Game Theory Economics

Basics of Game theory

1. Game Theory in Economics
Introduction
Game theory seeks to analyse competing situations which arise out of conflicts of interest. There is
conflict of interest between animals and plants in the consumption of natural resources. Animals compete
among themselves for securing food. Man competes with animals to earn his food. A man also competes
with another man. Every intelligent and rational participant in a conflict wants to be a winner but not all
participants can be the winners at a time. The situations of conflict gave birth to Darwin’s theory of the
‘survival of the fittest’. Game theory is a tool to examine situations of conflict so as to identify the
courses of action to be followed and to take appropriate decisions in the long run. Thus this theory assumes
importance from managerial perspectives. In the present day world, business organizations compete with
each other in getting the market share. In today’s world, decisions about many practical problems are made
in a competitive situation, where two or more opponents are involved under the conditions of competition
and conflict situations. The outcome does not depend on the decision alone but also the interaction between
the decision-maker and the competitor. The objective, in theory, of games is to determine the rules of
rational behaviour in game situations, in which the outcomes are dependent on the actions of the
interdependent players. The results obtained by the application of this theory can serve as an early
warning to the top level management in meeting the threats from the competing business organizations
and for the conversion of the internal weaknesses and external threats into opportunities and strengths,
thereby achieving the goal of maximization of profits. While this theory does not describe any procedure to
play a game, it will enable a participant to select the appropriate strategies to be followed in the pursuit of
his goals.
Definition: Game is defined as an activity between two or more persons involving moves by each
person according to a set of rules, at the end of which each person receive some benefit or
satisfaction or suffers loss. In a game, there are two or more opposite parties with conflicting
interests. They know the objectives and the rules of the game. An experienced player usually predicts with
accuracy how his opponent will react if a particular strategy is adopted. When one player wins, his
opponent losses.
The games are classified based on the following characteristics.
1. Chance or strategy: If in a game the moves are determined by chance, we call it a game of chance, if
they are determined by skill; it is a game of strategy. In general a game may involve partly strategy and
partly chance.
2. Number of Persons: A game is called an n-person game if the number of persons playing it is n.
3. Number of moves: The number of moves may be finite or infinite.
4. Number of alternative available to each person per move: These also may be finite of infinite. A finite
game has a finite number of moves, each involving a finite number alternative. Otherwise the game is
infinite.
5. Information available to players of the past moves of the other players: The two extreme cases are, (a)
no information at all (b) complete information available. There can be cases in between in which
information is partly available.
6. Pay off: It is a quantitative measure of satisfaction a person gets at the end of the play. It is real a valued
function of the variables in the game. Let Pi be the payoff to the person, Pi, i = 1, 2, …, n, in an n-person
game. Then if ∑ P i = 0, the game is said to be a zero-sum game.
Types of Games
Games can be of several types. Important ones are as follows:
(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 players remains only two. But games can
as well involve many people as active participants, each with his own set of choices for each play of the

2. game. In case of three players, it can be named as three-person game. Thus, in case of more than two
persons, the game is generally named as n person game.
(2) Zero sum and non-zero sum game. A zero sum game is one in which the sum of the payments to all
the competitors is zero for every possible outcome of the game. In other words, in such a game the sum of
the points won equals the sum of the points lost i.e., one player wins at the expense of the other (others).
Two-person matrix game is always zero sum game since one player loses what the other wins. But in a
non-zero sum game the sum of the payoffs from any play of the game may be either positive or negative
but not zero.
(3) Games of perfect information and games of imperfect information: A game of perfect information is
the one in which each player can find out the strategy that would be followed by his opponent. On the other
hand, a game of imperfect information is the one in which no player can know in advance what
strategy would be adopted by the competitor and a player has to proceed in his game with his guess
works only.
(4) Games with finite number of moves / players and games with unlimited number of moves: A
game with a finite number of moves is the one in which the number of moves for each player is
limited before the start of the play. On the other hand, if the game can be continued over an extended
period of time and the number of moves for any player has no restriction, then we call it a game with
unlimited number of moves.
(5) Constant-sum games: If the sum of the game is not zero but the sum of the payoffs to both players in
each case is constant, then we call it a constant sum game. It is possible to reduce such a game to a zero
sum game.
Assumptions for a Competitive Game
Game theory helps in finding out the best course of action for a firm in view of the anticipated
countermoves from the competing organizations. A competitive situation is a competitive game if the
following properties hold:
1. The number of competitors is finite, say N.
2. A finite set of possible courses of action is available to each of the N competitors.
3. A play of the game results when each competitor selects a course of action from the set of courses
available to him. In game theory we make an important assumption that all the players select their
courses of action simultaneously. As a result, no competitor will be in a position to know the choices
of his competitors.
4. The outcome of a play consists of the particular courses of action chosen by the individual
players. Each outcome leads to a set of payments, one to each player, which may be either positive, or
negative, or zero.
Managerial Applications of the Theory of Games
The techniques of game theory can be effectively applied to various managerial problems as
detailed below:
1) Analysis of the market strategies of a business organization in the long run.
2) Evaluation of the responses of the consumers to a new product.
3) Resolving the conflict between two groups in a business organization.
4) Decision making on the techniques to increase market share.
5) Material procurement process.
6) Decision making for transportation problem.
7) Evaluation of the distribution system.
8) Evaluation of the location of the facilities.
9) Examination of new business ventures and
10) Competitive economic environment.

