Theory of Games

INTRODUCTION

Many managerial decisions in business are made in the face of competitive action. As such, one of the factors to be considered is what competitors will dc f certain steps are taken. Reducing prices for example will not result in an increased. share of the market if competitors follow suit. Here, rather complex strategies may be required to meet competition effectively and the decision-maker is influenced rw the competitor's choice of strategy. A good deal of theoretical work has been done a deal with the situations in which a firm plays an active opponent or opponents. The area so developed is generally referred to as the Theory of Games or Game Theory- In is also called 'Decision- making under Conflicts.' The game theory originated in the late 1920s with the Hungary-1 mathematician and one of the three co-inventors of the United States hydrc bomb, John Von Neumann (1903-1957). The game theory, however, first achieved; wide audience when Neumann and Morgenstern published their book Theory Games and Economic Behaviour in 1944. The term 'games' is a generic term used to refer to conflict situations c: particular kind. In these situations, the success of one party tends to be at expense of the others; so they are in a rivalrous relationship, or in a conflict-games, the participants are competitors or contestants; they are called 'players':,, Each player of a game wants to win, if necessary, by outwitting other players player in a game is an autonomous decision-making unit. A player is "HI necessarily one person; it may be a group of individuals acting in an organizai_:n i.e., a firm.

Classifying Games The number of persons involved, that is players or opponents, is one of important criteria for classifying and studying a game. Thus, the simplest type i game is a two-person game. However, more elaborate games involving more two players are possible; such games are referred to as N-person games. Another criterion for classifying games is based on the pay-off, where pay-off ij defined as the amount won by a player as a result of the game considering winning as positive and losses as negative. If the algebraic sum of the pay-offs of competirsE! players after completion of the game is zero, the game is known as a zero-su: game. If not, the game is known as a non-zero sum game. To take an example j zero-sum game, suppose, there are two persons say, Ram and Shyam; if Ram win* Rs. 10, Shyam loses Rs. 10. In other words, Ram's winnings - Shyam's losses = 0 We may also write it as Ram's pay-off + Shyam's pay-off = Rs. 10 -10 = 0, adopting the convention that winnings are positive numbers and losses are negative numbers. Pay-off Matrix A simple way of showing a game situation is to arrange gains and losses (pay-ofs in matrix form showing who pays how much to whom under various possioiiur circumstances. The pay-off matrix shows in a tabular form the profits of the part! whose strategies are listed down on the left side of the table. The opponents strategies or acts are listed along the top of the table. The profits of the opponent! are not listed, since they are the negative values of the pay-offs shown. In the thecc of games, it is assumed that all players are aware of their own pay-off matrices artiL those of their own opponents. In other words, the pay-off matrix usually represeria-the pay-offs of the row player, ie., say, player X. The pay-offs of the column play a say, player V are just the negatives of the elements of the pay-off matrix because the zero sum nature of the game. An example of a pay-off matrix is given below ;

The above is the usual concise form in which the pay-off matrix is expressed, detailed explanation of the pay-off matrix is as follows:

X
Xwins 2 points

Y
X wins 4 points

Xwins 1 point

y wins 3 points

Thus, it can be said that if X plays strategy M and Y plays strategy 0, Xwins 2 ;; if X plays strategy M and Y" plays strategy R, X wins 4 points. If X plays Wand Y plays strategy Q, Xwins 1 point; if X plays strategy JV and Y" plays R, Y wins 3 points.

Ermine Strategy - 3hn von Neumann and Oskar Mongenstern proposed the use of the maximin D find the best strategy for each player. The maximin criterion refers to a strategy which maximises the minimum gain. The possible strategy which a decision-maker may adopt, he can assign the best or minimum possible pay-off. He would then select the strategy which yields Maximum among these minimum pay-offs, or in other words, the maximin explain, let us reproduce the game cited earlier: Maximum of the minimum. Le., maximin - ere, if X plays row 1, the minimum which he will earn is 2 (noted against row i" Again if he plays row two, the minimum which he will earn is -3 {that is a loss This again is noted against row two. So according to maximin strategy, the choose the maximum of the minimum values. That is between 2 and -3 (which .-.^ two minimum values), he shall choose 2 which is the highest or maximum. r. the game theory, it is assumed that each player plays to win. Under these -i stances, it is completely logical for the player to use the maximin criterion .:own as Wald criterion, after its exponent. _2 convention regarding the interpretation of the pay-off elements may be.virh advantage. The positive pay-off elements mean gain to player X (whose appears on the left hand side of the pay-off matrix) and loss to player Y r name appears on the top of the matrix). Similarly negative pay-off elements JDSS to player X and gain to player Y. Because of this interpretation of the _ elements, the conservative strategy recommended for player is known as the ~.:n strategy, which ensures him a minimum amount of gain which can not be incised any further irrespective of the course of the action adopted by player Y (or Hauler words, the maximum of the various possible minimum values). Similarly, the strategy recommended for player Y is known as the minimax (Minimum of the column maxima, or in other words, the minimum of the

various possible maximum values) which ensures for him a maximum loss which cannot be increased further irrespective of the course of action adopted by player X. Strictly speaking, the two terms are interchangeable. However, the convention is to use the term maximin strategy because the game is generally viewed from the standpoint of player X, though sometimes the term minimax is also used (which is the strategy from Y*s viewpoint).

Saddle Point The first step to solve a game is to look for the saddle point in the pay-off matrix. By solving the game, we mean determining the optimum strategies for both the players and the value of the game. By optimum strategy, we mean such strategy which maximizes the gain of one player and/or minimizes the loss of the other under all circumstances; any deviation from this strategy results in a decreased pay-off for the player. In other words, a course of action or plan which puts the player in the most preferred position irrespective of the strategy of his rivals is called an Optimal Strategy. Further, value of the game is the expected pay-off of play when all the players of the game follow their optimal strategies. When saddle point exists, the game is solved and complex calculations to determine optimum strategies and game value are no more necessary. If, however, a saddle point is not present, somewhat complex calculations become necessary to solve the game. Many pay-off matrices do not possess any saddle point. A saddle point of a pay-off matrix is that position in the matrix where the maximum of row minima coincide with the minimum of the column maxima. The pay-off at the saddle point is called the value of the game, and is obviously equal to the maximin and minimax values of the game. A saddle point, it must be remembered, is a point or position in the matrix where both the smallest numerical value in its row and the largest value in its column coincide. The name saddle point is on the analogy of a horse's saddle where the point P is both the maximum point on UU and the minimum point on the curve W (see Diagram). A saddle point is also known as equilibrium point. If a player's maximin strategy consequence is the same as the other player's minimax strategy consequence, then the consequence is naturally an equilibrium point. For it possesses, an element of stability in the sense that both players are satisfied at having followed their strategy successfully.

Rules for Determining a Saddle Point

X 2

7 1

After following rules 1 and 2, we get the following

So, row 1 and column 2 represents the saddle point.