Other Issues in Game Other Issues in Game TheoryTheory

BusinessBusinessNegotiationsNegotiations

ContractsContracts

Dominant Strategy• Regardless of whether FIRM 2 chooses strategy A,

B, or C, FIRM 1 is better off choosing “a”!• “a” is Player 1’s Dominant Strategy!• Dominance is a solution strategy --- but doesn’t

always lead to “resting point” --- so we introduce the Nash solution --- Nash equilibrium

FIRM 1

FIRM 2 A B C

A ------ 12, 11 11,12 14,13

b 11,10 10,11 12,12

c 11,15 10,13 13,14

FIRM 1

FIRM 2 A B C

A ------ 12, 11 11,12 14,13

b 11,10 10,11 12,12

c 11,15 10,13 13,14

Putting Yourself in your Rival’s Shoes

• What should FIRM 2 do?– 2 has no dominant strategy!– But 2 should reason that 1 will play “a”.– Therefore 2 should choose “C”.

FIRM 1

FIRM 2 A B C

A ------ 12, 11 11,12 14,13

b 11,10 10,11 12,12

c 11,15 10,13 13,14

The Outcome

• This outcome is called a Nash equilibrium:– “a” is player 1’s best response to “C”.– “C” is player 2’s best response to “a”.

A NASH EQUILIBRIUM IN WHICH EVERY PLAYER PLAYS A PURE STRATEGY IS CALLED A PURE STRATEGY NASH EQUILIBRIUM

ANY STRATEGY THAT IS NOT COMPLETELY DETERMINISTIC, BUT INSTEAD INVOLVES CHANCE (RANDOMIZATION), IS CALLED A MIXED STRATEGY --- SO A NASH EQUILIBRIUM IN WHICH AT LEAST ONE PLAYER PLAYS A MIXED STRATEGY IS CALLED A MIXED STRATEGY NASH EQUILIBRIUM

COORDINATION GAMES USUALLY RESULT IN MIXED STRATEGIES AS WELL AS PURE NASH

STRATEGIES ARE BASED ON 210

VOLT APPLIANCE VS

120 VOLT APPLIANCE

Firm 2

210 v 120 v

Firm 1

210 v 100, 100 0, 0

120 v 0,0 100,100

A MORE SIMPLIFIED COORDINATION GAME

Firm 2

210 v 120 v

Firm 1

210 v 100, 100 0, 0

120 v 0,0 100,100

There is 1 mixed strategy {1/2[210v], ½[120v]} for each firm

There are 2 pure Nash {both firms choose 210 v} and

{ both firms choose 120 v }

Both get

8 years

Funk gets 2 yrs

Naylor gets 10 years

Funk gets 10

Naylor gets 2

Both get

4 years

confess

mum

mum

confess

Naylor

Funk

Two stock brokers, Funk and alias, Naylor, are indicted by the N.Y. Attorney General for allegedly making use of illegal inside information --- but the evidence is weak

The attorney general brings them in to interrogate, one at a time

Both Funk and Naylor have two possible strategies, confess or remain mum

4 possible strategies are outlined in the normal form game below

Both get

8 years

Funk gets 2 yrs

Naylor gets 10 years

Funk gets 10

Naylor gets 2

Both get

4 years

confess

mum

mum

confess

Naylor

Funk

SO, WHAT’S GOING TO HAPPEN HERE? THE DOMINANT STRATEGY FOR BOTH BROKERS IS TO CONFESS!

BUT EACH IS DOING WORSE THAN IF THEY COULD TRUST EACH OTHER AND ONLY GET 4 YEARS BASED ON THE WEAK EVIDENCE BY REMAINING MUM (OH,OH, HERE COMES THE ROLE OF THE DEFENSE ATTORNEYS!)

THIS IS THE PRISONER’S DILEMMA GAME

(5, 5) (-2, 8)

(8, -2) (2, 2)

Price = $2,000

Price = $1,000

Price = $2,000

Price = $1,000

Israelsen

Simmons

Now recall the problems with the “Sweezy” or Kinked demand curve oligopoly case we introduced in Ch. 9

If one firm reduces price, the rival firm will match this action by also reducing price, but will not match price increases

Suppose now we have two firms, Simmons and israelsen who react on pricing Suppose now we have two firms, Simmons and israelsen who react on pricing of their product and reap the profits (in $millions) given in the normal form of their product and reap the profits (in $millions) given in the normal form given belowgiven below

(5, 5) (-2, 8)

(8, -2) (2, 2)

Price = $2,000

Price = $1,000

Price = $2,000

Price = $1,000

Israelsen

Simmons

What is going to happen here?What is going to happen here?

If Simmons and Israelsen form some sort of cartel --- then there is incentive to not follow through on the agreement, however supposed to be binding

The Nash is that both lower their price from $2,000 to $1,000 in hopes of capturing the market

(5, 5) (-2, 8)

(8, -2) (2, 2)

Price = $2,000

Price = $1,000

Price = $2,000

Price = $1,000

Israelsen

Simmons

Now suppose they offer a “most favored customer clause” , whereby a customer who buys early at a high price gets a rebate if price is later set at a lower price --- the rebate will lower profits ---- so the payoff is now given below

(5, 5) (-2, 8)

(8, -2) (2, 2)

Price = $2,000

Price = $1,000

Price = $2,000

Price = $1,000

Israelsen

Simmons

We now get two pure Nash {both firms set price at $2,000} and {both firms set price at $1,000}

And then we get a mixed strategy { both firms choose price = $2,000 with probability 4/5, and choose price = $1,000 with probability 1/5}

So the weight on the choice suggest the most favored customer clause provides incentives to hold at the $2,000 price --- a possible way out of the kinked demand curve

Suppose Dell and GE are considering engaging in a joint venture. Each will have to invest $12 million in assets that are of no value outside the project (specialized or specific investments and costs)

If both firms act in accord with their promises, the annual “economic profit” to each firm is $3 million

If one or both do not act in accord with promises, then the annual profit is as shown in the following normal form

(3, 3) (6, -2)

(-2, 6) (0, 0)

Dell

Accord

No accord

Accord No accord

GE

Economic profit in $millions

BOTH FIRMS HAVE THE OPTION TO NOT PLAY THE GAME

Will a contract be drawn up and signed by both parties?

Yes --- the Nash equilibrium is (3,3)

Without such a contract, each firm would have incentive to go their separate ways after investment ---- managers may be reluctant to take the risk of investment --- so trust is what binds the contract --- coordination

(3, 3) (6, -2)

(-2, 6) (0, 0)

Dell

Accord

No accord

Accord No accord

GE

Economic profit is profit above what could have been earned in alternative investment opportunities for the $12 million

THIS IS NOT A PRISONER’S DILEMMA GAME

WHEN IS A THREAT CREDIBLE?

Firms often signal to each other to indicate intentions --

(2, 3) (3, -1)

(7, 11) (11, 8)

Youngberg

Low price

High price

Low price High price

McKenna

Youngberg announces it is moving to lower price --- McKenna then intends to significantly lower its own price signaling willingness to engage in price war {see the payoff matrix below}

But McKenna’s threat is not credible --- profits at high price are more than profits at low price --- dominant strategy for McKenna is high price, irrespective of Youngberg’s price ---- Nash is (7,11)