If someone is nice to you, you feel good and may be inclined to be nice to somebody else. This every day experience is borne out by experimental games: the recipients of an act of kindness are more likely to help in turn, even if the person who benefits from their generosity is somebody else. This behaviour, which has been called 'upstream reciprocity', appears to be a misdirected act of gratitude: you help somebody because somebody else has helped you. Does this make any sense from an evolutionary or a game theoretic perspective? In this paper, we show that upstream reciprocity alone does not lead to the evolution of cooperation, but it can evolve and increase the level of cooperation if it is linked to either direct or spatial reciprocity. We calculate the random walks of altruistic acts that are induced by upstream reciprocity. Our analysis shows that gratitude and other positive emotions, which increase the willingness to help others, can evolve in the competitive world of natural selection.

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In this paper we study the problem of learning phylogenies and hidden Markov models. We call a Markov model nonsingular if all transition matrices have determinants bounded away from 0 (and 1). We highlight the role of the nonsingularity condition for the learning problem. Learning hidden Markov models without the nonsingularity condition is at least as hard as learning parity with noise, a wellknown learning problem conjectured to be computationally hard. On the other hand, we give a polynomial-time algorithm for learning nonsingular phylogenies and hidden Markov models.1. Introduction. In this paper we consider the problem of learning phylogenies and hidden Markov models, two of the most popular Markov models used in applications.Phylogenies are used in evolutionary biology to model the stochastic evolution of genetic data on the ancestral tree relating a group of species. More precisely, the leaves of the tree correspond to (known) extant species. Internal nodes represent extinct species, while the root of the tree represents the most recent ancestor to all species in the tree. Following paths from the root to the leaves, each bifurcation indicates a speciation event whereby two new species are created from a parent.The underlying assumption is that genetic information evolves from the root to the leaves according to a Markov model on the tree. This genetic information may consist of DNA sequences, proteins, and so on. Suppose,

The estimation of species trees using multiple loci has become increasingly common. Because different loci can have different phylogenetic histories (reflected in different gene tree topologies) for multiple biological causes, new approaches to species tree estimation have been developed that take gene tree heterogeneity into account. Among these multiple causes, incomplete lineage sorting (ILS), modeled by the multi-species coalescent, is potentially the most common cause of gene tree heterogeneity, and much of the focus of the recent literature has been on how to estimate species trees in the presence of ILS. Despite progress in developing statistically consistent techniques for estimating species trees when gene trees can differ due to ILS, there is substantial controversy in the systematics community as to whether to use the new coalescent-based methods or the traditional concatenation methods. One of the key issues that has been raised is understanding the impact of gene tree estimation error on coalescent-based methods that operate by combining gene trees. Here we explore the mathematical guarantees of coalescent-based methods when analyzing estimated rather than true gene trees. Our results provide some insight into the differences between promise of coalescent-based methods in theory and their performance in practice.