Download - Introduction to Game Theory Notes… · What is Game Theory? Decision Theory Unique decision maker Game Theory (Interactive Decision Theory) Many decision makers Strategic interdependence

N ik o l a o s Lio n is

U n i v e r s i t y O f A t h e n s

( R e v i s e d : O c t o b e r 2 0 1 4 )

Introduction to Game Theory 1

What is Game Theory?

Decision Theory Unique decision maker

Game Theory (Interactive Decision Theory) Many decision makers

Strategic interdependence

Note: For the interaction to become a strategic game, however, we need something more, namely the participants’ mutual awareness of the cross effects of their actions

2

Methodology of Game Theory 3

Strategic Problem

Game Model

Outcomes

Equilibrium Concepts

Equilibrium Outcomes

Predictions

F R O M A S T R A T E G I C P R O B L E M

T O A G A M E M O D E L

4

Constructing a Game

Pricing Problem 5

Two firms are about to price their products. They have to choose between high or low prices. The firm with the lowest price serves the whole market.

Their profits are:

If both set a high price, they will get profit 2.

If they both set a low price then they will get 1 as a profit.

If they differentiate their prices, i.e. one firm chooses a high price and the other chooses a low price, then the firm with the low price will get the whole market demand and will earn a profit of 3. The other firm earns 0.

What should the firms do?

Basic Elements of Games 6

1. The players

2. The actions

3. The order of play

4. The information

5. The outcomes

6. The payoffs

Who is involved? 7

The players

A finite set I = {1,2,…N}

… it can also be and an infinite set

Pricing Example I = { Firm 1 , Firm 2}

What can the players do?

Actions For each i I, a nonempty set Ai = {a1

i, a2i, …, aki

i}

Strategies

A strategy for player i is a complete plan of actions, i.e. a complete plan of how a player will play a game.

Importantly, a strategy determines the action a player will take for any situation he could face during the game.

For each i I, a nonempty set S i = {s1i, s2

i, …, smii }

Pricing example

A Firm 1 = { H, L } and A Firm 2 = { H, L } S Firm 1 = { H, L } and S Firm 2 = { H, L }

8

Who moves when?

Order of Play

Simultaneous moves (Static games) All players move at the same time

…. or without knowing their opponents’ moves

Sequential moves (Dynamic games) One player moves

Second player observes and then moves

9

What the players know when they move?

Information

Complete information All players’ payoff functions are common knowledge

Incomplete information At least one player is uncertain about another player’s payoff

function

10

What are the outcomes of the game?

Outcomes

For each possible set of strategies by the players, what is the outcome of the game?

An outcome s is a list (vector) of strategies chosen by each player i, i.e. s = (s1, s2, …, sN)

Pricing example s = ( (H, L), (H, H), (L, L), (L, H) )

11

What the players get?

Payoffs

The payoffs capture everything in the outcomes of the game that the player cares about

What are the players preferences over the possible outcomes?

A payoff function i for player i assigns a real number i(s) to every outcome of the game. Formally, i: S

12

What the players get?

Pricing example

Preferences over the possible outcomes Firm 1: (L, H) ≻ (H, H) ≻ (L, L) ≻ (H, L)

Firm 2: (H, L) ≻ (H, H) ≻ (L, L) ≻ (L, H)

Payoff functions Firm 1: Firm 1 (s) such that

Firm 1 ((L, H)) = 3 > Firm 1 ((H, H)) = 2 > Firm 1 ((L, L)) = 1 > Firm 1 ((H, L)) = 0

Firm 2: Firm 2 (s) such that

Firm 2 ((H, L)) = 3 > Firm 2 ((H, H)) = 2 > Firm 2 ((L, L)) = 1 > Firm 2 ((H, L)) = 0

13

Game Models 14

Order of Play

Static or Dynamic

Players’ Information

Complete or Incomplete

Game Models 15

Static Games

with

Complete Information

Dynamic Games

with

Complete Information

Static Games

with

Incomplete Information

Dynamic Games

with

Incomplete Information

Axiomatic Assumptions 16

: In any given situation a decision-maker always chooses the action which is the best according to his preferences, i.e. players play rational

: Rational play is common knowledge among all players in the game, i.e. each player knows that the other players play rational, knows that the other players’ know he is rational, etc. (ad infinitum)

