Game Theory

Elise Bonzon

[email protected]

LIPADE - Universit´e Paris Descartes http://www.math-info.univ-paris5.fr/

v

Game Theory

M. Osborne, A. Rubinstein, A Course in Game Theory

Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact. The basic assumptions that underlie the theory are that decision-makers pursue well defined exogenous objectives (that are rational) and take into account their knowledge or expectations of other decision-makers’

behavior (they reason strategically).

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Belote?

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

⇒ Belote?

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

⇒ Belote?

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

⇒ Belote?

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

Terminology: partial taxonomy

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

⇒ Belote?

2-players games / n-players games Static games / dynamic games

Zero-sum games / Non zero-sum games

Games with perfect information / Games with imperfect information Coalitional games / Non-coalitional games

⇒ Chess?

⇒ Belote?

Bibliography

J. Von Neumann, O. Morgenstein, Theory of Games and Economic Behavior, Princeton University Press, 1944.

D.Luce, H. Raiffa, Games and Decisions: Introduction and Critical Survey, Wiley, 1957.

D. Fudemberg, J. Tirole, Game Theory, MIT Press, 1991.

M. Osborne, A. Rubinstein, A Course in Game Theory, MIT Press, 1994.

M. Yildizoglu, Introduction ` a la th´ eorie des jeux, Dunod, 2003.

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

1

Formalisation of a game Definitions

Preferences

Strategic and extensive forms

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Definitions Preferences

Strategic and extensive forms

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Formalisation of a game

Formalisation of a game

A game is defined by:

Who? → Players

What? → Actions, strategies How? → Sequence of events

How much? → What is the payoff of each outcome for each player?

Definitions Preferences

Strategic and extensive forms

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Formalisation of a game

Utility

The players are rational: they want to obtain the best payoff as possible To do so, they need to know the preferences of each player over the different outcomes of the game

Utility: measurement of the “happiness” of a player for each outcome of the game. Allow us to define the preferences of players.

Example : Battle of the Sexes.

Anne and Charles wish to go together to a football game or the opera.

Their main concern is to go out together, but Anne prefers the opera and

Charles the football game.

Utility

The players are rational: they want to obtain the best payoff as possible To do so, they need to know the preferences of each player over the different outcomes of the game

Utility: measurement of the “happiness” of a player for each outcome of the game. Allow us to define the preferences of players.

Example : Battle of the Sexes.

Anne and Charles wish to go together to a football game or the opera.

Their main concern is to go out together, but Anne prefers the opera and

Charles the football game.

Formalisation of a game

Utility

Ordinal utility: ranking of the outcomes of the game Anne’s preferences: O

O

F

F

O

F



F

O

Cardinal utilities: associate to each outcome a numerical value Charles’ preferences: u

(F

, F

) = 2, u

(O

, O

) = 1, u

(O

, F

) = 0, u

(F

, O

) = 0

Problem: Size of the representation. Compact representation of the

preferences.

Utility

Ordinal utility: ranking of the outcomes of the game Anne’s preferences: O

O

F

F

O

F



F

O

Cardinal utilities: associate to each outcome a numerical value Charles’ preferences: u

(F

, F

) = 2, u

(O

, O

) = 1, u

(O

, F

) = 0, u

(F

, O

) = 0

Problem: Size of the representation. Compact representation of the

preferences.

1

Formalisation of a game Definitions

Preferences

Strategic and extensive forms

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Formalisation of a game

Strategic games

A strategic game is defined by:

A finite set of players N = {1, . . . n},

For each player i ∈ N a nonempty set of strategies S

= {s

, . . . , s

} (the set of actions available to player i )

For each player i ∈ N a utility function u

: S

× . . . × S

→ IR which associates a numerical value to each set of strategies.

H H H

H H 1

2 O

F

O

(2, 1) (0, 0)

F

(0, 0) (1, 2)

Formalisation of a game

Strategic games

A strategic game is defined by:

A finite set of players N = {1, . . . n},

For each player i ∈ N a nonempty set of strategies S

= {s

, . . . , s

} (the set of actions available to player i )

For each player i ∈ N a utility function u

: S

× . . . × S

→ IR which associates a numerical value to each set of strategies.

Example : Battle of the Sexes.

