We study an evolutionary game of chance in which the probabilities for different outcomes (e.g., heads or tails) depend on the amount wagered on those outcomes. The game is perhaps the simplest possible probabilistic game in which perception affects reality. By varying the reality map', which relates the amount wagered to the probability of the outcome, it is possible to move continuously from a purely objective game in which probabilities have no dependence on wagers, to a purely subjective game in which probabilities equal the amount wagered. The reality map can reflect self-reinforcing strategies or self-defeating strategies. In self-reinforcing games, rational players can achieve increasing returns and manipulate the outcome probabilities to their advantage; consequently, an early lead in the game, whether acquired by chance or by strategy, typically gives a persistent advantage. We investigate the game both in and out of equilibrium and with and without rational players. We introduce a method of measuring the inefficiency of the game and show that in the large time limit the inefficiency decreases slowly in its approach to equilibrium as a power law with an exponent between zero and one, depending on the subjectivity of the game.
There has been considerable past work on situations where subjective factors influence objective outcomes. Some examples include Hommes's studies of cobweb models @cite_8 @cite_3 , studies of increasing returns @cite_1 , Arthur's El Farol model and its close relative the minority game @cite_2 @cite_9 , Blume and Easley's model of the influence of capital markets on natural selection in an economy @cite_5 @cite_12 , and Akiyama and Kaneko's example of a game that changes due to the players' behaviors and states @cite_11 . The model we introduce here has the advantage of being very general yet very simple, providing a tunable way to study this phenomenon under varying levels of feedback.
@cite_8 @cite_9 @cite_1 @cite_3 @cite_2 @cite_5 @cite_12 @cite_11
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Abstract We investigate the dynamics of the cobweb model with adaptive expectations, a linear demand curve, and a nonlinear, S-shaped, increasing supply curve. Both stable periodic and chaotic price behaviour can occur. We investigate, how the dynamics of the model depend on the parameters. Both infinitely many period doubling and period halving bifurcations can occur, when the demand curve is shifted upwards. The same result holds with respect to the expectations weight factor. Abstract The price-quantity dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand curves is analysed. We prove that chaotic dynamical behaviour can occur, even if both the supply and demand curves are monotonic. The introduction of adaptive expectations into the cobweb model leads to price-quantity fluctuations with a smaller amplitude. However, at the same time the price-quantity cycles may become unstable and chaotic oscillations may arise. We present a geometric explanation why chaos can occur for a large class of nonlinear, monotonic supply and demand curves. Abstract In a conventional asset market model we study the evolutionary process generated by wealth flows between investors. Asymptotic behavior of our model is completely determined by the investors' expected growth rates of wealth share. Investment rules are more or less “fit” depending upon the value of this expectation, and more fit rules survive in the market at the expense of the less fit. Using this criterion we examine the long run behavior of asset prices and the common belief that the market selects for rational investors. We find that fit rules need not be rational, and rational rules not be fit. Finally, we investigate how the market selects over various adaptive decision rules. Evolutionary arguments are often used to justify the fundamental behavioral postulates of competive equilibrium. Economists such as Milton Friedman have argued that natural selection favors profit maximizing firms over firms engaging in other behaviors. Consequently, producer efficiency, and therefore Pareto efficiency, are justified on evolutionary grounds. We examine these claims in an evolutionary general equilibrium model. If the economic environment were held constant, profitable firms would grow and unprofitable firms would shrink. In the general equilibrium model, prices change as factor demands and output supply evolves. Without capital markets, when firms can grow only through retained earnings, our model verifies Friedman's claim that natural selection favors profit maximization. But we show through examples that this does not imply that equilibrium allocations converge over time to efficient allocations. Consequently, Koopmans critique of Friedman is correct. When capital markets are added, and firms grow by attracting investment, Friedman's claim may fail. In either model the long-run outcomes of evolutionary market models are not well described by conventional General Equilibrium analysis with profit maximizing firms. Submitted to Journal of Economic Theory. (This abstract was borrowed from another version of this item.) A theoretical framework we call dynamical systems game is presented, in which the game itself can change due to the influence of players’ behaviors and states. That is, the nature of the game itself is described as a dynamical system. The relation between game dynamics and the evolution of strategies is discussed by applying this framework. Computer experiments are carried out for simple one-person games to demonstrate the evolution of dynamical systems with the effective use of dynamical resources. © 2000 Elsevier Science B.V. All rights reserved. PACS: 02.50.Le