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Evolutionary_Game_Theory_Big_Bird_Replicator_Dynamic

by Jeff Russell (Submitted: 09/06/2007)

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WHAT IS IT?

A two-player, replicator dynamic model of evolutionary game theory is simulated in this program. The model contains two breeds of birds that randomly play a simultaneous game with other birds. Each breed is "hard-wired" to play their breed's strategy. The model allows the user to set different payoffs for each of the four possible matches between breeds. These payoffs determine each type's reproductive success. The model also allows the user to change the overall birth and death rate for all birds in the simulation. This allows a complete simulation of evolutionary models that include background levels of fitness as well as payoffs based on cooperation and competition.

HOW IT WORKS

The simulation contains two breeds of birds, Little birds and Big birds. Each breed is hard-wired to play their breedís strategy (a polymorphic population). Birds can fly in any direction and randomly fly a distance between one and 100 patches. The birds live in a swamp that consists of random black patches (tar pits), random green patches (nesting sites) and random white patches (neutral sand). Birds die if they land on a black patch, and they breed via cloning if they land on a green patch. After landing on a patch, the bird waits for all other birds to finish that round of movement before they react to their patch.
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"Setup" randomly scatters white, black, and green patches around the swamp based on the delta and beta settings. "Setup" also randomly scatters the initial number of birds around the swamp in the proportion determined by the "proportion big" setting.
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"Go" is clicked after setup to initiate the simulation. Clicking "Go" again pauses the simulation. If initial sliders are changed, "setup" needs to be re-clicked to reflect the changes.
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The swamp's carrying capacity limit is determined by the "carrying capacity" slider. If the population of birds exceeds this carrying capacity, disease and insufficient food will cause birds to randomly die off even if they don't land on a tar pit. This population limit highlights the importance of the "sufficiently large population" assumption associated with the replicator dynamic. If carrying capacity is set too low, the random nature of the simulation may cause a population to unexpectedly die out. If carrying capacity is set too high, the simulation will begin to run at a very slow speed, especially in older computers. In most situations, users should set carrying-capacity as high as possible to avoid unexpected population demise.
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The "beta" slider determines the proportion of the swamp that will be green nesting sites. The initial setting of 0.25 means one in four patches will be green. This proportion is equivalent to a 25% base-line birth rate in the simulation because any bird landing on a green patch duplicates itself.
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The "delta" slider determines the proportion of the swamp that will be black tar pits. The initial setting of 0.25 means one in four patches will be black. This proportion is equivalent to a 25% base-line death rate in the simulation because any bird landing on a black patch dies. If "beta" and "delta" are both set at 0.5, the swamp will be set up with an equal number of green and black patches and the only white patch will be the home patch in the middle of the swamp.
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The background level of fitness for the simulation is a function of the birth-rate minus the death-rate. If birth rate (beta) is set to be greater than death rate (delta), birds will die out. If "beta" is set to be less than "delta", birds should reach carrying capacity. If beta is set equal to delta, the outcome is random and the population will randomly walk away from the initial number of birds, eventually reaching carrying capacity, and/or eventually dying out. These results require all four of the "meet" sliders to be set to zero. This eliminates the additional cloning that is used to create payoffs later in the exercise.
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The "initial-birds" slider determines the starting number of birds. The "proportion_big" slider determines the initial mix of big and little birds. For example, if "initial-birds" is set at 100, and "proportion_big" is set at 0.25, the starting population will consist of 25 big and 75 little birds randomly scattered around the swamp.
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If a bird lands on a green, nesting patch, it will create two clones of itself, then die, resulting in a net increase of one of that breed. If exactly two birds are on a green patch, there is the possibility of additional cloning which is used to create payoffs for the two player game. The number of additional clones is determined by the four "meet" sliders which define payoffs for all four player combinations found in a simultaneous, two player game.
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The "BmeetB" slider determines the additional payoff received by each of (exactly) two Big birds that are on a green patch. The "BmeetL" slider determines the additional payoff received by a Big bird that encounters a Little bird on a green patch when there are exactly two birds on a green patch. The "LmeetB" slider determines the additional payoff received by a Little bird that encounters a Big bird on a green patch when there are exactly two birds on a green patch. The "LmeetL" slider determines the additional payoff received by each of (exactly) two Little birds that are on a green patch. For example, if a Big and a Little bird are on the same green patch, the Big bird can be assigned a 5 payoff and the Little bird a 1 payoff by setting the "BmeetL" at 5 and "LmeetB" at 1. This would result in the creation of five additional big bird clones and one additional little bird clone in addition to the base-line birth rate of one (net) additional clone.
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There are three meters that calculate at the end of each round: (1) the total number of birds in the population (count birds); (2) the cumulative average proportion of Big birds (cumulative prop big); and (3) the cumulative average proportion of Little birds (cumulative prop little). The simulation also provides a graph of the relative proportion of the two breeds (prop big and little), as well as a graph of the number of each breed (num. big (red) and little (blue)). Each unit of time on the graphs represents one round of play. The reference lines are at 1/3 and 2/3. A breed or population that reaches the carrying capacity ceiling as well as a breed or population that dies out can be easily seen on the graphs.
These meters reflect deaths in tar pits and cloning in nesting sites; however, they are calculated prior to the impact of a population exceeding the carrying capacity limit; consequently the (count birds) and the (num. big (red)) and (num. little (blue)) may "bounce" above the carrying capacity limit.
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SELF DIRECTED EXERCISE

