🗊Презентация Evolutionary games. (Lecture 7)

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Evolutionary games. (Lecture 7), слайд №1Evolutionary games. (Lecture 7), слайд №2Evolutionary games. (Lecture 7), слайд №3Evolutionary games. (Lecture 7), слайд №4Evolutionary games. (Lecture 7), слайд №5Evolutionary games. (Lecture 7), слайд №6Evolutionary games. (Lecture 7), слайд №7Evolutionary games. (Lecture 7), слайд №8Evolutionary games. (Lecture 7), слайд №9Evolutionary games. (Lecture 7), слайд №10Evolutionary games. (Lecture 7), слайд №11Evolutionary games. (Lecture 7), слайд №12Evolutionary games. (Lecture 7), слайд №13Evolutionary games. (Lecture 7), слайд №14Evolutionary games. (Lecture 7), слайд №15Evolutionary games. (Lecture 7), слайд №16Evolutionary games. (Lecture 7), слайд №17Evolutionary games. (Lecture 7), слайд №18Evolutionary games. (Lecture 7), слайд №19Evolutionary games. (Lecture 7), слайд №20Evolutionary games. (Lecture 7), слайд №21Evolutionary games. (Lecture 7), слайд №22Evolutionary games. (Lecture 7), слайд №23Evolutionary games. (Lecture 7), слайд №24Evolutionary games. (Lecture 7), слайд №25Evolutionary games. (Lecture 7), слайд №26Evolutionary games. (Lecture 7), слайд №27Evolutionary games. (Lecture 7), слайд №28Evolutionary games. (Lecture 7), слайд №29Evolutionary games. (Lecture 7), слайд №30Evolutionary games. (Lecture 7), слайд №31
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LECTURE 7 EVOLUTIONARY GAMES
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LECTURE 7 EVOLUTIONARY GAMES

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Classic game theory Lectures 1-6: “Classic game theory”, rational players: Players aim to maximize their payoffs, and they never make mistakes. Critiques of CGT: The assumption that players never make mistakes is unrealistic. To determine the optimal strategy may be difficult in many situations. How do we choose between the different equilibria? (e.g. coordination games have 2 PSNE and 1 MSNE)
Описание слайда:
Classic game theory Lectures 1-6: “Classic game theory”, rational players: Players aim to maximize their payoffs, and they never make mistakes. Critiques of CGT: The assumption that players never make mistakes is unrealistic. To determine the optimal strategy may be difficult in many situations. How do we choose between the different equilibria? (e.g. coordination games have 2 PSNE and 1 MSNE)

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Evolutionary game theory An Alternative approach Evolutionary game theory is an alternative approach: players are not fully rational, they make mistakes. Players’ behavior evolves overtime, systematic mistakes are eliminated in the long-run. What EGT achieves: Helps select between several Nash equilibria Provides an interpretation to the concept of mixed strategy
Описание слайда:
Evolutionary game theory An Alternative approach Evolutionary game theory is an alternative approach: players are not fully rational, they make mistakes. Players’ behavior evolves overtime, systematic mistakes are eliminated in the long-run. What EGT achieves: Helps select between several Nash equilibria Provides an interpretation to the concept of mixed strategy

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Evolution in biology Principles of evolution Animal behavior may be genetically predetermined, e.g. degree of aggressivity. Heterogeneity: different members of a group behave differently. Fitness: Some types of behavior are more successful. Selection: Animals pass their genes to the next generation. Animals with most successful types of behavior reproduce more quickly. e.g. if aggressive types are more successful, they will spread and eventually all animals within that species will be aggressive.
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Evolution in biology Principles of evolution Animal behavior may be genetically predetermined, e.g. degree of aggressivity. Heterogeneity: different members of a group behave differently. Fitness: Some types of behavior are more successful. Selection: Animals pass their genes to the next generation. Animals with most successful types of behavior reproduce more quickly. e.g. if aggressive types are more successful, they will spread and eventually all animals within that species will be aggressive.

