Stochastic discounting in repeated games: awaiting the almost inevitable

This paper studies repeated games with pure strategies and stochastic discounting under perfect information. We consider infinite repetitions of any finite normal form game possessing at least one pure Nash action profile. The period interaction realizes a shock in each period, and the cumulative shocks while not affecting period returns, determine the probability of the continuation of the game. We require cumulative shocks to satisfy the following: (1) Markov property; (2) to have a non-negative (across time) covariance matrix; (3) to have bounded increments (across time) and possess a denumerable state space with a rich ergodic subset; (4) there are states of the stochastic process with the resulting stochastic discount factor arbitrarily close to 0, and such states can be reached with positive (yet possibly arbitrarily small) probability in the long run. In our study, a player’s discount factor is a mapping from the state space to (0, 1) satisfying the martingale property. In this setting, we, not only establish the (subgame perfect) folk theorem, but also prove the main result of this study: In any equilibrium path, the occurrence of any finite number of consecutive repetitions of the period Nash action profile, must almost surely happen within a finite time window. That is, any equilibrium strategy almost surely contains arbitrary long realizations of consecutive period Nash action profiles

A Logic for Strategy Updates

Notion of strategy in game theory is static and presumably constructed before the game play. The static, pre-determined notion of strategies falls short analyzing perfect information games. Because, we, people, do not strategize as such even in perfect information games - largely because we are not logically omniscient, and we have limited computational power and bounded memory. In this paper, we focus on what we call move updates where some moves become unavailable during the game. Our goal here is to present a formal framework for move based strategy restrictions extending strategy logic which was introduced by Ramanujam and Simon. In this paper, we present a dynamic version of strategy logic, prove its completeness and decidability along with the decidability of the strategy logic which was an open problem so far. We also present an analysis of centipede by using our logic

Linear Codes from a Generic Construction

A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach. The first objective of this paper is to establish relationships among some classes of linear codes obtained with this approach, so that the parameters of some classes of linear codes can be derived from those of other classes with known parameters. In this way, linear codes with new parameters will be derived. The second is to present a class of three-weight binary codes and consider their applications in secret sharing.Comment: arXiv admin note: text overlap with arXiv:1503.06511, arXiv:1503.06512 by other author

Tensor Renormalization Group: Local Magnetizations, Correlation Functions, and Phase Diagrams of Systems with Quenched Randomness

The tensor renormalization-group method, developed by Levin and Nave, brings systematic improvability to the position-space renormalization-group method and yields essentially exact results for phase diagrams and entire thermodynamic functions. The method, previously used on systems with no quenched randomness, is extended in this study to systems with quenched randomness. Local magnetizations and correlation functions as a function of spin separation are calculated as tensor products subject to renormalization-group transformation. Phase diagrams are extracted from the long-distance behavior of the correlation functions. The approach is illustrated with the quenched bond-diluted Ising model on the triangular lattice. An accurate phase diagram is obtained in temperature and bond-dilution probability, for the entire temperature range down to the percolation threshold at zero temperature.Comment: Added comment. Published version. 8 pages, 7 figures, 1 tabl