Evolutionary Game Theory

An Introduction to...

Evolutionary Game Theory[without complex

mathematics]

Dr. Karobi MoitraBIOL 451 Evolution

What is Game Theory?

“Game theory is a study of how to mathematically determine the best

strategy for given conditions in order to optimize the outcome”

Find the best strategy to win!

RationalityAssumptions:

• humans are rational beings• humans always seek the best alternative in

a set of possible choices Why assume rationality?

• narrow down the range of possibilities• Predictability

When would you not choose the best alternative?

Game Theory Involves

• Finding acceptable, if not optimal, strategies in conflict situations.

• Game theory is highly mathematical eg: Strategy for playing chess

• Game theory assumes all human interactions can be understood and

navigated by presumptions.

Why is game theory important?

• All intelligent beings make decisions all the time.

• Helps us to analyze situations more rationally and formulate an acceptable alternative with respect to circumstance.

The Party Planner

• Shelby is planning a party, and is worried about whether it will rain or not

• The chances of rain are 33.3% on the day of the party • The party will be more fun outside• How does Shelby come to a decision?

The Party Planner

• Outside – 1 (1/3 rd) ~33.3%• Inside- 2 (2/3rds) ~66.6%

Where do you think Shelby will hold the party if she makes the decision based on the game theory?

The Party Planner

• She should hold it outside since there is a 66% chance that it will not rain

However, if Shelby is a cautious person she may hold it indoors or change the dateto account for the predicted weather

Basic Strategies

1. Plan ahead and look back 2. Use a dominating strategy if possible3. Look for any equilibrium4. Mix up the strategies

Game Theory?

Hunger Games Trailer

http://www.youtube.com/watch?v=4S9a5V9ODuY

Limitations & Problems

• Assumes players always maximize their outcomes

• Some outcomes are difficult to provide a utility for

• Not all of the payoffs can be quantified• Not applicable to all problems

Brief History of Game Theory

• 1913 - E. Zermelo provided the first theorem of game theory asserts that chess is strictly determined

• 1928 - John von Neumann proved the minimax theorem • 1944 - John von Neumann / Oskar Morgenstern’s wrote "Theory of Games and Economic Behavior”• 1950-1953, John Nash describes Nash equilibrium• 1972 - John Maynard Smith wrote

“Game Theory and The Evolution of Fighting”

http://william-king.www.drexel.edu/top/class/histf.html

John Nash Not John Nash

Definition: Nash Equilibrium

“If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium. “

Source: http://www.lebow.drexel.edu/economics/mccain/game/game.html

Nash Equilibrium

• Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy

• It is the best strategy(ies)

Evolutionary Game Theory

“Evolutionary game theory differs from classical game theory by focusing more on the dynamics of strategy change as influenced not solely by the quality of the various competing strategies, but by the effect of the frequency with which those various competing strategies are found in the population”

Rock- Paper- Scissors

“An evolutionary game that turns out to be a children’s game is rock-paper-scissors. Anyone who has ever played this simple game knows that it is not sensible to have any favored play – your opponent will soon notice this and switch to the winning counter-play. “

Rock-Paper-Scissors

“The best strategy (a Nash equilibrium) is to play a mixed random game with any of the

three plays taken a third of the time. This, in EGT terms, is a mixed strategy”

Game theory: when social conditions determine behavioral strategies

Evolutionary Game Theory

• Natural selection replaces rational behavior• Survival of the fittest

• http://www.youtube.com/watch?v=48EWLj3gIJ8

Game Theory in Evolution

1. Evolutionary Stable Strategy (ESS)2. Sexual Selection3. Social Interactions and the Evolution of Cooperation (Kin selection and altruism)

1. Evolutionary Stable Strategy

“It is a Nash equilibrium that is evolutionarily" stable: once it is fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from invading successfully. The theory is not intended to deal with the possibility of gross external changes to the environment that bring new selective forces to bear”

