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. Because the three firms do not know for sure whether there is mineral water under their tracts or not, the consequence of their actions depends on Nature (exogenous uncertainty) that beyond their knowledge and control. That is the outcome of the game depends on the state of the world chosen randomly by Nature (the state of the world) and the strategies chosen by the strategic players (TAI, CLA, and HIB). Suppose there possible states of the world: either there is a deposit of x billion gallon water under the land and any well will be a gusher (probability p), or there is no water under the land and any well will be a dry hole (probability q or 1 – p). The two states of the world (Nature) and the possible strategy profiles result in 54 possible outcomes. A three dimensional matrix (3×3×3) is suggested to represent the payoff matrix of p the mineral water drilling game when Nature is gushing (probability p). For example, M132 = ( T132,
p p C132, H132) means the profit for each firm if TAI chooses don’t drill, CLA chooses wide, and HIB p p p chooses narrow; M213 = ( T213, C213, H213) means the profit that TAI chooses narrow, CLA chooses don’t drill, and HIB chooses wide. Another matrix (3×3×3) represents the payoff matrix of the game when q q q Nature is dry well (probability q). For example, N131 = ( T131, C131, H131) means the profit for each firm if q q q TAI chooses don’t drill, CLA chooses wide, and HIB chooses don’t drill; N212 = ( T212, C212, H212) means the profit that TAI chooses narrow, CLA chooses don’t drill, and HIB chooses narrow. The expected payoff matrix Gijk for the mineral water drilling game is suggested. For example, in ((3×3×3) T C H T p q expected payoff matrix G312 = ( u312, u312, u312), the elements will be counted as u312 = T312 × p + T312 C p q H p q × q, u312 = C312 × p + C312 × q, and u312 = H312× p + H312 × q.
Pure Strategy
By using Excel, if input data into matrixes Mijk and Nijk the resulting Gijk will provide information for equilibrium judgment or advanced mixed strategy measurement. An empirical case study with digits will perform in the following. Table 1 and Table 2 are the data of payoff matrix of the mineral water drilling game if gashing and if dry well, respectively. Table 3 is the format of expected payoff matrix of the mineral water drilling game.
Table 1: The Payoff Matrix of the Mineral Water Drilling Game if Gushing Nature: Gushing (probability p) CLA HIB (Don’t drill) Don’t drill Narrow Wide Don’t drill 0 0 0 0 40 0 0 30 0 TAI Narrow 45 0 0 15 30 0 30 20 0 Wide 30 0 0 10 10 0 5 10 0
CLA HIB (Narrow) Don’t drill Narrow Wide Don’t drill 0 0 40 0 50 20 0 30 25 TAI Narrow 50 0 0 15 30 20 -5 20 12 Wide 30 0 0 20 -5 -10 5 5 10
CLA HIB (Wide) Don’t drill Narrow Wide Don’t drill 0 0 30 0 30 0 0 30 20 TAI Narrow 45 -5 0 15 15 2 -2 18 1 Wide 30 0 15 18 -2 0 5 5 3
Mixed Strategy If there is no pure strategy equilibrium, mixed strategy equilibrium will be figured out by the expected payoff and probabilities. Let pTD, pTN, and pTW denoted the probability of TAI choosing Don’t drill, Narrow, and Wide. Let pCD, pCN, and pCW denoted the probability of CLA choosing Don’t drill,
Narrow, and Wide. And, let pHD, pHN, and pHW denoted the probability of HIB choosing Don’t drill, Narrow, and Wide, respectively. The mixed strategy profile will be figured out by comparing these probabilities. The process to find the most possible probability set is first computing the average and standard deviation of all the elements to get two 3 × 3 matrixes; one is the average and the other is the standard
T deviation. For example, referring to Table 3, the average of u1jk (j and k = 1, 2, 3) is the element of H average matrix a11 and the standard deviation of u2jk (j and k = 1, 2, 3) is the element of standard deviation s23. Second, the standard normal random variable, Z, is calculated. Third, the probability of each Z is calculated. Finally, comparing the firm probability, the combination of probability for mixed strategy will be found.