3. Key concepts in the Theory of Games
Several of the key concepts used in the theory of games are described below:
 Players: The competitors or decision makers in a game are called the players of the game.
 Strategies: The alternative courses of action available to a player are referred to as his strategies.
 Pay off: The outcome of playing a game is called the payoff to the concerned player.
 Optimal Strategy: A strategy by which a player can achieve the best pay off is called the optimal
strategy for him.
 Zero-sum game: A game in which the total payoff to all the players at the end of the game is zero is
referred to as a zero -sum game.
 Non-zero sum game: Games with “less than complete conflict of interest” are called non-zero
sum games. The problems faced by a large number of business organizations come under
this category. In such games, the gain of one player in terms of his success need not be
completely at the expense of the other player.
 Payoff matrix: The tabular display of the payoffs to players under various alternatives is called the
payoff Matrix of the game.
 Pure strategy: If the game is such that each player can identify one and only one strategy
as the optimal strategy in each play of the game, then that strategy is referred to as the best
strategy for that player and the game is referred to as a game of pure strategy or a pure game.
 Mixed strategy: If there is no one specific strategy as the ‘best strategy’ for any player in a game,
then the game is referred to as a game of mixed strategy or a mixed game. In such a game, each
player has to choose different alternative courses of action from time to time.
 N-person game: A game in which N-players take part is called an N-person game.
 Maximin-Minimax Principle : The maximum of the minimum gains is called the maximin
value of the game and the corresponding strategy is called the maximin strategy. Similarly
the minimum of the maximum losses is called the minimax value of the game and the
corresponding strategy is called the minimax strategy. If both the values are equal, then that would
guarantee the best of the worst results.
 Negotiable or cooperative game: If the game is such that the players are taken to cooperate on any
or every action which may increase the payoff of either player, then we call it a negotiable or
cooperative game.
 Non-negotiable or non-cooperative game: If the players are not permitted for coalition then we
refer to the game as a non-negotiable or non-cooperative game.
 Saddle point: A saddle point of a game is that place in the payoff matrix where the maximum of
the row minima is equal to the minimum of the column maxima. The payoff at the saddle point is
called the value of the game and the corresponding strategies are called the pure strategies.
 Dominance: One of the strategies of either player may be inferior to at least one of the remaining
ones. The superior strategies are said to dominate the inferior ones.
 Value of the game (V): The value of the game is the maximum guaranteed gain to the maximizing
player A, if both the players use their best strategies. It is the expected pay off of a play when all the
players of the game follow their optimal strategies. The amount of payoff, at an equilibrium point is
known as the value of the game.
 Fair game: The game is said to be fair if the value of the game (V) = 0.
Two-Person Zero-Sum Game
In a game with two players, if the gain of one player is equal to the loss of another player, then the
game is a two person zero-sum game. Thus a game with only two players, say player A and player B,
and if the gain of the player A is equal to the loss of the player B, so that the total sum is zero, then
this is a two person zero-sum game. Such a game is also called rectangular game. The two person
zero sum game may be pure strategy game or mixed strategy game.

