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Researching and Further Studying of Game Theory Yuxi Weng, Xueying Liu, Lei Wang July, 22th, 2015 Abstract Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory provides general mathematical techniques for analyzing situations in which two or more individuals make decisions that will influence one another’s profit. As such, game theory offers insights of fundamental importance for scholars in all branches of the social sciences, as well as for practical decision-makers. Keyword Game theory, Prisoner’s dilemma, John Nash, Nash Equilibrium 1. Introductiong Game theory is an equal competition between 2 people, who want to make use of their opponent’s strategy to change them, so that they could achieve success. The thought of Game theory actually came from a long time ago, Master Sun's Art of War is a famous work at the history of China, and it is not only about war but also the earliest Game theory book. At the beginning, game theory focuses on the success or failure in chess, bridge and gamble, thus we can see that people’s controlling on gaming situation is just based on experiences, not for theorization. Game theory would think about the predictable action or actual one in games, and find the way to get better. The research of Game theory nowadays is beginning with Zemelo, Borel and Von Neumann. In 1928, Von Neumann proved that the basic principle of game theory, which declared the birth of game theory. In 1944, Von Neumann and Morgenstern write the An epoch-making masterpiece together with is called “Games and Economic Behavior”, and they developed the 2-people Game theory to Multiple Game theory, and apply Game theory to field of Economy, then they set the foundation and theory structure for this subject. From 1950 to 1951, John Forbes Nash Jr made use of fixed point theorem to prove the existence of Equantequation, so he made a strong foundation for game theory’s vague 1 generalization. Nash’s creative paper: “N game theory” (1950), “Non-cooperative Game” (1951) and so on, gave Nash concept of equilibrium and equilibrium existence theorem. Otherwise, Reinhard, hazel, John Harsanyi’s research did promotion for the development of Game theory. Now Game theory has developed nearly completed subject. Game theory mainly studies the formulation of interaction between the incentive structures. It is the way to study; something has mathematics problems with competitive phenomenon. Game theory considers the game's predictable behavior and actual behavior, and to study their optimization strategy. Biologists use game theory to understand and to predict some results of evolution. Game theory has become one of the standard analytic tools of economics. In biology, economics, international relations, computer science, politics, military strategy and many other disciplines are widely used. Included in the basic concept, all of players, action, information, strategy and yields balanced results. One player, policies, and income are the basic elements. Players, actions and results are referred to as game rules. 2. Game theory approaches 2.1 Prisoner’s dilemma Suppose that the police have arrested two people whom they know have committed a robbery together. Unfortunately, they lack enough evidence to get a jury to convict. However, they have enough evidence to send each prisoner away for two years of the car theft. The chief inspector now makes the following offer to each prisoner: If you confess to the robbery, and he doesn’t confess, then you'll go free and he'll get ten years in prison. If you both confess, you'll each get 5 years in prison. If neither of you confess, then you'll each get 2 years for the car theft. We can consider it as a game and represent the problem faced by both of players on a single matrix that has their separate choices interact; this is the form of their game: A B confess refuss confess (5,5) (0,10) refuse (10,0) (2,2) Table 1: Strategies which can be used for prisoner’s dilemma Each cell of the matrix gives the payoffs to both players for each combination of actions. A's payoff appears as the first number of each pair, B's as the second. So, if both players confess then they each get 5 years in prison. If neither of them confesses, they each get a payoff 2 years in prison. If A confesses and B doesn't then A goes free and B gets a payoff 10 years in prison. The reverse situation, in which B confesses and A 2 refuses, appears in the lower-left cell. Each one evaluates his or her two possible actions here by comparing their personal payoff in each column, because this shows which of their actions is preferable. So, let’s observe: If B confesses then A gets 5 years in prison by confessing and 10 years in prison by refusing. If B refuses, then A goes free by confessing and gets 2 years in prison by refusing. Therefore, A is better off confessing regardless of what B does. B, meanwhile, evaluates his actions by comparing his payoff in each row, and he comes to exactly the same conclusion that A does. Wherever one action for a player is superior to his other actions for each possible action by the opponent, we say that the first action strictly dominates the second one. In the Prisoner’s Dilemma, confessing strictly dominates refusing for both players. Thus the best situation is both players will confess, and both will go to prison for 5 years. 