CHAPTER SIX: GAME THEORY

6.1. INTRODUCTION
In general, the term ‘game’ refers to a situation of conflict and competition in which two or more competitors (or
participants) are involved in the decision-making process in anticipation of certain outcomes over a period of time.
The competitors are referred to as players. A player may be an individual, individuals, or an organization. A few
examples of competitive and conflicting decision environment, that involve the interaction between two or more
competitors are:
 Pricing of products, where sale of any product is determined not only by its price but also by the price
set by competitors for a similar product
 The success of any TV channel programme largely depends on what the competitors presence in
the same time slot and the programme they are telecasting.
 The success of a business strategy depends on the policy of internal revenue service regarding the
expenses that may be disallowed,
 The success of an advertising/marketing campaign depends on various types of services offered
to the customers.
For academic interest, theory of games provides a series of mathematical models that may be useful in explaining
interactive decision-making concepts, where two or more competitors are involved under conditions of conflict and
competition. However, such models provide an opportunity to a competitor to evaluate not only his personal
decision alternatives (courses of action), but also the evaluation of the competitor’s possible choices in order to
win the game.
Game theory came into existence in 20th Century. However, in 1944 John Von Neumann and Oscar
Morgenstern published a book named Theory of Games and Economic Behavior, in which they discussed how
businesses of all types may use this technique to determine the best strategies given a competitive business
environment. The author’s approach was based on the principle of ‘best out of the worst’.
The models in the theory of games can be classified based on the following factors:

Number of players If a game involves only two players (competitors), then it is called a two-person game.
However, if the number of players are more, the game is referred to as n-person game.

Sum of gains and losses If, in a game, the sum of the gains to one player is exactly equal to the sum of
losses to another player, so that, the sum of the gains and losses equals zero, then the game is said to be a
zero-sum game. Otherwise it is said to be non-zero sum game.

Strategy The strategy for a player is the list of all possible actions (moves, decision alternatives or courses of
action) that are likely to be adopted by him for every payoff (outcome). It is assumed that the players are aware of
the rules of the game governing their decision alternatives (or strategies). The outcome resulting from a particular
strategy is also known to the players in advance and is expressed in terms of numerical values (e.g. money, per
cent of market share or utility).
The particular strategy that optimizes a player’s gains or losses, without knowing the competitor’s
strategies, is called optimal strategy. The expected outcome, when players use their optimal strategy,
is
called value of the game.
Generally, the following two types of strategies are followed by players in a game:

(a) Pure Strategy A particular strategy that a player chooses to play again and again regardless of other
player’s strategy, is referred as pure strategy. The objective of the players is to maximize their gains or
minimize their losses.
(b) Mixed Strategy A set of strategies that a player chooses on a particular move of the game with some
fixed probability are called mixed strategies. Thus, there is a probabilistic situation and objective of the
each player is to maximize expected gain or to minimize expected loss by making the choice among pure

1
 strategies with fixed probabilities.
Mathematically, if pj ( j = 1, 2, . . ., n) is the probability associated with a pure strategy j to be chosen by a
player at any point in time during the game, then the set S of n non-negative real numbers
(probabilities) whose sum is unity associated with pure strategies of the player is written as: S = { p1, p2,
. . ., pn} where p1 + p2 + . . . + pn = 1 and pj  0 of all j.

Remark If a particular pj = 1 ( j = 1, 2, . . ., n) and all others are zero, the player is said to select pure
strategy j. A flow chart of using game theory approach to solve a problem is shown in Fig. 6.1.

6.2. TWO-PERSON ZERO-SUM GAMES

A game with only two players, say A and B, is called a two-person zero-sum game, only if one player’s gain is
equal to the loss of other player, so that total sum is zero.
Payoff matrix The payoffs (a quantitative measure of satisfaction that a player gets at the end of the play)
in terms of gains or losses, when players select their particular strategies (courses of action), can be represented in
the form of a matrix, called the payoff matrix. Since the game is zero-sum, the gain of one player is equal to the
loss of other and vice versa. In other words, one player’s payoff table would contain the same amounts in payoff
table of other player, with the sign changed. Thus, it is sufficient to construct a payoff table only for one of the
players.

