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 [18]. 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 [16]) and price of stability (Anshelevich et al. [2]) 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.

Congestion games with unrestricted latency functions are general enough to model the Prisoner’s Dilemma game, whose unique pure Nash equilibrium is known to perform arbitrarily bad with respect to the solution in which all players cooperate. Hence, in order to deal with significative bounds on the prices of anarchy and stability, some kind of regularity needs to be imposed on the latency functions associated with the resources. To this aim, lot of research attention has been devoted to the case of polynomial latency functions.

In particular, Awerbuch et al. [4] and Christodoulou and Koutsoupias [11] proved that the price of anarchy of congestion games is $5/2$ for linear latency functions and $d^{\Theta(d)}$ for polynomial latency functions of degree $d$. Subsequently, Aland et al. [1] obtained exact bounds on the price of anarchy for congestion games with polynomial latency functions. Still for linear latencies, Caragiannis et al. [7] proved that the same bounds hold for load balancing games as well, that is, for the restriction in which all possible strategies are singleton sets, while for symmetric load balancing games, that is load balancing games in which the players share the same set of strategies, Lücking et al. [17] proved that the price of anarchy is $4/3$. Moreover, the works of Caragiannis et al. [7] and Christodoulou and Koutsoupias [12] show that the price of stability of congestion games with linear latency functions is $1+1/\sqrt{3}$, while an exact characterization for the case of polynomial latency functions of degree $d$ has been recently given by Christodoulou and Gairing [10].

Hoefer and Skopalik [15] consider (atomic) linear congestion games with $\gamma_{i}$-altruistic players, where $\gamma_{i}\in[0,1]$, for each player $i$. According to their model, player $i$ aims at minimizing a function defined as $1-\gamma_{i}$ times her cost plus $\gamma_{i}$ 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 [15], Caragiannis et al. [8] 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 $\gamma_{i}\in[0,1]$, they model a $\gamma_{i}$-altruistic player $i$ as a player who aims at minimizing a function defined as $1-\gamma_{i}$ times her cost plus $\gamma_{i}$ times the sum of the costs of all the players in the game other than $i$¹¹1Note that each game with $\gamma_{i}$-altruistic players, where $\gamma_{i}\in[0,1]$, in the model of Hoefer and Skopalik [15] maps to a game with $\gamma^{\prime}_{i}$-altruistic players, where $\gamma^{\prime}_{i}\in[0,1/2]$, in the model of Caragiannis et al. [8].. In such a way, the more $\gamma_{i}$ increases, the more $\gamma_{i}$-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. [8] consider the basic case of $\gamma_{i}=\gamma$ for each player $i$ and show that the price of anarchy is $\frac{5-\gamma}{2-\gamma}$ for $\gamma\in[0,1/2]$ and $\frac{2-\gamma}{1-\gamma}$ for $\gamma\in[1/2,1]$ 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 $\frac{4(1-\gamma)}{3-2\gamma}$ for $\gamma\in[0,1/2]$ and $\frac{3-2\gamma}{4(1-\gamma)}$ for $\gamma\in[1/2,1]$, which shows that altruism has a beneficial effect as long as $\gamma\in[0,0.7]$. Note that, that for $\gamma=1/2$, 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 $\gamma$ approaches $1$, the price of anarchy again grows up to an unbounded value.

Recently, de Keijzer et al. [13] proposed a model for altruistic and spiteful behavior further generalizing the one of Caragiannis et al. [8]. According to their definition, each non-cooperative game with $n$ players is coupled with a real matrix $\Gamma=(\gamma_{ij})\in{\mathbb{R}}^{n\times n}$, where $\gamma_{ij}$ expresses how much player $i$ cares about player $j$. In such a framework, player $i$ wants to minimize the sum, for each player $j$ in the game (thus also counting $i$), of the cost of player $j$ multiplied by $\gamma_{ij}$. Thus, a positive (resp. negative) value $\gamma_{ij}$ expresses an altruistic (resp. spiteful) attitude of player $i$ towards player $j$. When considering linear congestion games with altruistic players, along the lines of the negative results of Caragiannis et al. [8], as soon as there are two players $i,j$ such that $\gamma_{ij}>\gamma_{ii}$, i.e., player $i$ cares more about player $j$ than about herself, the price of anarchy becomes unbounded. Therefore, Keijzer et al. [13] focus on the scenario, which they call restricted altruistic social context, in which $\gamma_{ii}\geq\gamma_{ij}$ for each pair of players $i$ and $j$. By extending the smoothness framework of Roughgarden [19], they show an upper bound of $7$ on the price of anarchy of coarse correlated equilibria, which implies the same upper bound also on the price of anarchy of correlated equilibria, mixed Nash equilibria and pure Nash equilibria (whenever the latter exist). Moreover, they prove that, when restricting to load balancing games with identical resources, such an upper bound decreases to $2+\sqrt{5}\approx 4.236$.