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Evolution in game theory Animal = Player Behavior = Strategy (not a choice variable) Behavior success = Payoff of strategy Successful strategies will spread by imitation or learning Firms observe which business practices work, and adopt them. e.g. if TFT dominates defect, then defectors will not survive in the long-term; and they will be replaced by TFT players.
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Evolution in game theory Animal = Player Behavior = Strategy (not a choice variable) Behavior success = Payoff of strategy Successful strategies will spread by imitation or learning Firms observe which business practices work, and adopt them. e.g. if TFT dominates defect, then defectors will not survive in the long-term; and they will be replaced by TFT players.

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Price competition Two firms compete on prices. The NE is to set low prices to gain market shares.
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Price competition Two firms compete on prices. The NE is to set low prices to gain market shares.

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Price competition Review of the pricing game Prisoner’s dilemma situation. A unique PSNE: (D,D). If the game is not repeated, cooperation cannot be sustained. If the game is repeated infinitely or indefinitely, cooperation may be sustained as long as the rate of return r is not too high. Classic game theory assumes that players make an informed choice to play cooperate (C) or defect (D) based on the payoffs.
Описание слайда:
Price competition Review of the pricing game Prisoner’s dilemma situation. A unique PSNE: (D,D). If the game is not repeated, cooperation cannot be sustained. If the game is repeated infinitely or indefinitely, cooperation may be sustained as long as the rate of return r is not too high. Classic game theory assumes that players make an informed choice to play cooperate (C) or defect (D) based on the payoffs.

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Player types EGT assumes that players have no choice between C and D. Each player is born with a predetermined trait. Suppose that there are two types of players: Cooperators (probability x). Defectors (1-x). Cooperators always cooperate; defectors always defect. Each player is “born” with a type. Suppose that players are randomly matched. The “other player” could be a cooperator or a defector.
Описание слайда:
Player types EGT assumes that players have no choice between C and D. Each player is born with a predetermined trait. Suppose that there are two types of players: Cooperators (probability x). Defectors (1-x). Cooperators always cooperate; defectors always defect. Each player is “born” with a type. Suppose that players are randomly matched. The “other player” could be a cooperator or a defector.

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Defectors are successful Expected payoff of cooperators: π(C)=324x+216(1-x) = 216+108x Expected payoff of defectors: π (D)=360x+288(1-x) = 288+72x π (D)-π (C)=72-36x  π (D)>π (C)  defectors have a higher payoff
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Defectors are successful Expected payoff of cooperators: π(C)=324x+216(1-x) = 216+108x Expected payoff of defectors: π (D)=360x+288(1-x) = 288+72x π (D)-π (C)=72-36x  π (D)>π (C)  defectors have a higher payoff

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ESS (evolutionary stable strategy) Thus, defectors are fitter than cooperators. This leads to an increase in the proportion of defectors from one “generation” to the next. E.g. suppose that x=0.4 initially. The proportion of defectors will increase gradually, as defection is more successful. At some point all players will adopt defection. The evolutionary stable strategy is the long-run outcome of the evolution process. The ESS is that all players defect. Only one type will remain. When a strategy is strictly dominant, it is the ESS.
Описание слайда:
ESS (evolutionary stable strategy) Thus, defectors are fitter than cooperators. This leads to an increase in the proportion of defectors from one “generation” to the next. E.g. suppose that x=0.4 initially. The proportion of defectors will increase gradually, as defection is more successful. At some point all players will adopt defection. The evolutionary stable strategy is the long-run outcome of the evolution process. The ESS is that all players defect. Only one type will remain. When a strategy is strictly dominant, it is the ESS.

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ESS The likely outcome is (D,D) Why do firms defect? Not because they choose to defect, but because those that don’t defect have a lower rate of survival
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ESS The likely outcome is (D,D) Why do firms defect? Not because they choose to defect, but because those that don’t defect have a lower rate of survival

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ESS
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ESS

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Repeated prisoners’ dilemma Suppose the game is repeated three times. Each pair of players plays the games 3 times in succession. Is cooperation possible? When the game is repeated, players can have more complex strategies. Suppose there are two types of strategies: Always defect (probability 1-x) Tit-for-tat (probability x) Players are randomly drawn against each other.
Описание слайда:
Repeated prisoners’ dilemma Suppose the game is repeated three times. Each pair of players plays the games 3 times in succession. Is cooperation possible? When the game is repeated, players can have more complex strategies. Suppose there are two types of strategies: Always defect (probability 1-x) Tit-for-tat (probability x) Players are randomly drawn against each other.