H H H

H H 1

2 O

F

O

(2, 1) (0, 0)

F

(0, 0) (1, 2)

Strategic games

s

is a pure strategy of the player i , that is an action plan which stipulates an action each times i has to play.

Ex: s

= O

, s

= F

s

is a winning strategy for the player i if it allows i to win whatever the other players do

s = {s

, . . . , s

}, where ∀i , s

∈ S

is a strategy profile. A strategy profile must include one and only one strategy for every player, and is a set of strategies for each player which fully specifies all actions in a game.

Ex: s = {O

F

}

s

represents the strategy profile s except the strategy of the player i : s

= {s

, . . . , s

, s

, . . . s

},

Ex: s = {F }

Formalisation of a game

Extensive games

An extensive game is defined by:

A finite set of players N = {1, . . . n}, A finite game-tree made up of

A set of nodes which represent the actions

A set of branches which represent the alternatives at each time Each terminal (leaf) node of the game tree has an n-tuple of payoffs, meaning there is one payoff for each player at the end of every possible play A partition of the non-terminal nodes of the game tree in n + 1 subsets, one for each player. Each player’s subset of nodes is referred to as the

“nodes of the player”.

Each set of nodes of a player is partitioned in information sets, which

make certain choices indistinguishable for the player when making a move

Extensive games - Battle of the sexes

A

C

(2, 1) O

(0, 0) F O

C

(0, 0) O

(1, 2) F F

Player 1

Player 2 Player 2

Formalisation of a game

Extensive games - Battle of the sexes

A

C

(2, 1) O

(0, 0) F O

C

(0, 0) O

(1, 2) F F

Player 1

Player 2 Player 2

Information sets: Anne: {A}; Charles: {{C (O)}, {C (F )}}

Sequential actions

Relation between extensive and strate- gic forms

Each game in extensive form corresponds to a game in strategic form in which players simultaneously choose their strategies

A game in strategic form can correspond to several games in extensive

form

Formalisation of a game

Relation between extensive and strate- gic forms

Strategic form:

H H H H H 1

2 O

F

O

(2, 1) (0, 0) F

(0, 0) (1, 2) Extensive form:

A

C

(2, 1) O

(0, 0) F O

C

(0, 0) O

(1, 2) F F Player 1

Player 2 Player 2

Is there another extensive form of this strategic game?

Relation between extensive and strate- gic forms

Strategic form:

H H H H H 1

2 O

F

O

(2, 1) (0, 0) F

(0, 0) (1, 2) Extensive form:

A

C O

C F Player 1

Player 2 Player 2

Formalisation of a game

Relation between extensive and strate- gic forms

Extensive form:

A

C

(2, 1) O

(0, 0) F O

C

(0, 0) O

(1, 2) F F Player 1

Player 2 Player 2

s

: O

(O

), O

(F

) ; s

: O

(O

), F

(F

) ; s

: F

(O

), O

(F

) ; s

: F

(O

), F

(F

).

Strategic form:

H H H

H H 1

2 s

s

s

s

O

(2, 1) (2, 1) (0, 0) (0, 0)

2

Solution concepts Dominated strategies Nash Equilibrium Pareto criterion Security levels Examples of games

Subgame perfect equilibrium

3

Repeated games

4

Two players zero-sum games

1

Formalisation of a game

2

Solution concepts Dominated strategies Nash Equilibrium Pareto criterion Security levels Examples of games

Subgame perfect equilibrium

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Dominated strategies

H H H H

H 1

2 A B

C (2, 1) (0, 1)

D (3, 1) (4, 2)

Solution concepts

Dominated strategies

H H H H

H 1

2 A B

C (2, 1) (0, 1)

D (3, 1) (4, 2)

Dominated strategies

H H H H

H 1

2 A B

C (2, 1) (0, 1) D (3, 1) (4, 2)

A strategy s

strictly dominates another strategy s

if s

is strictly better than s

for the player i whatever the strategies of the other players are:

∀s

∈ S

, u

(s

, s

) < u

(s

, s

)

Solution concepts

Dominated strategies

H H H H

H 1

2 A B

C (2, 1) (0, 1) D (3, 1) (4, 2)

A strategy s

weakly dominates another strategy s

if s

is at least as good than s

for the player i whatever the strategies of the other players are; and strictly better than s

for at least one combination of strategies:

