Lecture V: Game Theory

Zhixin Liu

Complex Systems Research Center, Complex Systems Research Center, Academy of Mathematics and Systems Academy of Mathematics and Systems

Sciences, CASSciences, CAS

In the last two lectures, we talked about

Multi-Agent SystemsAnalysisIntervention

In this lecture, we will talk about

Game theory complex interactions between

people

Start With A Game

Rock-paper-scissor

rock paper scissor

rock

paper

scissor

0,0

0,0

0,0

-1,1A

B

Other games: poker, go, chess, bridge, basketball, football,…

-1,1

-1,1

1,-1

1,-1

1,-1

Some hints from the games Rules Results (payoff) Strategies Interactions between strategies and payoff

Games are everywhere. Economic systems: oligarchy monopoly, market, trade … Political systems: voting, presidential election, international relations … Military systems: war, negotiation,…

Game theory the study of the strategic interactions among rational agents.

Rationality implies that each player tries to maximize his/her payoff

From Games To Game Theory

Not to beat the other players

History of Game Theory

1928, John von Neumann proved the minimax theorem

1944, John von Neumann & Oskar Morgenstern, 《 Theory of Games and Economic Behaviors 》

1950s, John Nash, Nash Equilibrium 1970s, John Maynard Smith, Evolutionarily stable s

trategy Eight game theorists have won Nobel prizes in econ

omics

Elements of A Game

Player:

Who is interacting? N={1,2,…,n} Actions/ Moves: What the players can do?

Action set : Payoff: What the players can get from the game

RAu inii 1:

iiliii aaaA ,,, 21

Strategy

Strategy: complete plan of actions Mixed strategy: probability distribution over the

pure strategies

Payoff:.2,1),,(),( 212121 jii j ji aaussssu

1,0),,,,(1

21

i

i

l

jijijiliiiii sssssssS

Pure strategy is a special kind of mixed strategies

An Example: Rock-paper-scissor Players: A and B Actions/ Moves: {rock, scissor, paper} Payoff: u1(rock,scissor)=1 u2(scissor, paper)=-1 Mixed strategies

s1=(1/3,1/3,1/3)

s2=(0,1/2,1/2) u1(s1, s2) = 1/3(0·0+1/2·(-1)+1/2·1)+

1/3(0·1+1/2·0+1/2·(-1))+1/3(0·(-1)+1/2·1+1/2·0) = 0

rock paper scissor

rock

paper

scissor

0,0

0,0

0,0

-1,1A

B

-1,1

-1,1

1,-1

1,-1

1,-1

Classifications of Games Cooperative and non-cooperative games Cooperative game: players are able to form binding commitments. Non cooperative games: the players make decisions independently Zero sum and non-zero sum games Zero sum game: the total payoff to all players is zero. E.g., poker, go,… Non-zero sum game: e.g., prisoner’s dilemma Finite game and infinite game Finite game: the players and the actions are finite. Simultaneous and sequential (dynamic) games Simultaneous game: players move simultaneously, or if they do not move

simultaneously, the later players are unaware of the earlier players' actions Sequential game: later players have some knowledge about earlier actions. Perfect information and imperfect information games Perfect information game: all players know the moves previously made

by all other players. E.g., chess, go,…

Perfect information ≠ Complete information

Every player know the strategies and payoffs of the other players but not necessarily the actions.

What is the solution of the game?

We will first focus on games: Simultaneous Complete information Non cooperative Finite

Assumption

Assume that each player knows the structure of the game attempts to maximize his payoff attempt to predict the moves of his

opponents. knows that this is the common knowledge

between the players

 Dominated Strategy 

Strategy s' of the player i is called a strictly dominated strategy if there exists a strategy s*, such that

S-i : the strategy set formed by all other players except player i

A strategy is dominated if, regardless of what any other players do, the strategy earns a player a smaller payoff than some other strategies.

iiiiii Ssssussu ),,(),( '*

Elimination of Dominated Strategies 

L M R

U

M

D

4,3

8,4

2,8

5,1

3,6

3,0

6,2

2,1

9,6

L R

U

M

D

4,3

2,8

3,6

3,0

6,2

2,1

L R

U 4,3 6,2

L

U 4,3

Example:

A dominant strategy may not exist!

(U,L) is the solution of the game.

Definition of Nash Equilibrium

(N, S, u) : a game Si: strategy set for player i : set of strategy profiles : payoff function s-i: strategy profile of all players except player i A strategy profile s* is called a Nash equilibrium if

where σi is any pure strategy of the player i.

isussu iiiiii ),,(),( ***

Nash Equilibrium (NE): A solution concept of a game

Remarks on Nash Equilibrium

A set of strategies, one for each player, such that each player’s strategy is a best response to others’ strategies

Best Response: The strategy that maximizes the payoff given

others’ strategies. No player can do better by unilaterally changing his

or her strategy A dominant strategy is a NE

Example

Players: Smith and Louis Actions: { Advertise , Do Not Advertise } Payoffs: Companies’ Profits

Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor

How to represent this game?

