an introduction to game theory - national …...game theory: what is a game? • there are a number...

AN INTRODUCTION

TO GAME THEORY

From Economics to Ecology and Evolution

NIMBioS Tutorial: Game Theoretical Modeling of Evolution in Structured Populations

Jonathan T. Rowell

University of North Carolina at Greensboro

April 25, 2016

GAME THEORY: What is a Game?


• There are a number of individuals Pi whom we call players

• Each player has a set of potential actions Sij called strategies

• Players are aware of each others strategies (Public Knowledge)

• Each player receives a payoff reward or penalty Ri(s) based on the strategies chosen above

• Strategies are implemented “simultaneously”

• The goal is for each player to maximize his or her payoff given the choices of the other players


Game

𝐺 = 𝑃, 𝑺, 𝑹

Players

𝑃 = {𝑃𝑖}

Player Strategy Set

𝑺𝑖 = {𝑆𝑖𝑗}

Strategies 𝑺 = 𝑺1 × 𝑺2 ×⋯× 𝑺𝑛

Game Play

𝒔 ∈ 𝑺

Payoff Vector

𝑹(𝒔)

GAME THEORY: Nash Equilibrium

Nash Equilibrium (Strong): given a game 𝐺 = 𝑃, 𝑺, 𝑹 , a Nash Equilibrium is

a game play strategy combination 𝒔∗ ∈ 𝑺 where no individual player may

unilaterally change strategies without decreasing their payoff.

𝑅𝑖 𝑠1∗, 𝑠2∗, … , 𝑠𝑖

∗, … , 𝑠𝑛∗ > 𝑅𝑖 𝑠1

∗, 𝑠2∗, … , 𝑠𝑖 , … , 𝑠𝑛

∗

Nash Equilibrium (Weak): given a game 𝐺 = 𝑃, 𝑺, 𝑹 , a Nash Equilibrium is a

game play strategy combination 𝒔∗ ∈ 𝑺 where no individual player may improve

their payoff by unilaterally change strategies.

𝑅𝑖 𝑠1∗, 𝑠2∗, … , 𝑠𝑖

∗, … , 𝑠𝑛∗ ≥ 𝑅𝑖 𝑠1

∗, 𝑠2∗, … , 𝑠𝑖 , … , 𝑠𝑛

∗

Sequential Games

• There is an order to which player decides or

implements their strategy at a given time.

• A player may have an incomplete knowledge of where he is in

relation to the whole game (information set).

• Payoffs are concluded after the game reaches a terminal

conclusion where no player has options remaining.

Sequential Games: Game Trees

• A Game Tree is a branching diagram representing all the possible routes for determining the outcome of a sequential game.

• Each branch point is called a decision node.

• There is an initial node at the top of a tree called the root.

• Each path of the tree ends at a terminal node where payoffs are assigned to the players.

• Players may control multiple nodes within the tree.

• Player strategies are a context – dependent, i.e. the strategy has a rule for choosing at each available node.

• Players may have an incomplete knowledge of their current position in relation to the whole game. Indistinguishable nodes are called information sets.

Nash Equilibria and Subgame Perfect Strategies

Subgame: A subgame 𝐺′of a sequential game 𝐺 is defined as a game whose

root node is equivalent to a node in game 𝐺, and which contains all branches

and daughter nodes of that node from the original game. Further, all terminal

nodes provide payoffs identical to the original games.

Subgame Perfect: Given a game 𝐺 = 𝑃, 𝑺, 𝑹 , a subgame perfect strategy is a

game play strategy combination 𝒔∗ ∈ 𝑺 that is a Nash Equilibrium for every

subgame of 𝐺.

Mutually Assured Destruction

Bananas Game

Bananas Game

• If Ape chose first, NE is (w,c)

• If Monkey chose first, NE is

(c,w)

• If they chose simultaneously,

the NE are (w,c), (c,w) and

(50%,50%) mixed strategy

Matrix Games-Simultaneous Actions

• In a two person game, the strategies of the first player are listed in a column, while those of the second player are listed in a row.

• The payoffs for both players are displayed in a matrix where each entry corresponds to the particular combination of strategies.

• In multi-player games, a collection of similar matrices are used.

