On Linear Congestion Games with Altruistic Social Context

Congestion games are, perhaps, the most famous class of non-cooperative games due to their capability to model several interesting competitive scenarios, while maintaining some nice properties. In these games there is a set of players sharing a set of resources, where each resource has an associated latency function which depends on the number of players using it (the so-called congestion). Each player has an available set of strategies, where each strategy is a non-empty subset of resources, and aims at choosing a strategy minimizing her cost which is defined as the sum of the latencies experienced on all the selected resources. Congestion games have been introduced by Rosenthal [11]. He proved that each such a game admits a bounded potential function whose set of local minima coincides with the set of pure Nash equilibria of the game, that is, strategy profiles in which no player can decrease her cost by unilaterally changing her strategic choice. This existence result makes congestion games particularly appealing especially in all those applications in which pure Nash equilibria are elected as the ideal solution concept. In these contexts, the study of the inefficiency of pure Nash equilibria, usually measured by the sum of the costs experienced by all players, has affirmed as a fervent research direction. To this aim, the notions of price of anarchy (Koutsoupias and Papadimitriou [9]) and price of stability (Anshelevich et al. [1]) are widely adopted. The price of anarchy (resp. stability) compares the performance of the worst (resp. best) pure Nash equilibrium with that of an optimal cooperative solution. To the best of our knowledge, Chen and Kempe [5] were the first to study the effects of altruistic (and spiteful) behavior on the existence and inefficiency of pure Nash equilibria in some wellunderstood non-cooperative games. They focus on the class of non-atomic congestion games, where there are infinitely many players each contributing for a negligible amount of congestion, and show that price of anarchy decreases as the degree of altruism of the players increases. Hoefer and Skopalik [7] consider (atomic) linear congestion games with γi-altruistic players, where γi ∈ [0, 1], for each player i. According to their model, player i aims at minimizing a function defined as 1−γi times her cost plus γi times the sum of the costs of all the players in the game (also counting player i). They show that pure Nash equilibria are always guaranteed to exist via a potential function argument, while, in all the other cases in which existence is not guaranteed, they study the complexity of the problem of deciding whether a pure Nash equilibrium exists in a given game. Given the existential result by Hoefer and Skopalik [7], Caragiannis et al. [4] focus on the impact of altruism on the inefficiency of pure Nash equilibria in linear congestion games with altruistic players. However, they consider a more general model of altruistic behavior: in fact, for a parameter γi ∈ [0, 1], they model a γi-altruistic player i as a player who aims at minimizing a function defined as 1 − γi times her cost plus γi times the sum of the costs of all the players in the game other than i. In such a way, the more γi increases, the more γi-altruistic players tend to favor the interests of the others to their own ones, with 1-altruistic and 0-altruistic players being the two opposite extremal situations in which players behave in a completely altruistic or in a completely selfish way, respectively. Caragiannis et al. [4] consider the basic case of γi = γ for each player i and show that the price of anarchy is 5−γ 2−γ for γ ∈ [0, 1/2] and 2−γ 1−γ for γ ∈ [1/2, 1] and that these bounds hold also for load balancing games. This result appears quite surprising, because it shows that altruism can only have a harmful effect on the efficiency of linear congestion games, since the price of anarchy increases from 5/2 up to an unbounded value as the degree of altruism goes from 0 to 1. On the positive side, they prove that, for the special case of symmetric load balancing games, the price of anarchy is 4(1−γ) 3−2γ for γ ∈ [0, 1/2] and 3−2γ 4(1−γ) for γ ∈ [1/2, 1], which shows that altruism has a beneficial effect as long as γ ∈ [0, 0.7]. Note that, that for γ = 1/2, that is when selfishness and altruism are perfectly balanced, the price of anarchy drops to 1 (i.e., all pure Nash equilibria correspond to socially optimal solutions), while, as soon as γ approaches