∀s

∈ S

, u

(s

, s

) ≤ u

(s

, s

) and

∃s

∈ S

t.q. u

(s

, s

) < u

(s

, s

)

Iterated elimination of dominated strategies

H H H

H H 1

2 G M D

H (4, 3) (5, 1) (6, 2)

M (2, 1) (8, 4) (3, 6)

B (3, 0) (9, 6) (2, 8)

Solution concepts

Iterated elimination of dominated strategies

H H H

H H 1

2 G M D

H (4, 3) (5, 1) (6, 2)

M (2, 1) (8, 4) (3, 6)

B (3, 0) (9, 6) (2, 8)

Iterated elimination of dominated strategies

H H H

H H 1

2 G M D

H (4, 3) (5, 1) (6, 2)

M (2, 1) (8, 4) (3, 6)

B (3, 0) (9, 6) (2, 8)

Solution concepts

Iterated elimination of dominated strategies

H H H

H H 1

2 G M D

H (4, 3) (5, 1) (6, 2)

M (2, 1) (8, 4) (3, 6)

B (3, 0) (9, 6) (2, 8)

Iterated elimination of dominated strategies

H H H

H H 1

2 G M D

H (4, 3) (5, 1) (6, 2)

M (2, 1) (8, 4) (3, 6)

B (3, 0) (9, 6) (2, 8)

Solution concepts

Iterated elimination of dominated strategies

H H H

H H 1

2 G M D

H (4, 3) (5, 1) (6, 2)

M (2, 1) (8, 4) (3, 6)

B (3, 0) (9, 6) (2, 8)

Iterated elimination of dominated strategies

The order of elimination of strictly dominated strategies does not affect the final result

The order of elimination of weakly dominated strategies can affect the final result

The process of iterated elimination of dominated strategies does not

necessarily lead to a unique solution

1

Formalisation of a game

2

Solution concepts Dominated strategies Nash Equilibrium Pareto criterion Security levels Examples of games

Subgame perfect equilibrium

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Pure strategy Nash equilibrium

H H H H

H 1

2 G M D

H (4, 3) (5, 1) (6, 2) M (2, 1) (8, 4) (3, 6) B (3, 0) (9, 6) (2, 8)

A pure strategy Nash equilibrium (PNE) is a strategy profile such that

each player’s strategy is an optimal response to the strategies of the other

players

Solution concepts

Pure strategy Nash equilibrium

H H H H

H 1

2 G M D

H (4, 3) (5, 1) (6, 2) M (2, 1) (8, 4) (3,6) B (3, 0) (9, 6) (2, 8)

A pure strategy Nash equilibrium (PNE) is a strategy profile such that

each player’s strategy is an optimal response to the strategies of the other

players

Pure strategy Nash equilibrium

H H H H

H 1

2 G M D

H (4, 3) (5, 1) (6, 2) M (2, 1) (8, 4) (3, 6) B (3, 0) (9, 6) (2, 8)

A pure strategy Nash equilibrium (PNE) is a strategy profile such that

each player’s strategy is an optimal response to the strategies of the other

players

Solution concepts

Pure strategy Nash equilibrium

H H H H

H 1

2 G M D

H (4, 3) (5, 1) (6, 2) M (2, 1) (8, 4) (3, 6) B (3, 0) (9, 6) (2, 8)

A pure strategy Nash equilibrium (PNE) is a strategy profile such that

each player’s strategy is an optimal response to the strategies of the other

players

Pure strategy Nash equilibrium

H H H H

H 1

2 G M D

H (4, 3) (5, 1) (6, 2) M (2, 1) (8, 4) (3, 6) B (3, 0) (9, 6) (2, 8)

A pure strategy Nash equilibrium (PNE) is a strategy profile such that

each player’s strategy is an optimal response to the strategies of the other

players

Solution concepts

Pure strategy Nash equilibrium

H H H H

H 1

2 G M D

H (4, 3) (5, 1) (6, 2) M (2, 1) (8, 4) (3, 6) B (3, 0) (9, 6) (2, 8)

A pure strategy Nash equilibrium (PNE) is a strategy profile such that each player’s strategy is an optimal response to the strategies of the other players

s = {s

, . . . , s

} is a PNE iff

∀i ∈ {1, . . . , n}, ∀s

∈ S

, u

(s) ≥ u

(s

, s

)

Pure strategy Nash equilibrium

A game can have several PNEs, or cannot have any A strictly dominated strategy cannot be in a PNE A weakly dominated strategy can be in a PNE

If the iterated elimination of strictly dominated strategies give a unique

solution, this solution is the unique PNE of the game

Solution concepts

Mixed strategies

Pure strategy: actions available for a player

Mixed strategy : strategy that defines the probabilities with which players choose each of their pure strategies

A mixed strategy for the player i is a probability distribution over S

. Σ

is the set of mixed strategy for i .

