Class Of Potential Games Research Articles

With Potential Games, Which Outcome Is Better?

Lower one- or two-dimensional coordination, or potential games, are popularly used to model interactive behavior, such as innovation diffusion and cultural evolution. Typically, this involves determining the “better” of competing solutions. However, examples have demonstrated that different measures of a “good” choice can lead to conflicting conclusions; a fact that reflects the history of game theory in equilibrium selection. This behavior is totally explained while extending the analysis to the full seven-dimensional class of potential games, which includes coordination games.

The structure of utility functions ensuring the existence of an exact potential in a strategic game

Strategic games in which the profit of each participant is the sum of local profits gained by means certain “objects” and shared by all players using these objects are considered. If each object satisfies a regularity condition, then the game has an exact potential. Both universal classes of potential games considered previously in the literature, overflow games and games with structured utility functions, are contained in the class under consideration and satisfy the regularity condition.

A survey of static and dynamic potential games

Potential games are noncooperative games for which there exist auxiliary functions, called potentials, such that the maximizers of the potential are also Nash equilibria of the corresponding game. Some properties of Nash equilibria, such as existence or stability, can be derived from the potential, whenever it exists. We survey different classes of potential games in the static and dynamic cases, with a finite number of players, as well as in population games where a continuum of players is allowed. Likewise, theoretical concepts and applications are discussed by means of illustrative examples.

Approximate Pure Nash Equilibria in Weighted Congestion Games

We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ --approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ . Do such equilibria exist for small values of ρ ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ -games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5/2 + O γ for arbitrarily small γ > 0; for latency functions with maximum degree d ≥ 2, it is d 2 d + o ( d ). The running time is polynomial in the number of bits in the representation of the game and 1/ γ . As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO ( d 2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ -games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications.

Potential Games Are Necessary to Ensure Pure Nash Equilibria in Cost Sharing Games

We consider the problem of designing distribution rules to share “welfare” (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., the Shapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantees the existence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class of scalable and separable games that includes a variety of applications such as facility location, routing, network formation, and coverage games. Given arbitrary local welfare functions 𝕨, we prove that a distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weighted Shapley value on some “ground” welfare functions 𝕨′, which can be distinct from 𝕨. However, if budget-balance is required in addition to the existence of a Nash equilibrium, then 𝕨′ must be the same as 𝕨. We also provide an alternate characterization of this space in terms of “generalized” marginal contributions, which is more appealing from the point of view of computational tractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games with any fixed local welfare functions, it is necessary to work within the class of potential games.

Two Classes of Potential Games and the Solving Method of the Equilibria

Using the definitions of potential functions and ordinal potential functions of strategic form games, we characterize two classes of potential games and give the solving method of the equilibria of finite potential games by testing and verifying potential games. The main contribution is proving that public goods games and social coordination games based on location strategic interaction with an exogenous network are potential games, and giving the equilibria by the method of solving potential games and ordinal potential games.

A Cross-Layer-Based Topology Control Framework for Wireless Multihop Networks

In this paper, we propose a distributed and asynchronous topology control framework for wireless multihop networks, aiming at improving the quality-of-service capabilities of such systems. The proposed mechanisms exploit cross-layer information such as physical-layer signal-to-interference-plus-noise ratio (SINR) values and medium-access-control (MAC)/network-layer connectivity to effectively allocate resources. Based on properly defined utility functions, we cast the transmission power allocation problem as a noncooperative game and then show that it belongs to the class of potential games. We obtain structural properties of the defined game and its variations and demonstrate their effectiveness. Through analysis and simulation, we explore the behavior of the proposed framework and compare it with the most relevant approach in the literature. It is shown that the proposed unifying game-theoretic framework achieves three key goals for multihop ad hoc networks, i.e., reduced transmission power, high-quality links, and connectivity maintenance.

Co-evolutionary dynamics and Bayesian interaction games

Recently there has been a growing interest in evolutionary models of play with endogenous interaction structure. We call such processes co-evolutionary dynamics of networks and play. We study a co-evolutionary process of networks and play in settings where players have diverse preferences. In the class of potential games we provide a closed-form solution for the unique invariant distribution of this process. Based on this result we derive various asymptotic statistics generated by the co-evolutionary process. We give a complete characterization of the random graph model, and stochastically stable states in the small noise limit. Thereby we can select among action profiles and networks which appear jointly with non-vanishing frequency in the limit of small noise in the population. We further study stochastic stability in the limit of large player populations.

Convergence and approximation in potential games

We study the speed of convergence to approximately optimal states in two classes of potential games. We provide bounds in terms of the number of rounds, where a round consists of a sequence of movements, with each player appearing at least once in each round. We model the sequential interaction between players by a best-response walk in the state graph, where every transition in the walk corresponds to a best response of a player. Our goal is to bound the social value of the states at the end of such walks. In this paper, we focus on two classes of potential games: selfish routing games, and cut games (or party affiliation games (Fabrikant et al. 2004 [12])).Other than bounding the price of anarchy of selfish routing games (Roughgarden and Tardos, 2002 [25], Awerbuch et al. 2005 [2], Christodoulou and Koutsoupias, 2005 [9]), there are many interesting problems about game dynamics in these games. It is known that exponentially long best-response walks may exist to pure Nash equilibria (Fabrikant et al. 2004 [12]), and random best-response walks converge to solutions with good approximation guarantees after polynomially many best responses (Goemans et al. 2005 [17]). In this paper, we study the speed of convergence on deterministic best-response walks in these games and prove that starting from an arbitrary configuration, after one round of best responses of players, the resulting configuration is a Θ(n)-approximate solution. Furthermore, we show that starting from an empty configuration, the solution after any round of best responses is a constant-factor approximation. We also provide a lower bound for the multi-round case, where we show that for any constant number of rounds t, the approximation guarantee cannot be better than nϵ(t), for some ϵ(t)>0.We also study cut games, that provide an illustrative example of potential games. The convergence of potential games to locally optimum solutions has been studied in the context of local search algorithms (Johnson et al. 1988 [19], Schaffer and Yannakakis, 1991 [28]). In these games, we consider two social functions: the cut (defined as the weight of the edges in the cut), and the total happiness (defined as the weight of the edges in the cut, minus the weight of the remaining edges). For the cut social function, we prove that the expected social value after one round of a random best-response walk is at least a constant factor approximation to the optimal social value. We also exhibit exponentially long best-response walks with poor social value. For the unweighted version of this cut game, we prove Ω(n) and O(n) lower and upper bounds on the number of rounds of best responses to converge to a constant-factor cut. In addition, we suggest a way to modify the game to enforce a fast convergence in any fair best-response walk. For the total happiness social function, we show that for unweighted graphs of sufficiently large girth, starting from a random configuration, greedy behavior of players in a random order converges to an approximate solution after one round.

Best-response potential games

Best-response potential games are introduced and characterized. Relations with other classes of potential games are indicated.