Bundle Divide Modify As-Is

Buy (10,5) (8,12) (8,4) (5,5)

Sell (5,8) (12,2) (7,3) (6,4)

Hold (1,7) (5,1) (13,2) (3,3)

• Normal Form Games: games in which players implement strategies simultaneously.

• Typically represented as a matrix or array illustrating all possible strategy combinations.

Iterated Dominance: Deciding Strategies

• Compare alternative strategies.

• If one strategy is always better

than another, then toss out the

inferior option.

• Each player continues to

reevaluate as others eliminate

possible choices.


Buy (10,5) (8,12) (8,4) (5,5)

Sell (5,8) (12,2) (7,3) (6,4)

Hold (1,7) (5,1) (13,2) (3,3)

Do Nash Equilibria Always Exist?

• If players are limited to pure strategies in which players must always follow a

specific strategy in 𝑺𝑖, then there may not always exist a Nash equilibrium.

• If players are permitted to randomize over their pure strategies (e.g. Buy 20%,

Sell 50% and Hold 30% of the time), all games will have at least one Nash

Equilibrium among mixed strategies.

• In games with mixed strategies, the choice of percentages becomes the

players actual strategy.

Π𝑖 = 𝜋1, 𝜋2, … , 𝜋𝑛𝑖

Dominance and Equilibrium

Dominance as Flow

• By following individual responses, we can eliminate additional strategies until we are left with either a single strategy combination or a cyclic flow of responses.

• For cyclic flows, the equilibrium is set where any surviving strategies are equally profitable given the mixed strategy of the other players.


Buy (10,5) (8,12) (8,4) (5,5)

Sell (5,8) (12,2) (7,3) (6,4)

Hold (1,7) (5,1) (13,2) (3,3)


Dominance as Flow

• By following individual responses, we can eliminate additional strategies until we are left with either a single strategy combination or a cyclic flow of responses.

• For cyclic flows, the equilibrium is set where any surviving strategies is equally profitable given then mixed strategy of the other players.


Buy (10,5) (8,12) (8,4) (5,5)

Sell (5,8) (12,2) (7,3) (6,4)

Hold (1,7) (5,1) (13,2) (3,3)


Mixed Strategy Solution

• Let a,(1-a) be the proportion of Buy and Sell choices by Player 1, and let b, (1-b) be the proportion of Bundle and Divide choices by Player 2.

• Payoff of Player 1 when using Buy: 10b + 8(1-b)

• Payoff of Player 1 when using Sell: 5b +12(1-b)

• Player 2 strategy that makes Player 1’s payoff equal: 10b + 8(1-b) = 5b+12(1-b), or b=2/3

• Similarly, Player 1 strategy that equalizes Player 2’s choices: 5a+8(1-a) = 12a+2(1-a), or a=6/13


Buy (10,5) (8,12) (8,4) (5,5)

Sell (5,8) (12,2) (7,3) (6,4)

Hold (1,7) (5,1) (13,2) (3,3)

Risk Aversion and Trembling Hand Perfection

Perturbed Game: A perturbed game 𝐺′of a game 𝐺 is defined as a game

where only totally mixed strategies are allowed, that is a random selection of

the strategies available in 𝐺 in which every possibility is played with non-trivial

probability. The frequency of deviation from the intended strategy is 𝜖 ∈ (0,1).

Trembling Hand Perfect: given a game 𝐺 = 𝑃, 𝑺, 𝑹 , a strategy is trembling

hand perfect if there is a sequence of perturbed games that converge to 𝐺 in

which there is a series of Nash Equilibrium that converges to that strategy.

The Match Game

Go Stay

Go (2,2) (-1,-1)

Stay (-1,-1) (1,1)

Two players are deciding whether to go out for the evening or stay at home. Disagreement leads to

unpleasantness.

There are two Nash Equilibria in the Match Game: (Go, Go) and (Stay, Stay).

Although (Go, Go) has a higher payoff than (Stay, Stay), both are strong Nash relative to individual

deviations from the strategy.

The Love and Hate Game

Love Hate

Love (1,-1) (-1,1)

Hate (-1,1) (1,-1)

A game with two players. Player 1 wants to match the emotions of Player 2, while Player 2 wants to be

contrarian and have opposite emotions.

There is no Nash Equilibria among pure strategies. One player will always have an incentive to change

emotional strategy.

The mixed strategy of 50% Love and 50% Hate from both players is weak Nash.