σ

: S

→ IR assigns to each pure strategy s

its probability to be played

A pure strategy s

corresponds to the mixed strategy σ

associated with

the probability of 1

Mixed strategy Nash equilibrium

A mixed strategy Nash equilibrium is a situation where all players choose their mixed strategies in order to maximise they expected utilities.

σ ∈ Σ is a mixed strategy Nash equilibrium iff

∀i ∈ N, ∀σ

∈ Σ

, u

(σ) ≥ u

(σ

, σ

)

Solution concepts

Mixed strategy Nash equilibirum

H H H

H H 1

2 P F

P (15, -15) (-15, 15) F (-15, 15) (15, -15)

No pure strategy Nash equilibrium

Mixed strategy Nash equilibrium(a)?

Mixed strategy Nash equilibrium

H H H

H H 1

2 H T

H (15, -15) (-15, 15) T (-15, 15) (15, -15)

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and

σ

(T ) = 1 − y

Solution concepts

Mixed strategy Nash equilibrium

H H H

H H 1

2 H T

H (15, -15) (-15, 15) T (-15, 15) (15, -15)

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 :

y (15x − 15(1 − x )) + (1 − y )(−15x + 15(1 − x ))

Mixed strategy Nash equilibrium

H H H

H H 1

2 H T

H (15, -15) (-15, 15) T (-15, 15) (15, -15)

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

Expected utility of 2 :

Solution concepts

Mixed strategy Nash equilibrium

H H H

H H 1

2 H T

H (15, -15) (-15, 15) T (-15, 15) (15, -15)

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

Mixed strategy Nash equilibrium

H H H

H H 1

2 H T

H (15, -15) (-15, 15) T (-15, 15) (15, -15)

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

Solution concepts

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =?

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =?

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =?

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =?

Solution concepts

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =?

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =?

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =?

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =?

Solution concepts

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =?

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =?

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =?

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =?

Solution concepts

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =?

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =?

Mixed strategy Nash equilibrium

Probability distribution: σ

(H) = x , σ

(T ) = 1 − x , σ

(H) = y and σ

(T ) = 1 − y

Expected utility of 1 : 15(x (4y − 2) − 2y + 1)

→ If 4y − 2 > 0 (y >

), then the best interest of 1 is to choose x = 1

→ If 4y − 2 < 0 (y <

), then the best interest of 1 is to choose x = 0

→ If 4y − 2 = 0 (y =

), then the best interest of 1 is to choose x =

Expected utility of 2 : 15(y (2 − 4x ) + 2x − 1)

→ If 2 − 4x > 0 (x <

), then the best interest of 2 is to choose y = 1

→ If 2 − 4x < 0 (x >

), then the best interest of 2 is to choose y = 0

→ If 2 − 4x = 0 (x =

), then the best interest of 2 is to choose y =

Solution concepts

Nash equilibrium

Every pure strategy Nash equilibrium is a mixed strategy Nash equilibrium

Every finite game has at least one mixed strategy Nash equilibrium

2

Solution concepts Dominated strategies Nash Equilibrium Pareto criterion Security levels Examples of games

Subgame perfect equilibrium

3

Repeated games

4

Two players zero-sum games

Solution concepts

Pareto criterion

H H H H

H 1

2 A B

C (4, 4) (0, 3)

D (3, 1) (4, 5)

Pareto criterion

H H H H

H 1

2 A B

C (4, 4) (0, 3)

D (3, 1) (4, 5)

Solution concepts

Pareto criterion

H H H H

H 1

2 A B

C (4, 4) (0, 3)

D (3, 1) (4, 5)

Pareto criterion

H H H H

H 1

2 A B

C (4, 4) (0, 3) D (3, 1) (4, 5)