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Repetition: payoffs A vs. A: 288+288+288 T vs. T: 324+324+324 A vs. T: 360+288+288 T vs. A: 216+288+288
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Repetition: payoffs A vs. A: 288+288+288 T vs. T: 324+324+324 A vs. T: 360+288+288 T vs. A: 216+288+288

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Repetition: Nash equilibrium Classic game theory. Suppose that players must decide in advance either T or A. Two pure strategy NE: {A,A}, {T,T} One mix strategy NE: Play A with probability p=1/3: 864p+936(1-p)=792p+972(1-p) Play T with probability 1-p=2/3 3 possible outcomes.
Описание слайда:
Repetition: Nash equilibrium Classic game theory. Suppose that players must decide in advance either T or A. Two pure strategy NE: {A,A}, {T,T} One mix strategy NE: Play A with probability p=1/3: 864p+936(1-p)=792p+972(1-p) Play T with probability 1-p=2/3 3 possible outcomes.

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Repetition: performance EGT expected payoffs: π(A)= 936x+864(1-x) = 864+72x π(T)= 972x+792(1-x) = 792+180x π(T)> π(A) if x>2/3 π(T)< π(A) if x<2/3 The performance of each type depends on the composition of the population Large % of type A  A is more successful Large % of type T  T is more successful
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Repetition: performance EGT expected payoffs: π(A)= 936x+864(1-x) = 864+72x π(T)= 972x+792(1-x) = 792+180x π(T)> π(A) if x>2/3 π(T)< π(A) if x<2/3 The performance of each type depends on the composition of the population Large % of type A  A is more successful Large % of type T  T is more successful

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Repetition: performance
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Repetition: performance

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Repetition: ESS If more than 2/3 of the population is T type, then T players are more successful, and their proportion will grow until it reaches 100% If less than 2/3 of the population is T type, then A players are more successful, and their proportion will grow until it reaches 100%  Two ESS: All A or all T
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Repetition: ESS If more than 2/3 of the population is T type, then T players are more successful, and their proportion will grow until it reaches 100% If less than 2/3 of the population is T type, then A players are more successful, and their proportion will grow until it reaches 100%  Two ESS: All A or all T

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Repetition: ESS “Monomorphic” outcome: all of the type. If everyone else is type A, types that don’t defect will not survive. If everyone else is type T, types that do defect will not survive. EGT can help select from a multiplicity of NE. In this example, only the PSNE are evolutionary stable, the MSNE is not. Thus, we can eliminate the MSNE on the ground that it is not evolutionary stable. Importance of the initial population mix of types.
Описание слайда:
Repetition: ESS “Monomorphic” outcome: all of the type. If everyone else is type A, types that don’t defect will not survive. If everyone else is type T, types that do defect will not survive. EGT can help select from a multiplicity of NE. In this example, only the PSNE are evolutionary stable, the MSNE is not. Thus, we can eliminate the MSNE on the ground that it is not evolutionary stable. Importance of the initial population mix of types.

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Repetition: ESS
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Repetition: ESS

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n-repetitions π(T)> π(A) if 324nx+(216+288(n-1))(1-x)>(360+288(n-1))x+288n(1-x) i.e. if x>2/n
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n-repetitions π(T)> π(A) if 324nx+(216+288(n-1))(1-x)>(360+288(n-1))x+288n(1-x) i.e. if x>2/n

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n-repetitions There are two ESS, one all T, one all A. The cut-off point depends on n: the higher n, the more likely that T types prevail. As n  very large, the cut-off point converges to x=0. Intuition: when the game is repeated more times, the long term benefits of cooperation outweigh the short term benefit of defection. Cooperation is more likely to be evolutionary stable if the game is repeated many times.
Описание слайда:
n-repetitions There are two ESS, one all T, one all A. The cut-off point depends on n: the higher n, the more likely that T types prevail. As n  very large, the cut-off point converges to x=0. Intuition: when the game is repeated more times, the long term benefits of cooperation outweigh the short term benefit of defection. Cooperation is more likely to be evolutionary stable if the game is repeated many times.

