Two-person Game Theory Research Articles

A Study of the Subjective Expectation Model of Real Estate Price Fluctuation Employing Two-person Game Theory

In the real estate market, the demanding of customers vary as their price expectations change. Conversely, changes in the needs of customers affect the price of real estate. This research proposes a model for expected alterations in real estate prices by analyzing variations using two-person game theory. The research also provides evidence for a psychological quantitative study of customer expectation.

The Parable of the Prisoners

Of the 78 possible strategic games in two-person game theory, one has acquired the most attention, and the most notoriety, from scholars and laymen alike. The so-called “Prisoner’s Dilemma,” or what we prefer to call the “Parable of the Prisoners,” is not only the most famous formal model of conflict and cooperation in the mathematical theory of games; it has also has generated extensive commentary in a wide variety of social sciences and other fields, including psychology, biology, politics, economics, law, and philosophy. In this paper, we shall revisit the origins of this popular parable and review a small but representative sample of this diverse literature, identifying common themes and ideas. We shall also present an opposing parable to show that the dilemma in the Prisoner’s Dilemma is unavoidable and inescapable in the one-shot version of the game, and we shall explain why this parable is more than just a story; it is an exemplar or mathematical “paradigm.”In summary, this paper is organized as follows: following this brief introduction, Part 2 reconstructs the origins of the Parable of the Prisoners. Part 3 then reviews various versions of the parable and the uses to which this parable has been put. By way of contrast, Part 4 presents a diametrically different model of behavior — the Altruist’s Dilemma — based on a suggestion by Schelling (1968), and Part 5 explains why the original Prisoner’s Dilemma is not just an instructive parable but also a scientific “paradigm.” Part 6 concludes.

The infinity Laplacian, Aronsson's equation and their generalizations

The infinity Laplace equation Δ ∞ u = 0 arose originally as a sort of Euler-Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional ess-sup U |Du|. The more general functional ess-sup U F(x, u, Du) leads similarly to the so-called Aronsson equation A F [u] = 0. In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to L ∞ variational problems, including two-person game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional L ∞ variational problems.

Measuring efficiency by a fuzzy statistical approach

Methods of measuring economic efficiency of input-output systems by employing a fuzzy statistical approach are explored here in the context of the recent technique of data envelopment analysis, which utilizes a nonparametric approach to measure efficiency. Three applied fuzzy measures are illustrated here, e.g., fuzzy linear programs, fuzzy linear regressions and finally the fuzzy entropy measures in the context of a two-person game theory model.

Comments on “Paradox Regained”

In the game of Roulette, the gambler has 36 strategies, and the wheel has 37 or 38, depending on whether there is only one zero or both a zero and a double zero. TO fix ideas, let us take the 36 x 37 In actual practice the well-balanced wheel plays a mixed strategy, i.e., chooses one of the 37 numbers with probability 1/37. To make Two-person game theory applicable, we must suppose that the wheel, like the gambler, chooses a strategy with a view to maximizing its, i.e., the casino's, expected gain. The Von Neumann-Morgenstern solution of this game prescribes a dominating strategy to the player if he ha one. Clearly the wheel has such a dominating strategy, namely Choose 0, since 0 wins regardless of the gambler's choice. A game like that would draw no customers. For this reason, the wheel's choice of strategy is restricted to a single one, namely the above mentioned mixed strategy. In a game for unit stakes, the wheel's expected gain is +1/37; the gambler's -1/37. Since, however, these are only expectations, not actual gains in single plays, gamblers are attracted to play the game in the hope of short-term gains or for the thrill of uncertainty. us now examine the meta-game. The strategy Choose the number that comes is not defined, because it cannot be realized if 0 comes up (the gambler cannot choose 0). The best he can do is choose a strategy like Choose the number that comes up if it is not '0'; if '0' choose m, where 7t = (1, 2, . . . or 36) . In this meta-game, the wheel, being the second player, chooses a meta-meta-strategy, that is, one conditioned on the gambler's meta-strategy. Again the wheel always wins! Suppose now that 0 is eliminated, so that Roulette becomes a game. Now, the expected gain of borh players is zero in the ordinary game, since the gambler loses 1 unit with probability 35/36 and gains 35 units with probability 1/36. Now the metastrategy Choose the number that comes wins against all possible meta-meta-strategies of the wheel. However, meta-game theory prescribes the interchange of roles, whereby the wheel chooses a meta-strategy, while the gambler chooses a meta-meta-strategy. Clearlv the wheel has a meta-strategy that always wins, namely, Let the number come up that the gambler has not chosen. If the game is played successively with roles interchanged, the gambler and the wheel will win alternately. Naturally, if the original odds are preserved, the meta-game will favor the gambler. To make the meta-game a fair game, the wheel must be given the role of the first player (choosing the meta-strategy rather than the meta-meta-strategy) 35 times out of 36. In my opinion, the meta-game theory presents no interest in the context of zerosum games. The theory was stimulated by paradoxes arising in some non-zerosum games. W e have seen how the paradox of the Prisoner's Dilemma is removed when the game is examined in its meta-game form, that is, beings that conceive all games in their metagame form would not be facing a paradox when playing Prisoner's Dilemma even once. W e have seen, however, that not all ambiguities are removed by the rneta-game theory. For instance, the ambiguity in the game of Chicken remains even when that game is put into meta-game form. This leads us to the question of what sort of paradoxes are removed by the meta-game model and what sort are not and whether the model can be further extended to remove remaining paradoxes. In conclusion, I can only reiterate that the concept of meta-strategy involves no contradiction any more than the concept of strategy. Except in trivial cases, a strategy is always stated in terms of choices conditional on (unknown!) choices of the other player or players. A meta-strategy is no more than an extension of the same idea. Once a game is reduced to normal form, it is always a refereed game, not in the sense that some third party makes decisions, but simply in the sense that a third party acts as a decoder of the strategic choices of the players, translating them into an actual outcome of the

