Selfishness Level of Strategic Games – p. 13/32

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. 14/32

Characterization Result Player i ’s appeal factor of s ′ i given the social optimum s : p i ( s ′ i , s − i ) − p i ( s i , s − i ) AF i ( s ′ i , s ) : = i , s − i ) . SW ( s i , s − i ) − SW ( s ′ Theorem The selfishness level of G is finite iff a stable social optimum s exists for which i ∈ U i ( s ) AF i ( s ′ α ( s ) : = max i ∈ N , s ′ i , s ) is finite, where U i ( s ) : = { s ′ i ∈ S i | p i ( s ′ i , s − i ) > p i ( s i , s − i ) } . If the selfishness level of G is finite, then it equals min s ∈ SSO α ( s ) , where SSO is the set of stable social optima. Selfishness Level of Strategic Games – p. 15/32

Some Observations If G is finite, then its selfishness level is finite iff it has a stable social optimum. Selfishness level can be unbounded. Theorem For each f : N → R + there exists a class of games G n for n players, such that the selfishness level of G n is f ( n ) . Selfishness Level of Strategic Games – p. 16/32

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. 17/32

Public Goods Game n players, b ∈ R + : fixed budget, c > 1 : a multiplier, S i = [ 0 , b ] , p i ( s ) : = b − s i + c n ∑ j ∈ N s j . 0 , 1 − c n � � Proposition Selfishness level is max . c − 1 Notes Free riding: contributing 0 (it is a dominant strategy). For fixed c temptation to free ride increases with n . For fixed n temptation to free ride decreases as c increases. Selfishness Level of Strategic Games – p. 18/32

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. 19/32

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 , i.e., each s i ⊆ E , 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. 20/32

Fair Cost Sharing Games (2) Singleton cost sharing game: for each s i , | s i | = 1 . c max : = max e ∈ E c e , c min : = min e ∈ E c e , L : = max i ∈ N , s i ∈ S i | s i | (maximum number of facilities that a player can choose). Proposition Selfishness level of 2 c max / c min − 1 , a singleton cost sharing game is ≤ 1 a fair cost sharing game with non-negative integer 2 Lc max − 1 . costs is ≤ 1 Note These bounds are tight. Selfishness Level of Strategic Games – p. 21/32

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 ) . Symmetric congestion game: S i = S j for all i , j . Selfishness Level of Strategic Games – p. 22/32

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 ∈ R + . ∆ max : = max e ∈ E ( a e + b e ) , ∆ min : = min e ∈ E ( a e + b e ) , L : = max i ∈ N , s i ∈ S i | s i | , λ max : maximum discrepancy between two facilities, a min : = min e ∈ E : a e > 0 a e . Proposition Selfishness level of a symmetric singleton linear congestion game is 2 ( ∆ max − ∆ min ) / (( 1 − λ max ) a min ) − 1 ≤ 1 2 , a linear congestion game with non-negative integer 2 ( L · ∆ max − ∆ min − 1 ) . coefficients is ≤ 1 Selfishness Level of Strategic Games – p. 23/32