A strategy profile s strongly Pareto dominates s

if s is strictly better than s

for all the players

∀s

∈ s, ∀s

∈ s

, u

(s) > u

(s

)

BD strongly Pareto dominates BC and AD

Solution concepts

Pareto criterion

H H H H

H 1

2 A B

C (4, 4) (0, 3) D (3, 1) (4, 5)

A strategy profile s weakly Pareto dominates s

if s is as good as s

for all the players, and strictly better than s

for at least one player

∀s

∈ s, ∀s

∈ s

, u

(s) ≥ u

(s

) and

∃s

∈ s, ∃s

∈ s

such that u

(s) > u

(s

)

BD weakly Pareto dominates AC

Pareto optimum

Pareto optimum:

State in which it is not possible to increase the well-being of an individual without decreasing the one of another individual

uniformly improvable: it is possible to increase the weel being of some individuals without decreasing the one of the others

non uniformly improvable: it is not possible to increase the weel being of some individuals without decreasing the one of the others

Pareto optimum = strategy profile non (weakly) Pareto dominated

Solution concepts

Pareto optimum

H H H H

H 1

2 A B

C (4, 4) (0, 3) D (3, 1) (4, 5)

Pure strategy Nash equilibria: CA, DB

Pareto optimum: DB

2

Solution concepts Dominated strategies Nash Equilibrium Pareto criterion Security levels Examples of games

Subgame perfect equilibrium

3

Repeated games

4

Two players zero-sum games

Solution concepts

Security levels

H H H

H H 1

2 A B

C (9, 9) (0, 8) D (8, 0) (7, 7)

The security level of a strategy s

for the player i is the minimum utility this player can obtain with this strategy whatever are the choices of the other players:

min

s−i

u

(s

, s

)

The security level of a player i is the maximal security level of all i ’s

strategies

Security levels

H H H

H H 1

2 A B

C (9, 9) (0, 8) D (8, 0) (7, 7)

The security level of a strategy s

for the player i is the minimum utility this player can obtain with this strategy whatever are the choices of the other players:

min

s−i

u

(s

, s

)

1

Formalisation of a game

2

Solution concepts Dominated strategies Nash Equilibrium Pareto criterion Security levels Examples of games

Subgame perfect equilibrium

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Prisoner dilemma

H H H

H H 1

2 C D

C (3, 3) (0, 5) D (5, 0) (1, 1)

Pure strategy Nash equilibrium(a)?

Dominated strategies?

Pareto optimum?

Solution concepts

Deadlock

H H H

H H 1

2 C D

C (1, 1) (0, 3) D (3, 0) (2, 2)

Pure strategy Nash equilibrium(a)?

Dominated strategies?

Pareto optimum?

Security levels of the players?

Friend or Foe

H H H

H H 1

2 Friend Foe

Friend (50, 50) (0, 100) Foe (100, 0) (0, 0)

Pure strategy Nash equilibrium(a)?

Dominated strategies?

Pareto optimum?

Solution concepts

Pure coordination game

H H H

H H 1

2 standard 1 standard 2

standard 1 (5, 5) (0, 0) standard 2 (0, 0) (3, 3)

Pure strategy Nash equilibrium(a)?

Dominated strategies?

Pareto optimum?

Security levels of the players?

Stag hunt

H H H

H H 1

2 stag hare

stag (10, 10) (0, 8) hare (8, 0) (7, 7)

Pure strategy Nash equilibrium(a)?

Dominated strategies?

Pareto optimum?

Solution concepts

Battle of the Sexes

H H H

H H 1

2 O F

O (1, 2) (0, 0) F (0, 0) (2, 1)

Pure strategy Nash equilibrium(a)?

Dominated strategies?

Pareto optimum?

Security levels of the players?

Chicken’s game

H H H

H H 1

2 stay swerve

stay (-100, -100) (1, -1) swerve (-1, 1) (0, 0)

Pure strategy Nash equilibrium(a)?

Dominated strategies?

Pareto optimum?

1

Formalisation of a game

2

Solution concepts Dominated strategies Nash Equilibrium Pareto criterion Security levels Examples of games

Subgame perfect equilibrium

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Dynamic games

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2)

F

B

Solution concepts

Dynamic game and backward induction

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

Backward induction Reasoning backward in time

Optimal choices on the terminal nodes

Using this information, one can then determine what to do at the second-to-last time of decision.