Greatly increased practical usefulness of two-person game theory by adoption of median criterion

Abstract : Considered is discrete two-person game theory where the players choose their strategies independently. Two approaches using a median criterion have been developed for obtaining optimum solution. In the first approach, the payoffs are ranked according to desirability separately for each matrix (with agreement between the players on these rankings). For the second approach, the possible outcomes of the game (pairs of payoffs, one to each player) are ordered according to desirability separately by each player. Only situations of players behaving competitively are considered for the first approach. Rather general situations can be considered for the second approach, which is usable for virtually all games. For both approaches, the payoffs can be of a very general nature (some payoffs may not even be numbers). In the paper, the two approaches are outlined and their application properties are discussed. (Author)

On Decision Theory Under Competition

Decision theory treats models in which a decision maker must choose an information function and a decision function, where the former transforms states of nature into messages, the latter transforms messages into actions, and the ultimate payoff depends upon the state of nature, the message, and the action. This paper treats similar models in which nature's role is played by a selfseeking rational opponent. These models fall into the purview of the theory of two-person games, and the main results concern the maximin and minimax theories of these games. The various games treated differ in the assumptions made about the decision maker's ability to probability mix his choices and about his opponent's information concerning these choices. Twenty different sets of assumptions are considered, but it is shown that the maximin and minimax theories of the resulting games are all reducible in substance to those of just three games: (i) in which the decision maker cannot probability mix his choices at all, and his opponent observes both choices; (ii) in which the decision maker can mix arbitrarily, and his opponent observes neither of his choices; or (iii) in which the decision maker can mix decision functions only, and his opponent observes only the information function. These games are represented in terms of matrix games, and the matrices are compressed by the deletion of dominated strategies for the decision maker. Further compressions are obtained for the case in which the cost of information is separable from the payoff resulting from the actions of the decision maker and his opponent. The game matrices are finally reduced to manageable proportions for the further specialized cases in which either the capacity of the decision maker's information channel is significantly limited and information cost is constant, or the capacity of the information channel is large and there is one low—cost “non-message” available. A numerical example is given, and the concluding comments point to directions for further research.

The Theory of Linear Economic Models.

In the past few decades, methods of linear algebra have become central to economic analysis, replacing older tools such as the calculus. David Gale has provided the first complete and lucid treatment of important topics in mathematical economics which can be analyzed by linear models. This self-contained work requires few mathematical prerequisites and provides all necessary groundwork in the first few chapters. After introducing basic geometric concepts of vectors and vector spaces, Gale proceeds to give the main theorems on linear inequalities—theorems underpinning the theory of games, linear programming, and the Neumann model of growth. He then explores such subjects as linear programming; the theory of two-person games; static and dynamic theories of linear exchange models, including problems of equilibrium prices and dynamic stability; and methods of play, optimal strategies, and solutions of matrix games. This book should prove an invaluable reference source and text for mathematicians, engineers, economists, and those in many related areas.

The Theory of Linear Economic Models

In the past few decades, methods of linear algebra have become central to economic analysis, replacing older tools such as the calculus. David Gale has provided the first complete and lucid treatment of important topics in mathematical economics which can be analyzed by linear models. This self-contained work requires few mathematical prerequisites and provides all necessary groundwork in the first few chapters. After introducing basic geometric concepts of vectors and vector spaces, Gale proceeds to give the main theorems on linear inequalities?theorems underpinning the theory of games, linear programming, and the Neumann model of growth. He then explores such subjects as linear programming; the theory of two-person games; static and dynamic theories of linear exchange models, including problems of equilibrium prices and dynamic stability; and methods of play, optimal strategies, and solutions of matrix games. This book should prove an invaluable reference source and text for mathematicians, engineers, economists, and those in many related areas.