Rules of the Game 17

Rules of the Game

Basic Elements • The players • The actions

• The order of play • The information

• The outcomes • The payoffs

Assumptions • Rational play

• Common Knowledge

N O R M A L F O R M

A N D

E X T E N S I V E F O R M

18

Representations of Games

Pricing Problem Normal Form Representation

19

Firm 2

Firm 1

Players


20

Firm 2

High Price Low Price

Firm 1

High Price

Low Price

Players, Actions/Strategies, Outcomes


21

Firm 2


Firm 1

High Price 2 , 2 0 , 3

Low Price 3 , 0 1 , 1

Players, Actions/Strategies, Outcomes, Payoffs

Pricing Problem Extensive Form

22

Firm 1

L H

First Decision Node: Player, Actions


23

Firm 1

Firm 2

L H L H

L H

Next Decision Nodes: Player, Actions


24

Firm 1

Firm 2

L H L H

L H

2 2 0 3 3 0 1 1

Players, Actions/Strategies, Order of Play, Information, Outcomes, Payoffs

Pricing Problem as a Static Game Extensive Form

25

Firm 1

Firm 2

L H L H

L H

2 2 0 3 3 0 1 1


Pricing Problem as a Dynamic Game 26

Firm 1

Firm 2

L H L H

L H

2 2 0 3 3 0 1 1


Actions and Strategies in Dynamic Games

Actions

For Firm 1: A1 = {H, L}

For Firm 2: A2 = {H, L}

Strategies

A strategy for each firm is a complete plan of actions

For Firm 1: S1 = {H, L}

For Firm 2: S2 = {(H,H), (H,L), (L,H), (L,L)}

27

F R O M O U T C O M E S

T O E Q U I L I B R I U M O U T C O M E S

28

Solving a Game

Methodology of Game Theory 29

Strategic Problem

Game Model

Outcomes


Equilibrium Outcomes

Predictions


An equilibrium is a strategy combination consisting of each player’s best strategy

Equilibrium in Dominant Strategies

Nash Equilibrium

Subgame Perfect Nash Equilibrium

Bayesian-Nash Equilibrium

Perfect Bayesian-Nash Equilibrium

…

30

Note: The main concept is Nash Equilibrium. All the rest are just equilibrium refinements of Nash Equilibrium.

Models Equilibrium Concept

1. Static Games with Complete Information

2. Dynamic Games with Complete Information

3. Static Games with Incomplete Information

4. Dynamic Games with Incomplete Information

Nash Equilibrium


Bayesian Nash Equilibrium

Perfect Bayesian Nash Equilibrium

31

Models and Equilibrium

S T A T I C G A M E S

W I T H

C O M P L E T E I N F O R M A T I O N

A N D

N A S H E Q U I L I B R I U M

32

Solving a Static Game

Nash Equilibrium

An outcome s = (s1, s2, …, s N) is said to be Nash Equilibrium if no player would find it beneficial to deviate provided that all other players are playing their Nash equilibrium strategies.

Formally,

for every i I, i (s i, s -i) i (s' i, s -i) for every s' i S i.

Comments Each player’s predicted strategy is the best response to the predicted strategies of

other players No incentive to deviate unilaterally Strategically stable or self-enforcing

33

Best Response Functions

The best response function of player i is the function R i (s -i), that for given strategies s -i of players 1, 2, …, i-1, i+1, …, N, assigns an action s i = R i (s -i) that maximizes player i’s payoff i (s i, s -i).

If s is a Nash Equilibrium outcome, then s i = R i (s -i) for every player i.

34

Pricing Problem Solving for Nash Equilibrium

35

Firm 2


Firm 1


Low Price 3 , 0 1 , 1


36

Firm 2


Firm 1


Low Price 3 , 0 1 , 1

You manage Firm 1:

• Suppose Firm 2 sets a High Price.


37

Firm 2

High Price

Firm 1

High Price 2 , 2

Low Price 3 , 0

You manage Firm 1:



38

Firm 2

High Price

Firm 1

High Price 2 , 2

Low Price 3 , 0

You manage Firm 1:


• What is the best response for you? High or Low Price?


39

Firm 2

High Price

Firm 1

High Price 2 , 2

Low Price 3 , 0

You manage Firm 1:




40

Firm 2

High Price

Firm 1

High Price 2 , 2

Low Price , 0

You manage Firm 1:




41

Firm 2


Firm 1


Low Price , 0 , 1

You manage Firm 1:

• Suppose Firm 2 sets a Low Price.



42

Firm 2


Firm 1

High Price 2 , 2 0 ,

Low Price , 0 , 1

You manage Firm 2:




43

Firm 2


Firm 1


Low Price , 0 ,

You manage Firm 2:

• Suppose Firm 1 sets a Low Price.



