Convergence in games with continua of equilibria

Networked noncooperative games are investigated, where each player (or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors' current information to decide its next strategy. By using sub-neighborhood, the dynamics of the evolution is obtained. Then a method for calculating Nash equilibriums from mixed strategies of multi-players is proposed. The relationship between local Nash equilibriums based on individual neighborhoods and global Nash equilibriums of overall network is revealed. Then a technique is proposed to construct Nash equilibriums of an evolutionary game from its one step static Nash equilibriums. The basic tool of this approach is the semi-tensor product of matrices, which converts strategies into logical matrices and payoffs into pseudo-Boolean functions, then networked evolutionary games become discrete time dynamic systems. Citation: Daizhan Cheng, Tingting Xu, Fenghua He, Hongsheng Qi. On dynamics and Nash equilibriums of networked games. IEEE/CAA Journal of Automatica Sinica, 2014, 1(1): 10-18

Social networks are the substrate upon which we make and evaluate many of our daily decisions: our costs and benefits depend on whether-or how many of, or which of-our friends are willing to go to that restaurant, choose that cellular provider, already own that gaming platform. Much of the research on the "diffusion of innovation," for example, takes a gametheoretic perspective on strategic decisions made by people embedded in a social context. Indeed, multiplayer games played on social networks, where the network's nodes correspond to the game's players, have proven to be fruitful models of many natural scenarios involving strategic interaction. In this paper, we embark on a mathematical and general exploration of the relationship between two-person strategic interactions (a "base game") and a "networked" version of that same game. We formulate a generic mechanism for superimposing a symmetric two-player base game M on a social network G: each node of G chooses a single strategy from M and simultaneously plays that strategy against each of its neighbors in G, receiving as its payoff the sum of the payoffs from playing M against each neighbor. We denote the networked game that results by M ⊕ G. We are broadly interested in the relationship between properties of M and of M ⊕ G: how does the character of strategic interaction change when it is embedded in a social network? We focus on two particular properties: the (pure) price of anarchy and the existence of pure Nash equilibria. We show tight results on the relationship between the price of anarchy in M and M ⊕ G in coordination games. We also show that, with some exceptions when G is bipartite, the existence or absence of pure Nash equilibria (and even the guaranteed convergence of best-response dynamics) in M and M ⊕G are not entailed in either direction. Taken together, these results suggest that the process of superimposing M on a graph is a nontrivial operation that can have rich, but bounded, effects on the strategic environment.

IEEE Transactions on Automatic Control, 2016

We prove that the piecewise linear best-response dynamical systems of strategic interactions are asymptotically convergent to their set of equilibria on any weighted undirected graph. We study various features of these dynamical systems, including the uniqueness and abundance properties of the set of equilibria and the emergence of unstable equilibria. We also introduce the novel notions of social equivalence and social dominance on directed graphs, and demonstrate some of their interesting implications, including their correspondence to consensus and chromatic number of partite graphs. Examples illustrate our results.

Theoretical Computer Science, 2012

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 [7]).

2015 IEEE Conference on Computer Communications (INFOCOM), 2015

We study the convergence properties of distributed network selection in HetNets with priority-based service. Clients in such networks have different priority weights (e.g., QoS requirements, scheduling policies, etc.) for different access networks and act selfishly to maximize their own throughput. We formulate the problem as a non-cooperative game, and study its convergence for two models: (i) A purely client-centric model where each client uses its own preference to select a network, and (ii) a hybrid client-network model that uses a combination of client and network preferences to arrive at pairings. Our results reveal that: (a) Pure client-centric network selection with generic weights can result in infinite oscillations for any improvement path (i.e., shows strongly cyclic behavior). However, we show that under several classes of practical priority weights (e.g., weights that achieve different notions of fairness) or under additional client-side policies, convergence can be guaranteed; (b) We study convergence time under client-centric model and provide tight polynomial and linear bounds; (c) We show that applying a minimal amount of network control in the hybrid model, guarantees convergence for clients with generic weights. We also introduce a controllable knob that network controller can employ to balance between convergence time and its networkwide objective with predictable tradeoff.

