A Brief Introduction To Game Theory & Nash Equilibrium

Abstract

The aim of this term paper is to introduce game theory and Nash equilibrium. The concepts discussed in this paper are very preliminary but nevertheless are very important and are very necessary for the higher study of game theory. We will be running through only one example for our whole paper : The Prisoners Dilemma, so that we can illustrate the concepts better. At the end of our paper, we will discuss two more examples.

Sec. 1 What is game theory?

Game theory is formally defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. In game theory, we study games which are formally defined as competitive activities where each competitor employs strategies to maximize their utility/payoff under a set of rules. In games, strategic interaction between various players takes place. Each player in a game has a possible set of choices or strategies, out of which he chooses one strategy while playing.

Examples of various games are Prisoner’s Dilemma, Duopoly Market etc.

Sec. 2 Basic Terminologies in Game Theory

To better illustrate the concepts involved we will be giving the example of Prisoner’s dilemma. We will be introducing the prisoner’s dilemma here.

2.1 Prisoner’s dilemma

In this game, there are two prisoners whose aim is to minimize the years of imprisonment. They have committed a crime jointly. Each prisoner is interviewed separately and there are not any contacts whatsoever between them. They decide individually to confess or deny the crime taking into account possible decisions of the other prisoner (strategic game).

Each prisoner chooses his dominant strategy, that is the behaviour giving the best result regardless of the decision of theOther prisoner.

The first number shows the years imprisonment of A, the second number of B. If for example A confesses and B denies, A gets 1 year imprisonment and B gets imprisonment of 4 years (field at the top right).Now, with the help of this example we will introduce some basic terms in game theory.

Sec. 2.2Theory of rational choice The basic idea of rational choice theory is that patterns of behaviour in societies reflect the choices made by individuals as they try to maximize their benefits. In game theory, we will assume that every individual makes a rational choice. In the case of prisoner’s dilemma this is also followed.

Strategic games As we have discussed above, in every game participant has to choose a strategy from the set of strategies available. For this sole reason, we call games in game theory as Strategic games. Prisoners dilemma is hence a strategic game.Every strategic game must have these components:

1. Set of players The participants or decision makers in a strategic game define the set of players. For example in Prisoners dilemma, there are two players prisoner A and prisoner B. They form the set of players.

2. Action Sets An action set Ai for a player i is defined as the set of choices/strategies available to player i. In the above example of prisoners dilemma every player has two choices either to confess or deny. If we represent confess by C and deny by D, then

Ai = {C,D} i=1,2 in the case of prisoners dilemma.

3. Action Profile/Strategy Profile In a general N-player game, the action profile A is defined as

A = (a1,a2,…………,an)Where ai belongs to Ai.In the case of prisoners dilemma, there are four possible action profiles B1,B2,B3,B4

B1 = {C,C}B2 = {C,D}B3 = {D,C}B4 = {D,D}

4. Set of Outcome Superset of action profiles is called Set of outcome.O = Ai

In the case of prisoners dilemma, the set O is defined asO = {B1,B2,B3,B4}Where B1,B2,B3,B4 are defined above.

5. Utility/Payoff of players Every player in a game corresponding to an action profile has some payoff/utility associated with it. For player i, this is denoted by Ui(ai,a-i)Where ai = strategy of player i a-i = strategy of all other players except i

In the case of prisoners dilemma, the utility for any action profile for any prisoner can be defined as the negative of years of

imprisonment one receives. So utility for prisoner A in the case of various action profiles will be U1 (C,C) = -3 U2 (C,D) = -1 U3 (D,C) = -4 U4 (D,D) = -2 Similarly, we can find utility for prisoner B also.

Remark Utility of players in game theory is taken to be ordinal i.e. we are only interested in the ranking of various action profiles for a player i according to utility gained by player i after various strategies are employed by players.

6. Payoff Matrix A payoff matrix of a game is a specification of players strategy spaces and payoff functions. A strategy space is nothing but the set of outcomes. We list payoffs for players according to various action profiles in a payoff matrix.

The payoff matrix for prisoners dilemma is given above.Remark We assume in game theory that actions by players are taken simultaneously i.e. they don’t have any modes of communication to communicate with each other.