theory of game (game theory)

GAME THEORY

Game theory and Strategic behaviour under oligopoly

Game theory is a technique used to analyse situations where individuals or organisations have conflicting objectives. Three important concepts are often used in game theory. (i)Players (ii)Strategy and (iii) Payoffs. Players are the decision makers (mangers). A strategy is a course of action to change price, develop new products, adopt new advertising, etc. The payoff is the outcome of the strategy. The players always try to optimise their strategy. This theory was first developed by Von Neuman (mathematician)and Oskar Morgenstern(economist). It explains the strategic interaction among the Oligopoly firms. Each player needs to adopt dominant strategy to maximise profit.

• Constant sum of game: In this case the total benefit of the players given each strategy, is a constant and the players have to share the profit

• Zero sum game: In this game the total benefit, given each strategy, is equal to zero. Under this game the gain of one player is the loss by the other player.

• Positive sum games: The total benefit of the players added together, given each strategy, is more than zero(+ve).

• Negative sum game: When the total benefit , given the strategy, is less than zero (-Ve)

• Co operative game: The games where the strategies of the players are coordinated or joint action.

Q.14.2

Types of games

• Role of InterdependenceThe essence of game is the interdependence of player strategies. It may be sequential game or a simultaneous game. In a sequential game, each player moves in succession, and each player is aware of all prior moves. A simultaneous game is one in which all players make decisions (or select a strategy) without knowledge of the strategies that are being chosen by other players. Even though the decisions may be made at different points of time, the decisions are made simultaneously. Simultaneous games are solved using the concept of Nash equilibrium.

The nature of Problems faced by the Oligopoly firms is best explained by the Prisoner’s Dilemma game.

Let two persons are involved with some illegal activities say; match fixing; were arrested and kept separately so that they cannot communicate each other. Four possible options were kept before them:(i) If both confess, each one will get 5 years of imprisonment; (ii) If Both deny, each one will be put in jail for 1 year;(iii) If A confesses and B denies, A will go free and B will get 10 years of imprisonment ;(iv) If B confesses and A denies, B will go free and A will get 10 years of imprisonment .

Individuals Startegies Do not confess

Do not confessIndividual A

Individual BConfess

Confess 5,5 0,10

1,110,0

Prisoner’s Dilemma: The Pay-off matrix With much of uncertainty, no one knows the action of each other, there is dilemma in taking a decision. The dominant strategy for A is to confess. The dominant strategy for B is also is to confess. Each one will end up with 5 years of imprisonment.It refers to a situation in which each individual firm adopts its dominant strategy and earns maximum profits.

Individuals Startegies Do not advertise

Do not advertise

Individual BAdvertise

Individual AAdvertise 4,3 5,1

2,5 3,2

Payoff Matrix for an advertising game

Firm A’s profit is always greater if it advertises than not advertising regardless of what B does and the dominant strategy for A is to advertise. The same is the case for B also. This is the final equilibrium as it is the optimal choice of both the players.

• In oligopoly, the business firm chooses its strategies to achieve equilibrium. There are actions, reactions and interactions to increase their prices to achieve the optimum profit.

• To analyse this type of situation, an American mathematician (John Nash)developed a technique which is known as Nash equilibrium. It is defined as a situation where each player chooses his/her optimal strategy, given the strategy chosen by other players. A game may have more than one Nash equilibrium.

Simple Two Persons, Zero Sum Game

• Each player knows both his and his opponent’s alternatives

• Preferences of all players are known

• Single period game• Sum of payoffs are zero• An Equilibrium (or

Nash Equilibrium) - if none of the participants can improve their payoff

ASSUMPTIONS

PLAYER 2

PLAYER 1a

b

c d

1, -1 3, -3

-2, 2 0, 0

Player 1 is the first number ineach pair. We will get to {a,c}which is an Equilibrium

Two Person Game, Non-Zero Sum Game:

• Each player can invade the territory of the other (no guard)or Guard his own territory

• Pak’s payoff is given first.

