GAME THEORY

Competition is the watch word of modern life. We say that a competitive solution exists if two or more individuals make decisions in a situation that involves conflicting interests, and in which the outcome is controlled by the decision of all the concerned parties. A competitive situation is called a game. The term represents a conflict between two or more parties. A situation is termed a game when it possesses the following properties.

(i) The number of competitors is finite.

(ii) There is conflict in interests between the participants.

(iii) 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 available to him.

(iv) Each of the participants has a finite set of possible courses of action.

(v) The outcome of the game is affected by the choices made by all the players.

(vi) The outcome for all specific set of choices by the entire player is known in advance and numerically defined.

The outcome of a play consists of a particular set of courses of action undertaken by the competitors. Each outcome determines a set of payments (+ve, -ve, or zero) one to each competitor.

DEFINITION

The term ‘strategy’ is defined as a complete set of plans of action specifying precisely what the player will do under every possible future contingency that might occur during the play of the game, i.e., strategy of a courses of action. Strategy can be classified as:

(i) Pure strategy

(ii) Mixed strategy

A strategy is called pure if one knows in advance of the play that is certain to be adopted irrespective of the strategy the other player might choose.

The optimal strategy mixture for each player may be determined by assigning to each strategy, its probability of being chosen. These strategies so determined are called mixed strategy because they are probabilistic combinations of the available choices of strategy. Mixed strategy is denoted by the sets

PAYOFF

Payoff is the outcome of playing the game. A payoff matrix is a table showing the amount received by the player named at the left hand side after all possible plays of the game. The payment is made by the player named at the top of the table.

If a player A has m-courses of action and player B has n-courses, then a payoff matrix may be constructed using the following steps.

(i) Row designations for each matrix are the course of action available to A.

(ii) Column designations for each matrix is the course of action available to B.

(iii) With a two person zero sum game, the cell entries in B’s payoff matrix will be the negative of the corresponding entries in A’s payoff matrix and the matrices will be as shown below.

TYPES OF GAMES

1. Two-person games and n-person games.

In two person games, the players may have many possible choices open to them for each play of the game but the number of player’s remains only two. Hence it is called a two person game. In case of more than two persons, the game is generally called n-person game.

2. Zero sum game.

A zero sum game is one in which the sum of the payment to all the competitors is zero for every possible outcome of the game in a game if the sum of the points won equals the sum of the points lost. I.e.

3. Two person zero sum game

A game with two players, where the gain of one player equals the loss to the other is known as a two person zero sum game. It also called rectangular form. The characteristics of such a game are.

(i) Only two players participate in the game.

(ii) Each player has a finite number of strategies to use.

(iii) Each specific strategy results in a payoff.

(iv) Total payoff to the two players at the end of each play is zero.

THE MAXIMIN-MIINMAX PRINCIPLE

This principle is used for the selection of optimal strategies by two players. Consider two players A and B. A is a player who wishes to maximize his gain while player B wishes to minimize his losses. Since A would like to maximize his minimum gain, we obtain for player A, the value called maximize value and the corresponding strategy is called maximize strategy.

On the other hand, since player B wishes to minimize his losses, a value called the Minimax value which is the minimum of the maximum losses is found. The corresponding strategy is called the minimax strategy. When these two are equal, the corresponding strategies are called optimal strategy and the game is said to have a saddle point. The value of the game is given by the saddle point.The selection of maximin and minimax strategies by A and B is based upon the so-called maximin –minimax principle which guarantees the best of the worst results.

SADDLE POINT

A saddle point is a position in payoff matrix where the maximum of row minima coincides with the minimum of column maxima. The payoff at the saddle point is called the value of the game.

1. Solve the game whose pay off matrix is given by

2. Determine which of the following two people zero sum games are strictly determinable and fair. Give the optimum strategy for each player in the case of strictly determinable games.

(a)

GAMES WITHOUT SADDLE POINTS (MIXED STRATEGIES)

A game without saddle point can be solved by various solution methods.

2*2 games without saddle point

Consider a 2*2 two-person zero sum game without any saddle point having the payoff matrix for player A.

6. In a game of matching coin with two players ,suppose A wins one unit of value when there are two heads, wins nothing when there are two tails and losses ½ unit of value when there are one head and one tail. Determine the payoff matrix, the best strategies for each player and the value of game to A.

Graphical method for 2*n or m*2 games

Consider the following 2*n games

Such that ,Now for each of the pure strategies available to B, expected payoff for player A would be as follows.

Dominance property

Sometimes it is observed that one of the pure strategies of either player is always inferior to at least one of the remaining ones. The superior strategies are said to dominate the inferior ones. In such cases of dominance, we reduce the size of the payoff matrix by deleting those strategies which are dominated by others. The general rules for dominance are:

(i) If all the elements of a row, saw row, are less than or equal to the corresponding elements of any other row say row, then row is dominated by the row.

(ii) If all the elements of a column, say column, are greater than or equal to the corresponding elements of other column, say row, then column, then column is dominated by the column.

(iii) Dominated rows and columns may be deleted to reduce the size of the pay-off matrix as the

optimal strategies will remain unaffected.

(iv) If some linear combinations of some rows dominates row, then the row will be deleted. Similar arguments follow for column.

Problem

9. Using the principle of dominance, solve the following game.

12. A labour union of a firm is negotiating a new 5-years settlement regarding payments with management. The option the union has are ‘ : aggressive bargaining ‘,’ : bargaining with reasoning’ and ‘ : conciliatory approach’. The likely mode of response from the management are ‘ : aggressive bargaining’,’ : bargaining with reasoning’,’ : legalistic approach’ and ‘ : conciliatory approach’. The gains to the union in each case are as follows.

What strategy would you suggest for the two sides? What is the value of the game?