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ESS vs. Nash equilibrium Two PSNE: They Correspond to ESS. An ESS must be a NE of the game played by rational players
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ESS vs. Nash equilibrium Two PSNE: They Correspond to ESS. An ESS must be a NE of the game played by rational players

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ESS vs. Nash equilibrium Backdoor justification for the NE Even if players are not rational, if the more successful strategies spread in the population, then the outcome must be the same as that resulting from consciously rational play. Thus, the NE can be reached even if players are not rational. Players who don’t play the successful strategy will die out.
Описание слайда:
ESS vs. Nash equilibrium Backdoor justification for the NE Even if players are not rational, if the more successful strategies spread in the population, then the outcome must be the same as that resulting from consciously rational play. Thus, the NE can be reached even if players are not rational. Players who don’t play the successful strategy will die out.

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ESS vs. Nash equilibrium One mixed strategy NE in which T is played with probability 2/3, and A 1/3: Does not correspond to ESS. The mixed strategy NE is “unstable”. Although all ESS are NE, not all NE are ESS. Number of NE ≥ number of ESS.
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ESS vs. Nash equilibrium One mixed strategy NE in which T is played with probability 2/3, and A 1/3: Does not correspond to ESS. The mixed strategy NE is “unstable”. Although all ESS are NE, not all NE are ESS. Number of NE ≥ number of ESS.

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Chicken game Quantity game: x is the proportion of H type. π(L)=0(1-x)-1x=-x π(H)=1(1-x)-2x=1-3x
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Chicken game Quantity game: x is the proportion of H type. π(L)=0(1-x)-1x=-x π(H)=1(1-x)-2x=1-3x

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Chicken game π(H)> π(L) if x<1/2 H is successful if the proportion of H is less than ½ L is successful if the proportion of L is less than ½ Each type is fitter when it is relatively rare! If most firms produce less, I am better off producing more. If most firms produce more, I am better off not producing less.
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Chicken game π(H)> π(L) if x<1/2 H is successful if the proportion of H is less than ½ L is successful if the proportion of L is less than ½ Each type is fitter when it is relatively rare! If most firms produce less, I am better off producing more. If most firms produce more, I am better off not producing less.

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Chicken game If x>1/2, L are more successful and x declines If x<1/2, H are more successful and x increases The ESS is at x=1/2 The ESS is that 50% of players play H, and 50% play L.
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Chicken game If x>1/2, L are more successful and x declines If x<1/2, H are more successful and x increases The ESS is at x=1/2 The ESS is that 50% of players play H, and 50% play L.

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Chicken game
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Chicken game

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Chicken game EGT provides an alternative interpretation of mixed strategies: With rational players, the 50-50 result suggest players randomize each time they play. In the evolutionary game, each player uses a pure strategy, but different players use different strategies. The distribution of those playing L and those playing H is 50-50.
Описание слайда:
Chicken game EGT provides an alternative interpretation of mixed strategies: With rational players, the 50-50 result suggest players randomize each time they play. In the evolutionary game, each player uses a pure strategy, but different players use different strategies. The distribution of those playing L and those playing H is 50-50.

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Summary Criticism of classic game theory: rationality; multiple equilibria; EGT does not assume rationality, and helps select between multiple NE. EGT provides a backdoor justification for the NE. All ESS are NE, not all NE are ESS.
Описание слайда:
Summary Criticism of classic game theory: rationality; multiple equilibria; EGT does not assume rationality, and helps select between multiple NE. EGT provides a backdoor justification for the NE. All ESS are NE, not all NE are ESS.

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