ESS

• An evolutionary stable strategy is a behavior that, when adopted by a population of players, cannot be invaded by an alternative strategy

Evolutionary Stable Strategy

Is the winning strategy…

Evolutionary Stable Strategy

• Introduced by Maynard Smith and Price (1973)• Strategy becomes stable throughout the

population• Mutations becoming ineffective

Hawk / Dove Game

Hawke – Dove Game

• The name "Hawk-Dove" refers to a situation in which there is a competition for a shared resource and the contestants can choose either conciliation or conflict; this terminology is most commonly used in biology and evolutionary game theory.

• HAWK: very aggressive, always fights for some resource.

• DOVE: never fights for a resource -- it displays in any conflict and if it is attacked it immediately withdraws before it gets injured.

• Thus, in any conflict situation, dove will always lose the resource to a hawk, but it never gets hurt (never sustains a decrease in fitness) when confronting a hawk, and therefore the interactions are neutral with respect to the dove's fitness

• Imagine a Hawk only population• Now imagine a Dove only population

• Hawk only population- fighting and unrest• Dove only population- too peaceful• None is the ideal situation• None are ESS or in Nash Equilibrium

Many animals react aggressively or not depending on their opponents size or correlated features

Male toads that clasps females before the eggs are laid and fertilized are often aggressively displaced by larger males but not by smaller ones. A male is unlikely to try to displace a mounted male that is larger than himself

ESS

Honest signals of the of the individual’s fighting ability or resource-holding potential

Deceptive signals indicating greater fighting ability than the individual actually has

Deceptive signals should be unstable in evolutionary time because selection would favor genotypes that ignored the signals, which having lost their utility would be lost in subsequent evolution

Dishonest signals however are not uncommon

Male fiddle crabUca annulipes

Evolutionary Stable Strategy

2. Sexual Selection

Sexual Selection by mate choice:The best mate should produce the best

offspring

Sexual SelectionSexual Selection by mate choice

Sensory Bias

Certain traits may be intrinsically stimulating and evoke a greater response simply because of the organization of the receptor’s sensory system

In some species of the fish genus Xiphophorus part of the male tail is elongated into a “sword”

Females preferred males with swords not only in Xiphophorus but also in the swordless genus Priapella

Sexual SelectionSexual Selection by mate choice

Direct Benefit to Choosy Females

Male provides a direct benefit to the female or her offspring: nutrition, superior territory, parental care

There is selection pressure on females to recognize males that are superior providers

The preference selects for males with the distinctive correlated character

Sexual SelectionSexual Selection by mate choice

Indirect Benefits of Mate Choice

Male provides no direct benefit to either the female or her offspring but contributes only his genes

Alleles affecting female mate choice increase or decrease in frequency depending on the fitness of the females’ offspring. Thus females may benefit indirectly from their choice of mates

Sexual SelectionSexual Selection by mate choice

GOOD GENES MODEL (HANDICAP HYPOTHESIS)

Because females risk substantial losses of fitness if their offspring do not survive or reproduce, one could argue that females should evolve to choose males with high genetic quality, so that their offspring will inherit “good genes” and so have a superior prospect of survival and reproduction

Any male trait that is correlated with genetic quality (an indicator of good genes) could be used by females as a guide to advantageous matings, so selection would favor a genetic propensity in females to choose mates on this basis

Stickelback Fish

“Explaining how altruism –which by definition reduces personal fitness– can arise by natural selection is a major problem, and the central

theoretical problem of sociobiology”

3. Social Interactions and the Evolution of Cooperation

3. Social Interactions and the Evolution of Cooperation

Darwin’s theory of natural selection is based on individual advantage. Cooperative interactions seem antithetical to evolution by natural selection and require explanation

Until 1960’s it was common for biologists to assume that cooperation had evolved because it benefited the population or species (group selection). The modern study of cooperation focus on individual selection.