This process continues backwards until one has determined the best action

Dynamic game and backward induction

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

Backward induction Reasoning backward in time

Optimal choices on the terminal nodes

Solution concepts

Dynamic game and backward induction

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

Backward induction Reasoning backward in time

Optimal choices on the terminal nodes

Using this information, one can then determine what to do at the second-to-last time of decision.

This process continues backwards until one has determined the best action

Dynamic game and backward induction

Every (finite) extensive game with perfect information has (at least) one pusre strategy Nash equilibrium

This equilibrium can be find using backward induction

Solution concepts

Dynamic game and subgames

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

A subgame of an extensive game is made up of A node

All the nodes that are successors of the initial node All the edges between these nodes

The utilities associated to the terminal nodes

Three subgames here

Dynamic game and subgames

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

A subgame of an extensive game is made up of A node

All the nodes that are successors of the initial node

Solution concepts

Dynamic game and subgames

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

A subgame of an extensive game is made up of A node

All the nodes that are successors of the initial node All the edges between these nodes

The utilities associated to the terminal nodes

Three subgames here

Dynamic game and subgames

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

A subgame of an extensive game is made up of A node

All the nodes that are successors of the initial node

Solution concepts

Subgame perfect equilibrium

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

A strategy profile is a Subgame perfect Nash equilibrium if it represents

a Nash equilibrium of every subgame of the original game

Subgame perfect equilibrium

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

Strategic form of this subgame

C D

Solution concepts

Subgame perfect equilibrium

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

Strategic form of this subgame:

E F

(4, 1) (3, 2)

This subgame has one pure strategy Nash equilibrium: F (B).

Subgame perfect equilibrium

Strategic form of the original game:

H H H H H 1

2 C (A), E (B) C (A), F (B) D(A), E (B) D(A), F (B)

A (3, 0) (3, 0) (5, 1) (5, 1)

B (4, 1) (3, 2) (4, 1) (3, 2)

This subgame has three pure strategy Nash equilibria (B, C (A)F (B))

(A, D(A)E (B))

(A, D(A)F (B))

Solution concepts

Subgame perfect equilibrium

Strategic form of the original game:

H H H H H 1

2 C (A), E (B) C (A), F (B) D(A), E (B) D(A), F (B)

A (3, 0) (3, 0) (5, 1) (5, 1)

B (4, 1) (3, 2) (4, 1) (3, 2)

This subgame has three pure strategy Nash equilibria

(B, C (A)F (B)) is not a subgame perfect Nash equilibrium: C (A) won’t be played in the subgame starting after A,

(A, D(A)E (B))

(A, D(A)F (B))

Subgame perfect equilibrium

Strategic form of the original game:

H H H H H 1

2 C (A), E (B) C (A), F (B) D(A), E (B) D(A), F (B)

A (3, 0) (3, 0) (5, 1) (5, 1)

B (4, 1) (3, 2) (4, 1) (3, 2)

This subgame has three pure strategy Nash equilibria

(B, C (A)F (B)) is not a subgame perfect Nash equilibrium: C (A) won’t be played in the subgame starting after A,

(A, D(A)E (B)) is not a subgame perfect Nash equilibrium: E (B) won’t be played in the subgame starting after B,

(A, D(A)F (B))

Solution concepts

Subgame perfect equilibrium

Strategic form of the original game:

H H H H H 1

2 C (A), E (B) C (A), F (B) D(A), E (B) D(A), F (B)

A (3, 0) (3, 0) (5, 1) (5, 1)

B (4, 1) (3, 2) (4, 1) (3, 2)

This subgame has three pure strategy Nash equilibria

(B, C (A)F (B)) is not a subgame perfect Nash equilibrium: C (A) won’t be played in the subgame starting after A,

(A, D(A)E (B)) is not a subgame perfect Nash equilibrium: E (B) won’t be played in the subgame starting after B,

(A, D(A)F (B)) is subgame perfect Nash equilibrium: D(A) and F (B) are

pure strategy Nash equilibria of the subgames associated

Subgame perfect equilibrium

1

2

(3, 0) C

(5, 1) D A

2

(4, 1) E

(3, 2) F B

This game has one subgame perfect Nash equilibrium: (A, D(A)F (B)).

