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Valerio Capraro

Publications and source records attributed to Valerio Capraro.

At least 73 records · Page 4Linked to original sources
arXiv2015

Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture

Sofic and hyperlinear groups are the countable discrete groups that can be approximated in a suitable sense by finite symmetric groups and groups of unitary matrices. These notions turned out to be very deep and fruitful, and stimulated in the last 15 years an impressive amount of research touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and more recently even quantum information theory. Several longstanding conjectures that are still open for arbitrary groups were settled in the case of sofic or hyperlinear groups. These achievements aroused the interest of an increasing number of researchers into some fundamental questions about the nature of these approximation properties. Many of such problems are to this day still open such as, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? A similar pattern can be found in the study of II_1 factors. In this case the famous conjecture due to Connes (commonly known as the Connes embedding conjecture) that any II_1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs in the understanding of II_1 factors, and stands out today as one of the major open problems in the field. The aim of these notes is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and the Connes embedding conjecture. The presentation is nonetheless self contained and accessible to any student or researcher with a graduate level mathematical background. An appendix by V. Pestov provides a pedagogically new introduction to the concepts of ultrafilters, ultralimits, and ultraproducts for those mathematicians who are not familiar with them, and aiming to make these concepts appear very natural.

math.GR↗
arXiv2014

Group size effect on cooperation in social dilemmas

Social dilemmas are central to human society. Depletion of natural resources, climate protection, security of energy supply, and workplace collaborations are all examples of social dilemmas. Since cooperative behaviour in a social dilemma is individually costly, Nash equilibrium predicts that humans should not cooperate. Yet experimental studies show that people do cooperate even in anonymous one-shot interactions. In spite of the large number of participants in many modern social dilemmas, little is known about the effect of group size on cooperation. Does larger group size favour or prevent cooperation? We address this problem both experimentally and theoretically. Experimentally, we find that there is no general answer: it depends on the strategic situation. Specifically, we find that larger groups are more cooperative in the Public Goods game, but less cooperative in the N-person Prisoner's dilemma. Theoretically, we show that this behaviour is not consistent with either the Fehr & Schmidt model or (a one-parameter version of) the Charness & Rabin model, but it is consistent with the cooperative equilibrium model introduced by the second author.

cs.GT↗
arXiv2014

The emergence of hyper-altruistic behaviour in conflictual situations

Situations where people have to decide between hurting themselves or another person are at the core of many individual and global conflicts. Yet little is known about how people behave when facing these situations in the lab. Here we report a large experiment in which participants could either take $x$ dollars from another anonymous participant or give $y$ dollars to the same participant. Depending on the treatments, participants could also exit the game without making any decision, but paying a cost. Across different protocols and parameter specifications, we provide evidence of three regularities: (i) when exiting is allowed and costless, subjects tend to exit the game; (ii) females are more likely than males to exit the game, but only when the cost is small; (iii) when exiting is not allowed, altruistic actions are more common than predicted by the dominant economic models. In particular, against the predictions of every dominant economic model, about one sixth of the subjects show hyper-altruistic tendencies, that is, they prefer giving $y$ rather than taking $x>y$. In doing so, our findings shed light on human decision-making in conflictual situations and suggest that economic models should be revised in order to take into account hyper-altruistic behaviour.

q-bio.PE↗
arXiv2014

Translucent Players: Explaining Cooperative Behavior in Social Dilemmas

In the last few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when non-cooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass (2013) call "translucent players". Typically, players are assumed to be "opaque", in the sense that a deviation by one player does not affect the strategies used by other players. But a player may believe that if he switches from one strategy to another, the fact that he chooses to switch may be visible to the other players. For example, if he chooses to defect in Prisoner's Dilemma, the other player may sense his guilt. We show that by assuming translucent players, we can recover many of the regularities observed in human behavior in well-studied games such as Prisoner's Dilemma, Traveler's Dilemma, Bertrand Competition, and the Public Goods game.

cs.GT↗
arXiv2014

Heuristics guide the implementation of social preferences in one-shot Prisoner's Dilemma experiments

