GAME THEORYGAME THEORYGAME THEORYGAME THEORY

Overview • Games capturing strategic decision-

making• Non-cooperative v/s cooperative games• Example of ‘Acquiring a Company’• Dominant Strategies• Nash Equilibrium• Maximin Strategies• Repeated games with finite/infinite

horizons• Sequential games• Advantage of moving first

Games and Strategic Decisions

• A game is any situation in which players (the participants) make strategic decisions – i.e. decisions that take into account each others actions and responses.

• Strategic decisions result in payoffs to the players – outcomes that generate rewards or benefits.

• A strategy is a rule or plan of action for playing the game.

– The optimal strategy is the one that maximizes the expected payoff.

Non-cooperative versus Cooperative Games

• Cooperative Game– Players negotiate binding contracts that allow

them to plan joint strategies• Example: Buyer and seller negotiating the price of

a good or service or a joint venture by two firms (e.g., Microsoft and Apple)

– Binding contracts possible to reach Pareto-superior position for both

• Non-cooperative Game– Negotiation and enforcement of a binding contract

are not possible• Example: Two competing firms take each other’s

likely behavior into account when independently setting pricing and advertising strategy to gain market share

Acquiring a Company (study yourself with clues given here)

• Company A: The Acquirer• Company T: The Target• A will offer cash for all of T’s shares• The value of T depends on the outcome of a

current oil exploration project.

– Failure: T’s value => $0

– Success: T’s value => $100/share

– All outcomes are equally likely• T’s value will be 50% greater with A’s

management.• A must submit the proposal before the exploration

outcome is known.• T will not choose to accept or reject until after the

outcome is known only to T.• How much should A offer?

Acquiring a Company (study yourself with clues given here)

Pdf = 1/100

0 100F

X

Suppose the amount offered for the firm is F. T accepts the offer if its value (X) is less than equal to F. T’s expected pay-off, as seen by A, is ½F. Under A’s management, F/2 is worth 1/2F.3/2=¾ F, which is still less than the offer amount F. So, A should not acquire the firm.

Myopic behavior under ‘Pay a Dollar Bill’ (study yourself with clues

given here)

• Rationale for a bid<$1.00 => as long as bid<1.00, net marginal gain>0

• Rationale for a bid>$1.00 => if a person has lost earlier, he may bid>1.00 in the hope that he would recoup a part of the accumulated loss if he wins the bid this time. The less risk-averse (or, more risk-loving) the person is, this situation is more likely to arise.

• However, the first time bid will generally be <1.00

Dominant Strategies

• One that is optimal, no matter what the opponent does.

• An Example

– A & B sell competing products

– They are deciding whether to undertake advertising campaigns

Payoff Matrix for Advertising Game

Firm A

AdvertiseDon’t

Advertise

Advertise

Don’tAdvertise

Firm B

10, 5 15, 0

10, 26, 8

• A: regardless of B, advertising is the best

• B: regardless of A, advertising is best

• Dominant strategy for A & B is to advertise, i.e (10,5)

• Do not worry about the other player

• Equilibrium in dominant strategy

• Note dominant strategy => Nash equilibrium as well

10, 5 15, 0

20, 26, 8

Firm A

AdvertiseDon’t

Advertise

Advertise

Don’tAdvertise

Firm B

Advertising Game – No Dominant Strategy

• A: No dominant strategy; depends on B’s actions

• B: Advertise as dominant strategy

• The optimal decision of a player without a dominant strategy will depend on what the other player does.

• (10,5) is a Nash equilibrium, though not a dominant strategy, because once it is reached, there will be no for either side to move away from it.

• So, Nash equilibrium is a much more general concept, of which dominant strategy constitutes only a sub-set

Dominant Strategies Equilibrium vs. Nash

Equilibrium • Dominant strategies are stable and self-

enforcing. • However, in many games one or more

players do not have a dominant strategy• Nash equilibrium is a more general

concept• A Nash equilibrium is a set of strategies

such that each player is doing the best it can given the actions of its opponents.– In the previous table, both firms advertise is

the Nash equilibrium.

• A dominant strategy equilibrium is a special case of a Nash equilibrium.

• Examples With A Nash Equilibrium– Two cereal companies– Operate in a market in which two new

types of cereal can be successfully introduced – crispy or sweet – only if each type is introduced by only one firm.

– Each firm only has the resources to introduce one cereal

– Each firm is indifferent about what it produces, as long as it does not introduce the same product as its competitor

– The firms behave in a non-cooperative way

Product Choice Problem

Product Choice Problem

Firm 1

Crispy Sweet

Crispy

Sweet

Firm 2

-5, -5 10, 10

-5, -510, 10

• There are two Nash equilibriums (even though no dominant strategy exists) – the bottom left and top right of the table (both arrows pointing to those two cells)

• Each is stable because once the strategies are chosen, no one will deviate

• Without more information, no way of knowing which equilibrium is likely to result.

Beach Location Game (study yourself with clues given here)

Where will the competitors locate (i.e. where is the Nash equilibrium)?

Ocean

0 B Beach A 200 yards

CY

Maximin Strategies (Best of a bad bargain!)

Firm 1

Don’t invest InvestFirm 2

0, 0 -10, 10

20, 10-100, 0

Don’t invest

Invest

Invest is a dominant strategy for firm 2. The outcome invest-invest is the only Nash equilibrium.

Firm 1’s managers must be sure that firm 2’s managers are rational. If firm 2 fails to invest, it would be very costly for firm 1.

If firm 1 is unsure about the rationality of firm 2 then it may play ‘don’t invest’. Then the worst that can happen is a loss of $10 mn, as opposed to a loss of $100mn.

Such a strategy is called MAXIMIN – maximizing the minimum gain that can be earned. A maximin strategy is conservative, and not profit maximizing.