Table 2: The Payoff Matrix of the Mineral Water Drilling Game if Dry Well Nature: Dry well (probability 1 − p) CLA HIB (Don’t drill) Don’t drill Narrow Wide Don’t drill 0 0 0 0 -20 0 0 -30 0 TAI Narrow -15 -5 0 -15 -10 0 -15 -20 0 Wide -30 0 0 -30 -5 -5 -30 -10 -10
CLA HIB (Narrow) Don’t drill Narrow Wide Don’t drill 0 0 0 0 -20 -25 0 -30 -20 TAI Narrow -15 -10 -20 -15 -10 -10 -15 -20 -5 Wide -30 0 -10 -30 -5 -5 -30 -10 0
CLA HIB (Wide) Don’t drill Narrow Wide Don’t drill 0 0 -30 0 -20 0 0 0 -15 TAI Narrow -15 -12 -5 -15 -15 2 -2 -16 -3 Wide -30 -5 -15 -15 -2 -30 -20 -30 -15
Table 3: The Expected Payoff Matrix of the Mineral Water Drilling Game CLA HIB (Don’t drill) Don’t drill Narrow Wide T C H T C H T C H Don’t drill u111 u111 u111 u121 u121 u121 u131 u131 u131 T C H T C H T C H TAI Narrow u211 u211 u211 u221 u221 u221 u231 u231 u231 T C H T C H T C H Wide u311 u311 u311 u321 u321 u321 u331 u331 u331
CLA HIB (Narrow) Don’t drill Narrow Wide T C H T C H T C H Don’t drill u112 u112 u112 u122 u122 u122 u132 u132 u132 T C H T C H T C H TAI Narrow u212 u212 u212 u222 u222 u222 u232 u232 u232 T C H T C H T C H Wide u312 u312 u312 u322 u322 u322 u332 u332 u332
CLA HIB (Wide) Don’t drill Narrow Wide T C H T C H T C H Don’t drill u113 u113 u113 u123 u123 u123 u133 u133 u133 T C H T C H T C H TAI Narrow u213 u213 u213 u223 u223 u223 u233 u233 u233 T C H T C H T C H Wide u313 u313 u313 u323 u323 u323 u333 u333 u333
RESULTS
Based on the principle and methodology, the study assumes asymmetrical date matrix (Table 1 and Table 2) and uses computer Excel to explore the pure strategy equilibrium and mixed strategy
equilibriums. The results are also explained by three tables (Tables 4, 5, and 6) to express the pure and mixed equilibrium.
Pure Strategies An example of asymmetrical data matrixes is suggested. After analysis the data with exogenous uncertainty (gushing probability p = 0.90, dry well probability q = 0.10) and the expected payoff matrix are listed in Table 4. The unique strategy for Nash (pure) equilibrium of the game is {Wide, Wide, T T T C C C H Narrow} because of u332 (1.5) > u132 (0) > u232 (-6), u333 (3.5) > u313 (0) > u323 (-5), and u332 (9) > H H u333 (1.2) > u331 (-1).
Table 4: The Expected Payoff Matrix of the Mineral Water Drilling Game (p = 0.90) CLA HIB (Don’t drill) Don’t drill Narrow Wide Don’t drill 0 0 0 0 34 0 0 24 0 TAI Narrow 39 -0.5 0 12 26 0 25.5 16 0 Wide 24 0 0 6 8.5 -0.5 1.5 8 -1
CLA HIB (Narrow) Don’t drill Narrow Wide Don’t drill 0 0 36 0 43 15.5 0 24 20.5 TAI Narrow 43.5 -1 -2 12 26 17 -6 16 10.3 Wide 24 0 -1 15 -5 -9.5 1.5 3.5 9
CLA HIB (Wide) Don’t drill Narrow Wide Don’t drill 0 0 24 0 25 0 0 27 16.5 TAI Narrow 39 -5.7 -0.5 12 12 2 -2 14.6 0.6 Wide 24 -0.5 12 14.7 -2 -3 2.5 1.5 1.2
If the range of gushing probability, p, is 1.0 ≥ p ≥ 0.8572, The expected payoff functions for
TAI-water (uT), CLA-water (uC), and HIB-water (uH) are Equation 3, 4, and 5.
