# Game Theory and its Applications in Various Fields

INTRODUCTION:

Game theory is a branch of mathematics that uses mathematical models to study strategic interactions with formalized incentive structures. Game theory is a body of reasoning, grounded in mathematics but readily understood intuitively as a reflection of how people may behave, particularly in situations that involve high stakes for them. It is part of a family of theories that assume people are rational, meaning that they do what they believe is in their best interest. Models of decision making such as prospect theory (Kahneman and Tversky, 1984; Kahneman and Miller, 1986) and operations research, for instance, examine rational choices in situations in which people confront constraints such as limited time, limited budget, incomplete or uncertain information, or other structural impediments. Game theory models examine choices under these constraints while also specifically attending to strategic interaction in which decision makers select their actions, taking into account expectations about how others will respond to them.

Brief History:

Game theory was originally developed by the Hungarian-born American mathematician John von Neumann and his Princeton University colleague Oskar Morgenstern, a German-born American economist, to solve problems in economics. In their book ‘The Theory of Games and Economic Behaviour’ (1944), von Neumann and Morgenstern asserted that the mathematics developed for the physical sciences, which describes the workings of a disinterested nature, was a poor model for economics. They observed that economics is much like a game, wherein players anticipate each other’s moves, and therefore requires a new kind of mathematics, which they called game theory.

MATHEMATICAL MODELS OF GAME THEORY:

There are three main mathematical models or forms used in the study of games, the extensive form, the strategic form and the coalitional form. These diﬀer in the amount of detail on the play of the game built into the model. The most detail is given in the extensive form, where the structure closely follows the actual rules of the game.

In the strategic form, many of the details of the game such as position and move are lost; the main concepts are those of a strategy and a payoﬀ. In the strategic form, each player chooses a strategy from a set of possible strategies. We denote the strategy set or action space of player i by Ai, for i =1 ,2,…,n. Each player considers all the other players and their possible strategies, and then chooses a speciﬁc strategy from his strategy set. All players make such a choice simultaneously, the choices are revealed and the game ends with each player receiving some payoﬀ. Each player’s choice may inﬂuence the ﬁnal outcome for all the players. We model the payoﬀs as taking on numerical values. The mathematical and philosophical justiﬁcation behind the assumption that each player can replace such payoﬀs with numerical values is discussed in the Utility Theory. We therefore assume that each player receives a numerical payoﬀ that depends on the actions chosen by all the players. Suppose player 1 chooses a1 ∈ A1, player 2 chooses a2 ∈ A2, etc. and player n chooses an ∈ An. Then we denote the payoﬀ to player j, for j =1 ,2,…,n , by fj(a1,a2,…,an), and call it the payoﬀ function for player j. The strategic form of a game is deﬁned then by the three objects: (1) the set, N ={1,2,…,n}, of players, (2) the sequence, A1,…,An, of strategy sets of the players, and (3) the sequence, f1(a1,…,an),…,fn(a1,…,an), of real-valued payoﬀ functions of the players. A game in strategic form is said to be zero-sum if the sum of the payoﬀs to the players is zero no matter what actions are chosen by the players. That is, the game is zero-sum if n  i=1 fi(a1,a2,…,an)=0 for all a1 ∈ A1, a2 ∈ A2,…, an ∈ An. In the ﬁrst four chapters of Part II, we restrict attention to the strategic form of ﬁnite, two-person, zero-sum games. Such a game is said to be ﬁnite if both the strategy sets are ﬁnite sets. Theoretically, such games have clear-cut solutions, thanks to a fundamental mathematical result known as the minimax theorem. Each such game has a value, and both players have optimal strategies that guarantee the value.

APPLICATIONS OF GAME THEORY:

Game theory has been applied to a wide variety of situations in which the choices interact to affect the outcome. In stressing the strategic aspects of decision making, or aspects controlled by the players rather than by pure chance, the theory both supplements and goes beyond the classical theory of probability. It has been used, for example, to determine what political coalitions or business conglomerates are likely to form, the optimal price at which to sell products or services in the face of competition, the power of a voter or a bloc of voters, whom to select for a jury, the best site for a manufacturing plant, and the behaviour of certain animals and plants in their struggle for survival. It has even been used to challenge the legality of certain voting systems.

It would be surprising if any one theory could address such an enormous range of subjects and in fact there is no single game theory. A number of theories have been proposed, each applicable to different situations and each with its own concepts of what constitutes a solution.

1.)    BIOLOGY:

Evolutionary game theory provides an analytic basis for explaining the emergence of subtle behaviours in animals over an evolutionary timescale.

Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled.

Evolutionary game theory has helped to explain the basis of altruistic behaviours in Darwinian evolution. It has in turn become of interest to economists, sociologists, anthropologists, and philosophers.

2.)    RESOURCE ALLOCATION AND NETWORKING:

Computer network bandwidth can be viewed as a limited resource. The users on the network compete for that resource. Their competition can be simulated using game theory models. No centralized regulation of network usage is possible because of the diverse ownership of network resources.

The problem is of ensuring the fair sharing of network resources. For example, ten students on the same local network need access to the Internet. Each person, by using their network connection, diminishes the quality of the connection for the other users. This particular case is that of a volunteer’s dilemma. That is, if one person abstains from using the network, the other people will be better off, but that person will be worse off. If a centralized system could be developed which would govern the use of the shared resources, each person would get an assigned network usage time or bandwidth, thereby limiting each person’s usage of network resources to his or her fair share. As of yet, however, such a system remains an impossibility, making the situation of sharing network resources a competitive game between the users of the network and decreasing everyone’s utility.

3.) COMPUTER SCIENCE:

There is almost an endless list of domains in computing where game theory has and will continue to gain traction. In fact, any application area involving automatic interaction and coordination of rational/intelligent agents, such as in robotics, cloud/distributed computing, spot pricing, network security, machine learning, social networks, recommendation systems and resource management

In cloud computing, game theory is used for modelling complex interactions between cloud providers–whose aim is to minimize cost while maximizing resource utilization–on one hand and  a number of service providers often with conflicting objectives of maximizing Quality of Service at minimal cost. In such a situation, a game is set up based on a utility function that will eventually steer game play towards an equilibrium state, the Nash Equilibrium, where no players could receive an incentive to change their strategy–that state where objectives of all players are balanced. There are many scenarios in cloud resource management involving spot pricing of cloud resource addressed by auction/bidding games.

4.) ECONOMICS:

Game theory is a major method used in mathematical economics and business for modelling competing behaviours 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, behavioural economics, information economics, industrial organization, and political economy.

5.)  POLITICAL SCIENCE:

The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

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