If player A has m strategies represented by the letters: A1, A2, . . ., Am and player B has n strategies represented
by the letters: B1, B2, . . ., Bn. The numbers m and n need not be equal. The total number of possible
outcomes is therefore m × n. It is assumed that each player not only knows his own list of possible strategies but
also of his competitor. For convenience, it is assumed that player A is always a gainer whereas player B a
loser. Let aij be the payoff that player A gains from player B if player A chooses strategy i and player B chooses
strategy j. Then the payoff matrix is shown in the Table 12.1.

Player A’s Player B’s Strategies

Strategies B1 B2 ... Bn
A1 a 11 a 12 ... a1 n
A2 a 21 a 22 ... a2 n
.. .. .. ..
Am a m1 a m2 ... a mn

Since player A is assumed to be the gainer, therefore he wishes to gain as large a payoff aij as possible, player
B on the other hand would do his best to reach as small a value of aij as possible. Of course, the gain to player
B and loss to A must be – aij.
Various methods discussed in this chapter to find value of the game under decision-making environment of
certainty are as follows:
Assumptions of the game
1. Each player has available to him a finite number of possible strategies (courses of
action). The list may not be the same for each player.
2. Players act rationally and intelligently.
3. List of strategies of each player and the amount of gain or loss on an individual’s
choice of strategy is known to each player in advance.
4. One player attempts to maximize gains and the other attempts to minimize losses.
5. Both players make their decisions individually, prior to the play, without direct
communication between them.

2
 6. Both players select and announce their strategies simultaneously so that neither
player has an advantage resulting from direct knowledge of the other player’s
decision.
7. The payoff is fixed and determined in advance.

6.3. PURE STRATEGIES (MINIMAX AND MAXIMIN PRINCIPLES):

GAMES WITH SADDLE POINT
The selection of an optimal strategy by each player, without the knowledge of the competitor’s strategy, is the
basic problem of playing games. Since the payoffs for either player provides all the essential information, therefore,
only one player’s payoff table is required to evaluate the decisions. By convention, the payoff table for the player
whose strategies are represented by rows (say player A) is constructed. The objective of the study is to know how
these players must select their respective strategies so that they are able to optimize their payoff. Such a decision-
making criterion is referred to as the minimax-maximin principle. Such principle in pure strategies game always
leads to the best possible selection of a strategy for both players.
Maximin principle For player A the minimum value in each row represents the least gain (payoff) to him, if
he chooses his particular strategy. These are written in the matrix by row minima. He will then select the strategy
that gives the largest gain among the row minimum values. This choice of player A is called the maximin
principle, and the corresponding gain is called the maximin value of the game.
Minimax principle For player B, who is assumed to be the looser, the maximum value in each column
represents the maximum loss to him, if he chooses his particular strategy. These are written in the payoff matrix by
column maxima. He will then select the strategy that gives the minimum loss among the column maximum values.
This choice of player B is called the minimax principle, and the corresponding loss is the minimax value of
the game.
Optimal strategy A course of action that puts any player in the most preferred position, irrespective of the
course of action his competitor(s) adopt, is called as optimal strategy. In other words, if the maximin value equals
the minimax value, then the game is said to have a saddle (equilibrium) point and the corresponding strategies are
called optimal strategies.
Value of the game This is the expected payoff at the end of the game, when each player uses his optimal
strategy, i.e. the amount of payoff, V, at an equilibrium point. A game may have more than one saddle points. A
game with no saddle point is solved by choosing strategies with fixed probabilities.
Remarks 1. The value of the game, in general, satisfies the equation: maximin value  V  minimax value.
2. A game is said to be a fair game if the lower (maximin) and upper (minimax) values of the game are equal
and both equals zero.
3. A game is said to be strictly determinable if the lower (maximin) and upper (minimax) values of the game are
equal and both equal the value of the game.

