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 Fangxu Ren International Journal of Business and Management , 2009, Abstract: This paper provides a dynamic model for two oligarch enterprises competing for prices and the conditions of their cooperation in repeated game, by analyzing the cooperation and competition strategies in Bertrand model. In conclusion, this paper suggests a range for effective adjustment of price at the common first-phase game.
 Computer Science , 2005, Abstract: We study and compare the learning dynamics of two universal learning algorithms, one based on Bayesian learning and the other on prediction with expert advice. Both approaches have strong asymptotic performance guarantees. When confronted with the task of finding good long-term strategies in repeated 2x2 matrix games, they behave quite differently.
 Computer Science , 2015, Abstract: Motivated by online advertising auctions, we consider repeated Vickrey auctions where goods of unknown value are sold sequentially and bidders only learn (potentially noisy) information about a good's value once it is purchased. We adopt an online learning approach with bandit feedback to model this problem and derive bidding strategies for two models: stochastic and adversarial. In the stochastic model, the observed values of the goods are random variables centered around the true value of the good. In this case, logarithmic regret is achievable when competing against well behaved adversaries. In the adversarial model, the goods need not be identical and we simply compare our performance against that of the best fixed bid in hindsight. We show that sublinear regret is also achievable in this case and prove matching minimax lower bounds. To our knowledge, this is the first complete set of strategies for bidders participating in auctions of this type.
 Academic Research International , 2011, Abstract: One of the main challenges facing Computer games is creating agents that are driven by artificial intelligence (AI) that interact with the player in reliable and entertaining ways. In the game world, it is being accepted that careful and considered use of learning makes it possible to come out with smarter and more robust AIs without the need to appropriate and counter every strategy that a player may adopt. It follows therefore, that rather than having all non-player character behaviours being determined by the time a game is produced, the game should instead evolve, learn, and adapt throughout the period the game is being played. The outcome of this is that the game grows with the player and is very difficult for the player to predict the next action, thus extending the play-life of the game. These learning techniques normally change the way that games are played, by forcing the player to continually search for new strategies to defeat the AI.This paper tries to highlight some of the learning paradigms for Game AI and the great potential they offer to the game world. It was discovered that each of the learning paradigm is suited to adifferent type of problem, and so the game developer has to be careful in the choice of a particular paradigm.
 Computer Science , 2014, Abstract: We consider eager-push epidemic dissemination in a complete graph. Time is divided into synchronous stages. In each stage, a source disseminates $\nu$ events. Each event is sent by the source, and forwarded by each node upon its first reception, to $f$ nodes selected uniformly at random, where $f$ is the fanout. We use Game Theory to study the range of $f$ for which equilibria strategies exist, assuming that players are either rational or obedient to the protocol, and that they do not collude. We model interactions as an infinitely repeated game. We devise a monitoring mechanism that extends the repeated game with communication rounds used for exchanging monitoring information, and define strategies for this extended game. We assume the existence of a trusted mediator, that players are computationally bounded such that they cannot break the cryptographic primitives used in our mechanism, and that symmetric ciphering is cheap. Under these assumptions, we show that, if the size of the stream is sufficiently large and players attribute enough value to future utilities, then the defined strategies are Sequential Equilibria of the extended game for any value of $f$. Moreover, the utility provided to each player is arbitrarily close to that provided in the original game. This shows that we can persuade rational nodes to follow a dissemination protocol that uses any fanout, while arbitrarily minimising the relative overhead of monitoring.
 Jacob W. Crandall Computer Science , 2014, Abstract: This paper addresses learning in repeated stochastic games (RSGs) played against unknown associates. Learning in RSGs is extremely challenging due to their inherently large strategy spaces. Furthermore, these games typically have multiple (often infinite) equilibria, making attempts to solve them via equilibrium analysis and rationality assumptions wholly insufficient. As such, previous learning algorithms for RSGs either learn very slowly or make extremely limiting assumptions about the game structure or associates' behaviors. In this paper, we propose and evaluate the notion of game abstraction by experts (Gabe) for two-player general-sum RSGs. Gabe reduces an RSG to a multi-armed bandit problem, which can then be solved using an expert algorithm. Gabe maintains many aspects of the original game, including security and Pareto optimal Nash equilibria. We demonstrate that Gabe substantially outperforms existing algorithms in many scenarios.
 Boyu Zhang PLOS ONE , 2013, DOI: 10.1371/journal.pone.0074540 Abstract: In the ultimatum game, two players divide a sum of money. The proposer suggests how to split and the responder can accept or reject. If the suggestion is rejected, both players get nothing. The rational solution is that the responder accepts even the smallest offer but humans prefer fair share. In this paper, we study the ultimatum game by a learning-mutation process based on quantal response equilibrium, where players are assumed boundedly rational and make mistakes when estimating the payoffs of strategies. Social learning is never stabilized at the fair outcome or the rational outcome, but leads to oscillations from offering 40 percent to 50 percent. To be precise, there is a clear tendency to increase the mean offer if it is lower than 40 percent, but will decrease when it reaches the fair offer. If mutations occur rarely, fair behavior is favored in the limit of local mutation. If mutation rate is sufficiently high, fairness can evolve for both local mutation and global mutation.
 Misha Perepelitsa Quantitative Finance , 2015, Abstract: We consider the dynamics of player's strategies in repeated market games, where the selection of strategies is determined by a learning model. Prior theoretical analysis and experimental data show that after large number of plays the average number of agents who decide to enter, per round of the game, approaches the market capacity and, after a longer wait, agents are being sorted into two groups: the agents in one group rarely enter the market, and in the other, the agents enter almost all the time. In this paper we obtain estimates of the characteristic times it takes for both patterns to emerge in the repeated plays of the game. The estimates are given in terms of the parameters of the game, assuming that the number of agents is large, the number of rounds of the game per unit of time is large, and the characteristic change of the propensity per game is small. Our approach is based on the analysis of the partial differential equation for the function $f(t,q)$ that describes the distribution of agents according to their level of propensity to enter the market, $q,$ at time $t.$
 Modern Applied Science , 2009, DOI: 10.5539/mas.v2n3p54 Abstract: This paper introduces repeated theory on the base of fuzzy cooperative game by Aubin etal in 1974 and then constructs repeated fuzzy games theory. It gives the conception of repeated convex fuzzy cooperative games and studies the property of repeated convex fuzzy games.
 Computer Science , 2002, Abstract: A valuation for a player in a game in extensive form is an assignment of numeric values to the players moves. The valuation reflects the desirability moves. We assume a myopic player, who chooses a move with the highest valuation. Valuations can also be revised, and hopefully improved, after each play of the game. Here, a very simple valuation revision is considered, in which the moves made in a play are assigned the payoff obtained in the play. We show that by adopting such a learning process a player who has a winning strategy in a win-lose game can almost surely guarantee a win in a repeated game. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1, strategies that are close to subgame perfect equilibrium are played after some time. A single player who adopts this procedure can guarantee only her individually rational payoff.
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