4. A Game in a competitive situation possesses the following properties
i. The number of players is finite.
ii. Each player has finite list of courses of action or strategy.
iii. A game is played when each player chooses a course of action (strategy) out of the available
strategies. No player is aware of his opponent’s choice until he decides his own.
iv. The outcome of the play depends on every combination of courses of action. Each outcome
determines the gain or loss of each player.
Assumptions for Two-Person Zero Sum Game
For building any model, certain reasonable assumptions are quite necessary. Some assumptions for
building a model of two-person zero sum game are listed below.
a) Each player has available to him a finite number of possible courses of action. Sometimes the
set of courses of action may be the same for each player. Or, certain courses of action may be available to
both players while each player may have certain specific courses of action which are not available to the
other player.
b) Player A attempts to maximize gains to himself. Player B tries to minimize losses to himself.
c) The decisions of both players are made individually prior to the play with no communication
between them.
d) The decisions are made and announced simultaneously so that neither player has an advantage
resulting from direct knowledge of the other player’s decision.
e) Both players know the possible payoffs of themselves and their opponents.
Payoff Matrix
In a game, the gains and the losses, resulting from different moves and countermoves, when represented in
the form of a matrix, is known as a pay off matrix. Each element of the pay off matrix is the gain of the
maximizing player when a particular course of action is chosen by him as against the course faction chosen
by the opponent.
Suppose we consider the decision problem in which two rational players, Player A and Player B,
each have a set of possible actions available to them. Suppose A has m strategies and B has n strategies.
Suppose players A and B have m actions a1, a2, …, am and n actions b1, b2, …, bn available, respectively.
The consequence of this decision by both players is a specific return or pay off. If the payoff is non-zero, it
represents a gain or loss, to player. We will consider the case where Player A’s gain is Player B’s loss and
conversely. No units enter or leave the game. This type of game is called a two person zero sum game.
In general, we will let aij represent the pay off from Player B to Player A if Player A chooses action
ai and Player B chooses action bj. If Player A chooses action ai, we will say that he plays row i. Likewise, if
Player B chooses action bj, we will say that he plays column j. Player A wishes to gain as large a payoff
aij as possible while player B will do his best to reach as small a value aij as possible where the gain to
player B and loss to player A be (-aij).

5. Pure & Mixed Strategies
If both players use their optimal minimax strategies, the resulting expected pay off is called the
value of the game. When the optimal minimax strategies are pure strategies, then the expected pay off
using these optimal minimax strategies is just the value of the game. So a pure strategy is a decision in
advance of all players, always to choose a particular course of action., it is a predetermined course of
action. The player knows it in advance.
The Saddle Point
The saddle point in a payoff matrix is one which is the smallest value in its row and the largest value in its
column. A saddle point of a game is that place in the payoff matrix where the maximum of the row
minima is equal to the minimum of the column maxima. The payoff at the saddle point is called the
Value of the Game and the corresponding strategies are called the Pure Strategies.
The saddle point is also known as equilibrium point in the theory of games. An element of a matrix
that is simultaneously minimum of the row in which it occurs and the maximum of the column in which it
occurs is a saddle point of the matrix game. In a game having a saddle point optimum strategy for player I
is always to play the row containing a saddle point and for the prayer II to play the column that contains a
saddle point. Saddle point also gives the value of such a game. Saddle point in a payoff matrix concerning
a game may be there and may not be there. If there is a saddle point we can easily find out the optimum
strategies and the value of the game but when saddle point is not there we have to use algebraic methods
for working out the solutions concerning game problems. If there is more than one saddle point there will
be more than one solution.
When there is no saddle point for a game problem, the minimax-maximin principle cannot be
applied to solve the problem. In those cases the concept of chance move is introduced. Here the choice
among a number of strategies is not the decision of the player but by some chance mechanism. That is,
predetermined probabilities are used for deciding the course of action. The strategies thus made are called
mixed strategies. Solution to a mixed strategy problem can be arrived at by any of the methods like
probabilities method or principle of dominance.
Pure Strategies: Game 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. The aim of the game is to determine how the players must select their
respective strategies such that the pay-off is optimized. The game theory helps to know how these players
must select their respective strategies, so that they may optimize their payoffs. Such a criterion of decision
making is referred to as Minimax-Maximin principle. The maximizing player arrives at his optimal
strategy on the basis of maximin criterion, while minimizing players’ strategy is based on the minimax
criterion. The game is solved by equating maximin value with minimax value.
In a pay-off matrix, the minimum value in each row represents the minimum gain for player A.
Player A will select the strategy that gives him the maximum gain among the row minimum values. The
selection of strategy by player A is based on maximin principle. Similarly, the same pay-off is a loss for
player B. The maximum value in each column represents the maximum loss for Player B. Player B will
select the strategy that gives him the minimum loss among the column maximum values. The selection of
strategy by player B is based on minimax principle. If the maximin value is equal to minimax value, the
game has a saddle point (i.e., equilibrium point). Thus the strategy selected by player A and player B are
optimal. The amount of payoff, i.e. V at an equilibrium point is known as the value of the game. The game
is said to be fair if the value of the game V = 0