44

Firm 2


Firm 1


Low Price , 0 ,

Nash Equilibrium is (Low Price, Low Price)

Pricing Problem Best Response Functions

What we have just done? Analyzed the pricing game using best response functions in our

way of thinking

Firm 1 best response function RFirm 1 (Firm 2 strategies) RFirm 1 (H) = L

Note that Firm 1 ((L, H)) = 3 > Firm 1 ((H, H)) = 2 Hence, if Firm 2 prices high then Firm’s 1 best response is L

RFirm 1 (L) = L

Firm 2 best response function RFirm 2 (Firm 1 strategies)

RFirm 2 (H) = L and RFirm 2 (L) = L

We argued that s = (L, L) is a N.E. outcome Since RFirm 1 (L) = L and RFirm 2 (L) = L

45

D Y N A M I C G A M E S

W I T H

C O M P L E T E I N F O R M A T I O N

A N D

S U B G A M E P E R F E C T N A S H E Q U I L I B R I U M

46

Solving a Dynamic Game


An outcome is said to be Subgame Perfect Nash Equilibrium if it induces a Nash Equilibrium in every subgame of the original game

A subgame is a decision node from the original game along with the decision nodes and terminal nodes directly following this node

47

Subgame 3

Pricing Problem as a Dynamic Game Subgames

48

Firm 1

Firm 2

L H L H

L H

2 2 0 3 3 0 1 1

Subgame 1 Subgame 2

Backward Induction 49

Backward induction is a procedure, which involves solving first for the optimal at the end of the game and then determining what optimal behaviour is earlier in the game given the anticipation of this later behaviour

It ensures that players’ strategies specify optimal behaviour at every decision node of the game

Pricing Problem as a Dynamic Game Solving for Subgame Perfect Nash Equilibrium

50

Firm 1

Firm 2

L H L H

L H

2 2 0 3 3 0 1 1


51

Firm 2

L H

2 2 0 3

Solving Subgame 1

Find Nash Equilibrium of Subgame 1

Only Firm 2 moves Trivial best response function Simple decision problem

Firm’s 2 payoffs Firm 2 (Η) = 2 < Firm 2 (L) = 3

Hence, “best response” for Firm 2 is to choose Low Price


52

L H

3 0 1 1


Only Firm 2 moves

Firm’s 2 payoffs Firm 2 (Η) = 0 < Firm 2 (L) = 1



53

Firm 1

Firm 2

L H L H

L H

2 2 0 3 3 0 1 1


Firm 1

L H


54

0 3 1 1

Subgame 1 can be simplified Firm 1 knows that Firm 2 will play its N.E. strategies in subgames 1 and 2

Only Firm 1 moves. Firm’s 1 payoffs are Firm 1 (Η) = 3 < Firm 1 (L) = 1




55

Firm 1

Firm 2

L H L H

L H

2 2 0 3 3 0 1 1

Nash Equilibria of all the subagmes consitutes the Subgame Perfect Nash Equilibrium

The Subgame Perfect Nash Equilibrium is … not !!!

56

EXAMPLES

C O L L E C T I V E C H O I C E P R O B L E M

C O O R D I N A T I O N F A I L U R E

57

Prisoner’s Dilemma

Prisoner’s Dilemma 58

Two suspects are arrested for armed robbery. They are immediately separated. If convicted, they will get a term of 10 years in prison. However, the evidence is not sufficient to convict them of more than the crime of possessing stolen goods, which carries a sentence of only 1 year.

The suspects are told the following: If you confess and your accomplice does not, you will go free. If you do not confess and your accomplice does, you will get 10 years in prison. If you both confess, you will both get 5 years in prison.

What should each prisoner do?


