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 s1s1 s2s2 s3s3 S1S1 1, 1-2, 04,-1 S2S2 0, 33, 15, 4 S3S3 1, 54, 25, 2

Dominance A strategy S 1 strictly dominates another strategy S 2 for Player 1 iff M 1 (S 1 ;s j ) > M 1 (S 2 ;s j ) for all s j. A strategy S 1 weakly dominates another strategy S 2 for Player 1 iff M 1 (S 1 ;s j ) ≥ M 1 (S 2 ;s j ) for all s j and M 1 (S 1 ;s j ) > M 1 (S 2 ;s j ) for some s j.

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 s1s1 s2s2 s3s3 S1S1 0, 1-2, 34,-1 S2S2 0, 33, 16, 4 S3S3 1, 54, 25, 2 CD C3, 31, 4 D4, 12, 2 LR L1, 10, 0 R -1, -1 LR L1, 10, 0 R 1, -1 LR U2, 2 D0, 03, 1

Prisoner’s Dilemma Player 2 CooperateDefect Player 1 Cooperate3,31,4 Defect4,12,2

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 ArmsArm Country 1 Freeze Arms3,31,4 Arm4,12,2

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 SwerveStraight Country 1 Swerve3,32,4 Straight4,21,1

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 BiteNot Bite Boy Hold the tail2,1 Cease holding the tail 1,44,2

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

Backwards Induction P1 C D 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 (3,3) (1,4) (4,1) (2,2) D D C C

Chicken Sequential Game Extensive Form P1 Swerve Straight P2 (3,3) (2,4) (4,2) (1,1) Swerve Straight Swerve Straight

Chicken Sequential Game Extensive Form P1 Swerve Straight P2 (3,3) (2,4) (4,2) (1,1) Swerve Straight Swerve Straight