6. We see that the maximum of row minima = the minimum of the column maxima. So the game has a saddle
point. The common value is 12. Therefore the value V of the game = 12.
Interpretation: In the long run, the following best strategies will be identified by the two players:
The best strategy for player A is strategy 4.
The best strategy for player B is strategy IV.
The game is favourable to player A.
Problem 2: Solve the game with the pay-off matrix for player A as given in Table below –

Solution: Find the smallest element in rows and largest elements in columns as shown in following Table

7. Select the largest element in row and smallest element in column. Check for the minimax criterion,
Max Min = Min Max
1 = 1
The value of the game is, V = 1
Therefore, there is a saddle point and it is a pure strategy.
Optimum Strategy Player A A2 Strategy
Player B B1 Strategy
Problem 3: Solve the game with the pay off matrix given in Table below and determine the best
strategies for the companies A and B and find the value of the game for them.

Solution: The matrix is solved using maximin criteria, as shown in Table below

Max Min = Min Max
2 = 2
Therefore, there is a saddle point.
Optimum strategy for company A is A1and Optimum strategy for company B is B1or B3
Problem 4: Check whether the following game is given in Table below, determinable and fair.

Solution: The game is solved using maximin criteria as shown in Table

8. The game is strictly neither determinable nor fair.
Strictly Determined Game
A game is strictly determined if it has at least one saddle point. The following statements are true about
strictly determined games.
A. All saddle points in a game have the same payoff value.
B. Choosing the row and column through any saddle point gives minimax strategies for both players.
In other words, the game is solved via the use of these (pure) strategies.
The value of a strictly determined game is the value of the saddle point entry. A fair game has value of
zero; otherwise it is unfair or biased.
Consider the following game.


Column Strategy

A B C

Row
Strategy
1

0 -1 1

2 0 0 2
3 -1 -2 3
In the above game, there are two saddle points. Since the saddle point entries are zero, this is a fair game.
Games without Saddle Point: Mixed Strategies
For any given pay off matrix without saddle point the optimum mixed strategies are shown in Table
below:-

We assume that the strategies are not pure strategies (saddle point does not exist). Since this game has no
saddle point, the following condition shall hold:

In this case, the game is called a mixed game. No strategy of Player A can be called the best strategy for
him. Therefore A has to use both of his strategies. Similarly no strategy of Player B can be called the best
strategy for him and he has to use both of his strategies.
Probability method: This method is applied when there is no saddle point and the pay of matrix has two
rows and two columns only. The players may adopt mixed strategies with certain probabilities. Here the
problem is to determine probabilities of different strategies of both players and the expected payoff of
the game.
Let p be the probability of A using strategy A1 and (1-p) is the probability for A using A2. Then we have
the equations (i) Expected gain of A if B chooses B1 = ap + c (1-p)
(ii) Expected gain of A if B chooses B2 = bp + d (1-p)

9. If the expected values in equations (i) and (ii) are different, Player B will prefer the minimum of the two
expected values that he has to give to player A. Thus B will have a pure strategy. This contradicts our
assumption that the game is a mixed one. Therefore the expected values of the pay-off to Player A in
equations (i) and (ii) should be equal. Thus we have the condition
ap + c (1-p) = bp + d (1-p)
Solving the equation ,we get

Similarly, let q and (1-q) be respectively probabilities for B choosing strategies B1 and B2, then
When A use his first strategy then we have
The expected value of loss to Player B with his first strategy = aq
The expected value of loss to Player B with his second strategy = b(1-q)
Therefore the expected value of loss to B = aq + b(1-q)………………………………..(iii)
When A use his second strategy
The expected value of loss to Player B with his first strategy = cq
The expected value of loss to Player B with his second strategy = d(1-q)
Therefore the expected value of loss to B = cq + d(1-q)……………………………………(iv)
If the two expected values are different then it results in a pure game, which is a contradiction.
Therefore the expected values of loss to Player B in equations (iii) and (iv) should be equal. Hence we
have the condition: aq + b(1-q) = cq + d(1-q)

