Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy

Subgame perfect equilibria are specific Nash equilibria in perfect information games in extensive form. They are important because they relate to the rationality of the players. They always exist in infinite games with continuous real-valued payoffs, but may fail to exist even in simple games with slightly discontinuous payoffs. This article considers only games whose outcome functions are measurable in the Hausdorff difference hierarchy of the open sets (\textit{i.e.} $\Delta^0_2$ when in the Baire space), and it characterizes the families of linear preferences such that every game using these preferences has a subgame perfect equilibrium: the preferences without infinite ascending chains (of course), and such that for all players $a$ and $b$ and outcomes $x,y,z$ we have $\neg(z <_a y <_a x \,\wedge\, x <_b z <_b y)$. Moreover at each node of the game, the equilibrium constructed for the proof is Pareto-optimal among all the outcomes occurring in the subgame. Additional results for non-linear preferences are presented.Comment: The alternative definition of the difference hierarchy has changed slightl

Phylogenetic Analyses of Vitis (Vitaceae) Based on Complete Chloroplast Genome Sequences: Effects of Taxon Sampling and Phylogenetic Methods on Resolving Relationships Among Rosids

Background The Vitaceae (grape) is an economically important family of angiosperms whose phylogenetic placement is currently unresolved. Recent phylogenetic analyses based on one to several genes have suggested several alternative placements of this family, including sister to Caryophyllales, asterids, Saxifragales, Dilleniaceae or to rest of rosids, though support for these different results has been weak. There has been a recent interest in using complete chloroplast genome sequences for resolving phylogenetic relationships among angiosperms. These studies have clarified relationships among several major lineages but they have also emphasized the importance of taxon sampling and the effects of different phylogenetic methods for obtaining accurate phylogenies. We sequenced the complete chloroplast genome of Vitis vinifera and used these data to assess relationships among 27 angiosperms, including nine taxa of rosids. Results The Vitis vinifera chloroplast genome is 160,928 bp in length, including a pair of inverted repeats of 26,358 bp that are separated by small and large single copy regions of 19,065 bp and 89,147 bp, respectively. The gene content and order of Vitis is identical to many other unrearranged angiosperm chloroplast genomes, including tobacco. Phylogenetic analyses using maximum parsimony and maximum likelihood were performed on DNA sequences of 61 protein-coding genes for two datasets with 28 or 29 taxa, including eight or nine taxa from four of the seven currently recognized major clades of rosids. Parsimony and likelihood phylogenies of both data sets provide strong support for the placement of Vitaceae as sister to the remaining rosids. However, the position of the Myrtales and support for the monophyly of the eurosid I clade differs between the two data sets and the two methods of analysis. In parsimony analyses, the inclusion of Gossypium is necessary to obtain trees that support the monophyly of the eurosid I clade. However, maximum likelihood analyses place Cucumis as sister to the Myrtales and therefore do not support the monophyly of the eurosid I clade. Conclusion Phylogenies based on DNA sequences from complete chloroplast genome sequences provide strong support for the position of the Vitaceae as the earliest diverging lineage of rosids. Our phylogenetic analyses support recent assertions that inadequate taxon sampling and incorrect model specification for concatenated multi-gene data sets can mislead phylogenetic inferences when using whole chloroplast genomes for phylogeny reconstruction

From winning strategy to Nash equilibrium

Game theory is usually considered applied mathematics, but a few game-theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e. the existence of a winning strategy in games that involve two players and two outcomes saying who wins. In a multi-outcome setting, the notion of winning strategy is irrelevant yet usually replaced faithfully with the notion of (pure) Nash equilibrium. This article shows that every determinacy result over an arbitrary game structure, e.g. a tree, is transferable into existence of multi-outcome (pure) Nash equilibrium over the same game structure. The equilibrium-transfer theorem requires cardinal or order-theoretic conditions on the strategy sets and the preferences, respectively, whereas counter-examples show that every requirement is relevant, albeit possibly improvable. When the outcomes are finitely many, the proof provides an algorithm computing a Nash equilibrium without significant complexity loss compared to the two-outcome case. As examples of application, this article generalises Borel determinacy, positional determinacy of parity games, and finite-memory determinacy of Muller games

Verbalizing phylogenomic conflict: Representation of node congruence across competing reconstructions of the neoavian explosion