If payoffs change from 1 and -1, the strategic percentages may differ from 50%.

Prisoner’s Dilemma

Cooperate Defect

Cooperate (R,R) (S,T)

Defect (T,S) (P,P)

Two players are being questioned about a crime that they committed. If they cooperate with one another,

they will escape justice and gain a share of their games R. If one person defects from the partnership

and confesses, they gain reward T, while the other player earns S (e.g. extra years in prison). If they

both confess, the receive equal payoffs T (e.g. prison terms).

The Dilemma arises when 𝑇 > 𝑅 > 𝑃 > 𝑆.

Under the Dilemma, it is always advantageous to defect from the partnership. There is single Nash

Equilibria to the game: (Defect, Defect).

This is core problem in trying to understand the origin of cooperation and honesty in various social

systems.

Hawk-Dove Game

Hawk Dove

Hawk ((V-C)/2, (V-C)/2) (V,0)

Dove (0,V) (V/2,V/2)

Also known as the Game of Chicken or the Snowdrift Game, two players decide their level of

aggressiveness. Hawk is a strong posture willing to fight, while Dove is a passive posture that flees

aggression.

The value of the object they seek to obtain is V. Doves will share the object with other Doves, while Hawks

will fight one another with cost C.

The structure of the Nash Equilibria depends on the value of the object relative to the fighting cost.

If 𝑉 > 𝐶, all players should choose Hawk.

If 𝑉 < 𝐶, both (Hawk, Dove) and (Dove, Hawk) are strategies. The players should differentiate themselves.

Non-Matrix Games and Other Ideas

Games of Intensity: Player strategy sets do not consist of discrete behaviors or strategies, but instead a continuum of possibilities. These include aggressiveness, vocalization, and auction bidding.

Games of Group Involvement: Many games involve multiple players where payoffs are set by the number of players choosing among different strategies.

Mammoth Hunt: Also called the Stag Hunt, Players can join a hunting party or remain behind for other duties. The risk of injury is equally spread among the hunters, but the spoils are equally shared among the entire group. A sufficient number of hunters are required for the expedition to be successful.

Tragedy of the Commons: Individual optimization of strategies (such as fishing a common lake) results in collective payoff less than that obtained by 1) a central organizer who determines how many people can do different activities, or 2) division of the common resource into individual ownership tracts.

Game Theory in a Biological Context

Economics

• Economics assumes the existence of rational, thinking agents (Homo economicus).

• Individuals have strategies, across which they can randomize.

• Different players can have different options.

• Economics assumes payoffs are of either a traditional financial nature (money, etc.) or a player specific valuation of the end result of the contest (seller vs. buyer of a piece of art).

• Seek to find Nash Equilibrium solutions (no one can change to improve)

• There is the possibility of conflict and cooperation within and between distinct groups or populations

Biology • Biology typically* rejects descriptions of studied organisms

with reasoned forethought as an anthropomorphic projection

• Individuals are treated as fixed in their characteristics (traits or morphs). They cannot randomize, but a population can be comprised of a mixture of different morphs.

• “Players” are often taken to possess identical options, with contests performed against conspecifics within your group.

• The primary biological payoff is relative reproductive success or resource items such as food or territory.

• Seek to find an evolutionarily stable strategy or ESS (cannot be invaded by alternate morphs or strategies)

• Conflict and cooperation within and between distinct groups or populations

Evolutionary Game Theory

Evolutionary Games: Games played within large populations where the

individual does not per se influence the current behavior within the population.

• The individual scores his strategy 𝑠𝑖𝑛𝑑 against the backdrop of the group

environment or state.

• The population shifts if novel individuals prove more fit than the average

individual drawn from the group.

𝑃(𝑠𝑖𝑛𝑑 , 𝑠𝑝𝑜𝑝)

Evolution and Selection

Biology

• Evolution is the change of trait frequencies over time.

• Genetic Drift is the selectively neutral change in trait frequency.

• Positive Selection increases trait frequency.

• Negative Selection decreases trait frequency.

• A monomorphic population consists of a single trait, and remains so absent mutation.

• A polymorphic population has individuals with different traits.

• Any model must be biologically plausible.