Therefore, the value V of the game is

Problem 5: Solve the following game

Solution: First consider the row minima

Maximum of {2, 1} = 2
Next consider the maximum of each column

Minimum of {4, 5}= 4
We see that, Max {Row Minima} ≠ Min {Column Maxima}
So the game has no saddle point. Therefore it is a mixed game.
Comparing this game with earlier game

10. We have a = 2, b = 5, c = 4 and d = 1.
Let p be the probability that player X will use his first strategy. We have


Let r be the probability that Player Y will use his first strategy. Then the probability that Y will use his
second strategy is (1-r). We have


Therefore, out of 2 trials, player X will use his first strategy once and his second strategy once. Again

Therefore, out of 3 trials, player Y will use his first strategy twice and his second strategy once.
The Principle Of Dominance
The principle of dominance states that if the strategy of a player dominates over another strategy in all
conditions, then the latter strategy can be ignored because it will not affect the solution in any way. A
strategy dominates over the other only if it is preferable in all conditions. In case of pay-off matrices larger
than (2×2) size, the dominance property can be used to reduce the size of the pay-off matrix by eliminating
the strategies that would never be selected. Principle of dominance is applicable to pure strategy and mixed
strategy problems. Mathematically speaking,

11. In a given pay -off matrix A, we say that the i
th
row dominates the k
th
row if
And

In such a situation player A will never use the strategy corresponding to k
th
row, because he
will gain less for choosing such a strategy.

And
In this case, the player B will loose more by choosing the strategy for the q
th
column than by choosing
the strategy for the p
th
column. So he will never use the strategy corresponding to the q
th
column.
When dominance of a row (or a column) in the pay-off matrix occurs, we can delete a row (or a column)
from that matrix and arrive at a reduced matrix. This principle of dominance can be used in the
determination of the solution for a given game.
Conditions
1. If all the elements in a row of a pay off matrix are less than or equal to the corresponding
elements of another row, then the latter dominates and so former is ignored.
Example 1 3 4 5
-2 1 4 0
Here every element of second row is less than or equal to the corresponding elements of row 1.Therefore
first row dominates and so second row can be ignored.
2. If all the elements in a column of a pay off matrix are greater than or equal to the corresponding
elements of another column, then the former dominates and so latter is ignored.
2 6
-3 2
4 4
2 3
Here the elements of second column are greater than or equal to the corresponding elements of first
column. So second column dominates and first column can be ignored.
3. If the linear combination of two or more rows or columns dominates a row or column, then the
latter is ignored. If all the elements of a row are less than or equal to average of the corresponding
elements of two other rows, then the former is ignored.
For example,
2 0 2
3 -1 3
1 -2 1
Here the elements of third column are greater than or equal to the average of corresponding
elements of two other columns, so third column dominates, and other columns can be ignored.
Problem 6: Solve the game given below in Table below after reducing it to 2 × 2 game

Solution: Reduce the matrix by using the dominance property. In the given matrix for player A, all the
elements in Row 3 are less than the adjacent elements of Row 2. Strategy 3 will not be selected by player
A, because it gives less profit for player A. Row 3 is dominated by Row 2. Hence delete Row 3, as shown
in Table below –

12. For Player B, Column 3 is dominated by column 1 (Here the dominance is opposite because Player B
selects the minimum loss). Hence delete Column 3. We get the reduced 2 × 2 matrix as shown below

Now, solve the 2 × 2 matrix, using the maximin criteria as shown below

Therefore, there is no saddle point and the game has a mixed strategy. Applying the probability formula,

Problem 7: Solve the game whose pay off matrix is given by

Solution: Applying principle of dominance, B3 is dominated by B1. So ignore B3 and the reduced matrix
is

13. Now A3 is dominated by A2. So ignore A3. The resulting matrix is

Let p and 1-p be the probabilities for A choosing A1 and A2. Let q and 1-q, be the probabilities for B
choosing B1 and B2. Then,

Probability that A choosing A1 = 2/5
Probability that A choosing A2 = 3/5

Probability that B choosing B1 = 1/2
Probability that B choosing B2 = ½

A’s mixed strategy = 2/5, 3/5
B’s mixed strategy = 1/2, ½
Problem 8: Solve the game whose pay off matrix is given by

Solution:
First consider the minimum of each row.

