Glossary

UNDER CONSTRUCTION

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.

Player

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 simplified, 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.

Graph

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.

Interaction

In the interaction relation your neighbors are the players with whom you could choose to play a game.

Replacement

In the replacement relation your neighbors are the players who might die to be replaced by your offspring, or who might imitate you. 

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.

Time

Infinite Populations

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.

Finite Populations

For finite populations time is simulated in units of Monte-Carlo steps (MC steps). If the population size is [math]\displaystyle{ N }[/math], then a single MC step consists of [math]\displaystyle{ N }[/math] individual player actions.

A player who acts may imitate a neighbor, they may reproduce (prompting a neighbor to die to make room for the offspring), they may die (prompting a neighboring player to reproduce to replace them), all of which may change someone's 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 MC time step, then use parameter nInteractions to indicate the average number of interactions. For instance, if in the graph you are using, each player has four neighbors, then nInteractions could be 2 (indicating that each action would play games with a randomly chosen 2 of the four neighbors) or 6 (the random choice is with replacement), and so on.

If you wish to model interaction as very frequent vis à vis reproduction, use parameter X 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.

lingering asynchronous time

The payoffs of the neighbors who play with the changed player are averaged in with their past interactions. The change builds on whatever fitness they had, modeling biology where the effects of all past interactions impact fitness.

memoryless asynchronous time

The payoffs of the players who play with the changed player are recalculated from scratch, modeling biology where earlier interactions have no residual effect on fitness.

synchronous time

All [math]\displaystyle{ N }[/math] 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.

asynchronous time

Under favorable climatic conditions animals may breed and reproduce throughout the year. To model this, during each MC step a sequence of [math]\displaystyle{ N }[/math] 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.

If a player is changed, that player's game payoffs are calculated by playing a round of games with their interaction neighbors.

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.

birth and death

To give birth is to be selected to reproduce.  Your offspring replaces someone in your replacement group. To die is to be drawn to be replaced by some neighbor's offspring, a neighbor that has you in their replacement group.  Which happens first is important, see birth-death and death-birth.

imitation

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...