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We study how an agent learns from endogenous data when their prior belief is misspecified. We show that only uniform Berk–Nash equilibria can be long-run outcomes, and that all uniformly strict Berk–Nash equilibria have an arbitrarily high probability of being the long-run outcome for some initial beliefs. When the agent believes the outcome distribution is exogenous, every uniformly strict Berk–Nash equilibrium has positive probability of being the long-run outcome for any initial belief. We generalize these results to settings where the agent observes a signal before acting.

This paper introduces an enhanced algorithm for computing generalized Nash equilibria for multiple player nonlinear games, which degenerates in a gradient algorithm for single player games (i.e., optimization problems) or potential games (i.e., equivalent to minimizing the respective potential function), based on the Rosen gradient algorithm. Analytical examples show that it has similar theoretical guarantees of finding a generalized Nash equilibrium when compared to the relaxation algorithm, while numerical examples show that it is faster. Furthermore, the proposed algorithm is as fast as, but more stable than, the Rosen gradient algorithm, especially when dealing with constraints and non-convex games. The algorithm is applied to an electricity market game representing the current electricity market model in Brazil.

We consider finite graphical n-person games with perfect information that have no Nash equilibria in pure stationary strategies. Solving these games in stationary mixed strategies, we introduce probability distributions in all non-terminal positions. The corresponding probability distributions on the set of plays can be defined in two different ways called the Markov and a-priori realizations. The former one guarantees existence of a uniform best response for each player in every situation. Nevertheless, Nash equilibrium may fail to exist even in stationary mixed strategies. The classical Nash’s theorem is not applicable, because in this case limit distributions and expected payoffs may be discontinuous. Although a-priori realizations do not share many nice properties of the Markov ones (for example, the existence of uniform best responses) but in return, Nash’s theorem is applicable. We illustrate both realizations in details by two examples with 2 and 3 players. We also survey some general results related to Nash-solvability, in pure and mixed stationary strategies, of stochastic n-person games with perfect information and n-person graphical games among them.

In this paper, we study distributionally robust optimization approaches for a one-stage stochastic minimization problem, where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions, which contains the true distribution and optimal decision is taken on the basis of the worst-possible distribution from that set. We consider the case when the distributional set (which is also known as the ambiguity set) varies and its impact on the optimal value and the optimal solutions. A typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size. The analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming. The discussion is extended briefly to a stochastic Nash equilibrium problem where each player takes a robust action on the basis of the worst subjective expected objective values.

Blockchain and cryptocurrency are a hot topic in today’s digital world. In this paper, we create a game theoretic model in continuous time. We consider a dynamic game model of the bitcoin market, where miners or players use mining systems to mine bitcoin by investing electricity into the mining system. Although this work is motivated by BTC, the work presented can be applicable to other mining systems similar to BTC. We propose three concepts of dynamic game theoretic solutions to the model: Social optimum , Nash equilibrium and myopic Nash equilibrium . Using the model that a player represents a single “miner” or a “mining pool”, we develop novel and interesting results for the cryptocurrency world.

The progress of science and technology and the expansion of the Internet of Things make the information transmission between communication infrastructure and wireless sensors become more and more convenient. For the power-limited wireless sensors, the life time can be extended through the energy-harvesting technique. Additionally, wireless sensors can use the unauthored spectrum resource to complete certain information transmission tasks based on cognitive radio. Harvesting enough energy from the environments, the wireless sensors, works as the second users (SUs) can lease spectrum resource from the primary user (PU) to finish their task and bring additional transmission cost to themselves. To minimize the overall cost of SUs and to maximize the spectrum profit of the PU during the information transmission period, we formulated a differential game model to solve the resource allocation problem in the cognitive radio wireless sensor networks with energy harvesting, considering the SUs as the game players. By solving the proposed resource allocation game model, we found the open loop Nash equilibrium solutions and feedback Nash equilibrium solutions for all SUs as the optimal control strategies. Ultimately, series numerical simulation experiments have been made to demonstrate the rationality and effectiveness of the game model.

The study on how equilibria behave when perturbations occur in the data of a game is a fundamental theme, since actions and payoffs of the players may be affected by uncertainty or trembles. In this paper, we investigate the asymptotic behavior and the variational stability of the subgame perfect Nash equilibrium (SPNE) in one-leader one-follower Stackelberg games under perturbations both of the action sets and of the payoff functions. To pursue this aim, we consider a general sequence of perturbed Stackelberg games and a set of assumptions that fit the usual types of perturbations. We study if the limit of SPNEs of the perturbed games is an SPNE of the original game and if the limit of SPNE-outcomes of perturbed games is an SPNE-outcome of the original game. We fully positively answer when the follower’s best reply correspondence is single-valued. When the follower’s best reply correspondence is not single-valued, the answer is positive only for the SPNE-outcomes; whereas the answer for SPNEs may be negative, even if the perturbation does not affect the sets and affects only one payoff function. However, we show that under suitable non-restrictive assumptions it is possible to obtain an SPNE starting from the limit of SPNEs of perturbed games, possibly modifying the limit at just one point.

Any symmetric mixed-strategy equilibrium in a Tullock contest with intermediate values of the decisiveness parameter (\"2 < R < ∞\") has countably infinitely many mass points. All probability weight is concentrated on those mass points, which have the zero bid as their sole point of accumulation. With contestants randomizing over a non-convex set, there is a cost of being \"halfhearted,\" which is absent from both the lottery contest and the all-pay auction. Numerical bid distributions are generally negatively skewed and exhibit, for some parameter values, a higher probability of ex-post overdissipation than the all-pay auction.

We consider a transportation station, where customers arrive according to a Poisson process. A transportation facility visits the station according to a renewal process and serves at each visit a random number of customers according to its capacity. We assume that the arriving customers decide whether to join the station or balk, based on a natural reward-cost structure. We study the strategic behavior of the customers and determine their symmetric Nash equilibrium strategies under two levels of information.

Following a brief review of the management of environmental externalities under strategic interactions in the traditional temporal domain, results are extended to the spatiotemporal domain. Conditions for spatial open-loop and feedback Nash equilibria, along with conditions for the benchmark cooperative solution, are presented and compared. A simplified numerical example illustrates the spatial patterns emerging at a steady state under Fickian diffusion and dispersal kernels, and the inefficiency of spatially flat emission taxes. This conceptual framework could provide new research areas.