Evolution of Game Theory

Game theory has its genesis with the publication of The Theory of Games and Economic Behavior by refugees John Von Neumann, a Hungarian mathematician, and Oskar Morgenstern, an Austrian economist, at Princeton. They took any economic problem, formulated it into a game, found a theoretical solution and linked the solutions back to the original problem. Though the original volume has less relevance in today’s world, the development of “prisoner’s Dilemma” and Nash’s work on equilibrium laid the foundations of non-cooperative game theory. By early 1950, all the game theory useful for the economists in the time to come has been developed. Game theory has been successful in recent years because of its practical application in almost all the spheres of life.

What is Game Theory?

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Game Theory is the study of multipurpose decision problems. It is concerned with the actions of the individuals who are conscious that their actions affect each other. It is the extension of the individual rational decision making to the broader arena of decision making of the organization due to the presence of significant interactions among the players. It is an interactive decision making tool. Game theory is the mathematical study of human interactions described by rules of play and alternative choices. Situations economists and mathematicians call games psychologists call social situations. While game theory has applications to “games” such as poker and chess, it is the social situations that are the core of modern research in game theory. Game theory has two main branches: Non-cooperative game theory models a social situation by specifying the options, incentives and information of the “players” and attempts to determine how they will play. Cooperative game theory focuses on the formation of coalitions and studies social situations axiomatically.

Game theory pertains to the conception of a complete model of a game situation, involving the following precise ‘restraining assumptions’

Rationality

Game theory assumes the rational behavior by all the players, that the players are perfect calculators and flawless followers of their best strategies. A consistent ranking will be given to all the possible out comes and the strategies are chosen accordingly. The rationality can be said to be in two different counts. Firstly, that the complete knowledge of his own interest exists. Secondly, meticulously calculated actions of what interests serve best are considered.

Common knowledge of Rules

Another important assumption is that the players will have some level of common understanding of all the rules of the game.

Some Terminology in Game Theory

Strategy

Strategies are simply the choices available to the players. It specifies how players play the game in every contingency. The word strategy generally denotes a longer term or larger scale plan of action. These are the decisions of players as a plan of action in response to evolving circumstances.

Dominant strategy

A strategy is a dominant strategy if it is a player’s strictly best response to any strategies the other players might pick resulting in the highest payoff than every other player. The player’s best strategy may not depend on the choice made by other players in the game and hence it becomes the dominant strategy. Nash equilibrium exists when one player has a dominant strategy, letting all other players to respond with their best alternatives. There are two types of dominant strategies:

* Strictly or strongly dominant strategy outperforms (in its payoffs) all other strategies, no matter what any opposing players do.

* Weakly dominant strategy is at least as good as all other strategies no matter what any opposing players do.

Dominated strategy

It is a strategy that yields lower payoff than any other strategy no matter what other players in the game do. When there is a dominated strategy the game gets simplified because one player will always use a dominant strategy.

Pure Strategy

Each participant selects one course of action.

Mixed Strategy

The alternatives of strategies are randomly mixed. The players mix various strategies to maximize the payoff. It is the probability distribution over or some of all the strategies available to the player

Payoffs

The objective of any player in a game is “To Win”. But many times the margin of victory matters; In business and social contexts there are few games that are purely zero sum or win-lose games. A player’s payoff is simply the number assigned by her ordinal utility function to the state of affairs corresponding to the outcome in question. So all conceivable outcomes of the game corresponding to the strategies they use will be numerically scaled and each number associated with each possible outcome will be called as player’s payoff.

Nash Equilibrium

The main underpinning of game theory is the Nash equilibrium proposed by John Nash in 1951, which describes a solution set for non-zero sum games in which no player has an incentive to deviate from best strategy, given that the other players don’t deviate from their best strategy. Nash equilibrium exists when each player strategy is a best response to the other player’s strategy.

The other terms are explained in the section “Battle of the Sexes” for better understanding.

Cooperative and Non Co-operative Games

The difference between the two games lies in their modeling approach. Both the games have rules, but they differ in the kinds of solutions concepts use. Cooperative games are axiomatic, and they generally appealing to Pareto-optimality, fairness and equity while the non cooperative games have the economic flavor where each player would like to maximize his or her own utility.

Co-operative Games

It is a game in which the players can make binding commitments with negotiating and enforcing agreements to a particular strategy

Non Co-operative Games

A non cooperative game is a game where absolutely no pre-play communication is permitted between the players and in which players are awarded their due payoff according to the rules of the game. Negotiation with other players is not possible for entering into some form of binding agreement. It can result in outcome, which is undesirable for participants as well as society. The common example of such type of game is Prisoners’ Dilemma.

Zero-Sum and Non-Zero-Sum Games

A zero-sum game is a game in which the sum of the payoffs of all the players is zero, whatever strategies they choose. A game, which is not zero-sum, is a non-zero-sum game. In a zero-sum game, what one-player gains, another player must lose.

Three classic equilibrium games

The Prisoner’s Dilemma

The two cooperating players have to decide whether to stop cooperating in exchange for an individual ‘pay-off’.

Chicken Game

The two competing players refuse to cooperate even though otherwise the ‘pay-off’ may be a fatal one

The Battle of the Sexes

The two competing players have to cooperate in order to bring about a mutually satisfactory ‘pay-off’

Battle Of The Sexes

There are many numerous situations in institutions and industries where they want to achieve a common objective but differ in the mechanics of achieving it. For example, two firms trying to set up some common standards or take some measures but disagree over how to go about it. This type of situation is characterized by a common objective but divergent views or ways of how to attain that objective. Battle of the sexes is one of the coordination games, which explain such situations by drawing a parallel between such situations with the predicament faced by a man and woman going on a date but prefer two different places. As mentioned before, it comes under the category of cooperation games, which have got many practical applications in our day-to-day life.

In the language of Game Theory, a cooperative game is a game in which the players can make binding commitments. These games allow the players to share the benefits from cooperation by making transfers among themselves, which would leave them better off. Because of the existence of interpersonal framework; players come together and through cooperation achieve the most favorable outcome for all the players. Games, which replicate such situations, are called negotiated games because the outcome is reached through negotiations and deliberations. And the outcome, which is the result of such negotiation, is called negotiated settlement. This game is the classic example of how cooperation can be achieved even when people are selfish. It shows how commonality of the objective can resolve conflict.

Battle of the sexes illustrates the conflict between a man who wants to go to a prizefight and a women who wants to go to a ballet. Though selfish, they are deeply in love and would, if need arises, sacrifice their preferences in order to be with each other. Here cooperation, not rivalry, works. This game has two Nash Equilibria, one of which is a strategy combination. Given that the man chooses prizefight, even the women chooses the same. If the woman chooses the ballet so does the man. Hence (Prize Fight, Prize Fight) and (Ballet, Ballet) are the two Nash Equilibria.