In statistical decision theory involving a single decision-maker, one says that an information structure is better than another one if for any cost function involving a hidden state variable and an action variable which is restricted to be only a function of some measurement, the solution value under the former is not worse than the value under the latter. For finite probability spaces, Blackwell's celebrated theorem on comparison of information structures leads to a complete characterization on when one information structure is better than another. For stochastic games with incomplete information, due to the presence of competition among decision makers, in general such an ordering is not possible since additional information can lead to equilibria perturbations with positive or negative values to a player. However, for zero-sum games in a finite probability space, Peski introduced a complete characterization of ordering of information structures. In this paper, we obtain an infinite dimensional (standard Borel) generalization of Peski's result. A corollary of our analysis is that more information cannot hurt a decision maker taking part in a zero-sum game in standard Borel spaces. During our analysis, we establish two novel supporting results: (i) a refined existence result for equilibria in zero-sum games with incomplete information when compared with the prior literature and ii) a partial converse to Blackwell's ordering of information structures in the standard Borel space setup.
We consider a privacy-signaling game problem in which a transmitter with privacy concerns and a receiver, which does not pay attention to these privacy concerns, communicate. In this communication scenario, the transmitter observes a pair of correlated random variables which are modeled as jointly Gaussian. The transmitter constructs its message based on these random variables with the aim to hide one of them and convey the other one. In contrast, the objective of the receiver is to accurately estimate both of the random variables so as to gather as much information as possible. These conflicting objectives are analyzed in a game theoretic framework where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg equilibria. We show that a payoff dominant (i.e., most desirable for both players) Nash equilibrium is attained by affine policies and we explicitly characterize these policies. In addition, the strategies at the characterized Nash equilibrium is shown to form also a Stackelberg equilibrium. Furthermore, we show that there always exists an informative Stackelberg equilibrium for the multidimensional parameter setup. We also revisit the information bottleneck problem within our Stackelberg framework under the mean squared error distortion criterion where the information bottleneck setup has a further restriction that only one of the parameters is observed at the sender. We fully characterize the Stackelberg equilibria under certain conditions and when these conditions are not met we establish the existence of informative equilibria.
It is common in online markets for agents to learn from other's actions. Such observational learning can lead to herding or information cascades in which agents eventually ``follow the crowd''. Models for such cascades have been well studied for Bayes-rational agents that choose pay-off optimal actions. In this paper, we additionally consider the presence of fake agents that seek to influence other agents into taking one particular action. To that end, these agents take a fixed action in order to influence the subsequent agents towards their preferred action. We characterize how the fraction of such fake agents impacts behavior of the remaining agents and show that in certain scenarios, an increase in the fraction of fake agents in fact reduces the chances of their preferred outcome.
We introduce the problem of strategic communication and find achievable rates for this problem. The problem consists of a sender that observes a source and a receiver that would like to recover the source. The sender can send messages to the receiver over a noiseless medium whose input space is as large as the space of source signals. However, unlike standard communication, the sender is strategic. Depending on the source signal it receives, the sender may have an incentive, measured by a utility function, to misreport the signal, whereby, not all signals are necessarily recoverable at the receiver. The dilemma for the receiver lies in selecting the right signals to recover so that recovery happens with high probability. We establish achievable rates associated with these settings.