6.3.1. Rules to Determine Saddle Point

The reader is advised to follow the following three steps, in this order, to determine the saddle point in the
payoff matrix.
1. Select the minimum (lowest) element in each row of the payoff matrix and write them under
‘row minima’ heading. Then, select the largest element among these elements and enclose it in a rectangle, .
2. Select the maximum (largest) element in each column of the payoff matrix and write them under ‘column
maxima’ heading. Then select the lowest element among these elements and enclose it in a circle, .
3. Find out the element(s) that is same in the circle as the well as rectangle and mark the position of such
element(s) in the matrix. This element represents the value of the game and is called the saddle (or
equilibrium) point.

3
For example. For the game with payoff matrix
Determine the optimal strategies for players A and B. Also determine the value of game. Is this game
(i) fair?
(ii) strictly determinable?

Player B
Player A B1 B2 B3
A1 –1 2 –2
A2 6 4 –6

Solution In this example, gains to player A or losses to player B are represented by the positive quantities,
whereas, losses to A and gains to B are represented by negative quantities. It is assumed that A wants to
maximize his minimum gains from B. Since the payoffs given in the matrix are what A receives, therefore, he is
concerned with the quantities that represent the row minimums. Now A can do no worse than receive one of these
values. The best of these values occurs when he chooses strategy A1. This choice provides a payoff of – 2 to A
when B chooses strategy B3. This refers to A’s choice of A1 as his maximum payoff strategy because this row
contains the maximum of A’s minimum possible payoffs from his competitor B.

Player B
Player A B1 B2 B3 Row minimum

A1 –1 2 – 2 – 2 ← Maximin
A2 6 4 – 6 – 6
Column maximum 6 4 – 2 ← Minimax

Similarly, it is assumed that B wants to minimize his losses and wishes that his losses to A be as small as
possible. The column maximums also represent the greatest payments B might have to make to A. The smallest of
these losses is – 2, which occurs when A chooses his course of action, A1 and B chooses his course of action, B3.
This choice of B3 by B is his minimax loss strategy because the amount of this column is the minimum of the
maximum possible losses.
In Table 12.2, the quantity – 2 in row A1 and column B3 is enclosed both in the box and the circle. That
is, it is both the minimum of the column maxima and the maximum of the row minima. This value is referred to as
saddle point.
The payoff amount in the saddle-point position is also called value of the game. For this game, value of the
game is, V = – 2, for player A. The value of game is always expressed from the point of view of the player
whose strategies are listed in the rows.
The game is strictly determinable. Also since the value of the game is not zero, the game is not fair.

Example 12.2 A company management and the labour union are negotiating a new three-year settlement. Each
of these has 4 strategies:
I: Hard and aggressive bargaining II: Reasoning and logical approach
III: Legalistic strategy IV: Conciliatory approach
The costs to the company are given for every pair of strategy choice.

4
 Company Strategies
Union Strategies I II III IV
I 20 15 12 35
II 25 14 8 10
III 40 2 10 5
IV – 5 4 11 0

What strategy will the two sides adopt? Also determine the value of the game.
Solution Applying the rule of finding out the saddle point, we obtain the saddle point that is enclosed both
in a circle and a rectangle, as shown in Table 12.3.
Company Strategies
Union Strategies I II III IV Row minimum

I 20 15 12 35 12 ← Maximin
II 25 14 8 10 8
III 40 2 10 5 2
IV – 5 4 11 0 – 5
Column maximum 40 15 12 35
↑ Minimax

As shown in Table 12.3, since Maximin = Minimax = Value of game = 12, therefore the company will always adopt
strategy III – Legalistic strategy and union will always adopt strategy I – Hard and aggressive.