The Influence of Nature on Outcomes of Three Players Game Dr. Sea-shon Chen, Department of Business Administration, Dahan Institute of Technology, Taiwan ABSTRACT This paper based on Nash equilibrium theory to explore the uncertainty of Nature that influences the outcome of the three player’s game. Nature has two phases; one is exogenous uncertainty or random and another is endogenous uncertainty or strategy choosing. Exogenous random is the environment or situation unpredictable. Endogenous uncertainty is the players’ unpredictable intentions that result in mixed strategy in the game. The paper expresses the methodology and utilize computer to explore discrete equilibrium of the game, then connect discrete equilibriums into continuous results. The result reveals that pure equilibriums and mixed equilibriums exist in different uncertainty range. The results also suggest, in the real world, the players know the information of Nature is important because that will influence decision making and payoffs. Keywords: Nature, exogenous uncertainty, endogenous uncertainty, Nash equilibrium INTRODUCTION Game theory is a modeling tool. It is an interdisciplinary and distinct approach to the study of human behavior and strategic management (Perea, et al., 2006; Rasmusen, 1995; Saloner, 1991). The disciplines most involved in game theory are mathematics, economics, social science, and behavioral science. The essential elements of a game are (1) players, (2) actions, (3) information, (4) strategies, (5) payoffs, (6) outcomes, and (7) equilibriums. Players are the individuals who make decision based on that they are absolutely rational in their economic choices. Game theory is based on the assumption that human beings are absolutely rational in their economic choices (von Neumann and Morgenstern, 2004). Specifically, the assumption is that each person tries to maximize her or his rewards (utilities, profits, incomes, or subjective benefits) in the circumstances that a player faces (Chen, 2008). This hypothesis serves a double purpose in the study of the allocation of resources. First, it narrows the range of possibilities; somewhat absolutely rational behavior is more predictable than irrational behavior. Second, it provides a criterion for evaluation of the efficiency of an economic system. Sometimes the irrational pseudo-player, Nature, takes random actions at specified points in the game with specified probabilities (Rasmusen, 1995). Nature is indifferent to the outcomes, but the strategic players care about the outcomes. Although the strategic players choose the strategies and the choices affect the outcomes, the state of the world chosen randomly by Nature who is either exogenous or endogenous uncertain probability and this information revealing basically affects the outcomes (Antonio, 2006; Bierman and Fernandez, 1998; Shmaya, 2006). The value of information about Nature is very important; players may or may not know the probability Nature will choose (Ponssard, 1976). In many cases, the result we predict immensely influences our decision. The misplay of decision is not due to shortage and/or error of information or the logic reasoning, but the phase we think is unique (Cruz & Simaan, 2000; Zhu, 2004). We always put our hands to the phase that we customized and overlook the changeful and uncertain reality. Eventually, our conservative and sealed vision results in faulty judgment and decision-making. One of the game theory players, Nature, does not commit the misplay of decision. Nature plays the game in many phases. Uncertainty or random is the reality of the player who is Nature. Exogenous uncertainty is that the strategic game players do not know the irrational pseudo-player has what kind of the state of the world. For example, three firms are deciding whether and how to drill wells into a spring water deposit that lies under their adjacent land tracts. The three firms do not know for sure whether there is water under their land or not, i.e. uncertainty of the state of the world or the phase of player Nature. Endogenous uncertainty is that the strategic game players choose among their action randomly, but sometimes with reasoning. In the non-cooperative game, when more than one player adopts a mixed strategy (a probability stated strategy) these players randomize independently of each other (Prasad, 2003; Ravikumar, 1987). Independence means that knowledge of the strategy chosen by one player provides no new information about the strategy that will be chosen by any other player who has adopted a mixed strategy in continuous or discontinuous game (Bierman and Fernandez, 1998; Reny, 1999). For example, TAI-water, CLA-water, and HIB-water firm may choose ‘don’t drill’, a ‘narrow’ well, or a ‘wide’ well on certain probability. The probabilities which the firms make decisions are endogenous