Maximum of {2, 3, 3} = 3
Next consider the maximum of each column

Minimum of {6, 4, 7, 6} = 4
The following condition holds: Max {row minima} ≠ min {column maxima}
Therefore we see that there is no saddle point for the game under consideration.
Compare columns II and III.

14. We see that each element in column III is greater than the corresponding element in column II. The choice
is for player B. Since column II dominates column III, player B will discard his strategy 3. Now we have
the reduced game

For this matrix again, there is no saddle point. Column II dominates column IV. The choice is for player B.
So player B will give up his strategy 4. The game reduces to the following:

This matrix has no saddle point.
The third row dominates the first row. The choice is for player A. He will give up his strategy 1 and retain
strategy 3. The game reduces to the following

15. Prisoner’s Dilemma
Prisoners’ dilemma is the best-known game of strategy in social science. The typical prisoner's dilemma is
set up in such a way that both parties choose to protect themselves at the expense of the other participant. It
helps us understand what governs the balance between cooperation and COMPETITION in business, in
politics, and in social settings. Prisoner’s Dilemma is the classic example of application of game theory to
oligopolistic market conditions. A classic example of game theory that explains the problem faced by
oligopoly is “Prisoner’s Dilemma”.
In the traditional version of this game, two Prisoners are accused of a joint crime and they are put in
two different jails. They cannot communicate with each other. They are asked to confess and the police are
interrogating them in separate rooms. Each can either confess, thereby implicating the other, or keep silent.
No matter what the other suspect does, each can improve his own position by confessing.
1. If both confess, both will get 5 years of imprisonment.
2. If no one confesses, both will get only 2 years of jail.
3. If one confesses and the other does not confess, the one who confesses will be jailed for one year and the
other - 10 years.
In this situation, most likely strategy will be both the prisoners would confess and get 2 years of
imprisonment, oligopolistic firms often find themselves in a prisoner’s dilemma.

16. They have to decide:
a) Whether to compete aggressively & capture larger market share.
b) Co-operate and compete more passively.
Actually both the firms would do better by co-operating and charging high price. But the firms are
in prisoners’ dilemma, where neither can trust its competitors to set a higher price.
Other Classic example of prisoner’s dilemma in the real world is encountered when two
competitors are battling it out in the marketplace. Many sectors of the economy have two main rivals. In
the U.S., for example, the fierce rivalry between Coca-Cola (NYSE:KO) and PepsiCo (NYSE:PEP) in soft
drinks. Consider the case of Coca-Cola versus PepsiCo, and assume that the former is thinking of cutting
the price of its iconic Coke drink. If it does so, Pepsi may have no choice but to follow suit for its Pepsi
Cola to retain its market share. This may result in a significant drop in profits for both companies. A price
drop by either company may therefore be construed as defecting, since it breaks an implicit agreement to
keep prices high and maximize profits. Thus, if Coca-Cola drops its price but Pepsi continues to keep
prices high, the former is defecting while the latter is cooperating (by sticking to the spirit of the implicit
agreement). In this scenario, Coca-Cola may win market share and earn incremental profits by selling more
Coke drinks.
The prisoner’s dilemma can be used to aid decision-making in a number of areas in one’s personal
life, such as buying a car, salary negotiations and so on. For example, assume you are in the market for a
new car, and you walk into a car dealership. The utility or payoff in this case is a non-numerical attribute,
i.e. satisfaction with the deal. You obviously want to get the best possible deal in terms of price, car
features, etc. while the car salesman wants to get the highest possible price to maximize his commission.
Cooperation in this context means no haggling; you walk in, pay the sticker price (much to the salesman’s
delight) and leave with a new car. On the other hand, defecting means bargaining; you want a lower price,
while the salesman wants a higher price. Assigning numerical values to the levels of satisfaction, where 10
means fully satisfied with the deal and 0 implies no satisfaction, the payoff matrix is as shown below:
Car Buyer vs. Salesman – Payoff Matrix
Salesman
Cooperate Defect
Buyer
Cooperate (a) 7, 7 (c) 0,10
Defect (b) 10, 0 (d) 3, 3
What does this matrix tell us? If you drive a hard bargain and get a substantial reduction in the car price,
you are likely to be fully satisfied with the deal, but the salesman is likely to be unsatisfied because of the
loss of commission (as can be seen in cell b). Conversely, if the salesman sticks to his guns and does not
budge on price, you are likely to be unsatisfied with the deal while the salesman would be fully satisfied
(cell c). Your satisfaction level may be less if you simply walked in and paid full sticker price (cell a). The
salesman in this situation is also likely to be less than fully satisfied, since your willingness to pay full price
may leave him wondering if he could have “steered” you to a more expensive model, or added some more
bells and whistles to gain more commission. Cell (d) shows a much lower degree of satisfaction for both
buyer and seller, since prolonged haggling may have eventually led to a reluctant compromise on the price
paid for the car.
The prisoner’s dilemma shows us that mere cooperation is not always in one’s best interests.
In fact, when shopping for a big-ticket item such as a car, bargaining is the preferred course of action from
the consumers' point of view. Otherwise the car dealership may adopt a policy of inflexibility in price
negotiations, maximizing its profits but resulting in consumers overpaying for their vehicles.
Understanding the relative payoffs of cooperating versus defecting may stimulate you to engage in
significant price negotiations before you make a big purchase.

