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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.
The simulation contains two breeds of birds, Little birds and Big birds. Each breed is hard-wired to play their breed’s 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.
The following questions allow users to develop a basic understanding of the replicator dynamic approach to evolutionary game theory.
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?
`
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.
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:
s* = the incumbent, Little bird strategy.
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:
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
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?
P( s, s) = BmeetB =_______ ```````````P( s, s*) = BmeetL = _____
P( s, s) = BmeetB =_________````````````P( s, s*) = BmeetL =___________
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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 = ___________
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).
12. Exploring the relationship between a mixed two-breed replicator dynamic stability and mixed Nash Equilibrium in a symmetric, two-player, simultaneous move game.
```````````````Big Bird``````````Little Bird
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.
b. Identify the stability condition equation that predicts Little bird stability based on these payoffs.
15. Now change the payoffs so that the mutant Big birds drive the Little birds to extinction.
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.
"Evolutionary Game Theory_Mayberry" is a much simpler approach to evolutionary game theory that models the stability concept of Evolutionarily Stable Strategies (ESS).
Dr. Jeffrey E. Russell
August 2007 |

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