2014

Various social contexts ranging from public goods provision to information collection can be depicted as games of strategic interactions, where a player's well-being depends on her own action as well as on the actions taken by her neighbors. Whereas much attention has been devoted to the identification and characterization of Bayes-Nash equilibria of such games, in this work we look at strategic interactions from an evolutionary perspective. Starting from a recent mean-field analysis of the evolutionary dynamics in these games, here we present results of numerical simulations designed to find out whether Nash equilibria are accessible by adaptation of players' strategies, and in general to find the attractors of the evolution. Simulations allow us to go beyond a global characterization of the cooperativeness of the equilibria and probe into the individual behavior. We find that when players imitate each other, the evolution does not reach Nash equilibria and, worse, leads to very unfavorable states in terms of welfare. On the contrary, when players update their behavior rationally, they self-organize into a rich variety of Nash equilibria, where individual behavior and payoffs are shaped by the nature of the game, the structure of the social network and the players' position within the topology. Our results allow us to assess the validity of the mean-field approaches and also show qualitative agreement with theoretical predictions for equilibria in the context of one-shot games under incomplete information.

In this chapter, we examine convergence behavior in simple bimatrix games. We classify the possible types of simple games, pick interesting examples of each type, and summarize convergence behavior under various information and player matching protocols. See Friedman (1996), Cheung and Friedman (1997) and Bouchez (1998) for more complete descriptions of the experiments.

Journal of Optimization Theory and Applications, 2014

In this paper, the problem of relations between closed loop and open loop Nash equilibria is examined in the environment of discrete time dynamic games with a continuum of players and a compound structure encompassing both private and global state variables. An equivalence theorem between these classes of equilibria is proven, important implications for the calculation of these equilibria are derived and the results are presented on models of a common ecosystem exploited by a continuum of players. An example of an analogous game with finitely many players is also presented for comparison.

2000

This paper deals with the approximation of Nash equilibria in m-player games. We present conditions under which an approximating sequence of games admits near-equilibria that approximate near-equilibria in the limit game. We apply the results to two classes of games:(i) a duopoly game approximated by a sequence of matrix games, and (ii) a stochastic game played under the S-adapted information structure approximated by games played over a sampled event tree.

Economics Letters, 2001

We study solution concepts for economic games that are resistant to local deviations. Strategy spaces are l subsets of R and local deviations are small in the Euclidean metric. We define local Nash equilibrium and local evolutionarily stable strategy, and present applications to Walrasian outcomes in Cournot games and separating outcomes in screening models.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2011

We consider a network of coupled agents playing the Prisoner's Dilemma game, in which players are allowed to pick a strategy in the interval [0, 1], with 0 corresponding to defection, 1 to cooperation, and intermediate values representing mixed strategies in which each player may act as a cooperator or a defector over a large number of interactions with a certain probability. Our model is payoff-driven, i.e., we assume that the level of accumulated payoff at each node is a relevant parameter in the selection of strategies. Also, we consider that each player chooses his/her strategy in a context of limited information. We present a deterministic nonlinear model for the evolution of strategies. We show that the final strategies depend on the network structure and on the choice of the parameters of the game. We find that polarized strategies (pure cooperator/defector states) typically emerge when (i) the network connections are sparse, (ii) the network degree distribution is heterogeneous, (iii) the network is assortative, and surprisingly, (iv) the benefit of cooperation is high.

Handbook of Experimental Economics Results, 2008

In this chapter, we examine convergence behavior in simple bimatrix games. We classify the possible types of simple games, pick interesting examples of each type, and summarize convergence behavior under various information and player matching protocols. See Friedman (1996), Cheung and Friedman (1997) and Bouchez (1998) for more complete descriptions of the experiments.

Proceedings of the 23rd ACM symposium on Parallelism in algorithms and architectures - SPAA '11, 2011

We present the first general bounds on the mixing time of logit dynamics for wide classes of strategic games. The logit dynamics describes the behaviour of a complex system whose individual components act "selfishly" and keep responding according to some partial ("noisy") knowledge of the system. In particular, we prove nearly tight bounds for potential games and games with dominant

SSRN Electronic Journal, 2019

Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public.

Proceedings of the 9th ACM conference on Electronic commerce - EC '08, 2008

... Market shar-ing games are a special case of profit maximizing congestion games and valid-utility games [26] that has been studied for the content distribution in service provider networks [16]. ... Nice Potential Games. Consider a potential game Λ. ...