The Splitting Game: Value and Optimal Strategies We introduce the dependent splitting game, a zerosum stochastic game in which the players jointly control a martingale. This game models the transmission of information in repeated games with incomplete information on both sides, in the dependent case: The state variable represents the martingale of posterior beliefs. We establish the existence of the value for any fixed, general evaluation of the stage payoffs, as a function of the initial state. We then prove the convergence of the value functions, as the evaluation vanishes, to the unique solution of the Mertensxe2x80x93Zamir system of equations is established. From this result, we derive the convergence of the values of repeated games with incomplete information on both sides, in the dependent case, to the same function, as the evaluation vanishes. Finally, we establish a surprising result: Unlike repeated games with incomplete information on both sides, the splitting game has a uniform value. Moreover, we exhibit a couple of optimal stationary strategies for which the stage payoff and the state remain constant.

A Survey on Entropy and Economic Behaviour Entropy plays a significant role in the study of games and economic behaviour in several ways. A decision maker faced with an nfold repetition of a decisionmaking problem needs to apply strategies that become increasingly complex as n increases. When several players are involved in selecting strategies in interactive games, bounds on the memories and cognitive capacities of the players can affect possible outcomes. A player who can recall only the last k periods of history is said to have bounded recall of capacity k. We present here a brief survey of results of games played by players with different bounded recall capacities, in particular those indicating surprisingly strong relations between memory and entropy in the study of the minmax values of repeated games with bounded recall. In addition, we consider uses of entropy in measuring the value of information of noisy signal structures, also known as experiments. These are represented by stochastic matrices, with the rows representing states of the world and the columns possible signals. The classic ordering of experiments, due to David Blackwell and based on decisionmaking criteria, is a partial ordering, which has led to attempts to extend this ordering to a total ordering. If a decision maker has a prior distribution over the states, receipt of a signal yields a posterior. The difference between the entropy of a prior and the expected entropy of the set of possible posteriors has been proposed as a natural extension of the Blackwell ordering. We survey this alongside the theory of rational inattention, which posits that, since individuals have limited attention, they do not always follow every single piece of economic news in planning their economic behaviour. By modelling attention limits as finite channel capacity in the sense of Shannon, economists have developed a theory that explains a range of observed economic behavioural phenomena well.