Frequency dependent fitness
by Christoph Hauert, Version 1.0, December 2004.
In different contexts it is certainly more reasonable to frequency dependent fitness rather than a static fitness concept where only the type of individual (resident versus mutant) determines its reproductive success. In a frequency dependent setting, the actual fitness of an individual is determined through interactions with other members of the population. The fitness of an individual now depends not only on its own type but also on the type of the interaction partners. This can be easily summarized in a payoff matrix. For the Moran process in well-mixed populations the frequency dependent fixation probabilities can be derived analytically. Unfortunately, however, for any kind of structured populations the dynamics is much more complicated and leaves many fascinating questions open for future research.
The following simulations illustrate the invasion dynamics of mutants focussing on two particular aspects: dominated strategies and super-stars. A strategy (or type) A is said to be dominated whenever it would be better to switch to strategy (type) B irrespective of the opponents types. In well mixed populations, dominated strategies are doomed and quickly vanish. Quite surprisingly, however, the evolutionary fate of such strategies changes dramatically for certain population structures and super-stars in particular. The structure of super-stars is simple enough that at least some analytical approximations can be made, if the payoff of an individual is fully determined by its downstream neighbor. Note that the simulations do not determine the probability to reach fixation for mutants but rather show the process of invading a resident population (if the mutant is capable) or reestablishing a homogenous resident population (if the mutant fails to invade). In order to get an idea of the fixation probability, several runs have to be carried out using the same parameter values.
The fitness of residents and mutants is frequency dependent and determined by interactions with other members of the population. The population structure determines the interaction neighborhood as well as the competition- or reproduction neighborhood of each individual. The results are preliminary for this complex setting but quite surprising results are obtained for dominated strategies.
Dominated strategies in well-mixed populations are doomed and are quickly driven to extinction. In order to visualize the extinction process, this simulation start of with 95% mutants (red) and just 5% 'residents' (blue) rather than a single mutant in a resident population. However, even for such favorable conditions for the mutant, it fails to reach fixation.
Star structures, where a single central hub is connected to peripherial leaf nodes and vice versa, have the surprising effect of amplifying selection in the frequency independent case of static fitnesses. For frequency dependent interactions these structures also exhibit surprising features. With the same parameter values as for the well-mixed case above, a single mutant adopting a dominated strategy (red) now has a probability of ~75% to reach fixation. This corresponds to the fixation probability in the original Moran process upon setting r = b2/(d c) in the limit of large N, where b, c, and d refer to the entries in the payoff matrix.
In the frequency independent case, super-stars act as evolutionary amplifiers that are able to guarantee the fixation of mutants with an arbitrarily small fitness advantage and thus suppress random drift entirely. In the frequency dependent case they exhibit similarly striking features in that they can guarantee fixation of a dominated strategy (at least if the payoff is determined through interactions with their downstream neighbors). In that case, the fixation probability is given the original Moran process with r=(b/d)(b/c)K-1, where b, c, and d again refer to the entries in the payoff matrix. The simulation illustrates the case where individuals interact with all neighbors (upstream and downstream) and their fitness is given by their average performance. In that case, dominated strategies also reach fixation with very high probabilities, however, it is another open question, whether fixation can be guaranteed.
In the frequency dependent case, the dynamics and fixation probabilities of mutants, in general, and of dominated strategies, in particular, is an open issue. However, the results for the frequency independent case suggest, that dominated strategies should be weakly favored on scale-free networks but to what extend will depend on the payoff values. This simulation considers a single mutant with the same parameter values as the simulations above. How does it compare to the well-mixed case and the different star-like structures?
Along the bottom of the VirtualLab are several buttons to control the execution and the speed of the simulations. Of particular importance are the Param button and the data views pop-up list on top. The former opens a panel that allows to set and change various parameters concerning the game as well as the population structure, while the latter displays the simulation data in different ways.
|New resident||New mutant|
Note: The yellow and green strategy colors are very useful to get an impression of the activitiy in the system.
|Params||Pop up panel to set various parameters.|
|Views||Pop up list of different data presentations.|
|Slider||Idle time between updates. On the right your CPU clock determines the update speed while on the left updates are made roughly once per second.|
|Mouse||Mouse clicks on the graphics panels generally start, resume or stop the simulations.|
|Structure - Strategy||Snapshot of the spatial arrangement of strategies. Mouse clicks cyclically change the strategy/type of the respective site for the preparation of custom initial configurations.|
|Mean frequency||Time evolution of the strategy frequencies.|
|Structure - Fitness||Snapshot of the spatial distribution of payoffs.|
|Mean Fitness||Time evolution of the mean payoff of each strategy together with the average population payoff.|
|Histogram - Fitness||Histogram of payoffs for each strategy.|
The list below describes only the parameters relevant for the frequency dependent Moran process. Follow the link for a complete list and detailed descriptions of all other parameters such as spatial arrangements or update rules on the player and population level.
- reward for mutual cooperation.
- temptation to defect, i.e. payoff the defector gets when matched with a cooperator.
- sucker's payoff which denotes the payoff the cooperator gets when matched with a defector.
- punishment for mutual defection.
- Init Coop, init defect:
- initial fractions of cooperators and defectors. If they do not add up to 100%, the values will be scaled accordingly. Setting the fraction of cooperators to 100% and of defectors to zero, then the lattice is initialized with a symmetrical configuration.