“Evolutionary Game Theory as a Framework for Studying Biological Invasions”
Pintor, L.M., Brown, J.S. and Vincent T.L. 2011. American Naturalist
We started our discussion of the paper “Evolutionary Game Theory as a Framework for Studying Biological Invasions” with a quick introduction to some basic game theory. To get everyone’s game theory juices flowing we played a few games. The following handout was given.
Game 1: A simple grade scheme for the class.
Read the following carefully. Without showing your neighbor what you are doing, put it in the box below either the letter Alpha or the letter Beta. Think of this as a grade bid. Dr. Reddy will randomly pair your form with another form and neither you nor your pair will ever know with whom you were paired. If you put α and you’re paired with β, then you will get an A and your pair a C. If you and your pair both put α, you’ll both get B-. If you put β and you’re paired with α, you’ll get a C and your pair an A. If you and your pair both put β, then you’ll both get B+.
Game 2: Best Strategy
Consider the two player-four strategy matrix below. The numbers in this game are your payoffs, with a higher number being the higher payoff. Choose a strategy A, B, C, or D. Consider both if your pair chooses randomly or the same strategy as you. Why did you choose that strategy?
A B C D
A 5 6 5 3
B 5 9 1 2
C 3 11 5 1
D 2 7 6 4
For game 1, we discussed the different reasons why people chose alpha or beta. Lindsay suggested that alpha would be a good choice in this game because you could potentially get the higher grade, while Matt chose beta because he is a nice guy and thought it would be better for the class as a whole to get B+’s. The matrix for this game would look as follows:
you α B-, B- A, C
β C, A B+, B+
In game theory it is common for information to be presented in matrix form. In this simple game the players are classmates, the strategies are alpha and beta, and the payoffs are grades obtained by you and your classmate. After discussing this game, we went on to look at a more complex strategy game. For each strategy A, B, C, and D there are possible reasons for each choice. The choice of strategy A is known as the Max-Min Strategy, which maximizes the lowest payoff possible. Strategy B is known as the Group Optimal Strategy, where the highest overall payoff given when all individuals use the same strategy. Strategy C is the Max – Max Strategy. They player in C plays the most desirable strategy for that individual. Also, row C has the highest average payoff if the other player selects at random. Finally, strategy D is the No Regret Strategy or Nash Equilibrium: If all individuals use Strategy D, then an individual has no incentive to unilaterally change his/her strategy. If an individual is free to change strategies a Nash Solution is equilibrium in the sense that no player should want to change their strategies.
So as you can imagine, if strategies are not finite like in our examples (A, B, C, or D) the theory can get pretty complex.
In evolutionary game theory the players are organisms, the strategies are heritable traits, and the payoffs are calculated in terms of fitness. By definition an Evolutionary Stable State (ESS) or Nash Equilibrium occurs when a strategy is resistant to invasion by another strategy. The ESS must be convergent and therefore evolutionarily stable. This definition leads to an important question of how evolutionary game theory can be applied to biological invasions.
Before addressing the application of game theory to biological invasions it is important to understand how one might model evolution. The G-Function or Fitness Generating Function is one of many approaches for modeling evolution of continuous traits. To create a fitness generating function you must choose an ecological model that best represents the population or species. In this paper the authors chose the Lotka-Volterra Model. The Lotka-Volterra model looks at predator –prey dynamics and competition.
After choosing an ecological model you look at strategies or strategy sets that may be continuous or discrete. Remember our strategies in this evolutionary game are heritable traits. These are determined from hypotheses concerning genetic, developmental, physiological, and physical constraints. Finally, to create the G-Function, hypothesize how the individual’s strategy, as well as the strategies in the population, influences the values of parameters in the ecological models of population dynamics.
The benefit to this approach is that it takes both ecological and evolutionary aspects into a fitness model. Ecologically, the G-Function will describe the change in population size over time. Evolutionarily, the G-Function will describe the change in the species’ mean strategy as an adaptive dynamic.
The results of Pintor et al. address what the evolutionary context of the invader is relative to the native community. This could be as a novel G-Function, or as the same G-Function. If the invader has a novel technology, it could be deemed as superior or non-superior. The authors address the possible outcomes of such hypotheses as being species coexistence or species replacement.
Next Pintor et al. discussed scenarios where the invader had a similar G-Function. In order for the invasion to occur they proposed that the community would either have to contain an empty niche, or a recipient community that is non-ESS or off of its fitness peak, in which case the resulting invasion would cause species coexistence or replacement accordingly. The author’s provide multiple graphs that show the adaptive landscape and possible outcomes in each scenario, as well as provide a well thought out flowchart for their work.
The application of game theory to invasion biology has many benefits. First of all, it allows for a formalized approach to modeling. The theory draws equal attraction to traits influencing competitive interactions, provides a priori hypotheses to invasion success and finally, provides an evolutionary mechanism to failed invasions. The application of game theory provides new insights to what would make a novel invader. The authors suggest that novel invaders do not contain the potential to evolve the same strategies. New insight is also obtained on the definition of an empty niche as, “an unoccupied peak in an existing adaptive landscape.” With this new definition the ecological consequence of an empty niche can only be coexistence.
At the end of the class we discussed short comings of this paper. We approached the idea that the Lotka-Volterra model might not be the best model to use for biological invasions. Other ecological models could be used in calculating G-functions and this could result in different outcomes than the authors found. Also, it is possible for an invader to have different invasion potential based on the life history of the organism (such as the zebra mussel, which has a pelagic and benthic stage to its life history).
There are many directions this paper can lead the current research in invasion biology. But most likely, the multiple hypotheses and assumptions that need to be made through the application of evolutionary game theory to biological invasions could prove to be difficult in its application.
For more information on game theory see the following link to free video lectures at Yale University: http://www.academicearth.org/lectures/introduction-to-game-theory
For more information on evolutionary game theory check out the book “Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics” by Thomas L. Vincent and Joel S. Brown. 2005.