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Bulletin of informatics and cybernetics

INFORMATION GAMES

Yu Kai

Published Mar 1, 1982• 0 citations
Publisher

Abstract

In a game each player holds a strategy based upon the actions that he can actually take. The quality and the quantity of the feasible actions are determined by his ability or power in the game. Generally every player has no information on the others' powers, so it is one of the most important problems for them to know the powers. We treat such unknown games, played periodically, from the view point of information theory. We discuss both non-cooperative and cooperative games and prove some sufficient conditions under which a player can get the full information on the other players' powers from sequential observations on the states. In two person games we show also optimal strategies with respect to the information.

DOI
10.5109/13336

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In a game each player holds a strategy based upon the actions that he can actually take. The quality and the quantity of the feasible actions are determined by his ability or power in the game. Generally every player has no information on the others' powers, so it is one of the most important problems for them to know the powers. We treat such unknown games, played periodically, from the view point of information theory. We discuss both non-cooperative and cooperative games and prove some sufficient conditions under which a player can get the full information on the other players' powers from sequential observations on the states. In two person games we show also optimal strategies with respect to the information.
Game theory is the study of strategic competition used in many branches of science. In general, a game is a match between two or more players and each wants to maximize his or her score. In a game, the player’s score depends on their strategies. The goal is to find the optimized decision, with regard to the payoff, in conformity with the rules of the game. This thesis considers the games with two players who know both their own and their rival’s payoff. However, they do not know the strategy chosen by the opponent.\nIn a classical game, the options of players are referred to as cooperation and defect. Quantum game theory is a generalization of classical game theory. In quantum games a linear combination of two options specify the strategy based on qubit. The quantum domain is characterized by entanglement. The quantum entanglement coefficient (y) varies from 0 to . The game is in the classical regime when y is smaller than a threshold. Quantum entanglement is intrinsically non-local.\nIn an iterated game tournament, 63 different strategies contest with four known strategies. Depending on the result, they are divided to a number of groups. The competition specifies which group of strategies can obtain a higher score.\nOther types of games are discussed in the thesis. They are known as incomplete information games. In these games one player has two payoff matrixes, and the rival does not know which of them is used. We discuss the advantage of using quantum strategy over the classical game, particular in avoiding the dilemma in classical games.
Non-local games are studied in quantum information because they provide a simple way for proving the difference between the classical world and the quantum world. A non-local game is a cooperative game played by 2 or more players against a referee. The players cannot communicate but may share common random bits or a common quantum state. A referee sends an input xix_ixi​ to the ithi^{th}ith player who then responds by sending an answer aia_iai​ to the referee. The players win if the answers aia_iai​ satisfy a condition that may depend on the inputs xix_ixi​. Typically, non-local games are studied in a framework where the referee picks the inputs from a known probability distribution. We initiate the study of non-local games in a worst-case scenario when the referee's probability distribution is unknown and study several non-local games in this scenario.
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
It is a well known phenomenon that communication between a group of players affects the set of equilibria of a repeated game. This communication can be costly. Its cost may be measured in terms of complexity, or payoffs. In particular, a payoff cost of communication occurs when players communicate through their actions, and one is interested in characterizing the optimal communication. We study a class of games that models the communication cost of online coordination. For example: A sequence X = (X(t)) of zeroes and ones is chosen at random, and announced to a player, the seer, but not to his teammate, the follower. A repeated game then proceeds in which the follower and the seer choose actions Y (t) and Z(t) in {0, 1} at stage t, and their (common) stage payoff is 1 whenever X(t) = Y (t) = Z(t). We address the question of the optimal communication level from the seer to the follower in order to maximize payoffs. We characterize this level using the notion of entropy, and provide construction of optimal strategies for the team: the seer and the follower. In particular, we prove that against a family of i.i.d. random variables (1/2, 1/2), the maximum expected payoff v to the team is the solution of the equation H(x) + log(3) (1 - x) = 1; where H is the entropy function and log is taken in basis 2. Moreover, we prove the existence of pure strategies of the prophet and the seer that guarantee v against all sequences X = X(t). Hence, the zero-sum game in which player I is the Team (no correlation is allowed between its members), player II is the opponent, and the payoff to I is as above, has a value, and this equals v. Our result has direct implications for repeated games played by finite automata, or by players with bounded recall.