2.2 Chicken game The other interesting game which can be used in construction management is chicken game. In this game, imagine two young drivers are driving by two fast cars toward each other in a narrow road. The probability of death for both young drivers is high, if none of them turn his direction. The priorities of these two drivers in this game are that they will not play the timid role. So the best payoff is to have your opponent be the chicken. The worst possible payoff is to crash to each other. So in the matrix for this game, this situation has the least value. We assign it 1. As mentioned before, the best payoff for each driver is to have his opponent be the chicken, so we assign it a value 4. The next worst possibility is to be the chicken, so we assign this a value 2. The last possibility is that both drivers swerve at the same time. We assign this a value 3. In this strategy they can maintain their pride and life, so this is preferable to being the chicken. But in these circumstances none of the players neither will be a loser nor will be a winner. Player 1 Player 2 Swerve Do not swerve Swerve (3,3) (4,2) Do not swerve (2,4) (1,1) Table 2: Strategies which can be used for chicken game According to the matrix above, the game has two Nash equilibrium points that are (swerve, do not swerve) and (do not swerve, swerve). In addition being the Nash equilibrium, these two options can be also Pareto optimal points. There is also another optimal Pareto point that is (swerve, swerve). At this point, both players reach to equality state and the play will have no loser or winner. 3. Further reading and further study 3 3.1 John Nash Nash was born on June 13, 1928, in Bluefield, W.Va., U.S, who is a mathematician. He earned a doctorate from Princeton University at 22. He began teaching at Massachusetts Institute of Technology in 1951 but left in the late 1950s because of mental illness; thereafter he was informally associated with Princeton. Beginning in the 1950s with his influential thesis “Non-cooperative Games,” Nash established the mathematical principles of game theory. His theory, known as the Nash solution or Nash equilibrium, attempted to explain the dynamics of threat and action among competitors. Despite its practical limitations, it was widely applied by business strategists. He shared the 1994 Nobel Prize in Economics with John C. Harsanyi and Reinhard Selten. A film version of his life, A Beautiful Mind, won an Academy Award for best picture. 3.2 Types of theory game Cooperative / Non-cooperative A game is cooperative if the players are able to form binding commitments. For instance, the legal system requires them to adhere to their promises. In non-cooperative games, this is not possible. Often it is assumed that communication among players is allowed in cooperative games, but not in non-cooperative ones. However, this classification on two binary criteria has been questioned, and sometimes rejected. Of the two types of games, non-cooperative games are able to model situations to the finest details, producing accurate results. Cooperative games focus on the game at large. Considerable efforts have been made to link the two approaches. The so-called Nash-program, which is the research agenda for investigating on the one hand axiomatic bargaining solutions and on the other hand the equilibrium outcomes of strategic bargaining procedures, has already established many of the cooperative solutions as non-cooperative equilibrium. Zero-sum / Non-zero-sum Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources. In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero more informally, and a player benefits only at the equal expense of others. Poker exemplifies a zero-sum game ignoring the possibility of the house's cut, because one wins exactly the amount one's opponents lose. Other zero-sum games include matching pennies and most classical board games including go and chess. Many games studied by game theorists, including the infamous prisoners' dilemma, are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another. Constant-sum games correspond to activities like theft and gambling, but not to the 4 fundamental economic situation in which there are potential game from trade. It is possible to transform any game into a possibly asymmetric zero-sum game by adding a dummy player is often called "the board" whose losses compensate the players' net winnings. 3.3 Some applied uses Computer science and logic Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations. Also, game theory provides a theoretical basis to the field of multi-agent systems. Separately, game theory has played a role in online algorithms. In particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games. Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms. The emergence of the internet has motivated the development of algorithms for finding equilibrium in games, markets, computational auctions, peer-to-peer systems, and security and information markets. Algorithmic game theory and within it algorithmic mechanism design combine computational algorithm design and analysis of complex systems with economic theory. Economics and business Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents. Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers & acquisitions pricing, fair division, duopolies, oligopolies, social network formation, agent-based computational economics, general equilibrium, mechanism design, and voting systems; and across such broad areas as experimental economics, behavioral economics, information economics, industrial organization, and political economy. This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibrium". A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing. The payoffs of the game are generally taken to represent the utility of individual players. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game is equilibrium of the appropriate type. Naturally one might wonder to what use this information should be put. 5 Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive. Reference Application of Game Theory Approach in Solving the Construction Project Conflicts Azin Shakiba Barougha, Mojtaba Valinejad Shoubia,*, Moohammad Javad Emami Skardib 6