basics of game theory. recap decision theory vs. game theory rationality completeness transitivity...

Basics of Game Theory

Recap

• Decision Theory vs. Game Theory• Rationality

• Completeness• Transitivity

• What’s in a game?• Players• Actions• Outcomes• Preferences• Beliefs• Constraints

Defining a Game

Are moves simultaneous or sequential?

•Normal/strategic form•Extensive form or game tree

Normal Form Game

• Normal (Strategic) Form• More general than extensive form• Less information that extensive form game• All finite extensive form games can be transformed into

normal form games• Reduces each player’s choice to the selection of a

complete plan (strategy) for playing the game

Elements of Normal Form• Players• Strategies for each players

• Strategy: complete plan of action for entire game that includes assignment of one move to each of i’s information sets

• n-dimensional array of players’ pure strategies• Players’ payoffs given all players’ strategies

Normal Form

s1 s2 s3

S1 1, 1 -2, 0 4,-1

S2 0, 3 3, 1 5, 4

S3 1, 5 4, 2 5, 2

Dominance• A strategy S1 strictly dominates another strategy S2 for Player 1 iff

M1(S1;sj) > M1(S2;sj) for all sj.

• A strategy S1 weakly dominates another strategy S2 for Player 1 iff

M1(S1;sj) ≥ M1(S2;sj) for all sj

and

M1(S1;sj) > M1(S2;sj) for some sj.

Dominance (in English)• A strategy, K, strictly dominates another strategy, L, for

Player 1 iff the payoff from playing K is greater than the payoff from playing L for all strategies of Player 2.

• A strategy, K, weakly dominates another strategy, L, for Player 1 iff the payoff from playing K is at least equal to the payoff from playing L for all strategies of Player 2 and greater than the payoff from playing L for some strategy of Player 2.

Exercises

s1 s2 s3

S1 0, 1 -2, 3 4,-1

S2 0, 3 3, 1 6, 4

S3 1, 5 4, 2 5, 2

C D

C 3, 3 1, 4

D 4, 1 2, 2

L R

L 1, 1 0, 0

R 0, 0 -1, -1

L R

L 1, 1 0, 0

R 0, 0 1, -1

L R

U 2, 2 2, 2

D 0, 0 3, 1

Prisoner’s DilemmaPlayer 2

Cooperate Defect

Player 1Cooperate 3,3 1,4

Defect 4,1 2,2

Prisoner’s Dilemma: OPEC--Organization of the Petroleum Exporting Countries

• Suppose Iran and Iraq choose whether to produce 2 mil barrels/day OR 4 mil barrels/day

• Market price/barrel is• $25/barrel if total output=4 mil barrels• $15/barrel if total output=6 mil barrels• $10/barrel if total output=8 mil barrels

• Extraction costs• Iran $2 mil/barrel• Iraq $4 mil/barrel

• Profits

U = output(price – cost)

IRAN-IRAQ oil cartel

Iraq

2 mil barrels(cooperate)

4 mil barrels(defect)

Iran

2 mil barrels(cooperate)

$46 million$42 million

$26 million$44 million

4 mil barrels(defect)

$52 million$22 million

$32 million$24 million

Arms Race

• Rank four outcomes• Both freeze (3,3)• Both arm (2,2)• 1 freezes, 2 arms (1,4)• 2 freezes, 1 arms (4,1)

Country 2

Freeze Arms Arm

Country 1Freeze Arms 3,3 1,4

Arm 4,1 2,2

Chicken Game

• Outcomes• Both swerve (3,3)• Both Straight (1,1)• C1 swerves, C2 straight (2,4)• C1 straight, C2 swerves (4,2)

• Example• Cuban Missile Crisis

Country 2

Swerve Straight

Country 1Swerve 3,3 2,4

Straight 4,2 1,1

Chicken key points• The game of chicken has no dominant strategy• If P2 goes straight, P1 would rather swerve. If P2

swerves, P1 would rather go straight• Main objective: If P1 wants to “win,” she must convince P2

that she is going to go straight. But P2 will also be trying to convince P1 that he will go straight

• How can P1 convince P2 that she is going to go straight?

Tiger by the Tail Game

• Preferences• Boy most prefers to let go and not get bitten and least prefers to let

go and get bitten. • Tiger most prefers that the boy lets go and so he can bite the boy.

He least prefers the boy holding on forever.

• Example• Foreign aid

Bear

Bite Not Bite

Boy

Hold the tail 2,1 2,1

Cease holding the tail

1,4 4,2

Tiger by the tail: key points • Bite is always a dominant strategy for the tiger if it

receives a move.• Because tiger cannot commit NOT to bite, the boy will

never let go and the tiger gets its worst outcome.• A credible commitment NOT to bite would make both the

tiger and the boy better off.• How can the tiger commit NOT to bite?

• To consider commitments, we need to understand sequential moves and extensive form games.

Sequential moves -- Extensive Form• Whose choice (move) is it at any particular point in time?• What alternative actions are available to each person at

any particular move?• What does each player know about other players’ prior

choices?• What are the alternative states of nature and their

likelihood?• What are each player’s preferences (utilities) over

outcomes?

Elements of Extensive Form• Game tree: representation of extensive form• Nodes: decision and terminal• Branches: extend from each node representing alternatives• Chance: nature makes each choice by chance from a specified

lottery over the alternative states • Information sets: represents what players know at decision

nodes• Set of outcomes• Set of utility functions• Common knowledge: each player knows and expects the other

players to know all details of the situation that the game presents (each players knows that the others know that he knows the tree, and so forth)

Game Tree

Solution: Backward Induction• Looking forward in time and reasoning backward to

determine the optimal move sequence.

PD Sequential Game Extensive Form

P1

C

D

P2

P2

(3,3)

(1,4)

(4,1)

(2,2)D

D

C

C

Backwards Induction

P1

C

D

P2

P2

(3,3)

(1,4)

(4,1)

(2,2)D

D

C

C

Some important points• Because actors are strategic, they do not always try to

obtain their most preferred outcome. Rather, they try to obtain the most preferred outcome they believe is possible

• In order for a conflict to be resolved peacefully two things must occur:

1. A settlement must be found that all actors prefer to fighting

2. A way must be found to make the agreements self-enforcing

Information• Perfect Information: if all information sets are singletons

(know the history of the game)• Complete information: if all players’ payoffs are known to

all players

Game with imperfect information

PD Simultaneous Move Game Extensive Form

P1

C

D

P2

P2

(3,3)

(1,4)

(4,1)

(2,2)D

D

C

C

Chicken Sequential Game Extensive Form

P1

Swerve

Straight

P2

P2

(3,3)

(2,4)

(4,2)

(1,1)

Swerve

Straight

Swerve

Straight

Chicken Sequential Game Extensive Form

P1

Swerve

Straight

P2

P2

(3,3)

(2,4)

(4,2)

(1,1)

Swerve

Straight

Swerve

Straight