Prisoner's DilemmaSocial Interactions and the Evolution of Cooperation

C S

Confess: C

3 1

Silent: S 4 2

Prisoner's Dilemma

The Prisoner's Dilemma game takes its name from the following scenario:

You and a criminal associate have been busted. Fortunately for you, most of the evidence was shredded, so you are facing only a year in prison. But the prosecutor wants to nail someone, so he offers you a deal: if you squeal on your associate –which will result in his getting a five year stretch– the prosecutor will see that you are set free. Which sounds good, until you learn your associate is being offered the same deal – which would get you five years

Social Interactions and the Evolution of Cooperation

To cooperate, or not cooperate? This simple question, expressed in an extremely simple game, is a crucial issue across a broad range of life

• Why shouldn't a barracuda eat the little fish that has just cleaned it of parasites

• Fig wasps collectively limit the eggs they lay in fig trees. But why shouldn't any one fig wasp cheat and leave a few more eggs than her rivals?

• At the level of human society, why shouldn't each of the villagers that share a common but finite resource try to exploit it more than the others?

Yet barracudas, fig wasps, and villagers all cooperate. It has been a constant problem in evolutionary studies to explain how such cooperation should evolve, let alone persist, in a world of self-maximizing egoists

Social Interactions and the Evolution of Cooperation

DIRECT BENEFITS: NON-ENFORCED

Cooperative behavior often evolves simply because it is advantageous to the individualEx Joining flock or herdEx Unrelated helpers and delayed benefit

Social Interactions and the Evolution of Cooperation

Pied kingfisherCeryle rudis

Social Interactions and the Evolution of CooperationINDIRECT BENEFITS: KIN SELECTION: KIN DISCRIMINATION AND CANNIBALISM

Many species of animals are cannibalistic preying on smaller individuals of the same species. Many such species discriminate kin from non-kin and are less likely to eat related individuals

Spadefoot ToadScaphiopus bombifrons

Tadpoles develop into:

• Detritus and plant omnivoresAssociate more with their siblings

• Cannibalistic carnivores Associate more with non-relativesCarnivores eat siblings much less frequently than unrelated individuals

Kin selection

Selection based on inclusive fitness

Hamilton’s Rule

An altruistic trait can increase in frequency if the benefit (B) received by the donor’s relatives weighted by their relatedness (r) to the donor, exceeds the cost (C) of this trait to the donor’s fitness:

rB>C

In evolutionary biology, an organism is said to behave altruistically when its behaviour

benefits other organisms, at a cost to itself. The costs and benefits are measured in

terms of reproductive fitness, or expected number of offspring. So by behaving

altruistically, an organism reduces the number of offspring it is likely to produce itself, but boosts the number that other

organisms are likely to produce

Social Interactions and the Evolution of CooperationSOCIAL INSECTS: ANT AND BEE COLONIES: VIDEO LINK- ALTRUISTIC WORKER BEESHTTP://WWW.YOUTUBE.COM/WATCH?V=VSCMSWOF8PY

Eusocial animals Animals in which nearly or completely sterile individuals (workers) rear the offspring of reproductive individuals , usually their parentsKnown in Naked mole rat, termites, many hymenoptera

Naked Mole RatHeterocephalus glaber

Where is game theory currently used?

– Ecology– Evolutionary Biology– Networks– Economics– Politics– Conflict– Business

Summary: Game Theory in Evolution

1. Evolutionary Stable Strategy (ESS)2. Sexual Selection3. Social Interactions and the Evolution of Cooperation (Kin selection and altruism)

References• Wikipedia• http://william-king.www.drexel.edu/top/class/his

tf.html•

http://www.lebow.drexel.edu/economics/mccain/game/game.html

• SYMBOLIC SYSTEMS 202: The Rationality Debate (3 units) Winter Quarter 2003-2004, Stanford University Instructor: Todd Davies

(handout)• Random .ppts from the internet- with no author names to acknowledge