- 1 THE BIG PICTURE (population is to player as graph is to vertex)
- 2 Graph
- 3 Time
- 4 Reproduction and Imitation
- 5 Game
THE BIG PICTURE (population is to player as graph is to vertex)
A player is an individual member of the population being modeled. If the population is structured, that is, modeled with a graph, each player occupies a vertex (a position, a node) in the graph, and each vertex is occupied by a player. This being so, we sometimes say "player" to refer to their "vertex" and vice versa.
A player's traits include their inherent fitness. If they reproduce, their offspring inherits a copy of their traits. If they imitate another player, they adopt the other player's traits. If mutation happens, their traits may be modified. If as usual an interaction game is being played, their traits determine their strategy which will impact their behavior, which will impact how much the game pays off for them, which will increase their fitness.
Sometimes when the situation is simple, like traits = strategy = behavior (for instance every players is either a Defector or a Cooperator and that's their sole trait and their strategy and their behavior) then the word "strategy" is often used for trait or behavior too.
A graph models a relation between players. It is a list, for each player p, of the players that p considers to be its neighbors in the relation modeled by the graph. Usually the graph is undirected (bidirected), i.e. whenever p is among v's neighbors, v is among p's neighbors. If the graph contains even one exception to that rule, it is called a directed graph. Unrequited neighborliness.
There are two logically distinct relations that are represented by a graph in Evoludo, called interaction and replacement. For the moment both concepts have the same graph, so that for every player, its interaction neighbors and its replacement neighbors are the same neighbors.
In the interaction relation your neighbors are the players with whom you could choose to play a game.
If the replacement graph is undirected, one could think of the graph vertices as territories in a territorial species, under the assumption that each territory is occupied by exactly one player. The replacement graph would then model which territories adjoin, under the assumption that territories are static and deaths are replaced by births from adjacent territories.
For infinite populations time is deterministically modeled. When using numerical integration of ordinary differential equations (ODE), the time increment is automatically chosen by the integrator (Fifth-order Runge-Kutta method with adaptive step size). When using numerical integration of partial differential equations (PDE), the time increment is an adjustable parameter.
For finite populations time is simulated in units of Monte-Carlo steps (MC steps). If the population size is \(N\), then a single MC step consists of \(N\) individual player actions. A player action is a potential transfer of traits, via reproduction or imitation of some neighbor.
An player action may cause a player to interact (play games) with some of its neighbors, or toimitate a neighbor, or to reproduce (prompting a neighbor to die to make room for the offspring), or to die (prompting a neighboring player to reproduce to replace them), all of which may change their (or a neighbor's) traits, i.e. game strategy and payoffs. Which of these actions occur, and in what order, is governed primarily by the population update settings.
If you wish to model that interactions occur only a few times per action, then use parameter nInteractions to indicate the number of interactions per action. For instance, if each player has four interaction neighbors, and nInteractions was 3 (or 7), then an action would play games with a randomly chosen 3 (or 7) of the four neighbors, with replacement, and the scores accumulated or averaged. Note that the chosen neighbors' scores will change too, since their scores reflect the additional games played.
If you wish to model interactions as much more frequent than actions, you can set the parameters so that whenever a vertex's traits change (via mutation, imitation, or death and replacement by someone else's offspring), the new player plays games with all their neighbors to arrive at the limiting case, as if nInteractions was infinite with averaged scores. Neighbors scores are also adjusted to reflect their score accounting for the change in their neighborhood.
All \(N\) players act simultaneously, in parallel, each MC step. This models situations where external influences such as diurnal rhythms or seasonal changes lead to a synchronization of the breeding season or where competitive interactions are synchronized under harsh environmental conditions such as occur in the arctic or in a desert.
Under favorable climatic conditions animals may breed and reproduce throughout the year. To model this, during each MC step a sequence of \(N\) players are chosen at random (with replacement) to act in turn. This implies that in a given time step, some players may not be chosen to act at all, and others may be chosen multiple times.
Reproduction and Imitation
Draw vs Selection for Fitness
In this documentation we attach special meaning to the terms draw and select. To draw a player is to chose a player at random with no regard for fitness. Whereas to select a player is to chose a player weighted by their fitness, i.e. fitter players have a greater chance of being chosen. Players are always selected to reproduce, and drawn to die; selected to be #Imitation:imitated, drawn to imitate; in other words selected to be the source of |traits, and drawn to take on those traits.
birth and death
To give birth is to be selected to reproduce. Your offspring replaces one of your replacement neighbors. To die is to be drawn to be replaced by some neighbor's offspring, a neighbor that has you in their replacement neighborhood. Which happens first is important, see birth-death and death-birth.
construction notes: uses interaction.out exclusively, never touches replacement (reproduction.out). Everybody evaluates change of strategy first (includes possibility of mutation), so one action can cause multiple strategy updates. memoryless. Assume pairwise: Since it resets(zeros) all scores in group, and recalcs scores solely based on games with me, it seems that the scores of the group reflect only on their games with me, not with other neighbors of theirs...