rock paper scissors - penn mathchhays/lecture23.pdf · to play rock i a best response to a strategy...
TRANSCRIPT
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What is the expected payoff of (13 ,13 ,
13) against (12 , 0,
12)?
I 0I Note that the expected payout is a weighted average of payouts
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What is the expected payoff of (13 ,13 ,
13) against (12 , 0,
12)?
I 0I Note that the expected payout is a weighted average of payouts
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What is the expected payoff of (13 ,13 ,
13) against (12 , 0,
12)?
I 0
I Note that the expected payout is a weighted average of payouts
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What is the expected payoff of (13 ,13 ,
13) against (12 , 0,
12)?
I 0I Note that the expected payout is a weighted average of payouts
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the average
I u((1, 0, 0), ( 12, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is
to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the average
I u((1, 0, 0), ( 12, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is
to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the average
I u((1, 0, 0), ( 12, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is
to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the averageI u((1, 0, 0), ( 1
2, 0, 1
2)) = 1
2
I u((0, 1, 0), ( 12, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is
to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the averageI u((1, 0, 0), ( 1
2, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is
to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the averageI u((1, 0, 0), ( 1
2, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is
to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the averageI u((1, 0, 0), ( 1
2, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is
to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the averageI u((1, 0, 0), ( 1
2, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I How can we raise the expected payout against (12 , 0,12)?
I By removing pure strategies that lower the averageI u((1, 0, 0), ( 1
2, 0, 1
2)) = 1
2I u((0, 1, 0), ( 1
2, 0, 1
2)) = 0
I u((0, 0, 1), ( 12, 0, 1
2)) = − 1
2
I Best strategy against ( 12 , 0,
12 ) is to play rock
I A best response to a strategy will consist of pure strategiesthat have the same (high) expected payout
Nash Equilibrium
I Mixed strategies (p1, . . . , pn) are a Nash equilibrium if pi is abest response to p−i
I Each player asks “if the other players stuck with theirstrategies, am I better off mixing the ratio of strategies?”
I If pi is a best response to p−i , the payouts of the purestrategies in pi are equal
I Note that pure Nash equilibria are still Nash equilibria
Nash Equilibrium
I Mixed strategies (p1, . . . , pn) are a Nash equilibrium if pi is abest response to p−i
I Each player asks “if the other players stuck with theirstrategies, am I better off mixing the ratio of strategies?”
I If pi is a best response to p−i , the payouts of the purestrategies in pi are equal
I Note that pure Nash equilibria are still Nash equilibria
Nash Equilibrium
I Mixed strategies (p1, . . . , pn) are a Nash equilibrium if pi is abest response to p−i
I Each player asks “if the other players stuck with theirstrategies, am I better off mixing the ratio of strategies?”
I If pi is a best response to p−i , the payouts of the purestrategies in pi are equal
I Note that pure Nash equilibria are still Nash equilibria
Nash Equilibrium
I Mixed strategies (p1, . . . , pn) are a Nash equilibrium if pi is abest response to p−i
I Each player asks “if the other players stuck with theirstrategies, am I better off mixing the ratio of strategies?”
I If pi is a best response to p−i , the payouts of the purestrategies in pi are equal
I Note that pure Nash equilibria are still Nash equilibria
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What should the (unique) Nash equilibrium be?
I When both players use ( 13 ,
13 ,
13 )
I Test this: what is the payoff of a pure strategy against( 13 ,
13 ,
13 )?
I 0
I Note that the expected payoff for each player is 0 (the gameis fair)
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What should the (unique) Nash equilibrium be?
I When both players use ( 13 ,
13 ,
13 )
I Test this: what is the payoff of a pure strategy against( 13 ,
13 ,
13 )?
I 0
I Note that the expected payoff for each player is 0 (the gameis fair)
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What should the (unique) Nash equilibrium be?I When both players use ( 1
3 ,13 ,
13 )
I Test this: what is the payoff of a pure strategy against( 13 ,
13 ,
13 )?
I 0
I Note that the expected payoff for each player is 0 (the gameis fair)
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What should the (unique) Nash equilibrium be?I When both players use ( 1
3 ,13 ,
13 )
I Test this: what is the payoff of a pure strategy against( 13 ,
13 ,
13 )?
I 0
I Note that the expected payoff for each player is 0 (the gameis fair)
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What should the (unique) Nash equilibrium be?I When both players use ( 1
3 ,13 ,
13 )
I Test this: what is the payoff of a pure strategy against( 13 ,
13 ,
13 )?