The following questions allow users to develop a basic understanding of the replicator dynamic approach to evolutionary game theory.
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The LEARNING OBJECTIVES for this exercise are:
(1) an understanding of the simulation's replicator dynamic mechanics through experimentation; (2) an understanding of the risk of extinction for a "too-small" population due to the random possibility of death, even if expected payoffs do not predict extinction; (3) practice in first determining asymptotic stability in a replicator dynamic model based on stability condition equations, then verifying that the simulation arrives at the same answer; and (4) an understanding of the equivalence between a stable, mixed population and the mixed Nash equilibrium for a two player, two strategy game.
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BASELINE BIRTH AND DEATH RATES

1. First, run the simulation with the "_meet_" sliders all left at 0 and "beta" and "delta" left at 0.25 each. Theoretically, with birth and death rates equal, the population should neither grow nor shrink. Observe that the random nature of the simulation causes a "random walk" pattern where sometimes the population eventually dies out, and sometimes it grows to carrying capacity. With these "meet" settings, there is no difference between payoffs for big versus little birds.

2. Change the "beta" setting to 0.26. Make sure the "_meet_" sliders are all left at 0. With birth rate slightly higher than death rate, what is the observed outcome? Does this make sense?
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3. Change the "delta" setting so that it is slightly above "beta". With birth rate slightly lower than death rate, what is the observed outcome? Does this make sense?
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LOW POPULATION, RANDOMNESS, AND EXTINCTION
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4. Set "beta" at 0.25 and "delta" at 0.24. Reduce the starting number of birds to ten. Run a number of simulations with different initial bird settings.
a. What seems to be the minimum number of starting birds to avoid random extinction?
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b. What does this imply regarding the random chance of extermination for a small population of an endangered species?
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ASYMPTOTIC STABILITY IN A REPLICATOR DYNAMIC MODEL

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Equilibrium in the replicator dynamic model is slightly different than the Evolutionarily Stable Strategy concept developed in the Mayberry simulation. A theoretical replicator dynamic model assumes an arbitrarily large population. With that assumption, over time, unsuccessful mutants are driven towards extinction; however, unlike the ESS concept, mutants never actually completely die out (e.g. if there is only one chance in a billion a mutant will survive, one mutant should survive if the population is assumed to be one billion or larger). If mutants are driven towards extinction, the incumbent population is said to be "asymptotically" stable, roughly the same thing as saying the incumbents are ESS in this two-player model. In this simulation with limited population size, asymptotic stability is inferred from the actual elimination of unsuccessful mutants. In other words, if the mutants die out, the incumbent population is stable.