For the games with perfect information, the subgame perfect Nash

equilibria correspond to the equilibria found with backward induction

1

Formalisation of a game

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Prisoner’s dilemma

Two men are arrested, but the police do not possess enough information for a conviction. Following the separation of the two men, the police offer both a similar deal:

If one testifies against his partner (defects / betrays), and the other remains silent (cooperates / assists), the betrayer goes free and the cooperator receives the full one-year sentence

If each ’rats out’ the other, each receives a three-month sentence If both remain silent, both are sentenced to only one month in jail for a minor charge

Each prisoner must choose to either betray or remain silent; the decision

Repeated games

Prisoner’s dilemma

Two men are arrested, but the police do not possess enough information for a conviction. Following the separation of the two men, the police offer both a similar deal:

If one testifies against his partner (defects / betrays), and the other remains silent (cooperates / assists), the betrayer goes free (+5) and the cooperator receives the full one-year sentence (0)

If each ’rats out’ the other, each receives a three-month sentence (+3) If both remain silent, both are sentenced to only one month in jail for a minor charge (+1)

Each prisoner must choose to either betray or remain silent; the decision

of each is kept quiet. What should they do?

Prisoner’s dilemma

H H H

H H 1

2 C D

C (3, 3) (0, 5) D (5, 0) (1, 1)

Nash equilibrium 6= Pareto optimum

Equilibrium: both prisoners choose to defect even though their individual utility would be greater if they cooperated

Each prisoner is incitated to betray after having promised to cooperate. It

Repeated games

Prisoner’s dilemma: real life examples

Some examples of the prisonner’s dilemma in real life:

Politics: two states engaged in an arms race

Environmental studies: global climate change. All countries will benefit from a stable climate, but any single country is often hesitant to curb CO2 emissions

Economics: two firms advertising the same product at the same time

Iterated game

Iterated game (or repeated game): some number of repetitions of some (two-players) base game

A player will have to take into account the impact of his current action on the future actions of other players

The threat of such retaliation is often enough: benefits related to the prospects for future earnings may outweigh the short-term gains achieved by breaking an agreement

Logic of r´ eciprocity

Repeated games

Iterated prisoner’s dilemma

You do not have the same musical tastes as your neighbor. He often listen to loud music. You may also (in retaliation) listen to your music at a more than reasonable sound level. This has for consequences that he will start again the next day. But you both enjoy the periods when none of you bother the other.

Suppose that we weighted your satisfaction:

You have a satisfaction of 5 to listen to your music very loud Your satisfaction is of 0 when your neighbor puts his music very loud A calm evening, without music, brings you a satisfaction of 3

The fact of hearing both your music and the one of your neighbor gives you a satisfaction of a 1

You know what was the behavior of your neighbor in previous days, what are

you doing today?

Iterated dilemma

Flood and Dresher, 1952 Non-zero sum game 2 simultaneaous players 2 strategies :

Cooperate (beeing “nice”), denoted C Deviate (beeing “mean”), denoted D

Utility of the players are denoted S , P, R and T . They are up to the game, but have to fulfil the following condition:

S < P < R < T

Repeated games

Iterated dilemma

Players meet each other several times

At each iteration, players know their previous actions They do not know when the game will stop

The final gain of a player is the sum of his gains obtained at each encounter

In order to promote the cooperation, we add the following constraint:

S + T < 2 × R

Iterated prisoner’s dilemma

H H H

H H 1

2 D C

D (P =1, P =1) (T = 5, S = 0) C (S = 0, T = 5) (R = 3, R = 3)

P: Punishment, punishment for mutual trahison S : Sucker’s payoff

R: Reward, reward for mututal cooperation

T : Temptation

Repeated games

Iterated prisoner’s dilemma: some strategies

all-C: always cooperate ([c]

) all-D: always deviate ([d ]

)

tit-for-tat: cooperate, and then do the same that the opponent did in the previous encounter

tester: deviate, and then do the same that the opponent did in the previous encounter

spiteful: cooperate until the opponent has defected, after that always defect.

random : cooperate with a probability of

.

per_ccd : plays periodically [c, c, d ]

per_ddc : plays periodically [d , d , c]

Encounters between strategies

1 2 3 4 5 6 7 8 9 10

Results of all-C 0 0 0 0 0 0 0 0 0 0 = 0

Actions of all-C C C C C C C C C C C

Actions of all-D D D D D D D D D D D

Results of all-D 5 5 5 5 5 5 5 5 5 5 = 50

Results of per_ddc 5 1 0 1 1 0 1 1 0 1 = 11

Actions of per_ddc D D C D D C D D C D

Actions of spiteful C D D D D D D D D D

Repeated games

What strategy to choose?

