Introduction to Game Theory

Xudong 2013/1/12

A:Playing Games

B:Concepts& Theory

C: Discussions

Game 1: Prisoners’ dilemma

Cooperate DefectCooperate (2 ,2) (-1,3)

Defect (3,-1) (0 ,0)

Player II

Player I

• The rational choice is defect• The payoff of defect is consistently better• No matter of the action of other players

Dominated strategy

• Definition– Suppose two strategies A and B– If the payoff of strategy A consistently smaller

than the payoff of strategy B, no matter what strategies other players use

– Then strategy A is dominated• Example

– Cooperation is a dominated strategy in prisoners’ dilemma game

Discussion

• Invisible hand v.s. Market failure– The wealth of the nation Adam Smith (1776)– Selfish motivation benefits the whole society

• Externality – Traffic jam in Beijing – Tragedy of commons – Pollution

• Negotiation – Transaction expense Coase, Ronald– The fable of the bee Cheung, S. N. (1973).

Definition of a game

• Players – Index i indicates different players– Two players in prisoner dilemma

• Action set – represent the action of player I– Players in PD can take two kinds of actions i.e.

cooperating or defecting • Payoffs

– The payoff of player i is represented by – which is a function the actions of all players i.e.

John Nash

There is at least an equilibrium in every finite game

Nash Equilibrium

• A state in a game • Nobody himself will gain • By making a deviation • Alone only player i deviates

,

It is a state in a game, in which nobody himself will gain by making a deviation alone

Game 2: Collective Investment

• A profitable project for investors

• If over x% of the population invest, then each investor will get $10

• Otherwise, the investor will lose his money

Discussion

• Two equilibriums in the investment– All of the players invest– None of the player invest

• Communication plays an important role• Bank runs

– Morgan’s role in saving the bank run in 1907• Getting out of Recession

– Encouraging public speech – Fiscal policy and Monetary policy

Game 3: 捡石子• Background

– Nim a more well-known name – It is said originated from china (wiki)

• Game description– Two players– Actions

• Two piles of stones with x and y stones in each pile • Player can pick the any number of stones from ONLY one

pile

– The player who take the final stone wins

捡石子• If the number of stones of the two piles are

even, the first mover will lose• Otherwise, the first mover will win• Backward induction

Extensive-form Games

• Using a tree to represent a game • 捡石子 as an example

(1,1)

(0,1)

(1,0)

(2,2)

(0,2)

(2,0)

(1,2)

(2,1)

…

…

(2,0)(1,1)

(0,1)

(2,0)(0,0)

(1,0)

…

…

……

Player blue Player red

Player blue wins Player red wins

…(2,0)

(0,1)

Backward induction for solving equilibrium

(2,2)

(0,2)

(2,0)

(1,2)

(2,1)

…

…

(1,1)

…

……

• Cutting off the branches which lead to worse payoffs comparing with other braches

• Backwards: from the leafs to the root

Example:• For the red player at the (2,1) node• the branches(dash line) leading to

(2,0) or (0,1) are cut off• because red player will win by

choosing (1,1), otherwise he will lose

Subgame

(2,3)

(0,3)

(2,0)

(2,2)

(0,2)

(2,0)

(1,2)

(2,1)

…

…

(2,0)(1,1)

(0,1)

……

………

• Subgame is – The subtree of original game tree– The information sets are self-contained

Game 4: Pirate Game

• 5 pirates (A,…,E)• Actions

– 100 Gold coins– Making proposal sequentially on how to

distribute the gold coins – The proposal will pass if more than half

of the pirates support it– Otherwise the proposer will be killed

• The pirates who are still alive will get the coins according to the approved proposal

Solving Pirate Game by Backward Induction • Two players left

– Payoff (E100, D-1)– E will take all gold and kill D, regardless of D’s decision

• Three players left– Payoff(E0,D0,C100)– D will agree, otherwise he will be killed in the next round after C is killed

• Four players left– Payoff(E1,D1,C0,B98)– E and D will agree, otherwise, their payoff will be 0 in the next round

• Five players left– Payoff(E2,D0,C1,B0,A97)– E and C will agree, because their payoff is better than the payoff next

round

Discussion

• First Mover – Facebook v.s. Google+– Google v.s. Bing

• Different solutions for query correction in Google and Bing

• Late Mover Advantage– The Chinese Miracle Lin, J. Y. et al (2003)

– Tencent

Game 5: Rock, Paper and Scissors

Rock Paper ScissorsRock (0,0) (-1,1) (1,-1)Paper (1,-1) (0,0) (-1,1)

Scissors (-1,1) (1,1) (0,0)

Equilibrium ??