To begin to develop this stability idea, mutant Big birds are introduced to the incumbent Little bird population. Set "beta" at 0.25 and "delta" at 0.24. Set "initial birds" at 200. Big birds will be the invading mutants, so they need to be a "small" number, so set "proportion_big" at 0.3. Set all "meet" sliders at 0.
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DEFINING STABILITY USING RELATIVE PAYOFFS

Whether or not a population will be stable against an invasion of mutants can be predicted by looking at the relative payoffs when the two breeds encounter each other. To make the stability condition more concise, a set of equations can be used. To that end, let:
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s* = the incumbent, Little bird strategy.
s = the mutant, Big bird strategy.
P = the payoff to the first breed when it encounters another bird.
For example, P( s*, s) represents the payoff to an incumbent Little bird when it encounters a mutant Big bird on a green patch.
Conversely, P( s, s*) represents the payoff to a mutant Big bird when it encounters an incumbent Little bird on a green patch.

Whether or not a population is stable can then be predicted as follows:

An incumbent (s*) population is asymptotically stable if the following equations are true:
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`````````P(s*,s*) > P(s,s*)`````````````````````````````````````````````(1)
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``````````````````If P(s*,s*) = P(s, s*)``````````````````````````````````(2)
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````````````````````````then P(s*, s) > P(s,s)
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In other words, if the payoff to an incumbent playing against another incumbent is greater than the payoff of a mutant playing against an incumbent (equation 1), then the incumbent population will successfully resist an invasion of mutants, regardless of the payoffs to the breeds when they play against mutants. Conversely, if the payoff to an incumbent playing against another incumbent is less than the payoff of a mutant playing against an incumbent, then equation (1) is not true and the incumbent population is not stable because an invasion of mutants will not be successfully driven towards extinction.

However, if equation (1) is an equality, then equation (2) determines if the incumbent population is stable. Now, if the payoff to an incumbent playing against a mutant is greater than the payoff of a mutant playing against another mutant (equation 2), then the incumbent population will successfully resist an invasion of mutants. Conversely, if the payoff to an incumbent playing against a mutant is less-than-or-equal-to the payoff of a mutant playing against another mutant, then equation (2) is not true and the incumbent population is not stable because an invasion of mutants will not be successfully resisted.

The payoffs used in equations (1) and (2) are determined by the four "meet" sliders in the program as follows:

P(s,s) = BmeetB`````````P( s, s*) = BmeetL
P( s*, s) = LmeetB````` P( s*, s*) = LmeetL

5. Given the initial "meet" settings of all 0, does equation (1) indicate that a population of Little birds will be stable against an invasion of mutant Big birds?
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6. If it is necessary to look to equation (2), does that equation indicate that a population of Little birds will be stable against an invasion of mutant Big birds?
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7. Based on your answers to 6 and 7, what do you expect to happen when you allow a small number of mutant big birds to invade the little bird population? To check your answer, set "beta" at 0.26, "delta" at 0.25, the "proportion big" at 0.3, click "setup", then run the simulation. Repeat a few times to get a sense of what typically happens. Do the graphs and readings on the cumulative proportion meters make sense?
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8. Change the meet slider payoffs so that Little birds is now stable
using the equation (1): P( s*, s*) > P( s, s*).
Identify the slider positions you use below:

P( s, s) = BmeetB =_______ ```````````P( s, s*) = BmeetL = _____
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P( s*, s) = LmeetB =______ ```````````P( s*, s*) = LmeetL =______
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9. Change the "meet" slider payoffs so that equation (1) is an equality; however, the meet sliders are set so that equation (2)
P( s*, s) > P( s, s) still keeps the Little bird population stable.
Identify the slider positions you use below:

P( s, s) = BmeetB =_________````````````P( s, s*) = BmeetL =___________
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P( s*, s) = LmeetB =_________```````````P( s*, s*) = LmeetL =___________

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IRRELEVANCE OF BIRTH AND DEATH RATE

Notice that the birth and death rate is not included in the stability condition equations. This implies that as long as the population enjoys an overall birth rate that is greater than the death rate, the actual settings of the birth and death rate DO NOT impact the stability calculations.

