Re: [Q] Neighbourhood on small toroidal grids

[Jason Alexander]

> What sort of biases do you get as a result of not worrying about
> repeated cells/self in your own neighbourhood?

I'm not sure how this fits into the framework you're working in, as the
example comes from my own area of research...

Suppose agents receive payoffs or rewards determined by their
interactions with neighboring agents, and these payoffs are interpreted
along the lines of "more is better".  For models with a
coordination-game like structure, not worrying about repeated cells or
your self in your own neighborhood can artificially inflate the payoffs
received by agents and, I suspect, skew the statistical behavior of the
model.

Consider the following example (N.B. I'm making this up as I type, so
take what follows with a grain of salt).  Suppose we have a spatially
distributed, sparsely populated world in which agents interact playing a
coordination game of the following form:

        A       B
A       (5,5)   (-6,-1)
B       (-1,-6) (2,2)

Suppose the model runs as follows.  Each generation, agents can move,
interact, and reproduce (in that order).  Agents reproduce when their
payoff gets above 10 and die when it reaches 0. When an agent dies, it
is removed from the world.

In this model, whether pure populations of A or B are susceptible to
invasion by mutants depends, in part, on whether agents are allowed to
interact with themselves or not.  Suppose self-interactions are
forbidden.  Then when a mutant arises, it will wander around until it
interacts with another agent.  Since these two agents are of
incompatible types, both will receive a negative score.  After a while,
the mutant will die.  (I assume the mutation rate is quite low, so the
chance of two mutants meeting is negligible.)  However, if agents can
self-interact and the world is sparsely populated, then mutants can
invade.  For if a mutation occurs, each round the mutant will be able to
interact with itself, increasing its score to the point where it can
replicate.  (At the same time, though, all of the original members of
the population will be replicating as well, so the long-term behavior
probably depends on the ratio of payoffs between strategies A and B.)

If we consider cases where the world begins with a polymorphism, I
suspect one would find that allowing/not allow self-interaction leads to
a difference in the size of the basins of attraction.

If anyone knows of cases in the literature which treat this (or similar
models), I'd appreciate a reference.

Cheers,

Jason Alexander
Logic & Philosophy of Science
University of California, Irvine


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