Pure Bayesian Nash Equilibrium Research Articles

Agent Transformation of Bayesian Games

A new transformation that converts a Bayesian game to a so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent game (a normal-form game) is proposed. Differently from the existing transformation proposed by R. Selten that changes a Bayesian game to an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interim</i> agent game (also a pure normal-form game), we prove that the new transformation preserves potentiality. In addition, there is a nonpotential Bayesian game whose <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent game is potential. We also prove that there is one-to-one correspondence between pure Bayesian Nash equilibria (BNE) of Bayesian games (if one exists) and pure Nash equilibria (NE) of the resulting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent games. Then, we provide a sufficient and necessary condition for a Bayesian game to have an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent potential game. By using these results, one can transform pure BNE seeking in Bayesian games to pure NE seeking in their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent games [by using the potential functions of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent games (if one exists)], where previously pure BNE seeking in Bayesian games by using potential functions can only be done in Bayesian potential games (BPGs). Particularly, we prove for two-player games that BPGs are exactly the Bayesian games having <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent potential games. Furthermore, by using the semi-tensor product of matrices, a potential equation for finite Bayesian games is developed. Based on the potential equation, algorithms for verifying potentiality and for searching pure BNE in finite Bayesian games are designed. Finally, the results are applied to a routing problem with incomplete information.

Rationalization and identification of binary games with correlated types

This paper studies the rationalization and identification of binary games where players have correlated private types. Allowing for type correlation is crucial in global games and in models with social interactions as it represents correlated private information and homophily, respectively. Our approach is fully nonparametric in the joint distribution of types and the strategic effects in the payoffs. First, under monotone pure Bayesian Nash Equilibrium strategy, we characterize all the restrictions if any on the distribution of players’ choices imposed by the game-theoretic model as well as restrictions associated with two assumptions frequently made in the empirical analysis of discrete games. Namely, we consider exogeneity of payoff shifters relative to private information, and mutual independence of private information given payoff shifters. Second, we study the nonparametric identification of the payoff functions and types distribution. We show that the model with exogenous payoff shifters is fully identified up to a single location-scale normalization under some exclusion restrictions and rank conditions. Third, we discuss partial identification under weaker conditions and multiple equilibria. Lastly, we briefly point out the implications of our results for model testing and estimation.

Congestion-Aware DNS for Integrated Cellular and Wi-Fi Networks

Intelligent network selection plays an important role in achieving an effective data offloading in the integrated cellular and Wi-Fi networks. However, previously proposed network selection schemes mainly focused on offloading as much data traffic to Wi-Fi as possible, without systematically considering the Wi-Fi network congestion and the ping-pong effect, both of which may lead to a poor overall user quality of experience. Thus, in this paper, we study a more practical network selection problem by considering both the impacts of the network congestion and switching penalties. More specifically, we formulate the users' interactions as a Bayesian network selection game (NSG) under the incomplete information of the users' mobilities. We prove that it is a Bayesian potential game and show the existence of a pure Bayesian-Nash equilibrium that can be easily reached. We then propose a distributed network selection (DNS) algorithm based on the network congestion statistics obtained from the operator. Furthermore, we show that computing the optimal centralized network allocation is an NP-hard problem, which further justifies our distributed approach. Simulation results show that the DNS algorithm achieves the highest user utility and a good fairness among users, as compared with the on-the-spot offloading and cellular-only benchmark schemes.

Existence of Equilibria in Joint Admission and Power Control for Inelastic Traffic

This letter considers the problem of admission and discrete power control, in the interfering-multiple-access channel, with rate constraints on admitted links. This problem is formulated as a normal form noncooperative game. The utility function models inelastic demand. An example demonstrates that in the fading channel, in some networks, a pure strategy equilibrium does not exist with strictly positive probability. Hence, the probability of existence of an equilibrium is analyzed and bounds are computed. To this end, the problem of finding equilibria is transformed into a constraint satisfaction problem. Next the letter considers the incomplete information setting, with compact convex channel power gains. The resulting Bayesian game is proven to possess at least one pure Bayesian–Nash equilibrium in on–off threshold strategies. Numerical results are presented to corroborate the findings.

Rationalization and Identification of Binary Games with Correlated Types

This paper studies the rationalization and identification of binary games where players have correlated private types. Allowing for correlation is crucial in global games and in models with social interactions as it represents correlated information and homophily, respectively. Our approach is fully nonparametric in the joint distribution of types and the strategic effects in the payoffs. First, under monotone pure Bayesian Nash Equilibrium strategy, we characterize all the restrictions if any on the distribution of players’ choices imposed by the game-theoretic model as well as restrictions associated with two assumptions frequently made in the empirical analysis of discrete games. Namely, we consider exogeneity of payoff shifters relative to private information, and mutual independence of private information given payoff shifters. Second, we study the nonparametric identification of the payoff functions and types distribution. We show that the model with exogenous payoff shifters is fully identified up to a single location–scale normalization under some exclusion restrictions and rank conditions. Third, we discuss partial identification under weaker conditions and multiple equilibria. Lastly, we briefly point out the implications of our results for model.

Selfish Routing with Incomplete Information

In his seminal work, Harsanyi (Manag. Sci. 14, 159–182, 320–332, 468–502, 1967) introduced an elegant approach to study non-cooperative games with incomplete information. In our work, we use this approach to define a new selfish routing game with incomplete information that we call Bayesian routing game. Here, each of n selfish users wishes to assign its traffic to one of m parallel links. However, users do not know each other’s traffic. Following Harsanyi’s approach, we introduce, for each user, a set of possible types. In our model, each type of a user corresponds to some traffic and the players’ uncertainty about each other’s traffic is described by a probability distribution over all possible type profiles. We present a comprehensive collection of results about our Bayesian routing game. Our main findings are as follows: Using a potential function, we prove that every Bayesian routing game has a pure Bayesian Nash equilibrium. More precisely, we show this existence for a more general class of games that we call weighted Bayesian congestion games. For Bayesian routing games with identical links and independent type distribution, we give a polynomial time algorithm to compute a pure Bayesian Nash equilibrium. We study structural properties of fully mixed Bayesian Nash equilibria for the case of identical links and show that they maximize Individual Cost. In general, there is more than one fully mixed Bayesian Nash equilibrium. We characterize fully mixed Bayesian Nash equilibria for the case of independent type distribution. We conclude with bounds on Coordination Ratio for the case of identical links and for three different Social Cost measures: Expected Maximum Latency, Sum of Individual Costs and Maximum Individual Cost. For the latter two, we are able to give (asymptotically) tight bounds using the properties of fully mixed Bayesian Nash equilibria we proved.