ECON6021 (Nov 2004)

Game Theory

Outlines

Nash Equilibrium Simultaneous games dominant and dominated strategies Prisoner dilemma Meeting-competition clauses Sequential games Subgame perfect equilibrium

What is a game?

A game consists ofA set of playersRules of the gamesA payoff function, one for each player, telling us his payoff given actions of all playersKnowledge of the players (viz., whether one firm’s MC is also known by the other, …)Remark: Unless o. w. stated, we assume common knowledge among players about bullet points 1 – 3. If not, we are in a domain of asymmetric information—a forthcoming topic in our class

John Nash

Nash equilibrium A Nash equil. is a complete prescription of what

each player will do in every possible contingency in such a way that no player will have incentive to unilaterally deviate from his own prescription. (John Nash--Nobel Laureate 1994)

Remarks: Nash equilibrium may not be unique (multiple equil.) Nash equilibrium can be mixed (penalty kick game) The game need not be symmetric. (Symmetric games

are special cases.)

Simultaneous Game

Output Decisions in Duopoly

2 firms producing the same product each can produce 0, 1, 2, 3 units of

output at a zero marginal cost the inverse market demand function

is as follows: Q. demanded 2 3 4 5

6 price 16 10 6 3

1.5

Output Decisions in Duopoly

firm 2's output1 2 3

1 (16, 16) (10, 20) (6. 18)firm 1's 2 (20, 10) (12, 12) (6, 9)output 3 (18, 6) (9, 6) (4.5, 4.5)

Payoff Table

Each producing 1 unit maximizes their joint profits

Each producing 2 units will be the outcome without a binding agreement

Cover story competition Each needs to determine

what cover story to use for the coming issue

70 percent of potential are interested only in EOC; the other 30 percent interested only in Harbor fest; If same story, buy either one with equal chance

Each cares only about its size of readership

The above is commonly known between Next and Oriental

Then, EOC being chosen by both is a Nash equilibrium

Oriental

Next

EOC Harbor fest

EOC 35, 35

70, 30

Harbor fest

30, 70

15, 15

Dominant & Dominated Strategies

A strategy is a dominant strategy for a player if that strategy gives a higher payoff than other strategies of his do, irrespective of the other player’s strategy.

A dominant strategy may not exist A strategy is a dominated strategy for a

player if that strategy gives a lower payoff than some other strategy of the player does, irrespective of the other player’s strategy.

A dominated strategy may not exist

How to find out a Nash equilibrium?

If you have a dominant strategy, use it.

If you have a dominated strategy, don’t use it.

Assume similar behavior on the part of your rivals

Cover story competition (cont.)

Next and Oriental each needs to determine what cover story to use for the coming issue

70 percent of potential are interested only in EOC; the other 30 percent interested only in Harbor fest

Same story, 60% among the interested readers will buy Next, the other 40% buy from Oriental

Oriental

Next EOC Harbor fest

EOC 42, 28 70, 30

Harbor fest

30, 70 18, 15

Prisoners’ Dilemma

Two suspects were prosecuted for robbery. Now they are under investigation separately and simultaneously. What will each of them do to minimize his/her years of imprisonment?

suspect 2confess not confess

suspect 1 confess 10, 10 0, 20not confess 20, 0 1, 1

Prisoners’ Dilemma

Neither to confess is best for both suspects when they are viewed as an entity (collective rationality).

But each to confess is optimal regardless of the other does (individual rationality)

Individual rationality may not lead to social optimum.

Much research in oligopoly is on collusion, I.e., how firms can resolve the prisoners’ dilemma.

Application: Meeting-Competition Clauses

firm 2low price high price

firm 1 low price 5, 5 15, 0high price 0, 15 10, 10

The equilibrium outcome is (low price, low price). Collusion of (high price, high price) is unsuccessful, at least for a one-shot relationship.

But collusion can be supported by meeting-competition clauses.

Meeting-Competition Clauses

A meeting-competition clause states that if another firm offers a lower price, the seller will match it or release the buyer from the contract.

With meeting-competition clauses, both firms choose high prices. Hence MCCs might be anti-competitive, rather than pro-competitive!!