Strategic Interactions

Ad

No Ad (50,50) (20,60)

(60,20) (30,30)AdLouis

Smith

No Ad

Best Responses

Best response for Louis: If Smith advertises: advertise If Smith does not advertise: advertise

The best response for Smith is the same. (Ad, Ad) is a dominant strategy! (Ad, Ad) is a NE! This is another Prisoners’ Dilemma!

Ad

No Ad (50,50) (20,60)

(60,20) (30,30)Ad

No Ad

Smith

Louis

Nash Equilibrium

NE may be a pair of mixed strategies. Example:

head (1,-1) (-1,1)

(-1,1) (1,-1)Tail

B

A

Matching Pennies

head Tail

(1/2,1/2) is the Nash Equilibrium.

Existence of NE

Theorem (J. Nash, 1950s)

For a finite game, there exists at least one Nash Equilibrium (Pure strategy, or mixed strategy).

Nash Equilibrium

NE may not be a good solution of the game, it is different from the optimal solution.

e.g.,

Ad

No Ad (50,50) (20,60)

(60,20) (30,30)Ad

No Ad

Smith

Louis

Nash Equilibrium

A game may have more than one NE.

e.g., The Battle of Sex

NE: (opera, opera), (football, football),

((2/3,1/3),(1/3, 2/3))

football

opera (2,1) (0,0)

(0,0) (1,2)football

opera

Husband

Wife

Nash Equilibrium

Zero sum games (two-person): Saddle point is a solution

),(maxminarg

),(minmaxarg

),(maxmin),(minmax),(

21*2

21*1

2121*2

*1

1122

2211

11222211

ssus

ssus

ssussussu

SsSs

SsSs

SsSsSsSs

Nash Equilibrium

Many varieties of NE: Refined NE, Bayesian NE, Sub-game Perfect NE, Perfect Bayesian NE …

Finding NEs is very difficult. NE can only tell us if the game reach such a st

ate, then no player has incentive to change their strategies unilaterally. But NE can not tell us how to reach such a state.

Iterated Prisoner’s Dilemma

Cooperation Groups of organisms:

Mutual cooperation is of benefit to all agents Lack of cooperation is harmful to them

Another types of cooperation: Cooperating agents do well Any one will do better if failing cooperate Prisoner’s Dilemma is an elegant embodiment

Prisoner’s Dilemma

C (3,3) (0,5)

(5,0) (1,1)D

Prisoner B

C

Prisoner A

The story of prisoner’s dilemma

Player: two prisoners

Action: {Cooperation, Defecti}

Payoff matrix

D

Prisoner’s Dilemma

No matter what the other does, the best choice is “D”.

(D,D) is a Nash Equilibrium. But, if both choose “D”, both will do worse

than if both select “C”

C (3,3) (0,5)

(5,0) (1,1)D

Prisoner B

C

Prisoner A

D

The individuals: Meet many times Can recognize a previous interactant Remember the prior outcome

Strategy: specify the probability of cooperation and defect based on the history P(C)=f1(History) P(D)=f2(History)

Iterated Prisoner’s Dilemma

Tit For Tat – cooperating on the first time, then repeat opponent's last choice.

Player A C D D C C C C C D D D D C…

Player B D D C C C C C D D D D C…

Strategies

Tit For Tat - cooperating on the first time, then repeat opponent's last choice. Tit For Tat and Random - Repeat opponent's last choice skewed by random se

tting.* Tit For Two Tats and Random - Like Tit For Tat except that opponent must ma

ke the same choice twice in a row before it is reciprocated. Choice is skewed by random setting.*

Tit For Two Tats - Like Tit For Tat except that opponent must make the same choice twice in row before it is reciprocated.

Naive Prober (Tit For Tat with Random Defection) - Repeat opponent's last choice (ie Tit For Tat), but sometimes probe by defecting in lieu of cooperating.*

Remorseful Prober (Tit For Tat with Random Defection) - Repeat opponent's last choice (ie Tit For Tat), but sometimes probe by defecting in lieu of cooperating. If the opponent defects in response to probing, show remorse by cooperating once.*

Naive Peace Maker (Tit For Tat with Random Co-operation) - Repeat opponent's last choice (ie Tit For Tat), but sometimes make peace by co-operating in lieu of defecting.*

True Peace Maker (hybrid of Tit For Tat and Tit For Two Tats with Random Cooperation) - Cooperate unless opponent defects twice in a row, then defect once, but sometimes make peace by cooperating in lieu of defecting.*

Random - always set at 50% probability.

Strategies

Always Defect Always Cooperate Grudger (Co-operate, but only be a sucker once) - Cooperate until the opponent

defects. Then always defect unforgivingly. Pavlov (repeat last choice if good outcome) - If 5 or 3 points scored in the last ro

und then repeat last choice. Pavlov / Random (repeat last choice if good outcome and Random) - If 5 or 3 p

oints scored in the last round then repeat last choice, but sometimes make random choices.*

Adaptive - Starts with c,c,c,c,c,c,d,d,d,d,d and then takes choices which have given the best average score re-calculated after every move.

Gradual - Cooperates until the opponent defects, in such case defects the total number of times the opponent has defected during the game. Followed up by two co-operations.