17. Nash Equilibrium
There are many games which don’t have a dominant strategy but still equilibrium can be attained.
Here the way is method developed by Nash. In game theory the optimal outcome of a game is one where
no player has an incentive to deviate from his chosen strategy after considering an opponent's choice. In
other words, each player plays a best-response strategy by assuming the other player’s moves.
In the Nash Equilibrium, each player's strategy is optimal when considering the decisions of other
players. It is the solution to a game in which two or more players have a strategy, and with each participant
considering an opponent’s choice, he has no incentive, nothing to gain, by switching his strategy. Every
player wins because everyone gets the outcome they desire. If every player in a game plays his dominant
pure strategy (assuming every player has a dominant pure strategy), then the outcome will be Nash
equilibrium. Nash Equilibria are self-enforcing; when players are at a Nash Equilibrium they have no
desire to move because they will be worse off.
When Nash equilibrium is reached, players cannot improve their payoff by independently changing
their strategy. For example, in the Prisoner's Dilemma game, confessing is a Nash equilibrium because it is
the best outcome, taking into account the likely actions of others. To quickly test if the Nash equilibrium
exists, reveal each player's strategy to the other players. If no one changes his strategy, then the Nash
Equilibrium is proven.
Necessary Conditions
The following game has payoffs defined
L R
T a,b c,d
B e,f g,h
In order for (T,L) to be an equilibrium in dominant strategies (which is also a Nash Equilibrium), the
following must be true:
 a > e
 c > g
 b > d
 f > h
In order for (T,L) to be a Nash Equilibrium, only the following must be true:
 a > or = e
 b > or = d
Example 1: considering below
P2
LEFT RIGHT
P1
UP 5, 4 3,10
DOWN 9, 2 0, 1
If P1 goes UP, P2 prefers right since 10>4. But if P1 goes down then P2 prefers left since 2>1. If P2 goes
left then P1 goes down since 9>5. If P2 goes right then P1 goes UP since 3>0. There are 2 outcomes which
are stable (UP,RIGHT) and (DOWN, LEFT) which are “stable”: neither player would wish to change his
action given the action of the other player. This is a NASH equilibrium.
Example 2: considering below

18. Player 1's best response against each of player 2's strategy = red circle

Player 2's best response against each of player 1's strategy = blue circle.

Overall considering following -
 Player 1's best response against each of player 2's strategy = red circle.
 Player 2's best response against each of player 1's strategy = blue circle.
Nash Equillibrium (NE) = (A,B),(B,A)

Example 3: Find Nash Equillibrium for the following game
P2
A B C
P1
A 1,1 10,0 -10,1
B 0,10 1,1 10,1
C 1,-10 1,10 1,1
To start, we find the best response for player 1 for each of the strategies player 2 can play.
P2
A B C
P1
A 1,1 10,0 -10,1
B 0,10 1,1 10,1
C 1,-10 1,10 1,1
Now we do the same for player 2 by underlining the best responses of the column player
P2
A B C
P1
A 1,1 10,0 -10,1
B 0,10 1,1 10,1
C 1,-10 1,10 1,1
Now a pure strategy Nash equilibrium is a cell where both payoffs are underlined, i.e., where both strategies are best
responses to each other. In the example, the unique pure strategy equilibrium is (A, A).