Convergence Issues in Competitive Games

NONCOOPERATIVE GAME ON SCHEDULING: THE SINGLE MACHINE CASE

Among other solution concepts, the notion of the pure Nash equilibrium plays a central role in Game Theory. Pure Nash equilibria in a game characterize situations with non-cooperative deterministic players in which no player has any incentive to unilaterally deviate from the current situation in order to achieve a higher payoff. Unfortunately, it is well known that there are games that do not have pure Nash equilibria. Furhermore, even in games where the existence of equilibria is guaranteed, their computation can be a computationally hard task. Such negative results significantly question the importance of pure Nash equilibria as solution concepts that characterize the behavior of rational players. Approximate pure Nash equilibria, which characterize situations where no player can significantly improve her payoff by unilaterally deviating from her current strategy, could serve as alternative solution concepts provided that they exist and can be computed efficiently. In this letter, we discuss recent positive algorithmic results for approximate pure Nash equilibria in congestion games.

Edge computing is an emerging novel computation paradigm, which plays a vital role in the 5G era. A basic application scenario in edge computing is that end users have the ability to choose edge server nodes, which are expected to participate in working on the tasks allocated by end users. At the same time, edge server nodes can freely make their own decisions on serving, which end user's task. End users have to compete with each other for edge server nodes’ service, through the payment they can offer to edge nodes for service. In this paper, we apply game theory to model the interaction between end users and edge server nodes as a noncooperative game. The end users are the players who compete for service, and the edge server nodes are the ones who provide service. We have proved the existence of Nash Equilibrium in the game, which suggests that there exists a stable system state in the edge computing environment. We have also shown the stability of the Nash Equilibrium in a best response dynamic game playing process, which further indicates that the stable system state can be reached in practice where end users dynamically adjusting their strategies in the process of competing for edge computing resources provided by server nodes.

Two noncooperative games between a coalition and its surrounding in a class of n-person games with constant sum

Convergence and approximation in potential games

Multi-person stopping games with players’ priorities are considered. Players observe sequentially offers Y1,Y2,… at jump times T1,T2,… of a Poisson process. Y1,Y2,… are independent identically distributed random variables. Each accepted offer Yn results in a reward Gn=Ynr(Tn), where r is a non-increasing discount function. If more than one player wants to accept an offer, then the player with the highest priority (the lowest ordering) gets the reward. We construct Nash equilibrium in the multi-person stopping game using the solution of a multiple optimal stopping time problem with structure of rewards {Gn}. We compare rewards and stopping times of the players in Nash equilibrium in the game with the optimal rewards and optimal stopping times in the multiple stopping time problem. It is also proved that presented Nash equilibrium is a Pareto optimum of the game. The game is a generalization of the Elfving stopping time problem to multi-person stopping games with priorities.

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash’s variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large. An important constituent of bargaining is the specification of the payoff vector when players fail to achieve an agreement. This payoff vector is called a disagreement point. With the major exception of Nash (1953), a disagreement point is assumed to be exogenously given. I) In some economic examples, exogenous disagreement points such as an endowment point in an exchange economy are naturally determined. In general game situations and in economic situations with externalities, however, we may not find a natural definition of an exogenously given disagreement point. This fact requires us to consider a bargaining model with an endogenous determination of a disagreement point. Nash (1953) gave such a model with two players. The purpose of this paper is to investigate the behaviour of this model with n players. We compare, from both axiomatic and noncooperative game theoretic viewpoints, the n-player extension of Nash’s model with an alternative model where disagreement points are also endogenously determined. Nash’s (1953) model - the first model in this paper - is as follows. An underlying environment is described by a noncooperative strategic game G. The players are allowed to cooperate for obtaining higher payoffs and bargain over possible cooperative payoffs, subject to the possibility of failing to achieve an agreement. If the players fail, they return to the original noncooperative game. For this possibility, each player has to choose a strategy in the case of disagreement, prior to bargaining.

Riccati equations are the basis to construct Nash equilibrium solutions for linear quadratic (LQ) non-cooperative games when players apply linear state feedback strategies. Diagonal-form solutions of the Riccati equation associated with a second-order two-player non-zero sum linear quadratic game is discussed and the Nash equilibrium can be implemented by decentralized control strategies. The result is also extended to a high order two-player linear quadratic game.

In this paper, energy efficient power control for small cells underlaying a macro cellular network is investigated. We formulate the power control problem in self-organizing small cell networks as a non-cooperative game, and propose a distributed energy efficient power control scheme, which allows the small base stations (SBSs) to take individual decisions for attaining the Nash equilibrium (NE) with minimum information exchange. Specially, in the non-cooperative power control game, a non-convex optimization problem is formulated for each SBS to maximize their energy efficiency (EE). By exploiting the properties of parameter-free fractional programming and the concept of perspective function, the non-convex optimization problem for each SBS is transformed into an equivalent constrained convex optimization problem. Then, the constrained convex optimization problem is converted into an unconstrained convex optimization problem by exploiting the mixed penalty function method. The inequality constraints are eliminated by introducing the logarithmic barrier functions and the equality constraint is eliminated by introducing the quadratic penalty function. We also theoretically show the existence and the uniqueness of the NE in the non-cooperative power control game. Simulation results show remarkable improvements in terms of EE by using the proposed scheme.