Confrontation of 100 encounters:

all-C all-D all-C 300, 300 0, 500 all-D 500,0 100,100

Each strategy is good against some strategies, and bad against others.

The key point is to note that a player does not know what strategy has

his opponent.

tit-for-tat: a good strategy

tit-for-tat: cooperate, and then do the same that the opponent did in the previous encounter

tit-for-tat does not ever win

In the best case, this strategy obtains the same results than its opponent

But in the worst case, this strategy has only 5 points less than its

opponent, whatever the length of the game is

Repeated games

The tournaments

Several strategies meet

The final gain of a strategy is the sum of all of its gains against each of its opponent

Every games have the same length, that no strategy can know

Repeated games

Example of one tournament

5 strategies, 10 iterations for each game

all-C all-D per_ddc tit-for-tat

all-C 30 50 44 30

all-D 0 10 7 9

per_ddc 9 22 16 18

tit-for-tat 30 14 23 30

Total 99 110 101 117

Ranking: 1. tit-for-tat - 2. all-D - 3. per_ddc - 4. all-C

http://www2.lifl.fr/IPD/

http://www.apprendre-en-ligne.net/jeux/dilemme/home.html

Repeated games

Example of one tournament

5 strategies, 10 iterations for each game

all-C all-D per_ddc tit-for-tat

all-C 30 50 44 30

all-D 0 10 7 9

per_ddc 9 22 16 18

tit-for-tat 30 14 23 30

Total 99 110 101 117

Ranking: 1. tit-for-tat - 2. all-D - 3. per_ddc - 4. all-C

More details about the iterated prisoner’s dilemma (websites in French):

http://www2.lifl.fr/IPD/

http://www.apprendre-en-ligne.net/jeux/dilemme/home.html

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Two players zero-sum games

Two players zero-sum games

Simplest games No coalitions Strictly competitives:

Players have preferences strictly opposites

For each strategy profile s, we have u

(s) + u

(s) = 0 Examples:

Board games (chess, draughts, ...) War

...

Two players zero-sum games

H H H H

H 1

2 E F G H

A (3, -3) (3, -3) (-5, 5) (5, -5)

B (-4, 4) (7, -7) (8, -8) (7, -7)

C (2, -2) (5, -5) (-18, 18) (2, -2)

D (5, -5) (6, -6) (7, -7) (5, -5)

Two players zero-sum games

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Player 1 tries to maximize his security level: v

= max

(min

(u

(s

, s

)))

Two players zero-sum games

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Player 1 tries to maximize his security level: v

= max

(min

(u

(s

, s

)))

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Player 1 tries to maximize his security level: v

= max

(min

(u

(s

, s

)))

Two players zero-sum games

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Player 1 tries to maximize his security level: v

= max

(min

(u

(s

, s

))) Player 2 tries to minimize player 1’s security level:

v

= min

(max

(u

(s

, s

)))

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Player 1 tries to maximize his security level: v

= max

(min

(u

(s

, s

))) Player 2 tries to minimize player 1’s security level:

v

= min

(max

(u

(s

, s

)))

Two players zero-sum games

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Player 1 tries to maximize his security level: v

= max

(min

(u

(s

, s

))) Player 2 tries to minimize player 1’s security level:

v

= min

(max

(u

(s

, s

)))

Two players zero-sum games

H H H H

H 1

2 E F G H

A 3 3 -5 5

B -4 7 8 7

C 2 5 -18 2

D 5 6 7 5

Player 1 tries to maximize his security level: v

= max

(min

(u

(s

, s

))) Player 2 tries to minimize player 1’s security level:

v

= min

(max

(u

(s

, s

)))

If v

= v

= v , then every couple of strategies (s

, s

), s

garanteeing v to

1

Formalisation of a game

2

Solution concepts

3

Repeated games

4

Two players zero-sum games

5

Bounds of games theory

Bounds of games theory

Complete preferences

Invariable preferences

Perfect knowledge

Bounded rationality