A derangement of a graph G = (V, E) is an injective function f :Not all graphs admit a derangement and previous results have characterized graphs with derangements using neighborhood conditions for subsets of V . We establish an alternative criterion for the existence of derangements on a graph. We analyze strict Nash equilibria of the biologically motivated Territorial Raider Game, a multi-player competition for resources in a spatially structured population based on animal raiding and defending behavior. We find that a graph G admits a derangement if and only if there is a strict Nash equilibrium of the Territorial Raider game on G.

Many organisms maintain collective territories and compete on behalf of the fitness of the overall group. Inspired by this concept, the territorial raider model is a graph-based resource competition in which populations have fixed home locations and a limited range of sites accessible for raiding. In our present extension of the model, groups control "colonies" or "armies" which can be divided across multiple locations. We present Nash equilibria for games played on both regular graphs and regular bipartite graphs, and we also examine differences that emerge when populations are composed of discrete units (pack scale) or when they are continuously divisible (colony scale). Reliance upon defense over aggressive raiding is greater here than in the original model where populations had to totally commit to a singular action. This defensive posture increases with the advantage of the local population and also varies with the degree of the graph's connectivity. When discrete units are employed, multiple strategies emerge.

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As with many chronic conditions, matching patients with schizophrenia to the best treatment option is difficult. Selecting antipsychotic medication is especially challenging because many of the medications can have burdensome side effects. Adjusting or tailoring medications based on patients' characteristics could improve symptoms. However, it is often not known which patient characteristics are most helpful for informing treatment selection. In this paper, we address the challenge of identifying and ranking important variables for tailoring treatment decisions. We consider a value-search approach implemented through dynamic marginal structural models to estimate an optimal individualized treatment rule. We apply our methodology to the Clinical Antipsychotics Trial of Intervention and Effectiveness (CATIE) study for schizophrenia, to evaluate if some tailoring variables have greater potential than others for selecting treatments for patients with schizophrenia (Stroup et al., 2003, Schizophrenia Bulletin, 29, 15-31).