Mathematics

• Given a state variable 𝑥, there is an

equation 𝑑𝑥

𝑑𝑡= 𝑓(𝑥, 𝑡)

• 𝑓(𝑥, 𝑡) has some stochastic/random component, defined as

𝑓 𝑥, 𝑡 − 𝐸 𝑓 𝑥, 𝑡 .

• 𝐸 𝑓 𝑥, 𝑡 > 0

• 𝐸 𝑓 𝑥, 𝑡 < 0

• 𝑥 𝑡 = 1 and 𝑥′ = 0

• 0 ≤ 𝑥 < 1

• 𝑥 ≥ 0 ∀𝑡 ≥ 0; 𝑥𝑖 = 1𝑖 or 𝑥 𝑑𝑥𝑀

𝐿= 1

Classical Game Theory in a Biological Context

Hawk Dove

Hawk [(V-C)/2, (V-C)/2] [V,0]

Dove [0,V] [V/2, V/2]

• Hawks are characteristically aggressive in encounters for resources.

• Hawks intensify their aggression against one another in a costly conflict.

• Doves are characteristically docile and flee confrontation.

• Doves either share with other doves, or the first one there gets the resource.

• If cost of conflict is not too great (V>C), then the Nash/ESS is to always play Hawk

• If the cost is greater (V<C), then there is a mixed strategy p = V/C which is a (weak) Nash Equilibrium

Evolutionary Game Theory

Nash Equilibrium (Weak): given a game 𝐺 = 𝑃, 𝑺, 𝑹 , a Nash Equilibrium is a game play strategy combination 𝒔∗ ∈ 𝑺 where no individual player may improve their payoff by unilaterally change strategies.

𝑅𝑖 𝑠1∗, 𝑠2∗, … , 𝑠𝑖

∗, … , 𝑠𝑛∗ ≥ 𝑅𝑖 𝑠1

∗, 𝑠2∗, … , 𝑠𝑖 , … , 𝑠𝑛

∗

Evolutionarily Stable Strategy (ESS): In an evolutionary game, an ESS is a strategy 𝒔∗ such that, if it is adopted by the whole popluation, cannot be invaded by a mutant strategy 𝒔𝑚.

𝑃 𝒔∗, 𝒔∗ > 𝑃 𝒔𝑚, 𝒔∗

Or

𝑃 𝒔∗, 𝒔∗ = 𝑃 𝒔𝑚, 𝒔∗ 𝑎𝑛𝑑 𝑃 𝒔∗, 𝒔𝑚 > 𝑃 𝑠𝑚, 𝒔𝑚

Evolutionarily Stable State: An evolutionarily stable state is a genetic or morph frequency distribution which will be restored by selection after a small perturbation. This is a DYNAMIC equilibrium.

Discrete Replicator Dynamics

• Let xi(t) be the frequency of the i-th version of an allele (trait) in a population in

generation t.

• The allele conveys a fitness to individuals who possess it equal to wi.

• The frequency of allele i in the next generation is determined by the relative

fitness of these individuals versus the whole population

𝑥𝑖 𝑡 + 1 =𝑤𝑖𝑤 𝑥𝑖(𝑡)

𝑤 = 𝑤𝑖𝑥𝑖(𝑡)

𝑛

𝑘=1

Discrete Replicator Dynamics

• Fitness wi may be a fixed constant value, or it may be a function of the composition of

the population (frequency dependence) or external conditions that change over time

(temporal dependence).

• The equilibrium condition for the replicators is that individual fitness must equal

average fitness

𝑤𝑖 = 𝑤 = 𝑤𝑖𝑥𝑖(𝑡)

𝑛

𝑘=1

• This is the haploid (single chromosome) version of the Hardy-Weinberg Law where

the presence of an allele p in a diploid population that randomly mates is

𝑝 𝑡 + 1 = (𝑤11𝑝2 +𝑤12𝑝 1 − 𝑝 )/𝑤

Continuous Replicator Dynamics

• Let xi(t) be the frequency of the i-th version of an allele (trait) in a population at time t.

• The allele conveys a fitness to individuals who possess it equal to wi.

