On Linear Congestion Games with Altruistic Social Context

Abstract

Abstract. We study the issues of existence and inefficiency of pure Nash equilibria in linear congestion games with altruistic social context, in the spirit of the model recently proposed by de Keijzer et al. [13]. In such a framework, given a real matrix Γ = (γij) specifying a particular social context, each player i aims at optimizing a linear combination of the payoffs of all the players in the game, where, for each player j, the multiplicative coefficient is given by the value γij. We give a broad characterization of the social contexts for which pure Nash equilibria are always guaranteed to exist and provide tight or almost tight bounds on their prices of anarchy and stability. In some of the considered cases, our achievements either improve or extend results previously known in the literature.

On the Robustness of the Approximate Price of Anarchy in Generalized Congestion Games

“…We introduced the primal-dual method in [4] as a tool for obtaining tight bounds on the inefficiencies caused by selfish behavior in weighted congestion games and their possible generalizations for a variety of solutions concepts. In particular the primal-dual method has been applied by Bilò, Flammini and Gallotti [7] to derive tight bounds on the worst-case price of anarchy of pure Nash equilibria in congestion games with affine latency functions under the assumption that the players' knowledge is restricted by the presence of an underlying social knowledge graph; by Bilò [5] to derive tight bounds on the worst-case price of stability of pure Nash equilibria in congestion games with affine latency functions and altruistic players; by Bilò and Paladini [9] to derive tight bounds on the approximation ratio of the solutions achieved after a one-round walk of ǫ-approximate bestresponses starting from any initial strategy profile in cut games, for any ǫ ≥ 0; by Bilò et al [8] to derive a surprising matching lower bound on the price of anarchy of subgame perfect equilibria in sequential cut games; and by Bilò, Fanelli and Moscardelli [6] to derive significant upper bounds on the price of anarchy of lookahead equilibria in congestion games with affine latency functions.…”

“…To this aim, for any function SF ∈ {SUM, MAX} 4 One could even relax the constraint β ∈ R n×n ≥0 and allow for negative entries in matrix β as long as i∈[n] βij ≥ 0 for each j ∈ [n] and i∈[n] βij > 0 for some j ∈ [n] which still guarantees either β-SUM(σ) > 0 and β-MAX(σ) > 0 for each σ ∈ Σ. 5 From now on, we will always assume that β is a non-null matrix. 6 Indeed, such a result implicitly holds for the worst-case ǫ-approximate price of anarchy of any kind of equilibrium.…”

“…One could even relax the constraint β ∈ R n×n ≥0 and allow for negative entries in matrix β as long as i∈[n] βij ≥ 0 for each j ∈ [n] and i∈[n] βij > 0 for some j ∈ [n] which still guarantees either β-SUM(σ) > 0 and β-MAX(σ) > 0 for each σ ∈ Σ 5. From now on, we will always assume that β is a non-null matrix 6.…”

Enhancing the Efficiency of Altruism and Taxes in Affine Congestion Games through Signalling

“…A fundamental question in this domain is whether better performances can be obtained when applying plausible changes to the game model. Among these, we consider two well-studied scenarios: partially altruistic players (Bilò 2014;Hoefer and Skopalik 2013) and resource taxation (Bilò and Vinci 2019;Paccagnan and Gairing 2021;Vijayalakshmi and Skopalik 2020).…”

“…In fact, the price of anarchy in games with θ-altruistic players is never smaller than 5/2 and equals 5/2 only when θ = 0. Although proved that a better price of anarchy holds in symmetric singleton games (i.e., games in which all players share the same strategy space which is made of single resources only) as long as θ < 0.7, and Bilò (2014) showed that the price of stability improves as long as θ < 17+ √ 3 26 ≈ 0.72, the question of whether other forms of altruism can yield a better price of anarchy in general (i.e., non-singleton and asymmetric) affine congestion games remained open.…”