EvoLudoLab: Rock-Paper-Scissors - IBS with Mutations
| Color code: | Rock | Scissors | Paper |
|---|---|---|---|
| New rock | New scissors | New paper |
| Payoffs: | Low High
|
|---|
Note: The gradient of the payoff scale is augmented by pale shades of the strategy colours to mark payoffs that are achieved in homogeneous populations of the corresponding type.
Individual based simulations (IBS) with mutations
The interior fixed point \(\hat x\) of the replicator dynamics is an unstable focus. Even without stochasticity all trajectories spiral away from \(\hat x\) towards the boundary of the simplex \(S_3\). However, due to mutations, the boundary is repelling, which results in a stochastic analog of a stable limit cycle. Larger mutation rates increasingly limit the range of values that can be attained by the mean frequencies of the three strategies. In particular, in the limit \(\mu\to1\) the game payoffs no longer affect the dynamics and the mean of all three strategies is simply \(1/3\).
The parameters are \(s = 0.2\), \(\mu=0.01\) and the population size is \(N=1000\).
Data views
| Snapshot of the spatial arrangement of strategies. | |
| Snapshot of the spatial arrangement of strategies in 3D. | |
| Time evolution of the strategy frequencies. | |
| Trajectories of strategy frequencies shown in the simplex \(S_3\). Double clicks in the interior of \(S_3\) set the initial frequencies of strategies. | |
| Snapshot of the spatial distribution of payoffs. | |
| Snapshot of the spatial distribution of payoffs in 3D. | |
| Time evolution of average population payoff bounded by the minimum and maximum individual payoff. | |
| Payoff distribution of each strategy in population. | |
| Degree distribution in structured populations. | |
| Displays messages, warnings and errors reported by the simulation engine plus information on the applet/application. |
Game parameters
The list below describes only the parameters related to the rock-scissors-paper game and the population dynamics. Follow the link for a complete list and descriptions of all other parameters such as spatial arrangements or update rules on the player and population level.
- --paymatrix <rr,rs,rp;sr,ss,sp;pr,ps,pp>
- general \(3\times 3\) payoff matrix for the three strategic types \(R, S, P\).
- --init <r,s,p>
- initial frequencies of \(R, S, P\), respectively. Frequencies that do not add up to 100% are scaled accordingly.
- --inittype <type>
- type of initial configuration:
- frequency <f0>,<f1>...
- random distribution with given trait frequencies, f0, f1,.... Note, only available for frequency based modules and models.
- density <d0>,<d1>...
- initial trait densities <d1,...,dn>. Note, only available for density based modules and models.
- uniform
- uniform random distribution, equal frequencies of all traits.
- monomorphic <t>[,<v>]
- monomorphic initialization with trait t. Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the monomorphic trait is initialized at its (estimated) carrying capacity.
- mutant <m>,<r>[,<v>]
- single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected uniformly at random (mutants arising through cosmic rays). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
- temperature <m>,<r>[,<v>]
- single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected proportional to the in-degree of nodes (temperature initialization, mutants arising through errors in reproduction). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
- stripes
- stripes of traits. Note, only available for 2D lattices.
- kaleidoscopes
- configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Note, only available for some modules.
Note, for modules that admit multiple species, the initialization types for each species can be specified as an array separated by ;. With more species than initialization types, they are assigned in a cyclical manner.