Game theory questions and solutions

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Game Theory : An Introduction Game Theory helps us understand situations in which decision-makers interact. It encompasses a wide range of applications some of which are listed below: firms competing for business Arms race between the countries political candidates competing for votes animals fighting over prey bidders competing in an auction penalty kicks in a football game And the list goes on…..

Like other sciences, game theory consists of a collection of models. These models provide insights when interactions affect incentives. Theory of Rational Choice The theory of rational choice is a component of many models in game theory. By assuming a decision maker to be rational, according to this theory a decision-maker chooses the best action among all the actions available to her. In any given situation the decision-maker is faced with a subset of A, from which she must choose a single element.

If the decision-maker prefers the action a to the action b, and the action b to the action c, then she prefers the action a to the action c. Payoff function associates a number with each action in such a way that actions with higher numbers are preferred. A person is faced with the choice of three vacation packages, to Havana, Paris, and Venice. She prefers the package to Havana to the other two, which she regards as equivalent. Her preferences between the three packages are represented by any payoff function that assigns the same number to both Paris and Venice and a higher number to Havana.

It is a relative measure and just tells the order of preferences of actions in a set. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other defects. If they both stay quiet, each will be convicted of the minor offense and spend one year in prison.

If one and only one of them defects, she will be freed and used as a witness against the other, who will spend four years in prison. If they both defect, each will spend three years in prison. States where the game has ended are called terminal states.

A utility function also called an objective function or payoff function , which gives a numeric value for the terminal states. Furthermore, given a choice, MAX will prefer to move to a state of maximum value, whereas MIN prefers a state of minimum value.

The minimax algorithm computes the minimax decision from the current state. It uses a simple recursive computation of the minimax values of each successor state, directly implementing the defining equations.

The recursion proceeds all the way down to the leaves of the tree, and then the minimax values are backed up through the tree as the recursion unwinds. What is the complexity of minimax algorithm?