To verify this, use the settings from question 9 above and change both the "delta" and "beta" to something over 0.40, and then set them both to something less than 0.10, and look for any changes in the outcome compared to question 9. (There should be none, unless a high death rate wipes out a low starting population). Write in the settings you experiment with below:

10. High beta = _______, High delta = ___________
````` Low beta = ______, Low delta = ___________

HAWKS AND DOVES

A game of "hawks and doves" is often used to demonstrate how asymptotic stability in a replicator dynamic model can include more than one breed (strategy).
The hawks and doves game is modeled in this simulation as follows:
If there are exactly two birds on a green nesting site, that site has additional resources. The two birds compete for these additional resources which result in the production of additional clones. The number of these additional clones is determined by the "_meet_" sliders.
Big birds exhibit aggressive, "hawk-like" behavior. Little birds exhibit a more passive and cooperative, "dove-like" behavior. If a Little bird and a Big bird land on a green nesting site at the same time, the Big bird aggressively takes most of the resources and creates 6 additional Big bird clones of itself while the Little bird is only able to create 2 additional Little bird clones. (Set BmeetL = 6, LmeetB = 2) If a Little bird meets a Little bird, they peacefully and equally split up the additional payoff between each other; consequently they each create 4 additional clones of Little birds (Set LmeetL = 4). If a Big bird meets another Big bird, they are both aggressive and waste energy fighting each other; consequently, both Big birds receive a smaller payoff than pairs of Little birds receive, equal to 1 in this scenario (Set BmeetB = 1).
With these payoffs, a small number of aggressive Big birds enjoy a significant advantage which translates into an increase in the proportion of Big birds. Eventually, the increased number of Big birds results in a large number of Big birds competing with each other for a low payoff and the more cooperative Little birds gain an advantage and increase their proportion. To see how this works, run the simulation a few times with these settings. (Set "proportion_big" at 0.5. Set "beta" at 0.26 and "delta" at 0.25. Set "initial birds" at 200. Click "setup".)
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11. After running the simulation with these settings, what proportions do the cumulative proportion monitors settle at?
cumulative prop big = _____________
cumulative prop little = ____________

12. Exploring the relationship between a mixed two-breed replicator dynamic stability and mixed Nash Equilibrium in a symmetric, two-player, simultaneous move game.
The above "meet" payoff settings can be converted to a normal form game theory matrix (bird species playing against itself using two strategies) as follows:

```````````````Big Bird``````````Little Bird
Big Bird`````````1,1````````````````6,2
Little Bird```````2,6````````````````4,4

Calculate the mixed Nash equilibrium given these payoffs.

Proportion of Big bird strategy = ___________________

13. How does this calculation compare to the cumulative proportion of Big birds that arises in the simulation?

14. Mixed stabilities are fairly sensitive to changes in payoffs.
Change ONE of the payoffs from question 11 so that the Big birds again die out.
Identify the slider positions you use below:
P( s, s) = BmeetB = _______ P( s, s*) = BmeetL =__________
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P( s*, s) = LmeetB =_______ P( s*, s*) = LmeetL = __________

b. Identify the stability condition equation that predicts Little bird stability based on these payoffs.
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15. Now change the payoffs so that the mutant Big birds drive the Little birds to extinction.
a. Identify the slider positions you use below:
BmeetB = ___________ BmeetL = ___________
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LmeetB = ___________ LmeetL = ___________

17. Develop a different story for Big and Little birds cooperating and competing. In this story, the LITTLE birds need to be the majority, yet the stable population must include both Big and Little birds.
NOTE: To maintain a stable mixed rather than a pure population, the relative values of payoffs need to be: BmeetL> LmeetL> LmeetB> BmeetB.
Indicate the slider positions you use to create the stable, mixed, majority-little-bird population below:
BmeetB =____________ BmeetL = _____________
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LmeetB = ___________ meetL = _____________
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18. Write out a brief narrative story supporting the above payoffs (e.g. why does LmeetL result in such a large payoff that is nonetheless lower than the BmeetL payoff, etc.):
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RELATED MODELS

"Evolutionary Game Theory_Mayberry" is a much simpler approach to evolutionary game theory that models the stability concept of Evolutionarily Stable Strategies (ESS).
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CONTACT THE AUTHOR

Dr. Jeffrey E. Russell
Department of Economics
Ashland University,
Ashland, OH
jrussell@ashland.edu

August 2007

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