lecture v: game theory zhixin liu complex systems research center, academy of mathematics and...
TRANSCRIPT
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
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?
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.
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.
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 ….