Cooperation in one-shot anonymous interactions is a widely documented aspect of human behaviour. Here we shed light on the motivations behind this behaviour by experimentally exploring cooperation in a one-shot continuous-strategy Prisoner's Dilemma (i.e. one-shot two-player Public Goods Game). We examine the distribution of cooperation amounts, and how that distribution varies based on the benefit-to-cost ratio of cooperation (b/c). Interestingly, we find a trimodal distribution at all b/c values investigated. Increasing b/c decreases the fraction of participants engaging in zero cooperation and increases the fraction engaging in maximal cooperation, suggesting a role for efficiency concerns. However, a substantial fraction of participants consistently engage in 50% cooperation regardless of b/c. The presence of these persistent 50% cooperators is surprising, and not easily explained by standard models of social preferences. We present evidence that this behaviour is a result of social preferences guided by simple decision heuristics, rather than the rational examination of payoffs assumed by most social preference models. We also find a strong correlation between play in the Prisoner's Dilemma and in a subsequent Dictator Game, confirming previous findings suggesting a common prosocial motivation underlying altruism and cooperation.

cs.GT↗
arXiv2014

Do good actions inspire good actions in others?

Actions such as sharing food and cooperating to reach a common goal have played a fundamental role in the evolution of human societies. These good actions may not maximise the actor's payoff, but they maximise the other's payoff. Consequently, their existence is puzzling for evolutionary theories. Why should you make an effort to help others, even when no reward seems to be at stake? Indeed, experiments typically show that humans are heterogeneous: some may help others, while others may not. With the aim of favouring the emergence of 'successful cultures', a number of studies has recently investigated what mechanisms promote the evolution of a particular good action. But still little is known about if and how good actions can spread from person to person. For instance, does being recipient of an altruistic act increase your probability of being cooperative with others? Plato's quote, 'Good actions give strength to ourselves and inspire good actions in others', suggests that is possible. We have conducted an experiment on Amazon Mechanical Turk to test this mechanism using economic games. We have measured willingness to be cooperative through a standard Prisoner's dilemma and willingness to act altruistically using a binary Dictator game. In the baseline treatments, the endowments needed to play were given by the experimenters, as usual; in the control treatments, they came from a good action made by someone else. Across four different comparisons and a total of 572 subjects, we have never found a significant increase of cooperation or altruism when the endowment came from a good action. We conclude that good actions do not necessarily inspire good actions in others, at least in the ideal scenario of a lab experiment with anonymous subjects.

physics.soc-ph↗
arXiv2014

Benevolent characteristics promote cooperative behaviour among humans

Cooperation is fundamental to the evolution of human society. We regularly observe cooperative behaviour in everyday life and in controlled experiments with anonymous people, even though standard economic models predict that they should deviate from the collective interest and act so as to maximise their own individual payoff. However, there is typically heterogeneity across subjects: some may cooperate, while others may not. Since individual factors promoting cooperation could be used by institutions to indirectly prime cooperation, this heterogeneity raises the important question of who these cooperators are. We have conducted a series of experiments to study whether benevolence, defined as a unilateral act of paying a cost to increase the welfare of someone else beyond one's own, is related to cooperation in a subsequent one-shot anonymous Prisoner's dilemma. Contrary to the predictions of the widely used inequity aversion models, we find that benevolence does exist and a large majority of people behave this way. We also find benevolence to be correlated with cooperative behaviour. Finally, we show a causal link between benevolence and cooperation: priming people to think positively about benevolent behaviour makes them significantly more cooperative than priming them to think malevolently. Thus benevolent people exist and cooperate more.

cs.GT↗
arXiv2013

Discrete Homology Theory for Metric Spaces

In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a `fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.

math.MG↗
arXiv2013

Cyclic Hilbert spaces and Connes' embedding problem

Let $M$ be a $II_1$-factor with trace $τ$, the linear subspaces of $L^2(M,τ)$ are not just common Hilbert spaces, but they have additional structure. We introduce the notion of a cyclic linear space by taking those properties as axioms. In Sec.2 we formulate the following problem: "does every cyclic Hilbert space embed into $L^2(M,τ)$, for some $M$?". An affirmative answer would imply the existence of an algorithm to check Connes' embedding Conjecture. In Sec.3 we make a first step towards the answer of the previous question.

math.OA↗
arXiv2013

A notion of continuity in discrete spaces and applications

We propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of A-theory for locally finite connected graphs. We use this notion of continuity to derive an analogue in Z^2 of the Jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the \ell^p-distortion of a metric space that has been recently proved by Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.

math.MG↗
arXiv2013

Existence of equilibria in countable games: an algebraic approach

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group and the payoffs are functions of the group operation. In order to obtain the existence of equilibria, finitely additive mixed strategies have to be allowed. This creates a problem of selection of a product measure of mixed strategies. We propose a family of such selections and prove existence of an equilibrium that does not depend on the selection. As a byproduct we show that if finitely additive mixed strategies are allowed, then Wald's game admits an equilibrium. We also prove existence of equilibria for nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we extend the main results to uncountable games.