59

Prisoner 2

Confess Not Confess

Prisoner 1

Confess -5 , -5 0 , -10

Not Confess -10 , 0 -1 , -1


60

Prisoner 2

Confess Not Confess

Prisoner 1

Confess -5 , -5 0 , -10

Not Confess -10 , 0 -1 , -1


61

Prisoner 2

Confess Not Confess

Prisoner 1

Confess -5 , -5 0 , -10

Not Confess -10 , 0 -1 , -1


62

Prisoner 2

Confess Not Confess

Prisoner 1

Confess 0 , -10

Not Confess -10 , 0 -1 , -1

Coordination failure

Prisoner’s Dilemma 63

Prisoner 1

Prisoner 2

NC C NC C

NC C

-5 -5 0 -10 -10 0 -1 -1

M U L T I P L E E Q U I L I B R I A

F I R S T M O V E R A D V A N T A G E

64

Battle of Sexes

Battle of Sexes 65

A couple deciding how to spend the evening

Wife would like to go for a movie

Husband would like to watch football on tv

Both however want to spend the time together

Battle of Sexes

66

Husband

Movie Football

Wife

Movie 2, 1 0 , 0

Football 0, 0 1 , 2

Battle of Sexes

67

Husband

Movie Football

Wife

Movie 2, 1 0 , 0

Football 0, 0 1 , 2

Battle of Sexes

68

Husband

Movie Football

Wife

Movie 2, 1 0 , 0

Football 0, 0 1 , 2

Battle of Sexes

69

Husband

Movie Football

Wife

Movie 2, 1 0 , 0

Football 0, 0 1 , 2

Multiple Equilibria

A Dynamic Version of Battle of Sexes 70

Wife

Husband

F M F M

F M

2 1 0 0 0 0 1 2

A Dynamic Version of Battle of Sexes 71

Wife

Husband

F M F M

F M

2 1 0 0 0 0 1 2

First Mover Advantage

S T R A T E G I E S I N D Y N A M I C G A M E S

I N C R E D I B L E T H R E A T S

72

Entry Deterrence Game

Entry Deterrence Game 73

The incumbent (Firm 1) first decides whether to expand the capacity of his plant (invest) or not.

A potential entrant observes incumbent’s investment decision and then decides whether to enter the market and compete against him or to stay out.

The capacity investment allows the incumbent to produce output at a lower cost. Entry is therefore profitable if and only if the incumbent does not invest.


Firm 1

Firm 2

S E S E

N I

3 -2 5 0 4 1 6 0


75

Firm 2

E S

Firm 1

I 3 , -2 5 , 0

N 4 , 1 6 , 0

Wrong normal form representation


76

Firm 2

E,E E,S S,E S,S

Firm 1

I 3 , -2 3 , -2 5 , 0 5 , 0

N 4 , 1 6 , 0 4 , 1 6 , 0

Correct normal form representation

Firm 2’s strategy set is S2 = {(E,E), (E,S), (S,E), (S,S)}


77

Firm 2

E,E E,S S,E S,S

Firm 1

I 3 , -2 3 , -2 5 , 0 5 , 0

N 4 , 1 6 , 0 4 , 1 6 , 0


78

Firm 2

E,E E,S S,E S,S

Firm 1

I 3 , -2 3 , -2 5 , 0

N 6 , 0 4 , 1 6 , 0

• Two Nash Equilibria: (N, (E,E)) and (I, (S,E))

• But the equilibrium (N, (E,E)) is not a sensible prediction for the game, i.e. the entrant’s strategy (E,E) is not credible.

• The incumbent should foresee that if he invests, the entrant will find it optimal to stay out.


Firm 1

Firm 2

S E S E

N I

3 -2 5 0 4 1 6 0

Two N.E. but only one S.P.N.E. (I, (S,E)) S.P.N.E. rules out any empty threats.

C O M M I T M E N T

80

Burning Bridge Game

Burning Bridge Game 81

Two armies are in conflict over an island, but both prefer to avoid fighting.

The island is situated between the land of Army 2 and the land of Army 1, a bridge connects each army's homeland with the island.

Army 2 is in currently possession of the island.

Bridge Bridge


Army 2 Homeland 1

Army 1 Homeland 2


Army 1

Army 2

Retreat Fight

No Attack Attack

-1 -1 5 5

0 10

Subgame Perfect Nash Equilibrium is (A, R) Fight is a not credible threat

Bridge Bridge Army 2 Homeland 1

Army 1 Homeland 2



Army 1

Army 2

Fight

No Attack Attack

-1 -1

0 10

Army 2 burns the bridge that connects the island with their homeland Fight is now a credible threat, hence S.P.N.E. is (F, NA)

Removing or limiting options can paradoxically improve payoffs.

86

EXTENSIONS

Mixed Strategies

Mixed strategy: A probability distribution over the pure strategies of a player Since probabilities are continuous, there are infinitely many mixed strategies

available to a player, even if their strategy set is finite. Pure strategies are a degenerate case of a mixed strategy in which that particular

pure strategy is selected with probability 1 and every other strategy with probability 0.

Example: Rock – Paper – Scissors Game Each player simultaneously forms his or her hand into the shape of either a rock,

a piece of paper, or a pair of scissors. Rule: rock beats (breaks) scissors, scissors beats (cuts) paper, and paper beats (covers) rock

No pure strategy Nash Equilibrium One mixed strategy Nash Equilibrium – each player plays rock, paper and

scissors each with 1/3 probability

87

Repeated Games

88

Cooperative Games

89

Mechanism Design

90

Download - Introduction to Game Theory Notes… · What is Game Theory? Decision Theory Unique decision maker Game Theory (Interactive Decision Theory) Many decision makers Strategic interdependence

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