We study evolutionary games on graphs. The individuals of a population occupy the vertices of the graph and interact with their neighbors to receive payoff. We consider finite population size, regular graphs, probabilistic death-birth updating and weak selection. There are two types of strategies, A and B, and a payoff matrix [(a, b),(c, d)]. The initial condition is given by an arbitrary configuration where each vertex is occupied by either A or B. The conjugate initial condition is obtained by swapping A and B. We ask: when is the fixation probability of A for the original configuration greater than the fixation probability of B for the conjugate configuration? The answer is a linear condition of the form σa+b > c+σd. We calculate σ for any initial condition. For large population size we obtain the well known result σ = (k + 1)/(k − 1), but now this result extends to any mixed initial condition. As a specific example we study evolution of cooperation. We calculate the critical benefit-to-cost ratio for natural selection to favor the fixation of cooperators for any initial condition. We obtain results that specify which initial conditions reduce and which initial conditions increase the critical benefit-to-cost ratio. Adding more cooperators to the initial condition does not necessarily favor cooperation. But strategic placing of cooperators in a network can enhance the takeover of cooperation.

We provide a model of a decentralized, dynamic auction market platform (e.g., eBay) in which a large number of buyers and sellers participate in simultaneous, singleunit auctions each period. Our model accounts for the endogenous entry of agents and the impact of intertemporal optimization on bids. Solving our model with a finite number of bidders is computationally intractable due to the curse of dimensionality, so we prove that a continuum version of our model provides a good approximation of an equilibrium in the finite model. We use the approximation to estimate the structural primitives of our model using Kindle sales on eBay. We find that just over one third of Kindle auctions on eBay result in an inefficient allocation with deadweight loss amounting to 13.5% of total possible market surplus. We also find that partial centralization - for example, runnng half as many 2-unit, uniform price auctions each day - would eliminate a large fraction of the inefficiency, but yield lower seller revenues. Our results highlight the importance of understanding platform composition effects—selection of agents into the market—in assessing the implications of market design.

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We present a simple model for the evolution of social behavior in family-structured, finite sized populations. Interactions are represented as evolutionary games describing frequency-dependent selection. Individuals interact more frequently with siblings than with members of the general population, as quantified by an assortment parameter r, which can be interpreted as “relatedness”. Other models, mostly of spatially structured populations, have shown that assortment can promote the evolution of cooperation by facilitating interaction between cooperators, but this effect depends on the details of the evolutionary process. For our model, we find that sibling assortment promotes cooperation in stringent social dilemmas such as the Prisoner's Dilemma, but not necessarily in other situations. These results are obtained through straightforward calculations of changes in gene frequency. We also analyze our model using inclusive fitness. We find that the quantity of inclusive fitness does not exist for general games. For special games, where inclusive fitness exists, it provides less information than the straightforward analysis.

Over time, a population acquires neutral genetic substitutions as a consequence of random drift. A famous result in population genetics asserts that the rate, K, at which these substitutions accumulate in the population coincides with the mutation rate, u, at which they arise in individuals: K = u. This identity enables genetic sequence data to be used as a “molecular clock” to estimate the timing of evolutionary events. While the molecular clock is known to be perturbed by selection, it is thought that K = u holds very generally for neutral evolution. Here we show that asymmetric spatial population structure can alter the molecular clock rate for neutral mutations, leading to either K<u or K>u. Our results apply to a general class of haploid, asexually reproducing, spatially structured populations. Deviations from K = u occur because mutations arise unequally at different sites and have different probabilities of fixation depending on where they arise. If birth rates are uniform across sites, then K
u. In general, K can take any value between 0 and Nu. Our model can be applied to a variety of population structures. In one example, we investigate the accumulation of genetic mutations in the small intestine. In another application, we analyze over 900 Twitter networks to study the effect of network topology on the fixation of neutral innovations in social evolution.