• The rate of change in frequency of allele i is determined by the difference in fitness between different alleles during mass-action encounters

𝑑

𝑑𝑡

𝑥1𝑥2= 𝑤1 −𝑤2

𝑥′ 𝑡 = 𝑤𝑥 −𝑤−𝑥 𝑥 1 − 𝑥

𝑥𝑖′ 𝑡 = 𝑤𝑖 −𝑤𝑗 𝑥𝑖𝑥𝑗

𝑗≠𝑖

Creating the Replicator Dynamics from a Game

The Replicator Equation 𝑑𝑥

𝑑𝑡= 12 𝑉 − 𝐶 𝑥 +

12𝑉 1 − 𝑥 𝑥 1 − 𝑥

𝑑𝑥

𝑑𝑡= 12 𝑉 − 𝐶𝑥 𝑥 1 − 𝑥

𝑉 > 𝐶 implies

𝑑𝑥

𝑑𝑡> 0 for all intermediate frequencies, and the population

goes to fixation for the Hawk Strategy.

𝑉 < 𝐶 implies 𝑑𝑥

𝑑𝑡> 0 only for small 𝑥. The population is driven to a mixed

(polymorphic) state.

Types of Selection

Selection Type

• Frequency Dependent Selection: the relative fitness of types chances with their frequencies

• Directional Selection: one trait is always better than the other

• Balancing Selection or Diversifying Selection: rare traits are favored

• Stabilizing Selection: genetic diversity decreases, usually in favor of a mean trait (continuum alleles)

• Disruptive Selection: intermediate forms/populations are selected against

Mathematical Interpretation

• Example: 𝑤1 𝑥 − 𝑤2 𝑥 = .5(V − Cx)

• 𝑤1 𝑥 > 𝑤2(𝑥) Example: .5 𝑉 − 𝐶𝑥 > 0 on [0,1] if 𝑉 > 𝐶

• 𝑤1 𝑥 > 𝑤2(𝑥) when 𝑥 close to 0 Example: .5 𝑉 − 𝐶𝑥 > 0 on only [0,.5] if 𝑉 = .5𝐶

• 𝑤1 𝑥 > 𝑤2(𝑥) when 𝑥 close to 1

• 𝑤1 𝑥∗ = 𝑤2 𝑥

∗ and d

dtw1 −w2 > 0

at 𝑥∗

Continuous Replicator Dynamics

• Monomorphic populations (only one allele or morph is present) are dynamic fixed points to the system, but may be subject to invasion (are dynamically unstable).

• Replicator dynamics can be extended to multiplayer scenarios where different roles have different choices (Defend or Flee vs. Invade or Ignore).

Three Alternative Male Morphs:

Rock-Paper-Scissors

Biological Setup

• Territorial Males: maintain a territory (e.g. a rockpile) and aggressively challenge all competing Territorial Males and Raiders.

• Helper Males: they assist in the upkeep of a site (e.g. brood care), and are tolerated by Territorial Males. Helper males use this to sneak copulations.

• Raiders: do not maintain territories, but periodically encroach on the sites of others to achieve copulations.

Territorial Helper Raider

Territorial [1,1] [.5, 1.5] [1.8,.2]

Helper [1.5,.5] [1,1] [.3,1.7]

Raider [.2, 1.8] [1.7,.3] [1,1]

𝑇′ = 1 − 1.5 𝑇 + .5 − 1 𝐻 + 1.8 − .3 𝑅 𝑇𝐻

+ 1 − .2 𝑇 + .5 − 1.7 𝐻 + 1.8 − 1 𝑅 𝑇𝑅

Three Alternative Male Morphs:

Rock-Paper-Scissors

Signaling Dynamics

Truthful Deceive

Believe [A,0] [0,B]

Disbelieve [0,C] [D,0]

Truthful Deceive

Believe v.

Disbelieve

A -D

The Game

The Differential Payoffs

Truthful vs.

Deceive

Believe -B

Disbelieve C

Signaling Dynamics

• 𝑥 is the frequency of belief

• 𝑦 is the frequency of honesty

𝑑𝑥

𝑑𝑡= 𝐴𝑦 − 𝐷 1 − 𝑦 𝑥 1 − 𝑥

𝑑𝑦

𝑑𝑡= −𝐵𝑥 + 𝐶 1 − 𝑥 𝑦 1 − 𝑦

• Compact Expression

𝑑𝑥

𝑑𝑡= 𝐷 + 𝐴 𝑦 − 𝐷 𝑥 1 − 𝑥

𝑑𝑦

𝑑𝑡= 𝐶 − 𝐵 + 𝐶 𝑥 𝑦(1 − 𝑦)

Signaling Dynamics

• If 𝐴, 𝐵, 𝐶, 𝐷 > 0, the frequencies of honesty and belief oscillate (BLUE). The interior fixed point is an unstable spiral.

