As Agreed 30

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Game Theory: Some

Extensions

BUS291 Microeconomics A

Session 11

Outline

• Extend out understanding of game theory

• Games of imperfect and incomplete information

– The prisoners’ dilemma

– Bargaining

• Repeated games allow us to get to the question of cheating and punishment

The State of Play

• So far we have modelled simple

sequential move games

• We have also implicitly assumed all

players can observe all actions by their

rivals

• Moreover, all payoffs are know to all

parties

Two New Situations

• A game of imperfect information: some

player is unable to observe the earlier or

simultaneous move of some other player

• A game of incomplete information: when a

player is unsure about some game

characteristic, eg another player’s payoffs

The Prisoners’ Dilemma: A

Game of Imperfect Information • Assume there are two prisoners, M and N

guilty of a serious crime

• The police have enough evidence to

convict them both on relatively minor

charges

• M and N are separated and each offered a

reduced sentence for the major crime if

they confess and dob in the other

Spill

Mum

Spill

Mum

Spill

Mum

(-20, -20)

(-5, -25)

(-25, -5)

(-10, -10)

Spill

Mum

Spill

Mum

Spill

Mum

(-20, -20)

(-5, -25)

(-25, -5)

(-10, -10)

Spill

Mum

Spill

Mum

Spill

Mum

(-20, -20)

(-5, -25)

(-25, -5)

(-10, -10)

Spill

Mum

Spill

Mum

Spill

Mum

(-20, -20)

(-5, -25)

(-25, -5)

(-10, -10)

Spill

Mum

Spill

Mum

Spill

Mum

(-20, -20)

(-5, -25)

(-25, -5)

(-10, -10)

Spill

Mum

Spill

Mum

Spill

Mum

(-20, -20)

(-5, -25)

(-25, -5)

(-10, -10)

Prisoners’ Dilemma:

Observations • A strategic situation where all players have

a dominant strategy, but playing this

strategy leads to an outcome worse for all

players than if they had cooperated and

played an alternative strategy

• The duopoly game we considered in the

last lecture can be turned into a dilemma

when it is a simultaneous move game

High

Low

High

Low

High

Low

(3, 3)

(6, 1)

(1, 6)

(5, 5)

High

Low

High

Low

High

Low

(3, 3)

(6, 1)

(1, 6)

(5, 5)

High

Low

High

Low

High

Low

(3, 3)

(6, 1)

(1, 6)

(5, 5)

High

Low

High

Low

High

Low

(3, 3)

(6, 1)

(1, 6)

(5, 5)

Pure and Mixed Strategies

• Up to this point we have focused on

games that lend themselves to pure

strategies, ie a specific action at each

decision point

• Some games have no equilibrium in pure

strategies

• Another possibility is a mixed strategy that

allows for randomised actions at some or

all decision points

Left

Right

Left

Right

Left

Right

(0, 1)

(1, 0)

(0, 1)

An Incomplete Information

Bargaining Game • A seller (S) has a once only chance to sell

an item to sell that cost $1 000

• The buyer (B) either values the item at

$1 500 or $2 000 but the seller does not

know which

• S has to decide wether to charge $1 499

or $1 999

High

Low

High

Low

Reject

(999, 1)

(0, 0)

(499, 501)

(0, 0)

(999, -499)

(0, 0)

(499, 1)

(0, 0)

High Value

Low Value



1 

High

Low

High

Low

Reject

(999, 1)

(0, 0)

(499, 501)

(0, 0)

(999, -499)

(0, 0)

(499, 1)

(0, 0)

High Value

Low Value



1 

High

Low

High

Low

Reject

(999, 1)

(0, 0)

(499, 501)

(0, 0)

(999, -499)

(0, 0)

(499, 1)

(0, 0)

High Value

Low Value



1 

High

Low

High

Low

Reject

(999, 1)

(0, 0)

(499, 501)

(0, 0)

(999, -499)

(0, 0)

(499, 1)

(0, 0)

High Value

Low Value



1 

An Incomplete Information

Bargaining Game: Observations • By offering the low price of $1 499 the

seller can gain $499 with certainty

• The seller will offer the higher price if their estimation of  is high enough so that the expect profit from the high price exceeds $499, ie

  1999  1000   1   0    999  499

  

499

999  0.5

Repeated Games

• In many situations players find themselves

repeatedly making the same decisions

• We will examine just one model but the

fact that a game is repeated can make a

huge difference to the outcome

• Let us have two firms A and B setting price

daily; simultaneously at the start of the

day, and per-day demand curve is D(p)

Repeated Games

• Each day’s choice is a Bertrand game (the

state game) within the overall game

• One possible outcome is that firms make

the same choice as for a single shot

version of this game

– Price equal to common marginal cost

– Nash equilibrium

– Both A and B earn zero profits

Grim-Trigger Strategy

• The Bertrand outcome makes no use of the two firms repeated interaction

• Suppose that A and B agree to charge a price ps, greater than marginal cost, on the understanding that any cheater will be punished forever by all firms setting price equal to marginal cost

• Question: can such a strategy be self- enforcing and credible

Some Preliminaries

• Each firm’s profit from sticking to the agreement is:

S  1

2 D pS   pS  c 

For every day that it escapes detection a cheater earns approximately:

C  D pS   pS  c 

Once punishment sets in:

P  0

Some Preliminaries

• Punishment is credible

• If one firm prices at marginal cost it is in

the other’s interests to do likewise

• If each firm expects the other to respond

to cheating by setting price equal to

marginal cost then it is in the firm’s interest

to do so as well

Benefits and Costs of Cheating

• Discounting required

• The benefit of cheating is the present

value of the extra profit earned before

punishment sets in

• If detection takes just one day the benefit

of cheating is:

C  S  S

Benefits and Costs of Cheating

• The cost of cheating is present value of

forgone profits when punishment takes

hold, which is:



S i

Cheating is only worthwhile if:

S 

S i

 i  1 

PS (1i)

 PS

(1i) 2 

(1i) 3

 ...

Repeated Games: Observations

• A grim-trigger strategy can support perfect

equilibrium collusive pricing so long as the

critical discount rate exceeds that of the

cheater

• This is independent of the particular value

of ps • The critical discount rate of course falls as

the time to detection increases

Finitely Repeated Games

• The effect of punishment unravels when the

game has a known end period

• Consider the example just given with firm B

known to be shutting down it’s operation in two

years time

• On the last day before B closes the game is a

one-shot Bertrand pricing game

• The unique Nash equilibrium is price at marginal

cost

Finitely Repeated Games

• Given that the result of the last iteration is known, at price equal marginal cost, the second last day becomes a one-shot Bertrand pricing game

• And so on …

• In general, if the state game has a unique Nash equilibrium, the unique perfect equilibrium for the finitely repeated game is the one-shot equilibrium in every period

Conclusion

• Gave some consideration to games of

imperfect and incomplete information

• Looked at some remarkable results when

games are infinitely repeated

• And possibly just as surprising we saw

how it can all unravel when the game has

a known end period