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What is a `game’ in game theory? What are the properties of a game? Explain the ``best strategy’’ on the basis of minimax criterion of optimality. Describe the maximin and minimax principles of game theory

Question: What is a `game’ in game theory? What are the properties of a game? Explain the ``best strategy’’ on the basis of mini max criterion of optimality. Describe the maximin and minimax principles of game theory

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In Game theory, game is a study of strategic decision making. More formally, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory. Game theory is mainly used in economics, political science, and psychology, as well as logic and biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant(s). Today, however, game theory applies to a wide range of class relations, and has developed into an umbrella term for the logical side of science, to include both human and non-humans, like computers. Classic uses include a sense of balance in numerous games, where each person has found or developed a tactic that cannot successfully better his results, given the other approach.

A model of optimality taking into consideration not only benefits less costs, but also the interaction between participants

Game theory attempts to look at the relationships between participants in a particular model and predict their optimal decisions.

One frequently cited example of game theory is the prisoner's dilemma.

Suppose there are two brokers accused of fraudulent trading activities: Dave and Henry. Both Dave and Henry are being interrogated separately and do not know what the other is saying. Both brokers want to minimize the amount of time spent in jail and here lies the dilemma. The sentences vary as follows:

1) If Dave pleads not guilty and Henry confesses, Henry will receive the minimum sentence of one year, and Dave will have to stay in jail for the maximum sentence of five years.

2) If nobody makes any implications they will both receive a sentence of two years.

3) If both decide to plead guilty and implicate their partner, they will both receive a sentence of three years.

4) If Henry pleads not guilty and Dave confesses, Dave will receive the minimum sentence of one year, and Henry will have to stay in jail for the maximum five years.

Obviously, pleading guilty is the most attractive should the other plead not guilty since the sentence is only one year. However, if the other party also chooses to plead guilty, both will have to serve three years. On the other hand, if both parties plead not guilty, they'd have to serve two years in jail. Consequently, the risk of pleading not guilty is a five-year sentence, should the other choose to confess.

A situation is termed a game when it possesses the following properties:

1. The number of competitors (players) is finite.

2. There is a conflict in interests between the

competitors.

3. Each of the competitors has generally a finite set of

possible courses of action.

4. The rules governing these choices are specified and

known to all players, a play of the game results when

each of the players chooses a single course of action

from the list of courses available to him.

5. The outcome of the game is affected by the choices

made by all the players.

6. The outcome for all specific set of choices by all the players is known in advance

and numerically defined.

Game theory is a type of decision theory which is based on reasoning in which the

choice of action is determined after considering the possible alternatives available to

the opponents playing the same game. Game theory has been projected as a

scientific approach to rational decision making, and rightly so.

Within game theory, there is a whole list of games that are classified and studied. These games are then used to study the interaction between individuals who are taking part in them. There are several common features that can be found across all of these games that include the number of players, the strategies per player, the number of pure Nash equilibria, sequential game, perfect information and constant sum.

The number of players is categorised by whether the participant in a game make their own choice or if they receive a payoff because of the choices made by another participant. In both of these circumstances, the participant is are available for the player to choose from. Sometimes this list of strategies will be the same for every player, if this is the case it will be listed at the start of the game. The number of pure Nash equilibria is the number of sets of strategies that represent the best mutual responses to other strategies available. This means that if a player is only using pure strategy, there are a number of Nash equilibria available. Players will have no incentive to change their strategy if every player is playing their part of a Nash equilibrium. The sequential game is one where each player performs his or her actions one after another. If the game is not played this way it is considered as a simultaneous move game. A game is classed as having perfect information if it is a sequential game and each player is aware of the strategy that the player that preceded him or her has used. Finally, a game is considered constant sum if the sum of the payoffs of every player are the same for every set of strategies that are available. This means that in a constant sum game a player can only gain if another player loses.

The minimax theory is a decision rule that is used within game theory to minimize the possible loss while maximizing the potential gain. It was originally formulated for the two player zero-sum game theory where it covers both when players move simultaneously and alternatively. Since it has been expanded to cover more complex games and to cover to decision making involved with uncertainty. The 'best strategy' in these circumstances is based on a rule. It is known as the criterion of optimality because players are expected to be rational in their approach. The player will list the possible outcomes for their action and choose the best action to achieve his or her objectives. This criterion of optimality is described as maximin for the maximizing player and minimax for the minimizing player. The maximin criteria within this principle sees that the maximizing player lists his minimum gains from each strategy and selects the strategy which gives the maximum out of these minimum gains. In comparison, the minimax criteria consider that the minimising player lists his maximum loss from each strategy and then selects the strategy that gives him the minimum loss from these maximum losses.

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