Phylogenomic research is accelerating the publication of landmark studies that aim to resolve deep divergences of major organismal groups. Meanwhile, systems for identifying and integrating the products of phylogenomic inference-such as newly supported clade concepts-have not kept pace. However, the ability to verbalize node concept congruence and conflict across multiple, in effect simultaneously endorsed phylogenomic hypotheses, is a prerequisite for building synthetic data environments for biological systematics and other domains impacted by these conflicting inferences. Here we develop a novel solution to the conflict verbalization challenge, based on a logic representation and reasoning approach that utilizes the language of Region Connection Calculus (RCC-5) to produce consistent alignments of node concepts endorsed by incongruent phylogenomic studies. The approach employs clade concept labels to individuate concepts used by each source, even if these carry identical names. Indirect RCC-5 modeling of intensional (property-based) node concept definitions, facilitated by the local relaxation of coverage constraints, allows parent concepts to attain congruence in spite of their differentially sampled children. To demonstrate the feasibility of this approach, we align two recent phylogenomic reconstructions of higher-level avian groups that entail strong conflict in the "neoavian explosion" region. According to our representations, this conflict is constituted by 26 instances of input "whole concept" overlap. These instances are further resolvable in the output labeling schemes and visualizations as "split concepts", which provide the labels and relations needed to build truly synthetic phylogenomic data environments. Because the RCC-5 alignments fundamentally reflect the trained, logic-enabled judgments of systematic experts, future designs for such environments need to promote a culture where experts routinely assess the intensionalities of node concepts published by our peers-even and especially when we are not in agreement with each other

The evolutionary radiation of Arvicolinae rodents (voles and lemmings): relative contribution of nuclear and mitochondrial DNA phylogenies

BACKGROUND: Mitochondrial and nuclear genes have generally been employed for different purposes in molecular systematics, the former to resolve relationships within recently evolved groups and the latter to investigate phylogenies at a deeper level. In the case of rapid and recent evolutionary radiations, mitochondrial genes like cytochrome b (CYB) are often inefficient for resolving phylogenetic relationships. One of the best examples is illustrated by Arvicolinae rodents (Rodentia; Muridae), the most impressive mammalian radiation of the Northern Hemisphere which produced voles, lemmings and muskrats. Here, we compare the relative contribution of a nuclear marker – the exon 10 of the growth hormone receptor (GHR) gene – to the one of the mitochondrial CYB for inferring phylogenetic relationships among the major lineages of arvicoline rodents. RESULTS: The analysis of GHR sequences improves the overall resolution of the Arvicolinae phylogeny. Our results show that the Caucasian long-clawed vole (Prometheomys schaposnikowi) is one of the basalmost arvicolines, and confirm that true lemmings (Lemmus) and collared lemmings (Dicrostonyx) are not closely related as suggested by morphology. Red-backed voles (Myodini) are found as the sister-group of a clade encompassing water vole (Arvicola), snow vole (Chionomys), and meadow voles (Microtus and allies). Within the latter, no support is recovered for the generic recognition of Blanfordimys, Lasiopodomys, Neodon, and Phaiomys as suggested by morphology. Comparisons of parameter estimates for branch lengths, base composition, among sites rate heterogeneity, and GTR relative substitution rates indicate that CYB sequences consistently exhibit more heterogeneity among codon positions than GHR. By analyzing the contribution of each codon position to node resolution, we show that the apparent higher efficiency of GHR is due to their third positions. Although we focus on speciation events spanning the last 10 million years (Myr), CYB sequences display highly saturated codon positions contrary to the nuclear exon. Lastly, variable length bootstrap predicts a significant increase in resolution of arvicoline phylogeny through the sequencing of nuclear data in an order of magnitude three to five times greater than the size of GHR exon 10. CONCLUSION: Our survey provides a first resolved gene tree for Arvicolinae. The comparison of CYB and GHR phylogenetic efficiency supports recent assertions that nuclear genes are useful for resolving relationships of recently evolved animals. The superiority of nuclear exons may reside both in (i) less heterogeneity among sites, and (ii) the presence of highly informative sites in third codon positions, that evolve rapidly enough to accumulate synapomorphies, but slow enough to avoid substitutional saturation