Noting that matrix $\Gamma$ implicitly represents the social context (for instance, a social network) in which the players operate, the model of de Keijzer et al. [13] falls within the scope of the so-called social context games. In these games, the payoff of each player is redefined as a function, called aggregating function, of her cost and of those of her neighbors in a given social context graph.

Social context games have been introduced and studied by Ashlagi, Krysta, and Tennenholtz [3] for the class of load balancing games, in the case in which the aggregating function is one among the minimum, maximum, sum and ranking functions, for which they gave an almost complete characterization of the cases in which existence of pure Nash equilibria is guaranteed. The model of de Keijzer et al. [13], hence, coincides with a social context game in which the social context graph has weighted edges and the aggregating function is a weighted sum. The issues of existence and inefficiency of pure Nash equilibria for the case of social context linear congestion games have been considered by Bilò et al. [6]. In particular, for the aggregating function sum, pure Nash equilibria are shown to exist for each social context graph via an exact potential function argument and the price of anarchy is shown to fall within the interval $[5;17/3]$.

Finally, the particular case of social context games in which the social context graph is a partition into cliques coincide with games in which static coalitions among players are allowed. These games have been considered by Fotakis, Kontogiannis and Spirakis [14] who focus on weighted congestion game defined on a parallel link graph when the aggregating function is the maximum function (i.e, the coalitional generalization of the KP-model of Koutsoupias and Papadimitriou [16]). Among their findings, they show that such games always admit a potential function which becomes an exact one in case of linear latency functions (even in the generalization to networks) and that the price of anarchy is $\Theta\left(\min\left\{k,\frac{\log m}{\log\log m}\right\}\right)$, where $m$ denotes the number of links and $k$ denotes the number of coalitions.

We then prove that, in the restricted altruistic social context, the price of anarchy is exactly $17/3$. Such a characterization is achieved by providing an upper bound of $17/3$ which holds for any matrix $\Gamma$ (even the ones for which pure Nash equilibria are not guaranteed to exist) and a matching lower bound which holds even in the special case in which $\Gamma$ is a boolean symmetric matrix and the game is a load balancing one. Such a result has two interesting interpretations: first, it shows that either the upper bound of $7$ given by de Keijzer et al. [13] for the price of anarchy of coarse correlated equilibria is not tight, or that the prices of anarchy of coarse correlated equilibria and pure Nash equilibria are different (the latter hypothesis would be an interesting one, since this situation does not happen in linear congestion games with selfish players); secondly, it proves that the assumption of having identical resources is essential in the upper bound of $2+\sqrt{5}$ given by de Keijzer et al. [13] for the case of load balancing games.

For the price of stability in the restricted altruistic social context, we give an upper bound of $2$ holding for each symmetric matrix $\Gamma$ and a lower bound of $1+1/\sqrt{2}\approx 1.707$ holding for the case in which $\Gamma$ is a boolean symmetric matrix.

Finally, we also consider the special case in which $\Gamma$ is such that $\gamma_{ij}=\gamma_{i}$ for each pair of indexes $i,j$ with $i\neq j$, which coincides with the general model of $\gamma_{i}$-altruistic players of Caragiannis et al. [8]. We show that pure Nash equilibria are always guaranteed to exist in any case via an exact potential function argument (this slightly improves the existential result by Hoefer and Skopalik [15] since they only proved the existence of a weighted potential function) and give an upper bound on the price of anarchy in the general case and an exact bound on the price of stability when $\gamma_{i}=\gamma$ for each player $i$.

