7. Two-Person, Zero-Sum GameA game is said to be zero-sum if for any outcome, the sum of the payoffs to all players is zero. In a two-person zero-sum game, one players gain is the other players loss, so their interests are diametrically opposed.

In game theory, a game with only a few strategies can be easily represented by a matrix showing the payoff for each player along with the strategy they use (Brook, 2007). This can be represented in the form of zero-sum games where there are two players and every set of payoffs adds to zero. According to (Binmore, 2007), zero-sum game is a mathematical representation of a situation in which a participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participant. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero, thus its called a zero-sum game. This can be represented in the form of zero-sum games where there are two players and every set of payoffs adds to zero. Zero sum games can also be viewed as a closed system, meaning everything that someone wins must be lost by someone else (Brook, 2007). (Turocy & Von Stengel, 2001) outlined that the extreme case of players with fully opposed interests is demonstrated in the class of two player zero-sum games. The theory of von Neumann and Morgenstern is mostly applied in games such as two-person zero-sum games, that is games with only two players in which one player wins what the other player loses. Mathematical description of the zero-sum games implies when a two-person zero-sum game, the payoff function of Player II is the negative of the payoff of Player I (Turocy & Von Stengel, 2001).

Two Person Zero-Sum Game - Prepared by Nazmul Hasan

7.1. Strategic form of Two-Person, Zero-Sum Game For a two-person zero-sum game, the payo function of Player II is the negative of the payo of Player I, so we may restrict attention to the single payo function of Player I, which is called here A.

The strategic form or the normal form, of a Two-Person Zero-Sum game is given by a triplet

(X, Y, A), where

(1) X is a nonempty set, the set of strategies of Player I

(2) Y is a nonempty set, the set of strategies of Player II

(3) A is a real-valued function dened on X Y .(Thus, A(x, y) is a real number for every x X and every y Y )

The interpretation is as follows. Simultaneously, Player I chooses x X and Player II chooses y Y , each unaware of the choice of the other. Then their choices are made known and I wins the amount A(x, y) from II. Depending on the monetary unit involved, A(x, y) will be cents, dollars, pesos, beads, etc. If A is negative, I pays the absolute value of this amount to II. Thus, A(x, y) represents the winnings of I and the losses of II.

Saddle Point:

In a two-person zero-sum game (X,Y,A), a pair of actions (x, y) is a saddle point if

A(x, y) = max x' X A(x', y) = min y' Y A(x, y'). (Duersch, Oechssler, & Schipper, 2010) In other words, an outcome in a matrix game is called a Saddle Point if the entry at that outcome is both less than or equal to any in its row, and greater than or equal to any entry in its column.

Ilustration of saddle point:

As per the conditions for saddle point stated above, the saddle point for the illustrative example is (x3,y1). At (x3,y1), A is maximum compared to (x1,y1) and (x2,y1) which satisfies A(x, y) = maxx' X A(x', y). Again at (x3,y1) A is minimum compared to (x3,y2) and (x3,y3) which satisfies A(x, y) = miny' Y A(x, y'). A Two-Person, Zero-Sum Game may involve one of the following cases:

(i) single saddle point

(ii) more than one saddle points

(iii) no saddle point.

Saddle point can be obtained easily with the help of Minimax Theorem stated by John von Neumann.

7.2. Minimax Theorem A two-person zero-sum game (X, Y, A) is said to be a nite game if both strategy sets X and Y are nite sets. The Minimax Theorem is a fundamental theorem of game theory stated by John von Neumann (1928) in his paper Zur Theorie Der Gesellschaftsspiele, for the situation encountered in two-person zero-sum games.

For every nite two-person zero-sum game-

(1) there is a number V , called the value of the game,

(2) there is a mixed strategy for Player I such that Is average gain is at least V no matter what II does,

(3) there is a mixed strategy for Player II such that IIs average loss is at most V no matter what I does.

If V is zero we say the game is fair. If V is positive, we say the game favors Player I, while if V is negative, we say the game favors Player II.

