Use of Graphical Method For Solving Game Without Saadle Point

The techniques of developing optimal strategies for dealing with conflicting and competitive situations (whenever these conflicts can be expressed in mathematical terms) have been termed as game theory. Key terms associated with the game theory [2, 3]- (i) Strategy - A strategy is a comprehensive plan of action, formulated by a player (an interested and active party in the game), who is well informed of all the alternatives available to him and to his adversary (competing player). A strategy can be good or bad. The only requirement is that it should be complete and cover all the possibilities. (ii) Finite game - when the total number of possible strategies in a game is finite, it is called a finite game. In the other situation, the game is an infinite game. (iii) Zero-sum game - Zero-sum games are those games in which one player gains exactly the same amount, which the other player(s) loose so that their net gains are equal to zero. (iv) Non zero-sum game - Zero-sum games are those games in which gain of one player is not necessarily equal to the loss of the other or vice versa. (v) Pay-off (game) matrix - A pay-off matrix is a tabular representation of the pay-offs of one competitor, which are associated with his strategies in response to the strategies of the other player.

Consider two players A and B playing a zero-sum game. Let A has m strategies numbered A1, A2… Am available to him and B has n strategies numbered B1, B2… Bn, available to him. Let the gain of A, when he chooses ith strategy in response to the jth strategy chosen by B be given by gij. Then the pay- offs ofA can be represented as follows: The matrix G is called the pay-off matrix of player A. If gij > 0, A has gained and if gij < 0, then A has lost an amount gij. Since the game is a zero-sum game, so whatever is the gain of A is loss of B.

possible strategies. For developing such criteria, we make some assumptions: (i) The number of players in the game is (in general) finite. (ii) The interests of the players clash and each player is choosing his strategy solely for his welfare. (iii) Each player is well aware of all the strategies available to him and to his opponents. (iv) All the players are making their moves simultaneously, without knowing the choices, which the other players have made. (v) The outcome of the game depends upon the moves made by different players; and (vi) All the players are rational players.

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. 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 pay off to the two players at the end of each play is zero.

In general, the games are zero-sum games. For the simplicity of presentation, we assume that the games are two- players’ games. We define some terms associated with the solution of the games.

Consider the following game (G1) This game matrix suggests that the two players A and B are playing a game with A having two (viz. A1and A2) and B having three (viz. B1, B2 and B3) strategies. In this case A would always opt for the strategy A1, as it would yield him better pay-offs than the pay-offs yielded by the strategy A2. We say that A1 is a dominant strategy. A strategy is said to be a dominant strategy if it always yields better (or at least equal) pay-offs than the other strategies irrespective of the strategies opted by the other player(s), i.e., superior strategies (resulting in higher pay-offs) dominate the inferior ones (resulting in lower pay-offs). In such situations, inferior strategies can always be strike off.

The algebraic method is a direct attempt to solve unknowns although this method becomes quite lengthy when there are more strategies (courses of action) for players. Such large games can be solved first by transforming the problem into a linear programming problem and then solving it by the simplex method on an electronic computer [8]. First, suppose that all inequalities given hold as equations. Then solve these equations for unknowns. Sometimes equations are not consistent. In such cases, one or more of the inequalities must hold as strict inequalities (with ‘>’ or ‘<‘signs). Hence, there will be no alternative except to rely on trial-and-error method for solving such games. Following important theorems will be helpful in making the computations easier.

In this paper we found that, the algebraic method is a direct attempt to solve unknowns although this method becomes quite lengthy when there are more strategies (courses of action) for players. Such large games can be solved first by transforming the problem into a linear programming problem and then solving it by the simplex method on an electronic computer.

1. Swarup K., Gupta P.K. and Mohan M. (2001), Operations Research, Sultan Chand and Sons, Delhi. 2. Watson J., (2002), Strategy: An Introduction to Game Theory, W.W. Nortan & Company 3. Osborne M.J., (2001) An Introduction to Game Theory, Oxford University Press. 4. http://www.egwald.ca/operationsresearch/gameintroduction.php 5. http.//www.univeralteacherpublications.com. univ/eBooks/or/Ch9/gamegrap.htm 7. Operation_Research.pdf by EIILM University 8. Mokhtar S. Bazaraa, John J. Jarvis, and Hanif D. Sherali, Linear programming and network flows, Wiley-Interscience, 2004.