TAI: uT = 35.008 p − 30.0071, (3)
CLA: uC = 14.992 p − 9.9926, (4)
and HIB: uH = 10.008 p − 0.0074. (5) Using the former stated rule, for the range of p is 0.75 ≥ p ≥ 0.50, the unique strategy for Nash (pure) equilibrium of the game is {Narrow, Narrow, Narrow}, and the expected payoff functions are Equation 6, 7, and 8.
TAI: uT = 30 p − 15, (6)
CLA: uC = 30 p − 10, (7)
and HIB: uH = 40 p − 10. (8) For the range of p is 0.49 ≥ p ≥ 0.4286, the unique strategy for Nash (pure) equilibrium of the game is {Don’t drill, Wide, Wide}, and the expected payoff functions are Equation 9, 10, and 11.
TAI: uT = 0, (9)
CLA: uC = 29.683 p − 0.1504, (10)
and HIB: uH = 35.008 p − 15.004. (11) For the range of p is 0.4285 ≥ p ≥ 0.3334, the unique strategy for Nash (pure) equilibrium of the game is {Don’t drill, Narrow, Don’t drill}, and the expected payoff functions are Equation 12, 13, and 14.
TAI: uT = 0, (12)
CLA: uC = 59.855 p − 19.953, (13)
and HIB: uH = 0. (14) For the range of p is 0.2308 ≥ p ≥ 0.001, the unique strategy for Nash (pure) equilibrium of the game is {Don’t drill, Don’t drill, Narrow}, and the expected payoff functions are Equation 15, 16, and 17. If gushing probability, p = 0.00, the equilibrium strategy is {Don’t drill, Don’t drill, Don’t drill}.
TAI: uT = 0, (15)
CLA: uC = 0, (16)
and HIB: uH = 39.994 p + 0.0003. (17)
Mixed Strategy If the gushing probability p is 0.857 ≥ p ≥ 0.751, then the pure strategy does not exist. For example, if the situation is gushing probability p = 0.80 and dry well probability q = 0.20, the expected probability matrix are listed in Table 5.
Table 5: The Expected Probability Matrix of the Mineral Water Drilling Game (p = 0.80) CLA HIB (Don’t drill) Don’t drill Narrow Wide Don’t drill 0.17 0.27 0.06 0.17 0.27 0.06 0.17 0.19 0.06 TAI Narrow 0.29 0.06 0.12 0.11 0.30 0.12 0.21 0.21 0.12 Wide 0.29 0.15 0.18 0.08 0.32 0.15 0.04 0.31 0.13
CLA HIB (Narrow) Don’t drill Narrow Wide Don’t drill 0.17 0.04 0.33 0.17 0.31 0.18 0.17 0.19 0.23 TAI Narrow 0.30 0.05 0.04 0.11 0.30 0.33 0.03 0.21 0.29 Wide 0.29 0.15 0.13 0.19 0.03 0.02 0.04 0.22 0.31
CLA HIB (Wide) Don’t drill Narrow Wide Don’t drill 0.17 0.04 0.25 0.17 0.21 0.06 0.17 0.24 0.20 TAI Narrow 0.29 0.02 0.09 0.11 0.17 0.16 0.04 0.20 0.12 Wide 0.29 0.11 0.32 0.21 0.08 0.06 0.06 0.08 0.16
Under the condition of gushing probability p = 0.8, the strategy for mixed equilibrium of the game is {Wide, Don’t drill, Wide}, because TAI (Wide: pTW = 0.29), CLA (Don’t drill: pCD = 0.11), and HIB
(Wide: pHW = 0.32) are the most possible combination of the probability matrix. If the gushing probability is 0.857 ≥ p ≥ 0.751, the mixed strategy is also {Wide, Don’t drill, Wide}, and the expected probability functions for TAI-water (pT), CLA-water (pC), and HIB-water (pH) are Equation 18, 19, and 20. 2 TAI: pT = – 0.4616 p + 0.981 p + 0.1996, (18)
CLA: pC = 0.1147, (19) 2 and HIB: pH = – 0.1911 p + 0.3988 p – 0.119. (20) If gushing probability is 0.333 ≥ p ≥ 0.230, the pure strategy does not exist. The mixed strategy is {Don’t drill, Narrow, Narrow}. For example, if p = 0.3, the expected probability matrix are listed in Table 6.
