8 research outputs found
Slow Emergence of Cooperation for Win-Stay Lose-Shift on Trees
We consider a group of agents on a graph who repeatedly play the prisoner’s dilemma game against their neighbors. The players adapt their actions to the past behavior of their opponents by applying the win-stay lose-shift strategy. On a finite connected graph, it is easy to see that the system learns to cooperate by converging to the all-cooperate state in a finite time. We analyze the rate of convergence in terms of the size and structure of the graph. Dyer et al. (2002) showed that the system converges rapidly on the cycle, but that it takes a time exponential in the size of the graph to converge to cooperation on the complete graph. We show that the emergence of cooperation is exponentially slow in some expander graphs. More surprisingly, we show that it is also exponentially slow in bounded-degree trees, where many other dynamics are known to converge rapidly
Discordant Voting Processes on Finite Graphs
We consider an asynchronous voting process on graphs called discordant voting, which can be described as follows. Initially each vertex holds one of two opinions, red or blue. Neighboring vertices with different opinions interact pairwise along an edge. After an interaction both vertices have the same color. The quantity of interest is the time to reach consensus, i.e., the number of steps needed for all vertices have the same color. We show that for a given initial coloring of the vertices, the expected time to reach consensus depends strongly on the underlying graph and the update rule (i.e., push, pull, oblivious)
On the Imitation Strategy for Games on Graphs
In evolutionary game theory, repeated two-player games are used to study
strategy evolution in a population under natural selection. As the evolution
greatly depends on the interaction structure, there has been growing interests
in studying the games on graphs. In this setting, players occupy the vertices
of a graph and play the game only with their immediate neighbours. Various
evolutionary dynamics have been studied in this setting for different games.
Due to the complexity of the analysis, however, most of the work in this area
is experimental. This paper aims to contribute to a more complete
understanding, by providing rigorous analysis. We study the imitation dynamics
on two classes of graph: cycles and complete graphs. We focus on three well
known social dilemmas, namely the Prisoner's Dilemma, the Stag Hunt and the
Snowdrift Game. We also consider, for completeness, the so-called Harmony Game.
Our analysis shows that, on the cycle, all four games converge fast, either to
total cooperation or total defection. On the complete graph, all but the
Snowdrift game converge fast, either to cooperation or defection. The Snowdrift
game reaches a metastable state fast, where cooperators and defectors coexist.
It will converge to cooperation or defection only after spending time in this
state which is exponential in the size, n, of the graph. In exceptional cases,
it will remain in this state indefinitely. Our theoretical results are
supported by experimental investigations.Comment: 32 page