Suspicious Tit For Tat - As for Tit For Tat except begins by defecting. Soft Grudger - Cooperates until the opponent defects, in such case opponent is

punished with d,d,d,d,c,c. Customised strategy 1 - default setting is T=1, P=1, R=1, S=0, B=1, always c

o-operate unless sucker (ie 0 points scored). Customised strategy 2 - default setting is T=1, P=1, R=0, S=0, B=0, always pla

y alternating defect/cooperate.

Strategies

The same players repeat the prisoner’s dilemma many times. After ten rounds

The best income is 50. A real case is to get 30 for each player. An extreme case is that each player selects “defection”, each player

can get 10. The most possible case is that each player will play with a mixing

strategy of “defect” and “cooperate” .

Iterated Prisoner’s Dilemma

C (3,3) (0,5)

(5,0) (1,1)D

Prisoner A

C D

Prisoner B

Which strategy can thrive/what is the good strategy?

Robert Axelrod, 1980s A computer round-robin tournament

Iterated Prisoner’s Dilemma

AXELROD R. 1987. The evolution of strategies in the iterated Prisoners' Dilemma. In L. Davis, editor, Genetic Algorithms and Simulated Annealing. Morgan Kaufmann, Los Altos, CA.

Strategies: 14 entries+ random strategyIncluding Markov process + Bayesian inference

Each pair will meet each other, totally there are 15*15 runs, each pair will play the game 200 times

Payoff: ∑S’ U(S,S’)/15 Tit For Tat wins (cooperation based on recipr

ocity)

The first round

Characters of “good” strategiesGoodness: never defect firstTFT vs. Naive prober

Forgiveness: may revenge, but the memory is short.TFT vs. Grudger

The first roundNaive Prober - Repeat opponent's last choice but sometimes probe by defecting in lieu of cooperating

Grudger - Cooperate until the opponent defects. Then always defect unforgivingly

Winning Vs. High Scores

This is not a zero sum game, there is a banker.

TFT never wins one game. The best result for it is to get the same result as its opponent.

“Winning the game” is a kind of jealousness, it does not work well

It is possible to arise “cooperation” in a “selfish” group.

Strategies: 62 entries+ random strategy “goodness” strategies “wiliness: strategies

Tit For Tat wins again “Win” or “lost” depends on the circumstance.

The second round

Characters of “good” strategies

Goodness: never defect first First round: the first eight strategies with “goodness” Second round: there are fourteen strategies with

“goodness” in the first fifteen strategies Forgiveness: may revenge, but the memory is

short. “Grudger” is not s strategy with “forgiveness”

“goodness” and “forgiveness” is a kind of collective behavior.

For a single agent, defect is the best strategy.

Evolve “good” strategies by genetic algorithm (GA)

Evolution of the Strategies

What is a “good” strategy?

TFT is a good strategy? Tit For Two Tats may be the best strategy in t

he first round, but it is not a good strategy in the second round.

“Good” strategy depends on the environment.

Evolutionarily stable strategy

Tit For Two Tats - Like Tit For Tat except that opponent must make the same choice twice in row before it is reciprocated.

Evolutionarily stable strategy (ESS)

Introduced by John Maynard Smith and George R. Price in 1973

ESS means evolutionarily stable strategy, that is “a strategy such that, if all member of the population adopt it, then no mutant strategy could invade the population under the influence of natural selection.”

ESS is robust for evolution, it can not be invaded by mutation.

John Maynard Smith, “Evolution and the Theory of Games”

Definition of ESS

A strategy x is an ESS if for all y, y x, such that

holds for small positiveε.

xyyxuxxuifyyuyxu

yyxuxxu

),,(),(),,(),()2(

),,(),()1(

))1(())1(( yxUyyxUx

ESS

ESS is defined in a population with a large number of individuals.

The individuals can not control the strategy, and may not be aware the game they played

ESS is the result of natural selection

Like NE, ESS can only tell us it is robust to the evolution, but it can not tell us how the population reach such a state.

ESS in IPD

Tit For Tat can not be invaded by the wiliness strategies, such as always defect.

TFT can be invaded by “goodness” strategies, such as “always cooperate”, “Tit For Two Tats” and “Suspicious Tit For Tat ”

Tit For Tat is not a strict ESS. “Always Cooperate” can be invaded by “Always

Defect”. “Always Defect ” is an ESS.

references

Drew Fudenberg, Jean Tirole, Game Theory, The MIT Press, 1991.

AXELROD R. 1987. The evolution of strategies in the iterated Prisoners' Dilemma. In L. Davis, editor, Genetic Algorithms and Simulated Annealing. Morgan Kaufmann, Los Altos, CA.

Richard Dawkins, The Selfish Gene, Oxford University Press.

Concluding Remarks

Tip Of Game theoryBasic ConceptsNash EquilibriumIterated Prisoner’s DilemmaEvolutionarily Stable Strategy

Concluding Remarks

Many interesting topics deserve to be studied and further investigated: Cooperative games Incomplete information games Dynamic games Combinatorial games Learning in games ….

Thank you!