• If 𝐴, 𝐵, 𝐷 > 0 > 𝐶, (a partial Prisoner’s Dilemma), the communication system collapses: all signals are dishhonest, and no one believes (RED). The point (0,0) is a globally stable attractor.

Multi-Listener Signaling:

Territorial Displays • Males of a species give a public

signal asserting control over a territory.

• These signalers may be large or small.

• Listeners (also large or small), perceive these signals and decide to Respect the territorial male’s claim or to Challenge.

• The outcome of fights are predicated on the size of individuals with the possibility of escalation among equally matched opponents.

Multi-Listener Signaling:

Territorial Displays

Small Respect Small Challenge

Big Respect 𝐵1, 𝐵2, 𝑇 𝐵1, −𝐹3, 𝑇 − 𝐹_3

Big

Challenge

𝑇

2− 𝐹1 − 𝐼, 𝐵2,

𝑇

2+ 𝐼 − 𝐹1

𝑇

2− 𝐹1 − 𝐼, 0,

𝑇

2+ 𝐼 − 𝐹1

Small Respect Small Challenge

Big Respect 𝐵1, 𝐵2, 𝑇 + 𝜖 𝐵1,𝑇 + 𝜖

2− 𝐹2 − 𝐼,

𝑇 + 𝜖

2+ 𝐹2𝐼

Big

Challenge

𝑇 − 𝐹3, 𝐵2, −𝐹3 𝑇 − 𝐹3, 0, −𝐹3

Big Signaler

Small Signaler

Sexual Selection

Sexual Reproduction

• Monogamy: an individual mates

with a single member of the

opposite sex

• Polygamy: an individual mates with

multiple individuals of the opposite

sex

• Polygyny: males mate with multiple

females

• Polyandry: females mate with

multiple males

Sexual Selection

• Assortative Mating: an individual mates with a single member of the opposite sex

• Trait Preference: an individual mates with multiple individuals of the opposite sex

• Prezyogtic: sexual selection occurs prior to the copulation and/or fertilization

• Postzyogtic: sexual selection occurs after fertilization

• Cryptic: the specific mechanism of selection occurs after copulation, but is not understood

Sexual Selection: Female Preferences

• We assume strict polygyny. All females are reproductively equally fit (V).

• Males have two alleles or morph variations for a particular trait (e.g. eye color).

• This trait may or may not be tied to the fitness of the male.

• Females have a preference for one of the two male trait options.

• Fitness is measured by the mean share of an individual’s contribution to the next generation.

• Let the current population of males with traits T1 and T2 be g and (1-g).

• Let the current population of females with preferences P1 and P2 be n and (1-n).

• Let the strength of preferences be 𝑠1and 𝑠2.

• The per capita fitness for males:

𝐹1 =

(1 + 𝑠1)

1 + 𝑠1 𝑔 + (1 − 𝑔)𝑛 +

1

𝑔 + 1 + 𝑠2 1 − 𝑔(1 − 𝑛)

𝐹2 =1

1 + 𝑠1 𝑔 + 1(1 − 𝑔)𝑛 +

(1 + 𝑠1)

𝑔 + 1 + 𝑠2 1 − 𝑔(1 − 𝑛)

Sexual Selection: Female Preferences

• If 𝑔 = 0, fitness scores are

𝐹2 = 𝑛 + 1 − 𝑛 = 1

𝐹1 = 1 + 𝑠1 𝑛 +(1 − 𝑛)

1 + 𝑠2

=1 + 𝑠1 + 𝑠2 + 𝑠1𝑠2 𝑛

1 + 𝑠2

𝐹1 = 1 + 𝑠1𝑛 −𝑠2(1 − 𝑛)

1 + 𝑠2

• Trait 1 can invade if

𝑠1𝑛 > 𝑠2(1 − 𝑛)

1 + 𝑠2

• Trait 2 can invade if

𝑠2(1 − 𝑛) > 𝑠1𝑛

1 + 𝑠1

Sexual Selection: Male Preferences

• We assume strict polygyny. All females are reproductively equally fit (V).