NO Equilibrium ??

Mixed strategy

• Rock, Paper and Scissors – Determinate– Pure strategy

• Mixed strategy– indeterminate– Assigning probabilities to the pure strategies– e.g. (1/3 Rock ， 1/3 Paper ， 1/3 Scissors) is

the Nash Equilibrium

Indifference Choices: A trick for solving equilibrium mixed strategy

• Suppose your opponent uses the equilibrium mixed strategy

• Your revenues under different pure strategies are identical

• Otherwise, your best response is pure strategy• Abbreviating it as ICTrick

An example of using ICTrick

• Consider the game: Rock, Paper and Scissors• Suppose the equilibrium mixed strategy of

player 2 is (p1, p2,p3)

Rock Paper Scissors

Rock (0,0) (-1,1) (1,-1)

Paper (1,-1) (0,0) (-1,1)

Scissors (-1,1) (1,1) (0,0)

Player 2 (p1, p2,p3)

Player 1

An example of using ICTrick cont.

• For Player 1, Revenues using pure strategies– Revenue of strategy Rock: – Revenue of strategy Paper: – Revenue of strategy Scissors:

• Due to indifference choice–

• And • We have

Game 6: Hawk and Dove

Hawk DoveHawk (V/2-C, V/2-C) (V,0)Dove (0,V) (V/2, V/2)

• A population• Each individuals have two choices when they

distribute the food with others– Hawk(aggressive, fight with others)– Dove(mild, equally sharing with others)

• The fraction of Hawk and Dove?

Equilibrium of Hawk and Dove Game

• Using the ICTrick, the equilibrium of mixed strategy – Hawk V/(2C)– Dove 1 – V/(2C)

• The probabilities of the mixed strategy– Not only represents the odds of the certain pure

strategy taken by an individual – But also the proportion of certain population

Evolutionary Stable Strategy(ESS): Historical reviews

• Smith, J. M., & Price, G. R. The logic of animal conflict (1973)

• Richard Dawkins Selfish Gene (1976)• Axelrod, R. The evolution of cooperation (1984)

ESS: Definition

• Provided that all individuals in a population use strategy S• For any strategy T (T S)• Suppose there is a small fraction of mutants using strategy T

–If the mutants die out finally, then strategy S is Evolutionary Stable Strategy

–Otherwise it is not ESS

New fitness & Original fitness

• Payoffs– E(S,S) origins v.s. origins– E(S,T) origins v.s. mutants– E(T,T) mutants v.s. mutants

• Small amount of mutation• S: 1-, T:

• Original fitness • E(S,S)

• New fitness

Evolutionary stable strategy criterions

• New fitness < Original fitness– New fitness - Original fitness

– First order term smaller than 0

– Or first order term equal to 0 and second order term smaller than 0

• and

• ESS Criterions– or– and

Relationship with Nash Equilibrium

• ESS Criterions– or– and

• Nash Equilibrium Criterions

Discussions

• Revisiting ESS criterions– When E(S,S) > E(S,T), strategy S is equilibrium

stable, even when E(T,T) > E(S,S) – Progressive reform fail to jump out the local

minimum – Shock Therapy for the reform of Russia in 1990s

• Is the mixed strategy in Hawk and Dove Game a evolutionary stable strategy?

Game 7: Migration • Two towns: East and West Town Strategies• Two Types of people: Black and White Players • Choose which town they will live

(0.5, 1)

(1, 0.5)

payoff

Proportion of same type people

Equilibrium

• Equilibrium I– All white in one town, all black in the other town– Racial segregation, even it is not the willingness of

anybody Schelling, T. C. (1969)

– Stable• Equilibrium II

– Half white and half black in the one town– Unstable, need additional force to achieve better

equilibrium– Randomization improve the outcomes

Schelling, T. C. (1969). Models of segregation. The American Economic Review, 59(2), 488-493.

Discussion

• The segregation is a stable equilibrium, even both the black and the white are willing to live together with each other

• Other scenarios?– Boy and girl segregation in high school?

Useful links

• http://gametheory101.com/• http://v.163.com/special/gametheory/• http://en.wikipedia.org/wiki/Game_theory