Application of minimax theorem in solving Two-Person Zero-Sum Games:

7.2.1 Two-Person, Zero-Sum Game with single saddle point:

Since, Maximin value = Minimax value, the game has a saddle point. Thus, it can be solved by pure strategy. Thus optimal strategy for Player 1 is A2 and for Player 2 is B3. The value of the game is 1 and the game is not fair since a game that has a value of 0 is said to be a fair game.

7.2.2 Two-Person, Zero-Sum Game with two saddle points:

Since, there are two Maximin values being equal to one Minimax value, the game has two saddle points, at A1-B1 and A3-B3 positions. Thus optimal strategy for Player 1 is either A1 or A3 and for Player 2 is B1. The value of the game is 3 and the game is not fair since a game that has a value of 0 is said to be a fair game.

7.3. Dominance in Two-Person, Zero-Sum Game Since all players are assumed to be rational, they make choices which result in the outcome they prefer most, given what their opponents do. In the extreme case, a player may have two strategies A and B so that, given any combination of strategies of the other players, the outcome resulting from A is at least as good as the corresponding outcome in B, and at least one outcome in A is strictly better than the corresponding outcome in B. Then strategy A is said to dominate strategy B and strategy B is called dominated strategy.

Dominance Principle: A rational player would never play a dominated strategy.

Solving Two-Person, Zero-Sum Game by proporty of Dominance:

Here, column B4 is dominated by columns B1 and B2.

Hence, by Dominance Principle column B4 is eliminated.

Since, rows A1, A2 and A4 all are dominated by row A3.

Hence, by Dominance Principle rows A1, A2 and A4 all are eliminated.

Since, column B1 and B3 is dominated by column B2.

Hence, by Dominance Principle column B1 and B3 are eliminated.

In addition, it can be illustrated that "The Minimax Criterion" would yield the same result.

7.4. Two-Person, Zero-Sum Game with no saddle point:

Since, Maximin value Minimax value, the game has no saddle point. Thus, it can't be solved by pure strategy, rather it need to be solved by mixed strategy.

A mixed strategy problem can be solved by Graphical method (El-Kareem, 2010). But, for that purpose, initially we need to apply Dominance property to eliminate any Dominated Strategy if there is any.

In our example, row A3 is dominated by row A2.

Hence, by Dominance Principle row A3 is eliminated.

Again, column B3 is dominated by column B2.

Hence, by Dominance Principle column B3 is eliminated.

Probability distribution of different strategies for Player 1 and Player 2:

For each of the pure strategies available to Player 2, the expected payoff for player 1 will be :

(El-Kareem, 2010)

7.4.1 Graphical Solution to obtain Mixed Strategy Probabilities and Expected Payoff:

From graph: x2 = 1217

Therefore, x1 = 11217

= 517

Expected value : 10 10 ( 1217

) = 5017

2 + 7 ( 1217

) = 5017

Finding optimal mixed strategy for Player 2:

Solving the equations: 10 1 + (2) y2 = 5017

0 1 + 5 y2 = 5017

we get, y1 = 717

; y2 = 1017

Thus, the optimal mixed strategy for Player 1 (x1, x2, x3) = ( 517

, 1217

, , 0 ) and the optimal mixed strategy for Player 2 (y1, y2, y3) = ( 7

17 , 1017

, 0 ). The expected value or, the value of game, V = 50

17 .

7.4.2 Another approach to find the Mixed Strategy Probabilities:

3. Definition of Games4. Dominance5. Nash Equilibrium6. Mixed Strategy7. Two-Person, Zero-Sum Game7.1. Strategic form of Two-Person, Zero-Sum Game7.2. Minimax Theorem7.3. Dominance in Two-Person, Zero-Sum Game7.4. Two-Person, Zero-Sum Game with no saddle point:7.4.2 Another approach to find the Mixed Strategy Probabilities:

8. Game theory in Industrial Organization8.1. Entry and Exit Decisions:8.2. Price and Quantity Competition:

9. Conclusion:Bibliography