• Inida always ranks Guard above no guard, so India has a Dominant Strategy

• Knowing what India will do, Pak decides to Guard as well.

• An Equilibrium--none of the participants can improve their payoff

ASSUMPTIONS

India

PAK Guard

No guard

Guard no guard

Better, 1st Worst, 4th

Worse, 2nd Best, 3rd

We will get to {Guard, Guard}which is an Equilibrium

Unstable Games: No Equilibrium Is Found

• Suppose KIM thinks that the solution is going to be: {b, c}

• Then, KIM has an incentive to switch to strategy-a

• Then JOHN has an incentive to switch to strategy-d, etc., etc.

Johnc d

KIMa

b

3, - 3 1, - 1

2, - 2 4, - 4

There is no, single stable equilibriumEach player may elect a random strategy

PLAYER 2

PLAYER 1

a

b

c d

1, -1 3, -3

-2, 2 0, 0

• For Player 1, strategy (a) is a dominant strategy - an action that maximizes the decision maker’s welfare independent of the actions of the other players.

– Also, strategy (b) is a dominated strategy, which is worst regardless of what others do

• Player 2 also has a dominant strategy of (c).

• Dominant strategies make games easy to solve.

Dominant strategy and domonated startegy

With dominant strategies of (a) for Player 1 and ( c) for Player 2, the solution will be {a, c},

which is an Equilibrium.


Do not advertise


Individual AAdvertise

4,3 5,12,5 8,2

•In real world the one player or both of them may not have a dominant strategy as shown in the matrix. Here firm B has the dominant strategy is to advertise whether firm A advertises or not, because the payoffs for B remain the same.• Firm A has no dominant strategy now. Because if B advertises, A earns a profit of 4 if it advertises and 2 if it does not. Thus if B advertises, firm A should advertise.•A earns a profit of 5 if it advertises and 8 if it does not advertise. Here the assumption is that A uses an expensive advertisement and it adds to its cost than revenue. It would shift the burden of increased cost to its consumers and get the larger share. It indicates that if B does not advertise, it is better for A not to advertise and get the larger share of the market. Hence A does not have a dominant strategy.


Do not advertise


Individual AAdvertise

4,3 5,12,5 8,6

B gets better payoff by advertising than not advertising. But if A does not advertise, B will not advertise as advertisement leads to lower payoffs to B than not advertising when A does not advertise. Hence the decision of B to advertise or not depends on whether A advertises or not. In other words, B does not have a dominant strategy on his own. B gets a share of 6 if it does not advertise when A also does not advertise.

No one can choose a dominant strategy independently of the other firm. When each player chooses its optimal strategy given the strategy of the other player, we will have a Nash equilibrium. Hence a dominant strategy equilibrium is always a Nash equilibrium, but a Nash equilibrium is not necessarily a dominant strategy.Problem 14.3 for assignment

• Every dominant strategy equilibrium is also a Nash equilibrium.

• Nash equilibrium can exist where there is no dominant strategy equilibrium.

• In Nash bargaining, two competitors or players "bargain" over some item of value. In a simultaneous-move, one-shot game, the players have only one chance to reach an agreement.

• A. Yes, the dominant strategy for firm A is “up.” If firm B chooses “left,” the highest payoff of $5 million can be achieved if Firm A chooses “up.” On the other hand, if firm B chooses “right,” the highest payoff of $7.5 million can be achieved if firm A again chooses “up.” No matter what firm B chooses, the highest payoff results for firm A occurs if A chooses “up.” Therefore, “up” is a dominant strategy for firm A.

B. No, there is no dominant strategy for firm B. If firm A chooses “up,” the highest payoff of $10 million can be achieved if firm B chooses “left.” On the other hand, if firm A chooses “down” the highest payoff of $5 million can be achieved if firm B chooses “right.” Therefore, there is no dominant strategy for firm B. The profit-maximizing choice by firm B depends upon the choice made by firm A.

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theory of game (game theory)