Cross-silo federated learning (FL) is a distributed learning approach where clients of the same interest train a global model cooperatively while keeping their local data private. While there has been some work on incentivizing clients to join FL, the analysis of clients long-term selfish participation behaviors in cross-silo FL remains largely unexplored. In this paper, we model clients long-term selfish participation behaviors as an infinitely repeated game, with the stage game being a selfish participation game in one cross-silo FL process (SPFL). For the stage game SPFL, we derive the unique Nash equilibrium (NE), and propose a distributed algorithm for each client to calculate its equilibrium participation strategy. We show that at the NE, some clients may choose to be free riders. The existence of free riders has a detrimental effect on sustaining other clients long-term participation. For the long-term interactions among clients, we derive the optimal SPNE which minimizes the number of free riders while maximizing the amount of local data for model training. Simulation results show that our derived optimal SPNE can effectively reduce the number of free riders by up to 99.3% and increase the amount of local data for model training by up to 82.3%.

We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a directed network in which each edge possesses a latency function describing the common latency incurred by all traffic on the edge as a function of the edge congestion. Given a rate of traffic between each pair of nodes in the network, we aspire toward an assignment of traffic to paths in which the sum of all travel times (the total latency) is minimized; however, in many settings network users are free to route their traffic in a selfish manner, without regard to the total latency. We therefore assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a selfishly motivated assignment of traffic to paths (a Nash equilibrium) will not minimize the total latency; hence, selfish behavior carries the cost of decreased network performance. We quantify this degradation in network performance via the price of anarchy, defined as the worst possible ratio between the total latency of a Nash equilibrium and of a minimum-latency routing of the traffic.In this paper, we show that the underlying network topology plays no role in the determination of the price of anarchy. Specifically, we show that under weak hypotheses on the class of allowable edge latency functions, the worst-case ratio between the total latency of a Nash equilibrium and of a minimum-latency routing for any multicommodity flow network is achieved by a single-commodity instance on a set of parallel links. In the special case where the class of allowable latency functions includes all of the constant functions, we prove that a network with only two parallel links suffices to achieve the worst-possible ratio. Informally, these results imply that the inefficiency inherent in a flow at Nash equilibrium stems from the inability of selfish users to discern which of two competing routes is superior and not from the topological complexity arising from the diverse intersections of many paths belonging to different commoditie.Our proof techniques also give powerful methods for computing the price of anarchy with respect to an arbitrary class of latency functions. We apply these methods to function classes that have been well studied in the literature (such as degree-bounded polynomials and functions of the form $\ell(x) = (u—x)—1} that are used to model edges with capacity u), thereby achieving the first tight analyses of the price of anarchy for significant classes of latency functions outside the class of linear functions.

Reciprocity, fairness and learning in medium access control games

This chapter is devoted to the notion of a skeptical ability. Though skepticism is characterized by an ability (PH 1.8) and the skeptic is the person who has this ability (PH 1.11), the idea of a skeptical ability is restricted to philosophical investigation. The skeptical way of reasoning leads the Pyrrhonist to suspension of judgment by exercising his ability under the direction of the main principle. Some scholars claim that this ability is basically a reactive capacity to counterattack what has been put forward by dogmatists. Arguing against this view, I propose that the skeptical ability consists in the capacity of actively arguing on both sides of an opposition. The abilities required to argue more dogmatically on the positive side are different from the abilities required to argue more skeptically or aporetically on the negative side. Simply put: whereas in the positive or dogmatic side of the opposition, the abilities required are the capacity to interpret and use arguments already available, the chief abilities in the negative or skeptical side of the opposition are to invent new arguments by applying some patterns and combining them in a persuasive counterattack. What is important to my interpretation is that we realize how many things one must do in order to be a (good) skeptic.KeywordsAbilityOppositionDogmatic argumentsSkeptical argumentsArgumentative patternsModes of suspensionSkeptical counterattack

Learning in Games by Random Sampling

I consider games with ambiguous payoffs played by non-Expected Utility decision makers. Three equilibrium solutions are studied. Nash equilibrium in which equilibrium mixed strategies must be best responses, Crawford equilibrium in beliefs and pure equilibrium in beliefs in which equilibrium strategies are mixtures of best responses, with the latter restricting best responses to pure actions. I study the interactions between ambiguity preferences on one side and equilibrium properties on the other. I show how the equilibrium concepts differ, computing necessary and sufficient conditions for existence and equivalence. I also show how these solution concepts fare against two fundamental principles of Nash equilibrium in standard games: the principle of indifference and the reduction principle. Given both are central to the computation of Nash equilibrium in games with Expected Utility players, their failure indicates how relaxing the Expected Utility hypothesis may disrupt standard game theoretic results such as the characterization of equilibria in two-player games.