cs.GT↗
arXiv2013

Groups associated to $II_1$-factors

We extend recent work of the first named author, constructing a natural Hom semigroup associated to any pair of II$_1$-factors. This semigroup always satisfies cancelation, hence embeds into its Grothendieck group. When the target is an ultraproduct of a McDuff factor (e.g., $R^ω$), this Grothendieck group turns out to carry a natural vector space structure; in fact, it is a Banach space with natural actions of outer automorphism groups.

math.OA↗
arXiv2013

A Model of Human Cooperation in Social Dilemmas

Social dilemmas are situations in which collective interests are at odds with private interests: pollution, depletion of natural resources, and intergroup conflicts, are at their core social dilemmas. Because of their multidisciplinarity and their importance, social dilemmas have been studied by economists, biologists, psychologists, sociologists, and political scientists. These studies typically explain tendency to cooperation by dividing people in proself and prosocial types, or appealing to forms of external control or, in iterated social dilemmas, to long-term strategies. But recent experiments have shown that cooperation is possible even in one-shot social dilemmas without forms of external control and the rate of cooperation typically depends on the payoffs. This makes impossible a predictive division between proself and prosocial people and proves that people have attitude to cooperation by nature. The key innovation of this article is in fact to postulate that humans have attitude to cooperation by nature and consequently they do not act a priori as single agents, as assumed by standard economic models, but they forecast how a social dilemma would evolve if they formed coalitions and then they act according to their most optimistic forecast. Formalizing this idea we propose the first predictive model of human cooperation able to organize a number of different experimental findings that are not explained by the standard model. We show also that the model makes satisfactorily accurate quantitative predictions of population average behavior in one-shot social dilemmas.

cs.GT↗
arXiv2013

A solution concept for games with altruism and cooperation

Over the years, numerous experiments have been accumulated to show that cooperation is not casual and depends on the payoffs of the game. These findings suggest that humans have attitude to cooperation by nature and the same person may act more or less cooperatively depending on the particular payoffs. In other words, people do not act a priori as single agents, but they forecast how the game would be played if they formed coalitions and then they play according to their best forecast. In this paper we formalize this idea and we define a new solution concept for one-shot normal form games. We prove that this \emph{cooperative equilibrium} exists for all finite games and it explains a number of different experimental findings, such as (1) the rate of cooperation in the Prisoner's dilemma depends on the cost-benefit ratio; (2) the rate of cooperation in the Traveler's dilemma depends on the bonus/penalty; (3) the rate of cooperation in the Publig Goods game depends on the pro-capite marginal return and on the numbers of players; (4) the rate of cooperation in the Bertrand competition depends on the number of players; (5) players tend to be fair in the bargaining problem; (6) players tend to be fair in the Ultimatum game; (7) players tend to be altruist in the Dictator game; (8) offers in the Ultimatum game are larger than offers in the Dictator game.

cs.GT↗
arXiv2012

Amenability, locally finite spaces, and bi-lipschitz embeddings

We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein, Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible applications of the property SN in the study of embedding a metric space into another one. In particular, we propose three results: we prove that a certain class of metric graphs that are isometrically embeddable into Hilbert spaces must have the property SN. We also show, by a simple example, that this result is not true replacing property SN with amenability. As a second result, we prove that \emph{many} spaces with \emph{uniform bounded geometry} having a bi-lipschitz embedding into Euclidean spaces must have the property SN. Finally, we prove a Bourgain-like theorem for metric trees: a metric tree with uniform bounded geometry and without property SN does not have bi-lipschitz embeddings into finite-dimensional Hilbert spaces.

math.MG↗
arXiv2012

Optimal strategies for a game on amenable semigroups

The semigroup game is a two-person zero-sum game defined on a semigroup S as follows: Players 1 and 2 choose elements x and y in S, respectively, and player 1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on S, to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.

cs.GT↗
arXiv2012

Solution of Wald's game using loadings and allowed strategies

We propose a new interpretation of the strange phenomena that some authors have observed about the Wald game. This interpretation is possible thanks to the new language of \emph{loadings} that Morrison and the author have introduced in a previous work. Using the theory of loadings and allowed strategies, we are also able to prove that Wald's game admits a \emph{natural} solution and, as one can expect, the game turns out to be fair for this solution. As a technical tool, we introduce the notion of \emph{embedding a game into another game} that could be of interest from a theoretical point of view. \emph{En passant} we find a very easy example of a game which is loadable in infinitely many different ways.

cs.GT↗