Game Theory : An Introduction
Game Theory helps us understand situations in which decision-makers interact. A game in the everyday sense—“a competitive activity . . . in which players contend with each other according to a set of rules. 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.
Basic Ingredients of a Model
are the ones who make the decisions in a game/model.
A set A consisting of all the actions that, under some circumstances, are available to the decision-maker, and a specification of the decision-maker’s preferences. 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. This preference is decided on the basis of “payoffs/utilities”. Payoff function associates a number with
each action in such a way that actions with higher numbers are preferred.
Let “u” be a payoff function, “a” & “b” be two actions in set A, then action “a” is preferred over “b” if (and only if) u(a) >u(b)
If u(a)=u(b) then the player is said to be indifferent.
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. For example, we can set u(Havana) = 1 and u(Paris) = u(Venice) = 0, or u(Havana) = 10 and u(Paris) = u(Venice) = 1, or u(Havana) = 0 and u(Paris) = u(Venice) = −2.
A high payoff doesn’t convey in any way the intensity of preference.
It is a relative measure and just tells the order of preferences of actions in a set.
Apart from the above there exists other ingredients (information, strategies) depending upon the model being followed.
Different Forms of Games
Normal Form
Extensive Form
Repeated Games
Bayesian Games
In this tutorial we will cover the Normal Form.
Normal Form
Consists of:
• a set of players
• for each player, a set of actions
• for each player, preferences over the set of action profiles.
Example 1: The Prisoner’s Dilemma
Two suspects in a major crime are held in separate cells. 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.
Players: The two suspects.
Actions: Each player’s set of actions is {Quiet, Defect}.
Preferences Suspect 1’s ordering of the action profiles, from best to worst, is
(Defect, Quiet) (he defects and suspect 2 remains quiet, so he is freed), (Quiet,
Quiet) (he gets one year in prison), (Defect, Defect) (he gets three years in prison),
(Quiet, Defect) (he gets four years in prison). Suspect 2’s ordering is (Quiet, Defect),
(Quiet, Quiet), (Defect, Defect), (Defect, Quiet).
All of the above data can be represented in tabular form.