A congestion game is a tuple ${\cal G}=\langle[n],E,S_{i\in[n]},\ell_{e\in E}\rangle$, where $[n]:=\{1,\ldots,n\}$ is a set of $n\geq 2$ players, $E$ is a set of resources, $\emptyset\neq S_{i}\subseteq 2^{E}$ is the set of strategies of player $i$, and $\ell_{e}:{\mathbb{N}}\rightarrow{\mathbb{R}_{\geq 0}}$ is the latency function of resource $e$. The special case in which, for each $i\in[n]$ and each $s\in S_{i}$, it holds $|s|=1$ is called load balancing congestion game. Denoted by ${\cal S}:=\times_{i\in[n]}S_{i}$ the set of strategy profiles in $\cal G$, that is, the set of outcomes of $\cal G$ in which each player selects a single strategy, the cost of player $i$ in the strategy profile $S=(s_{1},\ldots,s_{n})\in{\cal S}$ is defined as $c_{i}(S)=\sum_{e\in s_{i}}\ell_{e}(n_{e}(S))$, where $n_{e}(S):=|\{j\in[n]:e\in s_{j}\}|$ is the congestion of resource $e$ in $S$, that is, the number of players using $e$ in $S$.

Given a strategy profile $S=(s_{1},\ldots,s_{n})$ and a strategy $t\in S_{i}$ for a player $i\in[n]$, we denote with $S_{-i}\diamond t$ the strategy profile obtained from $S$ by replacing the strategy played by $i$ in $S$ with $t$. A pure Nash equilibrium is a strategy profile $S$ such that, for any player $i\in[n]$ and for any strategy $t\in S_{i}$, it holds $c_{i}(S_{-i}\diamond t)\geq c_{i}(S)$.

The function ${\sf{SUM}}:{\cal S}\rightarrow{\mathbb{R}_{\geq 0}}$ such that ${\sf{SUM}}(S)=\sum_{i\in[n]}c_{i}(S)$, called the social function, measures the social welfare of a game. Given a congestion game $\cal G$, let ${\cal NE}({\cal G})$ denote the set of its pure Nash equilibria (such a set has been shown to be non-empty by Rosenthal [18]) and $S^{*}$ be the strategy profile minimizing the social function. The price of anarchy (PoA) of $\cal G$ is defined as $\max_{S\in{\cal NE(G)}}\left\{\frac{{\sf{SUM}}(S)}{{\sf{SUM}}(S^{*})}\right\}$, while the price of stability (PoS) of $\cal G$ is defined as $\min_{S\in{\cal NE(G)}}\left\{\frac{{\sf{SUM}}(S)}{{\sf{SUM}}(S^{*})}\right\}$.

A linear congestion game is a congestion game such that, for each $e\in E$, it holds $\ell_{e}(x)=\alpha_{e}x+\beta_{e}$, with $\alpha_{e},\beta_{e}\geq 0$. For these games, the cost of player $i$ in the strategy profile $S=(s_{1},\ldots,s_{n})$ becomes $c_{i}(S)=\sum_{e\in s_{i}}\left(\alpha_{e}n_{e}(S)+\beta_{e}\right)$, while the social value of $S$ becomes ${\sf{SUM}}(S)=\sum_{i\in[n]}\sum_{e\in s_{i}}\left(\alpha_{e}n_{e}(S)+\beta_{e}\right)=\sum_{e\in E}\left(\alpha_{e}n_{e}(S)^{2}+\beta_{e}n_{e}(S)\right)$.