I 0
I Note that the expected payoff for each player is 0 (the gameis fair)
Rock Paper Scissors
I Consider Rock Paper Scissors:
0, 0
1, -1
-1, 1
0, 0
1, -1
-1, 1
-1, 1 1, -1 0, 0R P S
R
P
S
I What should the (unique) Nash equilibrium be?I When both players use ( 1
3 ,13 ,
13 )
I Test this: what is the payoff of a pure strategy against( 13 ,
13 ,
13 )?
I 0
I Note that the expected payoff for each player is 0 (the gameis fair)
Nash Equilibrium
I We saw that there are not always pure Nash equilibrium
I Can we guarantee a mixed Nash equilibrium?
Theorem (Nash)Suppose that:
I a game has finitely many playersI each player has finitely many pure strategiesI we allow for mixed strategies
Then the game admits a Nash equilibrium
Nash Equilibrium
I We saw that there are not always pure Nash equilibrium
I Can we guarantee a mixed Nash equilibrium?
Theorem (Nash)Suppose that:
I a game has finitely many playersI each player has finitely many pure strategiesI we allow for mixed strategies
Then the game admits a Nash equilibrium
Nash Equilibrium
I We saw that there are not always pure Nash equilibrium
I Can we guarantee a mixed Nash equilibrium?
Theorem (Nash)Suppose that:
I a game has finitely many playersI each player has finitely many pure strategiesI we allow for mixed strategies
Then the game admits a Nash equilibrium
TennisI You’re playing tennis, and returning the ball
I Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?
I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?
I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or right
I Opponent can anticipate where you will hit the ball (to theirleft or right)
I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?
I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)
I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?
I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?
I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?
I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?I No pure Nash equilibrium
I Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)
I Idea: your opponent’s pure strategies must return the sameexpected payoff (for them) against (p, 1 − p)
I Strategies in Nash equilibrium are (.7, .3)(p = .7) and(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)
I Strategies in Nash equilibrium are (.7, .3)(p = .7) and(.6, .4)(q = .6)
TennisI You’re playing tennis, and returning the ballI Options:
I You can hit the ball to either the opponent’s left or rightI Opponent can anticipate where you will hit the ball (to their
left or right)I Payoffs are:
L RL
R
50,50
90,10 20,80
80,20You
Opponent
I What are the Nash equilibrium?I No pure Nash equilibriumI Assume that strategies are (p, 1 − p) and (q, 1 − q)I Idea: your opponent’s pure strategies must return the same
expected payoff (for them) against (p, 1 − p)I Strategies in Nash equilibrium are (.7, .3)(p = .7) and
(.6, .4)(q = .6)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?
I Pure Nash equilibria occur when you both go to the samemovie
I For mixed Nashed equilibria, write out the strategies as(p, 1 − p) and (q, 1 − q)
I (Same) trick: to find p, consider your date’s pure strategies:the payouts for both strategies must be the same
I So 2p = p + 3(1 − p)I p = 3
4
I Similarly, q = 14
I Note that both you and your date’s expected payout is
32
(between the original payouts of the Nash equilbrium)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?I Pure Nash equilibria occur when you both go to the same
movieI For mixed Nashed equilibria, write out the strategies as
(p, 1 − p) and (q, 1 − q)
I (Same) trick: to find p, consider your date’s pure strategies:the payouts for both strategies must be the same
I So 2p = p + 3(1 − p)I p = 3
4
I Similarly, q = 14
I Note that both you and your date’s expected payout is
32
(between the original payouts of the Nash equilbrium)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?I Pure Nash equilibria occur when you both go to the same
movieI For mixed Nashed equilibria, write out the strategies as
(p, 1 − p) and (q, 1 − q)I (Same) trick: to find p, consider your date’s pure strategies:
the payouts for both strategies must be the same
I So 2p = p + 3(1 − p)I p = 3
4
I Similarly, q = 14
I Note that both you and your date’s expected payout is
32
(between the original payouts of the Nash equilbrium)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?I Pure Nash equilibria occur when you both go to the same
movieI For mixed Nashed equilibria, write out the strategies as
(p, 1 − p) and (q, 1 − q)I (Same) trick: to find p, consider your date’s pure strategies:
the payouts for both strategies must be the sameI So 2p = p + 3(1 − p)
I p = 34
I Similarly, q = 14
I Note that both you and your date’s expected payout is
32
(between the original payouts of the Nash equilbrium)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?I Pure Nash equilibria occur when you both go to the same