• Females have two alleles or morph variations for a particular trait (e.g. eye color).

• This trait may or may not be tied to the fitness of the female.

• Males either have a preference for female trait T1 or no preference.

• Fitness is measured by the mean share of an individual’s contribution to the next generation.

• Let the current population of males with and without preference for trait T1 be g and (1-g).

• Let the current population of females with traits T1 and T2 be n and (1-n).

• Strength of preference is 𝜌

• Ignoring non-preferred females interpretation

𝐹1 =1

𝑔 + (1 − 𝑔)𝑛 +

(1 − 𝜌)

(1 − 𝜌)𝑔 + 1 − 𝑔(1 − 𝑛)

𝐹2 =1

𝑔 + (1 − 𝑔)𝑛 +

1

(1 − 𝜌)𝑔 + 1 − 𝑔(1 − 𝑛)

𝑭𝟏 < 𝑭𝟐

Salvaging Male Preference:

Ideal Free Distribution • Consider each type of female a patch of real value 𝑛𝑉 and 1 − 𝑛 𝑉.

• Biased Males perceive the patch values as 𝑛𝑉(1 + 𝜌) and 1 − 𝑛 𝑉.

• Males are allowed to relocate to contest for females at a time scale faster than the one on which evolution is action. (Ecological vs. Evolutionary times scales).


Ideal Free Distribution • Value of pursuing females is the expected mating success

𝑢𝑛𝑖𝑡 𝑓𝑒𝑚𝑎𝑙𝑒 𝑣𝑎𝑙𝑢𝑒

𝑐𝑜𝑚𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑚𝑎𝑙𝑒𝑠 + 𝑐𝑜𝑢𝑟𝑡𝑠ℎ𝑖𝑝 𝑡𝑖𝑚𝑒

𝑛𝑉

𝑎1𝑔𝑝 + 𝑎2 1 − 𝑔 𝑞 + 𝑛𝑐


Ideal Free Distribution • Genetic Equation

•𝑑𝑔

𝑑𝑡= 𝑘𝑔 𝑅1

∗− 𝑅2

∗𝑔(1 − 𝑔)

• Courtship Efforts for Biased (p) and Unbiased Males (q)

•𝑑𝑝

𝑑𝑡= 𝑘𝑠 𝑅11 𝑉11 − 𝑅12 𝑉12 𝑝 1 − 𝑝

𝑑𝑞

𝑑𝑡= 𝑘𝑠 𝑅21 𝑉21 − 𝑅22 𝑉22 𝑝 1 − 𝑝

Sex-Linked Traits to

Population Genetics • Individuals are haploids with two di-

allelic loci.

• There are four genotypes available to both males and females: P1T1, P2T1, P1T2, and P2T2.

• Expression of traits and preferences is sex-specific.

• Genes can recombine during reproduction (e.g. P from father, but T from mother).

• The replicator equations reflect both the selection process and recombination.

Conclusions

• Game Theory began as a means of understanding optimal decision making

against the strategies of others.

• The Nash Equilibrium is a strategy combination from which no individual

player may independently change to improve their payoff in the game.

• Game Theory was adopted for use in biology to study questions of ecology

and evolution.

• It was later expanded into a new evolutionary game theory concept that

integrated game theory with dynamic equations governing the change of

frequencies within a population.

Conclusions

• A replicator equation describes the rate of change in the frequency of a trait or

morph in a population.

• It is obtained by looking at the difference in fitness of each potential morph,

subjected to the rules of selection for the problem. As such, it can be tied with

standard matrix-style strategy games.

• The replicator equation can be used within a population or role (traditional

evolutionarily stable states) or in the context of multiple actors with different

roles and options.

• By including multiple listeners in signaling dynamics, varied dynamic

outcomes are achievable in terms of the maintaining signal fidelity and

credibility.

Conclusions

• Female preferences for arbitrary male preferences could promote the

preferred trait depending on the strength of preference.

• Under the equivalent treatment, male preferences ought to be selected

against.

• If the courtship/contest stage occurs at a faster time scale than selection, the

redistribution of males with a preference can initiate a complete shift in the

male population. Unbiased males abandon overly competitive females for

those non-preferred.

• Male preference therefore strongly encourages assortative mating in the

population.

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