A linear congestion game with an altruistic social context is a pair $({\cal G},\Gamma)$ such that $\cal G$ is a linear congestion game with $n$ players and $\Gamma=(\gamma_{ij})\in{\mathbb{R}}^{n\times n}$ is a real matrix such that $\gamma_{ij}\geq 0$ for each $i,j\in[n]$. The set of players and strategies is defined as in the underlying linear congestion game $\cal G$, while, for any strategy profile $S$, the cost of player $i$ is $S=(s_{1},\ldots,s_{n})$ is defined as $\widehat{c}_{i}(S)=\sum_{j\in[n]}\left(\gamma_{ij}\cdot c_{j}(S)\right)=\sum_{j\in[n]}\left(\gamma_{ij}\left(\alpha_{e}n_{e}(S)+\beta_{e}\right)\right)=\sum_{e\in E}\left(\left(\alpha_{e}n_{e}(S)+\beta_{e}\right)\sum_{j\in[n]:e\in s_{j}}\gamma_{ij}\right)$, where $c_{j}(S)$ is the cost of player $j$ in $S$ in the underlying linear congestion game $\cal G$. The special case in which $\gamma_{ii}\geq\gamma_{ij}$ for each $i,j\in[n]$, is called restricted altruistic social context. Note that, in such a case, as pointed out by de Keijzer et al. [13], it is possible to assume without loss of generality that $\gamma_{ii}=1$ for each $i\in[n]$²²2This claim follows from the fact that both the set of pure Nash equilibria and the social value of any strategy profile do not change when dividing all the entries in row $i$ of $\Gamma$ by the value $\gamma_{ii}$..

In this section, we provide a complete characterization of the social contexts for which pure Nash equilibria are guaranteed to exist, independently of which is the underlying linear congestion game.

For a strategy profile $S=(s_{1},\ldots,s_{n})$, a player $i\in[n]$ and a strategy $t\in S_{i}$, for the sake of brevity, let us denote with $x_{e}:=n_{e}(S)$ and with $x^{\prime}_{e}=n_{e}(S_{-i}\diamond t)$. It holds

On the positive side, we show that, for restricted altruistic social contexts such that $\Gamma$ is symmetric, pure Nash equilibria do always exist.

In order to prove that the characterization given in Theorem 3.1 is tight, we provide the following two non-existential results. In the first one, although preserving the property that $\Gamma$ is symmetric, we relax the constraint that the game is played in a restricted altruistic social context: in particular, we allow $\gamma_{ii}=0$ for some player $i\in[n]$.

In the second result, although preserving the property that the game is played in a restricted altruistic social context, we relax the constraint that $\Gamma$ is symmetric.

In this section, we give bounds on the prices of anarchy and stability of linear congestion games with restricted social context. These bounds are achieved by applying the primal-dual technique that we introduced in [5]. To this aim, we recall that it is possible to assume without loss of generality that $\beta_{e}=0$ for each $e\in E$ as long as we are not interested in load balancing games. For a given linear congestion game with altruistic social context $({\cal G},\Gamma)$, we denote with $K=(k_{1},\ldots,k_{n})$ and $O=(o_{1},\ldots,o_{n})$, respectively, a Nash equilibrium and a social optimum of $({\cal G},\Gamma)$ and we use $K_{e}:=n_{e}(K)$ and $O_{e}:=n_{e}(O)$ to denote the congestion of resource $e$ in $K$ and $O$, respectively.

The primal-dual method aims at formulating the problem of maximizing the ratio $\frac{{\sf SUM}(K)}{{\sf SUM}(O)}$ via linear programming. The two strategy profiles $K$ and $O$ play the role of fixed constants, while, for each $e\in E$, the values $\alpha_{e}$ defining the latency functions are variables that must be suitably chosen so as to satisfy two constraints: the first, assures that $K$ is a pure Nash equilibrium, while the second normalizes to $1$ the value of the social optimum ${\sf SUM}(O)$. The objective function aims at maximizing the social value ${\sf SUM}(K)$ which, being the social optimum normalized to $1$, is equivalent to maximize the ratio $\frac{{\sf SUM}(K)}{{\sf SUM}(O)}$. Let us denote with $LP(K,O)$ such a linear program, which, in our scenario of investigation becomes

