Intermediate Micro (Econ 311) Game Theoryjkennan/teaching/Rasmus_games.pdf · Intermediate Micro...

Intermediate Micro (Econ 311)

Game Theory

Prof. Rasmus Lentz

War Games

Econ 311 - Game Theory 2 / 13

● Player B moves after seeing Player A’s move.

A

B

no fight

fight

conventional

response

massive

retaliation

(

1, 1)

(

2, 0)

(

−1,−1)

● A’s strategy set: SA � {n , f }. B’s strategy set: SB � {c ,m}.

● 2 Nash Equilibria:{

(n ,m), ( f , c)}

. Only one subgame perfect: ( f , c).

War Games...


● But player B likes the (n ,m) equilibrium better. Maybe change the game...

A

B

no fight

fight

massive

retaliation

(

1, 1)

(

−1,−1)

conventional

response(

2, 0)

War Games...



A

B

no fight

fight

massive

retaliation

(

1, 1)

(

−1,−1)

conventional

response(

2, 0)

● Take away the conventional response option.

War Games...



A

B

no fight

fight

massive

retaliation

(

1, 1)

(

−1,−1)

conventional

response(

2, 0)

● Take away the conventional response option.

● One subgame perfect equilibrium: (n ,m).

Commitment


● B is better off if committed to modified game without conventional response option.

Repeated games


● Prisoner’s dilemma,

Suspect 2

Fink Silent

Suspect 1Fink (1, 1) (3, 0)

Silent (0, 3) (2, 2)

● Single Nash: (fink,fink).

● If the 2 players play this game repeatedly, could they support (silent,silent) by reputation?

Finitely repeated games


● If the game is repeated T times, answer is no, regardless of how big T is.

● By backward recursion:

● In the the last game, play is (fink,fink) regardless of what came before.

● In second to last game, play must then also be (fink,fink), since the final game will be

(fink,fink) regardless.

● And so on...

● This is a general result for stage games with unique Nash equilibria.

● If the stage game has multiple Nash equilibria, can sustain some corporation

● In second to last stage game: "if you cooperate, we will play the good Nash equilibrium

in the final stage game. If you don’t, we play the bad Nash."

Infinitely repeated games


● Trigger strategies. If a player deviates from corporation, revert to Nash equilibrium play

forever (grim strategy) or for a finite period.

● Let δ be the discount factor.

● Can (silent,silent) be supported as a subgame perfect equilibrium by use of a grim trigger:

If a player ever deviates, then revert to (fink,fink) forever on.

● Discounted payoff stream from cooperation,

V eq�

∞∑

t�0

2δt�

2

1 − δ.

● Compare to value of deviation,

V dev� 3 + δ

∞∑

t�0

1δt� 3 +

δ

1 − δ.

Infinitely repeated games...


● To sustain cooperation, it must dominate the value of deviating:

V eq ≥ V dev

m

2

1 − δ≥ 3 +

δ

1 − δ

m

2 − δ ≥ 3(1 − δ)

m

2δ ≥ 1

m

δ ≥1

2.

● That is, if the players are sufficiently patient.

Incomplete information - Signaling


● Sequential game where informed individuals move first and possibly relay information to

the uninformed player.

● In order to realize gains from trade, informed individuals may attempt to provide

information to the uninformed party.







● Classic example is the Spence signaling model of college as a signal of ability.








● Two types of individuals: Able and unable.









● Able individuals have marginal productivity a2 in the labor market.










● Unable individuals have marginal productivity a1 in the labor market. a1 < a2.











● In the labor force, the fraction b are able and the fraction 1 − b are unable.












● Employers cannot tell the productivity of the worker and so expects a randomly selected

worker to have productivity (1 − b)a1 + ba2.














● Suppose individuals are willing to work for any wage w > 0.














● Suppose individuals are willing to work for any wage w > 0.

● Pooling Equilibrium: Everybody is employed at wage wp � (1 − b)a1 + ba2.

College Signaling...


● High ability individuals would like to transmit information.




● Suppose individuals can acquire some chosen amount of education e (say years of

education).





education).

● Furthermore, suppose able individuals have education cost c2 per year of education.





education).


● And let unable individuals have education cost c1 per year of education. c1 > c2.





education).


● And let unable individuals have education cost c1 per year of education. c1 > c2.

● Make the stark assumption that education has no impact on productivity.

Payoff


● Payoff is wage minus education cost.

College Signaling... separating equilibrium


● Let’s think of an equilibrium where the able individuals take an education level, e, and

employers believe that individuals with education e are able. Perfect competition in input

market gives educated individuals a wage of a2.






● Unable individuals choose no education and employers believe that individuals without

education are unable. Perfect competition in input market gives uneducated individuals a

wage of a1.








wage of a1.

● Incentive compatibility constraints








wage of a1.


● Able individuals prefer to take education,

a2 − ec2 ≥ a1.








wage of a1.



a2 − ec2 ≥ a1.

● Unable individuals prefer not to take education,

a1 ≥ a2 − ec1.








wage of a1.



a2 − ec2 ≥ a1.

● Unable individuals prefer not to take education,

a1 ≥ a2 − ec1.

● Which combined yields,a2 − a1

c2

≥ e ≥a2 − a1

c1

.



● Hence, if e is set so as to satisfy the IC constraints, individuals that take education level eare rewarded with a higher wage a2 relative to those who do not take education, who get

a wage of a1.




a wage of a1.

● And this without education adding anything to an individual’s productivity!




a wage of a1.


● In fact, education is strictly wasteful.




a wage of a1.



● But that is often the point of attrition games.




a wage of a1.



● But that is often the point of attrition games.

● You can signal your type by engaging in wasteful activity to the point where the other type

finds it optimal not to follow you.

Intermediate Micro (Econ 311) Game Theoryjkennan/teaching/Rasmus_games.pdf · Intermediate Micro...

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