If the sum of payments to all players is zero, then the

game is called a zero-sum game. This means that any

player’s win must be balanced by a loss for some

other players.

The simplest type of game is a two-player zero-sum

game. The choice of actions and the possible results of

the game can be summarized in a “payoff matrix” A, in

which the rows represent the possible actions one

player, player R, can take; the columns the possible

actions of the second player, player C; and the (i,j)-th

entry of the matrix Aij represents the amount player C

must pay to player Rif Rselects row iand Cselects

column jin the play of the game. (If Aij is negative,

then player Rpays Ca certain amount.) For example,

the following table represents a two-person zero-sum

game for which each player can take one of three possi-

ble actions:

As a first attempt to analyze this game, consider

the actions that player Rcan take. She hopes that the

game will result in the largest positive number possible

(in this case 5), for this is the amount player Cmust

pay Rif this is the result of the game. Thus player Ris

tempted to select action 1 (especially since Ris aware

that player C’s optimal outcome of –3 lies in column

1). Of course, player Cis aware that player Ris likely

to think this way, and would be loath to select column

1 as her action for fear of having 5 be the outcome of

the game. Player Ris aware that player Cwill think

this way, and will suspect then that player Cwill

choose column 2, knowing the she will likely select row

1, and so R, to foil this plan, is tempted to choose an

action different from 1 after all. And so on. As one

sees, one can quickly enter a never-ending cycle of sec-

ond guessing.

Instead of aiming to maximize her profit from the

game, another approach player Rcould adopt is to

minimize her losses. For example, if player Rtakes

action 1, she could potentially lose 1 point (if Chap-

pens to choose column 2). If Rchooses row 2, the

worst outcome would be no gain or loss, and if she

chooses row 3, she could potentially be down three

points. Thus the maximal minimum outcome for player

Roccurs with the choice of row 2. This line of reason-

ing is called the “maximin strategy.” Similarly, the

choice of column 2 results in the minimal maximum

outcome in player C’s favor. Thus following a “mini-

max strategy,” player Cwould choose column 2, and

the outcome of the game is consequently 0.

The entry 0 in the above payoff matrix is called a

saddle point. It is a minimum in its row and a maxi-

mum in its column. If a payoff matrix contains a saddle

point, then the optimal strategy for each player is to

take actions corresponding to the saddle point. The

value of the saddle point, for a game that possesses

such a point, is called the value of the game.

Not every payoff matrix for a two-person game pos-

sesses a saddle point. For instance, the following game

corresponding to two possible actions for each player

has no saddle point, and following the minimax or the

maximin strategies is not optimal for either player.

(The maximin strategy for player Rsuggests that she

select row 1, and the minimax strategy for player C

that she take column 1 yielding the result 1 for the

game. Player C, however, can anticipate this and is

tempted then to change choice to column 2 to obtain

the preferable result of –2. Player R, of course, is aware

that player Cwill likely do this, and so will change her

choice to row 2 to obtain the outcome 4, and so forth.

The two players again are trapped in an endless cycle

of second guessing.)

The appropriate strategy in such a game lacking a

saddle point is a mixed strategy, in which each player

decides to select an action by random choice, appropri-

ately weighted so as to maximize her expected profit

from the game. For instance, suppose player Rdecides

to select row 1 with

PROBABILITY

pand row 2 with

probability 1 – p. Under this strategy, if player C

chooses column 1, then the expected value of player R’s

profit is E1= 1 ×p+ (–3) ×(1 – p) = 4p– 3. If, on the

other hand, player Cwere to select column 2, then the

expected profit for player Ris E2= (–1) ×p+ 4 ×(1 –

p) = 4p– 5p. These expected outcomes are equal if we

choose p= 7/9. Thus, by selecting row 1 with this

12

11–2

2–3 4

123

15–11

2104

3–3 –2 2

game theory 217