The Frequency of Convergent Games under Best-Response Dynamics

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

Analyzing the Dynamic Data Sponsoring in the Case of Competing Internet Service Providers and Content Providers

With a sponsored content plan on the Internet market, a content provider (CP) negotiates with the Internet service providers (ISPs) on behalf of the end-users to remove the network subscription fees. In this work, we have studied the impact of data sponsoring plans on the decision-making strategies of the ISPs and the CPs in the telecommunications market. We develop game-theoretic models to study the interaction between providers (CPs and ISPs), where the CPs sponsor content. We formulate the interactions between the ISPs and between the CPs as a noncooperative game. We have shown the existence and uniqueness of the Nash equilibrium. We used the best response dynamic algorithm for learning the Nash equilibrium. Finally, extensive simulations show the convergence of a proposed schema to the Nash equilibrium and show the effect of the sponsoring content on providers’ policies.

Walrasian Dynamics in Multi-Unit Markets

In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set of buyers that have monetary endowments. While a Walrasian equilibrium does not always exist in this model, natural relaxations of the concept that retain its desirable fairness properties do exist. We study the dynamics of (Walrasian) envy-free pricing mechanisms in this environment, showing that for any such pricing mechanism, the best response dynamic starting from truth-telling converges to a pure Nash equilibrium with small loss in revenue and welfare. Moreover, we generalize these bounds to capture all the (reasonable) Nash equilibria for a large class of (monotone) pricing mechanisms. We also identify a natural mechanism, which selects the minimum Walrasian envy-free price, in which for n=2 buyers the best response dynamic converges from any starting profile. We conjecture convergence of the mechanism for any number of buyers and provide simulation results to support our conjecture.

Analyzing the Customers' Dynamic Confusion in Telecommunication Networks Share Game

In this article, the authors have developed a simple oligopolistic model to formalize the interactions between service providers and end-users by considering that the rationality of customers varies over time. This article assessed the impact of the dynamics of consumer confusion on the competition and profitability of service providers who are considered rational and competitive with one another to maximize their respective gains in the face of a confused fraction of consumers while others are not confused. This article shows the existence and uniqueness of the Nash equilibrium. The authors used the best response dynamic algorithm for learning Nash equilibrium. We have shown that when the number of confused customers is large, the ISP interest this and they offer moderately high prices with low quality of service. On the other hand, over time, rationality increases, forcing the ISPs to change their strategies by offering better services so that their demand increases. We also add that when customer behavior changes quickly, the ISPs follow clearer strategies with customer satisfactory services.

A Customer Confusion Environment in Telecommunication Networks: Analysis and Policy Impact

In this paper, we have studied the impact of customer confusion on the decision-making strategies of Internet service providers (ISP) in the network and telecommunications market. This confusion may come from several factors, e.g. incomplete information on the offer, non-transparent advertising, the ability of the analysis, etc.; but that sure varies over time since yesterday’s customer is no longer today’s. In this work, we have developed a simple oligopolistic model, using non-cooperative game theory, to formalize the interactions between service providers and end-users by considering that the rationality of customers varies over time. We assessed the impact of the dynamics of consumer confusion on the competition and profitability of service providers who are considered rational and competitive with one another to maximize their respective gains in the face of a confused fraction of consumers while others are not confused. We have shown the existence and uniqueness of the Nash equilibrium. We used the best response dynamic algorithm for learning Nash equilibrium. On the one hand, we have shown that when the number of confused customers is large, the ISP is interested in that and they offer moderately high prices with low quality of service. On the other hand, over time, rationality increases, forcing the ISPs to change their strategies by offering better services so that their demand increases. We also add that when customer behavior changes quickly, the ISPs follow clearer strategies with customer satisfactory services.

Crime, punishment, and evolution in an adversarial game

We examine the game theoretic properties of a model of crime first introduced by Short et al. (2010 Phys. Rev. E82, 066114) as the SBD Adversarial Game. We identify the rationalizable strategies and one-shot equilibria under multiple equilibrium refinements. We further show that SBD's main result about the effectiveness of defecting-punishers (“Informants”) in driving the system to evolve to the cooperative equilibrium under an imitation dynamic generalizes to a best response dynamic, though only under certain parameter regimes. The nature of this strategy's role, however, differs significantly between the two dynamics: in the SBD imitation dynamic, Informants are sufficient but not necessary to achieve the cooperative equilibrium, while under the best response dynamic, Informants are necessary but not sufficient for convergence to cooperation. Since a policy of simply converting citizens to Informants will not guarantee success under best response dynamics, we identify alternative strategies that may help the system reach cooperation in this case, e.g., the use of moderate but not too severe punishments on criminals.

Research on Power Producer’s Bidding Behavior Based on the Best-Response Dynamic Model

As China’s electricity market is facing many problems, the research on power producer’s bidding behavior can promote the healthy and sustainable development of China’s electricity market. As a special commodity, the “electricity” possesses complicated production process. The instable market constraint condition, nonsymmetric information, and a lot of random factors make the producer’s bidding process more complex. Best-response dynamic is one of the classic dynamic mechanisms of the evolutionary game theory, which applies well in the repeated game and strategy evolution that happen among a few bounded rational players with a quick learning capability. The best-response dynamic mechanism is employed to study the power producer’s bidding behavior in this paper, the producer’s best-response dynamic model is constructed, and how the producers would engage in bidding is analyzed in detail. Taking two generating units in South China regional electricity market as the example, the producer’s bidding behavior by following the producer’s best-response dynamic model is verified. The relationships between the evolutionarily stable strategy (ESS) of power producer’s bidding and the market demand, and ceiling and floor price as well as biding frequency are discussed in detail.

ANTI-COORDINATION GAMES AND DYNAMIC STABILITY

We introduce the class of anti-coordination games. A symmetric two-player game is said to have the anti-coordination property if, for any mixed strategy, any worst response to the mixed strategy is in the support of the mixed strategy. Every anti-coordination game has a unique symmetric Nash equilibrium, which lies in the interior of the set of mixed strategies. We investigate the dynamic stability of the equilibrium in a one-population setting. Specifically we focus on the best response dynamic (BRD), where agents in a large population take myopic best responses, and the perfect foresight dynamic (PFD), where agents maximize total discounted payoffs from the present to the future. For any anti-coordination game we show (i) that, for any initial distribution, BRD has a unique solution, which reaches the equilibrium in a finite time, (ii) that the same path is one of the solutions to PFD, and (iii) that no path escapes from the equilibrium in PFD once the path reaches the equilibrium. Moreover we show (iv) that, in some subclasses of anti-coordination games, for any initial state, any solution to PFD converges to the equilibrium. All the results for PFD hold for any discount rate.