rock paper scissors - penn math

Rock Paper Scissors

I Consider Rock Paper Scissors:

-1, 1 1, -1 0, 0R P S

I What is the expected payoff of (13 ,13 ,

13) against (12 , 0,

I 0I Note that the expected payout is a weighted average of payouts

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

13) against (12 , 0,

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

13) against (12 , 0,

I Note that the expected payout is a weighted average of payouts

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

13) against (12 , 0,

Rock Paper Scissors

-1, 1 1, -1 0, 0R 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

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

-1, 1 1, -1 0, 0R P S

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

12 ) is

to play rock

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

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

12 ) is

to play rock

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

I By removing pure strategies that lower the averageI u((1, 0, 0), ( 1

2, 0, 1

2)) = 1

I u((0, 1, 0), ( 12, 0, 1

2)) = 0

I u((0, 0, 1), ( 12, 0, 1

2)) = − 1

12 ) is

to play rock

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

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

12 ) is

to play rock

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

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

12 ) is

to play rock

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

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

12 ) is

to play rock

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

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

12 ) is to play rock

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

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

12 ) is to play rock

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

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

I What should the (unique) Nash equilibrium be?

I When both players use ( 13 ,

I Test this: what is the payoff of a pure strategy against( 13 ,

I Note that the expected payoff for each player is 0 (the gameis fair)

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

I What should the (unique) Nash equilibrium be?

I When both players use ( 13 ,

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

I What should the (unique) Nash equilibrium be?I When both players use ( 1

3 ,13 ,

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

3 ,13 ,

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

3 ,13 ,

Rock Paper Scissors

-1, 1 1, -1 0, 0R P S

3 ,13 ,

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

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:

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:

90,10 20,80

80,20You

Opponent

(.6, .4)(q = .6)

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:

90,10 20,80

80,20You

Opponent

(.6, .4)(q = .6)

left or right)

I Payoffs are:

90,10 20,80

80,20You

Opponent

(.6, .4)(q = .6)

90,10 20,80

80,20You

Opponent

(.6, .4)(q = .6)

90,10 20,80

80,20You

Opponent

(.6, .4)(q = .6)

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

(.6, .4)(q = .6)

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)

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)

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

(.6, .4)(q = .6)

Going to the Movies

0, 0 2, 3

1, 1You

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

I Similarly, q = 14

I Note that both you and your date’s expected payout is

(between the original payouts of the Nash equilbrium)

Going to the Movies

0, 0 2, 3

1, 1You

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

I Similarly, q = 14

Going to the Movies

0, 0 2, 3

1, 1You

(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

I Similarly, q = 14

Going to the Movies

0, 0 2, 3

1, 1You

the payouts for both strategies must be the sameI So 2p = p + 3(1 − p)

I p = 34

I Similarly, q = 14

Going to the Movies

0, 0 2, 3

1, 1You

the payouts for both strategies must be the sameI So 2p = p + 3(1 − p)I p = 3

I Similarly, q = 14

Going to the Movies

0, 0 2, 3

1, 1You

I Similarly, q = 14

Going to the Movies

0, 0 2, 3

1, 1You

I Similarly, q = 14

Going to the Movies

0, 0 2, 3

1, 1You

I Similarly, q = 14

I Note that both you and your date’s expected payout is 32

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?

I An example: ants may or may not help defend the nest

I This creates a game such as:

Defend Not

3, 1 0, 0

1, 3Defend

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

Defend Not

3, 1 0, 0

1, 3Defend

Defend Not

3, 1 0, 0

1, 3Defend

Defend Not

3, 1 0, 0

1, 3Defend

Defend Not

3, 1 0, 0

1, 3Defend

rock paper scissors - penn math

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