I’s Min

-10

-100

II’s Min 0 10

Maximin for II=10=invest

Maximin for I=-10=no invest

Maximizing the Expected Payoff

• If firm 1 is unsure of what firm 2 will but can assign probabilities to each possible action of firm 2 then it can maximize its expected payoff.

• Firm 1’s strategies depend upon its assessment of the probabilities of different actions of firm 2 in the face of uncertainties over market conditions, future costs, competitor behavior etc.

Prisoner’s Dilemma

Prisoner A

Confess Don’t Confess

Confess

Don’tConfess

Prisoner B

-5, -5 -1, -10

-2, -2-10, -1

Confessing (-5, -5) is a dominant strategy for each prisoner.

Dominant strategies are also maximin strategies. So confess-confess is both a Nash equilibrium and a maximin

solution.

A’s Min

-5

-10

B’s Min -5 -10

Repeated Games• Oligopolistic firms play a repeated

game of making output and pricing decisions.

• With each repetition of the Prisoners’ Dilemma, firms can develop reputations about their behavior and study the behavior of their competitors.

• Firms search for the strategy that is best in a series of repeated games.

Example of a Repeated Game – Pricing Problem

Firm 1

Low Price High Price

Low Price

High Price

Firm 2

10, 10 100, -50

50, 50-50, 100

• Tit-for-tat strategy works best under

• Infinite repetitions of game – cooperative behavior is the rational response

– Cumulative loss of profits from under-cutting outweighs any short term gain from first time under cutting

Example of a Repeated Game – Pricing Problem

– Even if competitor unsure of tit-for-tat strategy, cooperation is still rational in an infinite period game, because expected gains from cooperation outweigh those from undercutting, even if probability of competitor playing tit-for-tat is small.

• Finite repetitions– Non-cooperation is the rational

outcome, with each one charging a low price every month.

– Outcome arises because each one strives to be the first to undercut price and make a windfall gain.

Tit-for-tat Strategy • The mere possibility that you play tit-for-tat

is sufficient for competitor to cooperate if the time horizon is long enough.

• Most managers don’t know how long they will be competing with their rivals, serving to make cooperation a good strategy, except near the end (called end game problem).

• Thus in a repeated game, prisoner’s dilemma can have a cooperative outcome. Industries where only a few firms compete under stable demand and cost conditions may cooperate even though no contractual arrangements are made. E.g. water meters.

• Failure to cooperate is the result of rapidly shifting demand or cost conditions, e.g. airlines.

Sequential Games

• Players move in turn

• Players must think through the possible actions and rational reactions of each player

• Examples– Responding to a competitor’s ad

campaign– Entry decisions– Responding to regulatory policy

• Scenario– Two new (sweet, crispy) cereals– Successful only if each firm produces one

cereal– Sweet will sell better– Both still profitable with only one producer

The Extensive Form of a Game

Firm 1

Crispy Sweet

Crispy

Sweet

-5, -5 10, 20

-5, -520, 10

Firm 2

Nash equilibrium

Nash equilibrium

• Assume that Firm 1 will introduce its new cereal first (a sequential game).

• Using a decision tree, work backward from the best outcome for Firm 1.

The Extensive Form of a Game

Crispy

Sweet

Crispy

Sweet

-5, -5

10, 20

20, 10

-5, -5

Firm 1

Crispy

Sweet

Firm 2

Firm 2

In this product-choice game, there is a clear advantage to moving first. By introducing the sweet cereal first, firm 1 creates a fait

accompli that forces firm 2 to introduce the crispy one.

The Advantage of Moving First

• Assume: Duopoly

Firm/100 10 and 10

0

Production Total

30

21

21

PQQ

MC

QQQ

QP

The Advantage of Moving First

• Duopoly

25.56 50.112

50.7 and 5.7 15

rg)(StackelbeFirst Moves Firm

Firm/50.112 15 and 5.7

CollusionWith

21

21

21

PQQ

PQQ

Choosing Output

Firm 1

7.5

Firm 2

112.50, 112.50 56.25, 112.50

0, 0112.50, 56.25

125, 93.75 50, 75

93.75, 125

75, 50

100, 100

10 15

7.5

10

15

• This payoff matrix illustrates various outcomes– Move

together, both produce 10

– Question• What if Firm

1 moves first?

Collusion

Counot

Stackelberg

Stackelberg

A Re-look at the same Example

Firm 1

a=7.5

Firm 2

112.50, 112.50 56.25, 112.50

0, 0112.50, 56.25

125, 93.75 50, 75

93.75, 125

75, 50

100, 100

b=10 c=15

a=7.5

b=10

C=15

• There is no dominant strategy for Firm 1, nor for Firm 2, as directions of arrows indicate

• For Firm 1, c is a dominated strategy – dominated by strategy b, as directions of red arrows indicate. So, for finding out profit-maximizing strategies firm 1’s c strategy can be deleted.

• Similarly, firm 2’s c strategy, which is dominated by strategy b, can be deleted.

• Thus, one can find out profit-maximizing strategies by merely concentrating on strategies a & b only of both firms (i.e., at 2x2 matrix).

• As both red & green arrows are pointed towards (100,100) cell, it is a Nash equilibrium.

• However, (112.5,112.5) constitutes a maximin strategy, as arrows in the last row & column, made out of the entire matrix, indicate.

Firm 1

Firm 1’s min

56.25

50

0

Firm 2’s min 56.25 50

0

An Additional Example

Prisoner A

No price rise Price rise

Price rise

Firm B

10,10 100,-30

140,35-20,30

•No dominant strategy

• (10,10) & (140,35) are Nash equilibria

•(10,10) is also maximin strategy

A’s Min

10

-20

B’s Min 10 -30

No price rise