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.


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Abi Ainscough Profile
Abi Ainscough answered
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.
Maria Anna Profile
Maria Anna answered

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Amanda Wells Profile
Amanda Wells answered
The game is another name for strategy and trying to gain an advantage. In this sense 'playing games' is all about trying to defeat an opponent through tactics and exploiting their weaknesses, minimising their chances of success etc. You may find useful ideas here - there are also some good articles on line, which you can access by putting the words ,game theory minimax maximin' in a search engine.

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