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...
Econ 311 - Game Theory 3 / 13
● 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...
Econ 311 - Game Theory 3 / 13
● 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)
● Take away the conventional response option.
War Games...
Econ 311 - Game Theory 3 / 13
● 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)
● Take away the conventional response option.
● One subgame perfect equilibrium: (n ,m).
Commitment
Econ 311 - Game Theory 4 / 13
● B is better off if committed to modified game without conventional response option.
Repeated games
Econ 311 - Game Theory 5 / 13
● 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
Econ 311 - Game Theory 6 / 13
● 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
Econ 311 - Game Theory 7 / 13
● 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...
Econ 311 - Game Theory 8 / 13
● 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
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
Incomplete information - Signaling
Econ 311 - Game Theory 9 / 13
● 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.
● Pooling Equilibrium: Everybody is employed at wage wp � (1 − b)a1 + ba2.
College Signaling...
Econ 311 - Game Theory 10 / 13
● High ability individuals would like to transmit information.
College Signaling...
Econ 311 - Game Theory 10 / 13
● High ability individuals would like to transmit information.
● Suppose individuals can acquire some chosen amount of education e (say years of
education).
College Signaling...
Econ 311 - Game Theory 10 / 13
● High ability individuals would like to transmit information.
● Suppose individuals can acquire some chosen amount of education e (say years of
education).
● Furthermore, suppose able individuals have education cost c2 per year of education.
College Signaling...
Econ 311 - Game Theory 10 / 13
● High ability individuals would like to transmit information.
● Suppose individuals can acquire some chosen amount of education e (say years of
education).
● Furthermore, suppose able individuals have education cost c2 per year of education.
● And let unable individuals have education cost c1 per year of education. c1 > c2.
College Signaling...
Econ 311 - Game Theory 10 / 13
● High ability individuals would like to transmit information.
● Suppose individuals can acquire some chosen amount of education e (say years of
education).
● Furthermore, suppose able individuals have education cost c2 per year of 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
Econ 311 - Game Theory 11 / 13
● Payoff is wage minus education cost.
College Signaling... separating equilibrium
Econ 311 - Game Theory 12 / 13
● 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.
College Signaling... separating equilibrium
Econ 311 - Game Theory 12 / 13
● 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.
College Signaling... separating equilibrium
Econ 311 - Game Theory 12 / 13
● 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.
● Incentive compatibility constraints
College Signaling... separating equilibrium
Econ 311 - Game Theory 12 / 13
● 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.
● Incentive compatibility constraints
● Able individuals prefer to take education,
a2 − ec2 ≥ a1.
College Signaling... separating equilibrium
Econ 311 - Game Theory 12 / 13
● 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.
● Incentive compatibility constraints
● Able individuals prefer to take education,
a2 − ec2 ≥ a1.
● Unable individuals prefer not to take education,
a1 ≥ a2 − ec1.
College Signaling... separating equilibrium
Econ 311 - Game Theory 12 / 13
● 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.
● Incentive compatibility constraints
● Able individuals prefer to take education,
a2 − ec2 ≥ a1.
● Unable individuals prefer not to take education,
a1 ≥ a2 − ec1.
● Which combined yields,a2 − a1
c2
≥ e ≥a2 − a1
c1
.
College Signaling... separating equilibrium
Econ 311 - Game Theory 13 / 13
● 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.
College Signaling... separating equilibrium
Econ 311 - Game Theory 13 / 13
● 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.
● And this without education adding anything to an individual’s productivity!
College Signaling... separating equilibrium
Econ 311 - Game Theory 13 / 13
● 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.
● And this without education adding anything to an individual’s productivity!
● In fact, education is strictly wasteful.
College Signaling... separating equilibrium
Econ 311 - Game Theory 13 / 13
● 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.
● And this without education adding anything to an individual’s productivity!
● In fact, education is strictly wasteful.
● But that is often the point of attrition games.
College Signaling... separating equilibrium
Econ 311 - Game Theory 13 / 13
● 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.
● And this without education adding anything to an individual’s productivity!
● In fact, education is strictly wasteful.
● 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.