movieI For mixed Nashed equilibria, write out the strategies as
(p, 1 − p) and (q, 1 − q)I (Same) trick: to find p, consider your date’s pure strategies:
the payouts for both strategies must be the sameI So 2p = p + 3(1 − p)I p = 3
4
I Similarly, q = 14
I Note that both you and your date’s expected payout is
32
(between the original payouts of the Nash equilbrium)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?I Pure Nash equilibria occur when you both go to the same
movieI For mixed Nashed equilibria, write out the strategies as
(p, 1 − p) and (q, 1 − q)I (Same) trick: to find p, consider your date’s pure strategies:
the payouts for both strategies must be the sameI So 2p = p + 3(1 − p)I p = 3
4
I Similarly, q = 14
I Note that both you and your date’s expected payout is
32
(between the original payouts of the Nash equilbrium)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?I Pure Nash equilibria occur when you both go to the same
movieI For mixed Nashed equilibria, write out the strategies as
(p, 1 − p) and (q, 1 − q)I (Same) trick: to find p, consider your date’s pure strategies:
the payouts for both strategies must be the sameI So 2p = p + 3(1 − p)I p = 3
4
I Similarly, q = 14
I Note that both you and your date’s expected payout is
32
(between the original payouts of the Nash equilbrium)
Going to the Movies
C DC
D
3, 2
0, 0 2, 3
1, 1You
Date
I What are the Nash equilibria?I Pure Nash equilibria occur when you both go to the same
movieI For mixed Nashed equilibria, write out the strategies as
(p, 1 − p) and (q, 1 − q)I (Same) trick: to find p, consider your date’s pure strategies:
the payouts for both strategies must be the sameI So 2p = p + 3(1 − p)I p = 3
4
I Similarly, q = 14
I Note that both you and your date’s expected payout is 32
(between the original payouts of the Nash equilbrium)
Evolutionarily Stable Strategies
I Simplification of theory due to John Maynard Smith
I Idea: some small percentage of a population develops amutation
I This creates a competing ‘strategy’, compared to animalswithout the mutation
I Will those with the mutation thrive or die?
Evolutionarily Stable Strategies
I Simplification of theory due to John Maynard Smith
I Idea: some small percentage of a population develops amutation
I This creates a competing ‘strategy’, compared to animalswithout the mutation
I Will those with the mutation thrive or die?
Evolutionarily Stable Strategies
I Simplification of theory due to John Maynard Smith
I Idea: some small percentage of a population develops amutation
I This creates a competing ‘strategy’, compared to animalswithout the mutation
I Will those with the mutation thrive or die?
Evolutionarily Stable Strategies
I Simplification of theory due to John Maynard Smith
I Idea: some small percentage of a population develops amutation
I This creates a competing ‘strategy’, compared to animalswithout the mutation
I Will those with the mutation thrive or die?
Evolutionarily Stable Strategies
I An example: ants may or may not help defend the nest
I This creates a game such as:
Defend Not
2, 2
3, 1 0, 0
1, 3Defend
Not
I Suppose that ε % (some really small percent) of the ants havethe mutation, and 100 − ε % don’t
I Payoff for a mutant is bigger when ε is small
I Percent of population with mutation will grow until Nashequilibrium is reached
Evolutionarily Stable Strategies
I An example: ants may or may not help defend the nest
I This creates a game such as:
Defend Not
2, 2
3, 1 0, 0
1, 3Defend
Not
I Suppose that ε % (some really small percent) of the ants havethe mutation, and 100 − ε % don’t
I Payoff for a mutant is bigger when ε is small
I Percent of population with mutation will grow until Nashequilibrium is reached
Evolutionarily Stable Strategies
I An example: ants may or may not help defend the nest
I This creates a game such as:
Defend Not
2, 2
3, 1 0, 0
1, 3Defend
Not
I Suppose that ε % (some really small percent) of the ants havethe mutation, and 100 − ε % don’t
I Payoff for a mutant is bigger when ε is small
I Percent of population with mutation will grow until Nashequilibrium is reached
Evolutionarily Stable Strategies
I An example: ants may or may not help defend the nest
I This creates a game such as:
Defend Not
2, 2
3, 1 0, 0
1, 3Defend
Not
I Suppose that ε % (some really small percent) of the ants havethe mutation, and 100 − ε % don’t
I Payoff for a mutant is bigger when ε is small
I Percent of population with mutation will grow until Nashequilibrium is reached
Evolutionarily Stable Strategies
I An example: ants may or may not help defend the nest
I This creates a game such as:
Defend Not
2, 2
3, 1 0, 0
1, 3Defend
Not
I Suppose that ε % (some really small percent) of the ants havethe mutation, and 100 − ε % don’t
I Payoff for a mutant is bigger when ε is small
I Percent of population with mutation will grow until Nashequilibrium is reached