Dynamic games with side information in economic networks

Abstract

We consider a dynamic game with payoff exter nalities. Agents' utility depends on an unknown true state of the world and actions of everyone in the network. Each agent has an initial private information about the underlying state and repeatedly observes actions of its neighbors. We analyze the asymptotic behavior of agents' actions and beliefs in a connected network when it is common knowledge that the agents are myopic and rational. Given a quadratic payoff function, we provide a new proof for an existing result that claims almost sure consensus in actions asymptotically. Given consensus in actions, we show that agents have the same mean estimate of the true state of the world in the limit. We justify these results in a numerical example motivated by a socio economic scenario. I. IN TRODUCTIONThe model discussed in this paper belongs to the class of repeated games of incomplete information. Agents repeatedly play a game where the payoffs depend on a parameter ("state of the world") as well as the actions taken by other agents. In a game of incomplete information the payoff-relevant parameter is unknown to the agents; rather, they make private noisy observations about the parameter that can be used when deciding on an action. In such a setting, on the one hand, agents' optimal actions depend on their private observations as well as how they expect others to play, and on the other hand, by selecting certain actions agents are revealing perhaps unwillingly-pieces of private information about the unknown parameter [1]. As a result, the actions chosen by "rational" agents are influenced by both an information externality pertaining to the flow of information about the unknown parameter and a payoff externality corresponding to the dependence of the payoff on actions taken by other agents. Rational agents need to select actions that are optimal given their information while also taking into account the effect of their decisions on the future play. They also need to try the leam about the underlying parameter as much as possible by incorporating the new information revealed to them optimally, i.e., using the Bayes rule. This kind of strategic leaming is relevant to the vast literature on learning in games; refer to [2] and references therein.There exists an extensive literature on learning over net works. Bayesian leaming stands as the normative behavioral model for agents in social networks; however, it is often computationally intractable even for networks with small IEEE 520 number of agents. This is since a Bayesian update requires an agent to infer not only about the information of her neighbors but also that of the neighbors of her neighbors and so on. Because of such computational intractability, Bayesian leaming models often focus on asymptotic char acterizations of agents' behavior [3]- [5]. Only under some structural assumptions on the network or distribution of information, Bayesian updating can be performed tractably [6]- [8]. The mathematical intractability of Bayesian learning in the general case motivate...

Learning in network games with incomplete information: asymptotic analysis and tractable implementation of rational behavior

“…From here it follows that beliefs are propagated as per the LMMSE filter summarized in (31)- (36). A simple set of substitutions allows us to conclude that (20) is true at time t 1 + with the vector k , i t 1 + propagated as per (38) and the matrix L , i t 1 + propagated as in (38).…”

“…In this section we present results from [12] and [38] that focus on symmetric, supermodular, and diagonally dominant games. The game defined by the utility function in (9) is symmetric when the pairwise influences ij b are equal for any pair; that is,…”

Information aggregation in a beauty contest game

“…We remark that these results extend beyond the beauty contest game payoffs to quadratic symmetric payoffs with strategic complementarities [1]. Next, we show that consensus in actions implies that agents agree on their individual estimates on the state of the world [2,3]. However, this does not necessarily mean that the agreed estimate is optimal.…”

“…4,5] for a discussion of different equilibrium concepts. In Section 3, we survey some of the results presented in [1][2][3] that show consensus in actions and ex-ante payoffs. We remark that these results extend beyond the beauty contest game payoffs to quadratic symmetric payoffs with strategic complementarities [1].…”

“…The agents thus can benefit from learning from others. This paper provides a survey of results from [1][2][3][4]. We first present a recent result that states repeated play of the game by myopic but Bayesian agents, who observe the actions of their neighbors over a connected network, eventually yield coordination on a single action.…”