Table 6: The Expected Probability Matrix of the Mineral Water Drilling Game (p = 0.30) CLA HIB (Don’t drill) Don’t drill Narrow Wide Don’t drill 0.17 0.08 0.06 0.17 0.08 0.06 0.17 0.05 0.06 TAI Narrow 0.11 0.06 0.12 0.06 0.12 0.12 0.08 0.08 0.12 Wide 0.09 0.15 0.18 0.04 0.20 0.16 0.03 0.18 0.14
CLA HIB (Narrow) Don’t drill Narrow Wide Don’t drill 0.17 0.04 0.16 0.17 0.10 0.07 0.17 0.05 0.09 TAI Narrow 0.11 0.05 0.06 0.06 0.12 0.19 0.04 0.08 0.16 Wide 0.09 0.15 0.14 0.06 0.09 0.11 0.03 0.14 0.23
CLA HIB (Wide) Don’t drill Narrow Wide Don’t drill 0.17 0.04 0.08 0.17 0.06 0.06 0.17 0.08 0.08 TAI Narrow 0.11 0.04 0.10 0.06 0.08 0.13 0.05 0.08 0.11 Wide 0.09 0.12 0.20 0.08 0.12 0.08 0.04 0.05 0.14
Table 6 showed that TAI (Don’t drill: pTD = 0.17), CLA (Narrow: pCN = 0.10), and HIB (Narrow: pHN = 0.07) are the most probable situation combination for the gushing probability p = 0.30. Actually, in the range (0.333 ≥ p ≥ 0.230), the expected probability functions are Equation 21, 22, and 23.
TAI: pT = 0.50, (21) 2 CLA: pC = 0.9943 p + 0.842 p + 0.039, (22) 2 and HIB: pH = 0.3588 p + 0.3467 p + 0.1071. (23)
DISCUSSION AND CONCLUSION
The results reveal that Nature influences the outcome of the games and makes the results to be very complex especially if the players are three or more than three (Daskalakis and Papadimitriou, 2005). In the results, the pure equilibrium strategy profile (Wide, Wide, Narrow) exists if gushing probability is in the range of 1.0 ≥ p ≥ 0.8572. The pure strategy does not exist if 0.857 ≥ p ≥ 0.751; but, the mixed equilibrium strategy profile is (Wide, Don’t drill, Wide). The pure equilibrium strategy profile (Narrow, Narrow, Narrow) exists if 0.75 ≥ p ≥ 0.50. If 0.49 ≥ p ≥ 0.4286, the pure equilibrium strategy profile is (Don’t drill, Wide, Wide). If 0.4285 ≥ p ≥ 0.3334, the pure equilibrium strategy profile is (Don’t drill, Narrow, Don’t drill). If 0.333 ≥ p ≥ 0.230, the pure strategy does not exist; but, the mixed strategy is (Don’t drill, Narrow, Narrow). If 0.2308 ≥ p ≥ 0.001, the strategy profile for pure equilibrium of the game is (Don’t drill, Don’t drill, Narrow). If gushing probability p = 0.00, the equilibrium strategy profile is (Don’t drill, Don’t drill, Don’t drill). To sum up, Table 7 lists all the situations.
Table 7: The Gushing Probability Range and Equilibrium Strategy Profile with Type Gushing probability Strategy profile (TAI, CLA, HIB) Equilibrium profile 1.0 ≥ p ≥ 0.8572 (Wide, Wide, Narrow) Pure 0.857 ≥ p ≥ 0.751 (Wide, Don’t drill, Wide) Mixed 0.75 ≥ p ≥ 0.50 (Narrow, Narrow, Narrow) Pure 0.49 ≥ p ≥ 0.4286 (Don’t drill, Wide, Wide) Pure 0.4285 ≥ p ≥ 0.3334 (Don’t drill, Narrow, Don’t drill) Pure
0.333 ≥ p ≥ 0.230 (Don’t drill, Narrow, Narrow) Mixed 0.2308 ≥ p ≥ 0.001 (Don’t drill, Don’t drill, Narrow) Pure p = 0.00 (Don’t drill, Don’t drill, Don’t drill) Pure
In the real world, Nature has two faces those are exogenous uncertainty and endogenous uncertainty. Both of them will influence the process and results of games. The information is important, if the probability of gashing is known, the firms may choose the best strategy to make profits. In the mixed equilibrium situations, the endogenous uncertainty of decision making can be reduced by the gushing information, too. The special of this study is the resulting equilibrium expectation utilities or probabilities are expressed in the form of equations, i.e. the solutions are segmental continuous. With the aid of computer these results of games are easier to obtain. Further research may use the other software or computer programming and suggest more complex situation to explore the game theory.
