math 180 - game theorychhays/lecture24.pdf · i an introduction to game theory (osborne) (contains...

Administration

There are three more books on reserve:

I Strategies and Games: Theory and Practice (Dutta)(also available online)

I An Introduction to Game Theory (Osborne)(contains evolutationarily strategic strategies)

I Thinking Strategically (Dixit and Nalebuff)

Administration

Find a real world application of the quantitative reasoning thatwe’ve covered in class

By Friday, November 8. turn in:

I topic

I one paragraph summary

I at least one reference

Administration

Find a real world application of the quantitative reasoning thatwe’ve covered in classBy Friday, November 8. turn in:

I topic

I one paragraph summary

I at least one reference

Evolutionarily Stable Strategies

I Idea: some small percentage of a population develops amutation

I This creates a competing ‘strategy’, compared to animalswithout the mutation

I A strategy is a genetic dispositionI The payoff corresponds to the likelihood of offspring

Evolutionarily Stable Strategies

I Idea: some small percentage of a population develops amutation

I This creates a competing ‘strategy’, compared to animalswithout the mutation

I A strategy is a genetic dispositionI The payoff corresponds to the likelihood of offspring

Evolutionarily Stable Strategies

I Idea: some small percentage of a population develops amutation

I This creates a competing ‘strategy’, compared to animalswithout the mutation

I A strategy is a genetic disposition

I The payoff corresponds to the likelihood of offspring

Evolutionarily Stable Strategies

I Idea: some small percentage of a population develops amutation

I This creates a competing ‘strategy’, compared to animalswithout the mutation

I A strategy is a genetic dispositionI The payoff corresponds to the likelihood of offspring

Evolutionarily Stable StrategiesI An example: a species cooperates while hunting

I A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defect

I This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2ε

I u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2ε

I Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thrive

I Cooperation is not evolutionarily stable

Evolutionarily Stable StrategiesI An example: a species cooperates while huntingI A mutation causes some to defectI This creates a game such as:

C D

2, 2

3, 0 1, 1

0, 3C

D

I Question: is cooperation evolutionarily stable?(will the defecting mutation die out?)

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (1 − ε, ε)(ε is the proportion with the mutation)

I u(C , (1 − ε, ε)) = 2 − 2εI u(D, (1 − ε, ε)) = 3 − 2εI Defectors will thriveI Cooperation is not evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defect

I Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)I u(C , (ε, 1 − ε)) = 2εI u(D, (ε, 1 − ε)) = 1 + 2εI Cooperators will not thriveI Defection is evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defectI Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)I u(C , (ε, 1 − ε)) = 2εI u(D, (ε, 1 − ε)) = 1 + 2εI Cooperators will not thriveI Defection is evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defectI Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)I u(C , (ε, 1 − ε)) = 2εI u(D, (ε, 1 − ε)) = 1 + 2εI Cooperators will not thriveI Defection is evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defectI Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)

I u(C , (ε, 1 − ε)) = 2εI u(D, (ε, 1 − ε)) = 1 + 2εI Cooperators will not thriveI Defection is evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defectI Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)I u(C , (ε, 1 − ε)) = 2ε

I u(D, (ε, 1 − ε)) = 1 + 2εI Cooperators will not thriveI Defection is evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defectI Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)I u(C , (ε, 1 − ε)) = 2εI u(D, (ε, 1 − ε)) = 1 + 2ε

I Cooperators will not thriveI Defection is evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defectI Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)I u(C , (ε, 1 − ε)) = 2εI u(D, (ε, 1 − ε)) = 1 + 2εI Cooperators will not thrive

I Defection is evolutionarily stable

Evolutionarily Stable Strategies

C D

2, 2

3, 0 1, 1

0, 3C

D

I Now suppose that the default behavior was to defectI Is defection evolutionarily stable?

I Need to compare payoffs of cooperation and defection againstrandom animal in population

I Look at payoffs of C and D against (ε, 1 − ε)I u(C , (ε, 1 − ε)) = 2εI u(D, (ε, 1 − ε)) = 1 + 2εI Cooperators will not thriveI Defection is evolutionarily stable

Definition

In a 2-player symmetric game, a strategy s is evolutionarily stableif for sufficiently small numbers ε > 0, and any other strategy s∗,

(1 − ε) u(s, s) + ε u(s, s∗) > (1 − ε) u(s∗, s) + ε u(s∗, s∗)

This is saying that the utility of s in a mixed population (1 − ε, ε)is better than the utility of s∗ is the mixed population

Definition

In a 2-player symmetric game, a strategy s is evolutionarily stableif for sufficiently small numbers ε > 0, and any other strategy s∗,

(1 − ε) u(s, s) + ε u(s, s∗) > (1 − ε) u(s∗, s) + ε u(s∗, s∗)

This is saying that the utility of s in a mixed population (1 − ε, ε)is better than the utility of s∗ is the mixed population

Evolutionarily Stable Strategies

I See Handout #7

I Morals:

I evolutionarily stable strategies give Nash equilibriaI (symmetric) Nash equilibria are not necessarily evolutionarily

stableI evolutionarily stable strategies are not strictly dominated by

other strategiesI evolutionarily stable strategies do not necessarily strongly

dominate other strategies

Evolutionarily Stable Strategies

I See Handout #7I Morals:

I evolutionarily stable strategies give Nash equilibriaI (symmetric) Nash equilibria are not necessarily evolutionarily

stableI evolutionarily stable strategies are not strictly dominated by

other strategiesI evolutionarily stable strategies do not necessarily strongly

dominate other strategies

Evolutionarily Stable Strategies

I See Handout #7I Morals:

I evolutionarily stable strategies give Nash equilibria

I (symmetric) Nash equilibria are not necessarily evolutionarilystable

I evolutionarily stable strategies are not strictly dominated byother strategies

I evolutionarily stable strategies do not necessarily stronglydominate other strategies

Evolutionarily Stable Strategies

I See Handout #7I Morals:

I evolutionarily stable strategies give Nash equilibriaI (symmetric) Nash equilibria are not necessarily evolutionarily

stable

I evolutionarily stable strategies are not strictly dominated byother strategies

I evolutionarily stable strategies do not necessarily stronglydominate other strategies

Evolutionarily Stable Strategies

I See Handout #7I Morals:

I evolutionarily stable strategies give Nash equilibriaI (symmetric) Nash equilibria are not necessarily evolutionarily

stableI evolutionarily stable strategies are not strictly dominated by

other strategies

I evolutionarily stable strategies do not necessarily stronglydominate other strategies

Evolutionarily Stable Strategies

I See Handout #7I Morals:

I evolutionarily stable strategies give Nash equilibriaI (symmetric) Nash equilibria are not necessarily evolutionarily

stableI evolutionarily stable strategies are not strictly dominated by

other strategiesI evolutionarily stable strategies do not necessarily strongly

dominate other strategies