uncertainty or psychological Nature. Some papers or books introduce or investigate exogenous and endogenous uncertainty in static games which limited in a case or study two games separately. This paper studies two kinds of Nature, exogenous and endogenous uncertainty, impact on the equilibrium of games with computer. By using computer to study, the processes are easy and the results can be continuous and illustrated. PRINCIPLE Exogenous and endogenous uncertainty in static games may result in mixed strategy. In game theory a mixed strategy is a strategy which chooses randomly among possible moves. The strategy has some probability distribution which corresponds to how frequently each move is chosen. A totally mixed strategy is a mixed strategy in which the player assigns strictly positive probability to every pure strategy. A mixed strategy should be understood in contrast to a pure strategy where a player plays a single strategy with probability. Pure strategy Nash equilibriums are Nash equilibriums where all players are playing pure strategies. Mixed strategy Nash equilibriums are equilibriums where at least one player is playing a mixed strategy. During the 1980s, the concept of mixed strategies came under heavy fire for being intuitively problematic. Randomization, central in mixed strategies, lacks behavioral support (Shachat, and Swarthout, 2004). Seldom do people make their choices following a lottery. This behavioral problem is compounded by the cognitive difficulty that people are unable to generate random outcomes without the aid of a random or pseudo-random generator (Aumann, 1985). However, rational strategies exist for finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (Zimper, 2007). Game theorist Rubinstein (1991) points out two alternative ways of understanding the concept: one is to imagine that the game players stand for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents. The other, called purification is to suppose that the mixed strategies interpretation merely reflects our lack of knowledge of the agent's information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors. However, it is unsatisfying to have results that hang on unspecified factors, and this dismisses the possibility of a mixed-strategies analysis to have any predictive power. Arguing that those factors are simply other players' beliefs about a player's strategy, hence adopting a mixed strategy is the best response to a player playing mixed strategies, gives a credible interpretation, but does not restore predictive power to the concept of mixed equilibriums. Although economist’s attitude towards mixed strategies-based results has been ambivalent, mixed strategies are still widely used for their capacity to provide Nash equilibrium in any game concerning minimum monetary regret or maximum profits (Bade, 2005; Chong and Benli, 2005). The following is an example. Let (S, f) be a game, where Si is the strategy set for player i, S = S1 × S2 … × Sn is the set of strategy profiles and f = (f1(x), ..., fn(x)) is the payoff function. Let x−i be a strategy profile of all players except for player i. When each player i belongs to {1, ..., n} chooses strategy xi resulting in strategy profile x = (x1, ..., xn), then player i obtains payoff fi(x). The payoff depends on the strategy profile chosen, i.e. on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile x* belongs to S is a Nash equilibrium if no unilateral deviation in strategy by any single player is profitable for that player, that is * * * * i, xi Si , xi xi : fi (xi , xi ) fi (xi , xi ). (1) In order to investigate mixed strategies, noting pi is the payoff converted probability and p = (p1(x), ..., pn(x)) is the function of the payoff converted probability. A strategy profile x* belongs to S is a mixed equilibrium if no unilateral deviation in strategy by any single player has the most probable of the payoff converted probability for that player, that is * * * * i, xi Si , xi xi : pi (xi , xi ) pi (xi , xi ). (2) A game can have a pure strategy Nash equilibrium (NE) or a mixed NE in its mixed extension that of choosing a pure strategy stochastically with a fixed frequency. Nash explained that, if we allow mixed strategies, i.e. players choose strategies randomly according to the most probable probabilities then every n-player game in which every player can choose from finitely many strategies admits at least one Nash equilibrium. METHODOLOGY A good model in game theory has to be realistic in the sense that it provides the perception of real life social phenomena (Rubinstein, 1991). The paper based on this argument and the principles to investigate pure strategy and possible mixed strategy. A situation is three firms TAI, CLA, and HIB are deciding whether and how to drill wells into a mineral water resources that lie under their adjacent tracts.