Moranův proces s nesmrtelnými jedinci

Fof, Martin

We study the immortal Moran process, a modification of the standard Moran process on graphs in which certain types cannot be overwritten. We focus on the case of two immortal types, which we call the World Conquest process, and analyze it through the lens of a two-player game. We compute the game value exactly for several standard graph families. In the case of complete graphs, we establish an exact correspondence with generalized diagonal Pólya urns. Our main result shows that neither player is inherently dominant. For any fitness values and any ε, δ > 0, there exists a graph on which the second player wins at least 1 − δ of all vertices with probability at least 1 − ε, regardless of the first player's choice. The construction achieving this is a new family of graphs, which we call weighted super simplex graphs. Combined with the star graph, where the first player wins all but one vertex with certainty, this demonstrates that the game value can be pushed arbitrarily close to either extreme.

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