REFERENCES
Antonio, J.M. (2006). A model of interim information sharing under incomplete information, International Journal of Game Theory, 34, 425-442. Aumann, R. (1985). What is Game Theory Trying to accomplish? In K. Arrow & S. Honkapohja (Eds.), Frontiers of Economics (pp. 909-924). Basil Blackwell, Oxford. Bade, S. (2005). Nash equilibrium in games with incomplete preferences, Economic Theory, 26, 309–332. Bierman, H.S. & Fernandez, L. (1998). Game Theory with Economic Applications. Reading, Massachusetts: Addison-Wesley. Chen, S.S. (2008). Comparing Cournot output and Bertrand price duopoly game. The Journal of Global Business Management, 4(2), 67-75. Chen, S.S., You, J.Y., & Lin, S.L. (2007). Using computer restudies Nash equilibrium of Bertrand price competition game. 2007 Enterprise Environment and Management Symposium, Taiwan: Dahan Institute of Technology, 1st June 2007, 204-17. Chong, P.S. & Benli, Ö .S. (2005). Consensus in team decision making involving resource allocation. Management Decision, 43(9), 1147-60. Cruz, J.B. & Simaan, M.A. (2000). Ordinal games and generalized Nash and Stackelberg solutions. Journal of Optimization theory and Applications, 107(2), 205-22. Daskalakis, C. & Papadimitriou, C.H. (2005). Three-Player Games Are Hard. Retrieved November 3, 2008, from http://www.cs.berkeley.edu/~christos/papers/3players.pdf. Gibbons, R. (1992). A Primer in Game Theory. Harlow, England: Prentice Hall. Khan, M.A., Rath, K.P., & Sun, Y. (2006). The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games. Int. J. Game Theory, 34, 91–104. Perea, A., Peters, H., Schulteis, T., & Vermeulen, D. (2006). Stochastic dominance equilibria in two-person noncooperative games. Int. J. Game Theory, 34, 457–473. Ponssard, J.P. (1976). On the concept of the value of information in competitive situations. Management Science, 22(22), 739-47. Prasad, K. (2003). Observation, measurement, and computation in finite games. Internal Journal of Game Theory, 32, 455–470. Rasmusen, E. (1995). Games and Information. Cambridge, MA: Blackwell Pub. Inc. Ravikumar, K. (1987). The relationship between mixed strategies and strategic groups. Managerial and Decision Economics, 8(3), 235-242. Reny, P.J. (1999). On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica, 67(5), 1029-56.
Rubinstein, A. (1991). Comments on the interpretation of Game Theory. Econometrica, 59(4), 909-914. Saloner, G. (1991). Modeling, game theory, and strategic management. Strategic Management Journal, 12, 119-36. Shachat, J. & Swarthout, J.T. (2004). Do we detect and exploit mixed strategy play by opponents? Mathematical Method of Operational Research, 59, 359–373. Shmaya, E. (2006). The value of information structure in zero-sum games with lack of information on one site. International Journal of Game Theory, 34, 155-65. Von Neumann, J. & Morgenstern, O. (2004). Theory of Games and Economic Behavior, (p. 11). Woodstock, Oxfordshire: Princeton University. Zhu, K. (2006). Information transparency of business-to business electronic markets: A game theoretic analysis. Management Science, 50(5), 670-685. Zimper, A. (2007). Strategic games with security and potential level players. Theory and Decision, 63, 53-78.