Meeting-Competition Clauses

5, 5 15, 00, 15 10, 10

firm 2low price high price

firm 1 low price 5, 5 5, 5high price 5, 5 10, 10

With MCC

Most-Favored-Nation Clauses

A most-favored-Nation clause guarantees the buyer that the seller would not offer a lower price to any other buyer, current or future, without offering the same price decrease to the initial buyer.

This rebate system creates a penalty for cheating on the cartel:

If either firm deviates from the agreement by cutting its price, it would have to cut prices to all previous buyers as well!

Sequential Game

Entry Game

New Firm

Enter

Incumbent

Accommodate

Fight Price War

$100,000 to New Firm$100,000 to Incumbent

-$200,000 to New Firm-$100,000 to Incumbent

Keep Out

$0 to New Firm$300,000 to Incumbent

Entry Game

Traditionally, a sequential game is represented in a matrix form like this (as if it is a simultaneous game):

incumbent

accommodate fight price war

new firm

enter 100k,100k -200k,-100k

keep out

0,300k 0,300k

Although (keep out, fight price war) is a Nash equilibrium. But we wonder if the incumbent will really implement the fighting plan if the new firm has really entered. It is not in its interest to do so…

Subgame Perfection (Selten)

For sequential games, an equilibrium consists of instructions regarding players’ prescribed actions even for the point which is never reached should the equilibrium strategies are followed.

Refinement/Restriction: the agents at the point is still required to act optimally even if the equilibrium does not prescribe the point to be reached. Equilibria survive such a refinement is called subgame perfection. Subgame perfect equilibria rule out incredible threats.

Poison Pills against Hostile takeovers

Sharon – 1 million shares, $2 each Shareholder rights plan – allowing each

shareholder to buy from the company new shares equal to his current stake at half a price.

If any party acquires 100,000 or more Sharon’s shares, then management can activates the plan – but the one whose acquisition activated the rights, however, is depleted of the rights

It increases the costs of acquisition, making successful hostile takeover more difficult

Sharon

Hilda

acquires 100,000shares

doesn’t acquire

does not

activates rights

Hilda loses on initial stake + cost of takeover rises

Poison Pills

Poison Pills (cont)

Suppose the raider – Hilda – has acquired 100,000 shares and the management activates the plan. Then price of share adjusts from $2 to $v.

Suppose all eligible shareholders use their rights It is dominant strategy for each shareholder to exercise

his rights--the market price is v, while you can buy it at v/2

The worth of the firm = $2M + 900,000(v/2)

The number of share = 1.9M

Hence, each share is worth (2 + 0.45v)/1.9 =v. Hence, v = 1.38.

Poison Pills

Hostile takeover is more difficult to succeed for two reasons Acquiring the whole firm is more costly --

After the plan is activated, the remaining shares is worth 1.38 x 1.8m = $2.48m, rather than 2 x 0.9m = $1.8m – the worth in the absence of poison pills

The worth of the 100,000 shares is reduced – Hilda paid 2 x 100,000 = $0.2m for it. But after the plan is activated, it is worth 1.38 x 100,000 = $0.138m only.

Are Poison Pills beneficial to Shareholders?

Poison Pills are takeover defenses

They remove the ability of the owners (shareholders) of the firm to sell their shares to a buyer that their nominal agents – management and the board – do not like.

Takeover defenses, if not strong enough to make takeovers impossible, may improve the stockholders’ bargaining position and raise the price they ultimately receive

Judicial Procedures

A defendant’s fate rests in the hands of three Judges Three possible outcomes: acquittal, life in prison,

and death sentence

Judge A’s ranking

Judge B’s ranking

Judge C’s ranking

Best Death Sentence

Life in Prison

Acquittal

Middle Life in Prison

Acquittal Death Sentence

Worst Acquittal Death Sentence

Life in Prison

Judicial Procedures (cont.)

The so called status quo Judicial Procedure is used:

Stage 1: determine (by majority vote) innocence or guilt by majority rule

Stage 2: If innocence, acquittal automatically; if guilty, determine (by majority rule) the appropriate punishment.

What will the judges do in equilibrium?

Judicial Procedures (cont.)

What will the judges do under different judicial procedures?

Roman Tradition: First decide if the death penalty should be imposed for this case. If not, then decide whether a life sentence is justified. If, after proceeding down the list, no sentence is imposed, then the defendant is acquitted.

Mandatory Sentencing: First specify the sentence for the crime. Then determine whether the defendant should be convicted.