Let $DLP(K,O)$ be the dual program of $LP(K,O)$. By the Weak Duality Theorem, each feasible solution to $DLP(K,O)$ provides an upper bound on the optimal solution of $LP(K,O)$. Hence, by providing a feasible dual solution, we obtain an upper bound on the ratio $\frac{{\sf SUM}(K)}{{\sf SUM}(O)}$. Anyway, if the provided dual solution is independent on the particular choice of $K$ and $O$, we obtain an upper bound on the ratio $\frac{{\sf SUM}(K)}{{\sf SUM}(O)}$ for any possible pair of profiles $K$ and $O$, which means that we obtain an upper bound on the price of anarchy of pure Nash equilibria. The dual program $DLP(K,O)$ is

When compared to the upper bound of $7$ for the price of anarchy of coarse correlated equilibria given by de Keijzer et al. [13], our upper bound implies that either the one for coarse correlated equilibria is not tight, or the prices of anarchy of coarse correlated equilibria and pure Nash equilibria are different. Such a latter case would be quite significant since this does not happen in linear congestion games with selfish players.

We now give a marching lower bound which holds even in the special case in which $\Gamma$ is a symmetric boolean matrix and the underlying linear congestion game is a load balancing one. The basic idea of our construction, suitably extended to comply with our altruistic scenario, is borrowed from Caragiannis et al. [7].

Note that such a lower bound implies that the the assumption of identical resources in crucial in the upper bound of $2+\sqrt{5}$ given by de Keijzer et al. [13] for load balancing games with restricted altruistic social context.

We now turn our attention to the study of the price of stability. By exploiting the potential function defined in the previous section and the fact that there exists a pure Nash equilibrium $K$ such that $\Phi(K)\leq\Phi(O)$, we easily obtain the following upper bound.

In this case, we are only able to provide a lower bound of $1+\frac{1}{\sqrt{2}}\approx 1.707$.

In this section, we focus on the special case given by model of Caragiannis et al. [8] in which, for each $i\in[n]$, it holds $\gamma_{ii}=1-\gamma_{i}$ and $\gamma_{ij}=\gamma_{i}$ for each $j\neq i\in[n]$, where $\gamma_{i}\in[0,1]$. In such a model, the restricted altruistic social context coincides with the case in which, for each $i\in[n]$, it holds $\gamma_{i}\leq 1/2$. Caragiannis et al. [8] show that, when $\gamma_{i}=\gamma$ for each $i\in[n]$, the price of anarchy is exactly $\frac{2-\gamma}{1-\gamma}$ for general altruistic social contexts and $\frac{5-\gamma}{2-\gamma}$ in the restricted one.

First of all, we prove that pure Nash equilibria are always guaranteed to exist via an exact potential function argument. An existential result had already been given by Hoefer and Skopalik [15], nevertheless, their proof makes use of a weighted potential function. So, our result is slightly stronger and, more importantly, provides a better potential function to be subsequently exploited in the derivation of an upper bound on the price of stability of these games.

Let ${\cal V}_{n}:=[0,1]^{n}$ be the set of $n$-dimensional vectors whose entries belong to the interval $[0,1]$. Given a vector $V=(v_{1},\ldots,v_{n})\in{\cal V}_{n}$, denote with $\Gamma_{V}$ the $n\times n$ matrix $\Gamma$ such that, for each $i\in[n]$, it holds $\gamma_{ii}=1-v_{i}$ and $\gamma_{ij}=v_{i}$ for each $j\neq i\in[n]$.

By exploiting the potential function defined above, we obtain an upper bound on the price of stability for the case in which $v_{i}=v$ for each $i\in[n]$ as follows.

The fact that there exists a pure Nash equilibrium $K$ such that $\Phi(K)\leq\Phi(O)$ easily implies the following inequality (where, as usual, we have removed the terms $\beta_{e}$ from the latency functions):

where we have used the equalities $\sum_{j:e\in k_{j}}v_{j}=vK_{e}$ and $\sum_{j:e\in o_{j}}v_{j}=vO_{e}$.

By exploiting the inequality $\widehat{c}_{i}(K)-\widehat{c}_{i}(K_{-i}\diamond o_{i})\leq 0$, we obtain that, for each $i\in[n]$, it holds

which implies

Using both inequalities (5) and (6), the primal formulation $LP(K,O)$ becomes the following one.

The dual program $DLP(K,O)$ is

We now show matching lower bounds.