Institute for Mathematical Behavioral Sciences

Investigating the interactions between universal and culturally specific influences on color categorization across individuals and cultures has proven to be a challenge for human color categorization and naming research. The present article simulates the evolution of color lexicons to evaluate the role of two realistic constraints found in the human phenomenon: (i) heterogeneous observer populations and (ii) heterogeneous color stimuli. Such constraints, idealized and implemented as agent categorization and communication games, produce interesting and unexpected consequences for stable categorization solutions evolved and shared by agent populations. We find that the presence of a small fraction of color deficient agents in a population, or the presence of a "region of increased salience" in the color stimulus space, break rotational symmetry in population categorization solutions, and confine color category boundaries to a subset of available locations. Further, these heterogeneities, each in a different, predictable, way, might lead to a change of category number and size. In addition, the concurrent presence of both types of heterogeneity gives rise to novel constrained solutions which optimize the success rate of categorization and communication games. Implications of these agent-based results for psychological theories of color categorization and the evolution of color naming systems in human societies are discussed.

Trust and trustworthiness are crucial to the survival of online markets, and reputation systems that rely on feedback from traders help sustain trust. However, in current online auction markets only half of the buyers leave feedback after ransactions, and nearly all of it is positive. In this paper, I propose a mechanism whereby sellers can provide rebates to buyers contingent on buyers provision of reports. Using a game theoretical model, I show how the rebate incentive mechanism can increase reporting. In both a pure adverse selection model, and a model with adverse selection and moral hazard, there exists a pooling equilibrium where both good and bad sellers choose the rebate option, even though their true types are revealed through feedback. In the presence of moral hazard, the mechanism induces bad sellers to improve the quality of the contract.

David Lewis (1969) introduced sender-receiver games as a way of investigating how meaningful language might evolve from initially random signals. In this report I investigate the conditions under which Lewis signaling games evolve to perfect signaling systems under various learning dynamics. While the 2-state/2-term Lewis signaling game with basic urn learning always approaches a signaling system, I will show that with more than two states suboptimal pooling equilibria can evolve. Inhomogeneous state distributions increase the likelihood of pooling equilibria, but learning strategies with negative reinforcement or certain sorts of mutation can decrease the likelihood of, and even eliminate, pooling equilibria. Both Moran and APR learning strategies (Bereby-Meyer and Erev 1998) are shown to promote successful convergence to signaling systems. A model is presented that illustrates how a language that codes state-act pairs in an order-based grammar might evolve in the context of a Lewis signaling game. The terms, grammar, and the corresponding partitions of the state space co-evolve to generate a language whose structure appears to reflect canonical natural kinds. The evolution of these apparent natural kinds, however, is entirely in service of the rewards that accompany successful distinctions between the sender and receiver. Any metaphysics grounded on the structure of a natural language that evolved in this way would track arbitrary, but pragmatically useful distinctions.

Interpersonal interaction over short time scales is frequently understood in terms of actions, which can be thought of as discrete events in which one individual emits a behavior directed at one or more other entities in his or her environment (possibly including him or herself). Here, we introduce a highly flexible framework for modeling actions within social settings, which permits likelihood-based inference for behavioral mechanisms with complex dependence. The utility of the framework is illustrated via an application to dynamic modeling of responder radio communications during the early hours of the World Trade Center disaster.

Exponential family models for random graphs (ERGs, also known as p∗ models) are an increasingly popular tool for the analysis of social networks. ERGs allow for the parameterization of complex dependence among edges within a likelihood-based framework, and are often used to model local influences on global structure. This paper introduces a family of cycle statistics, which allow for the modeling of long-range dependence within ERGs. These statistics are shown to arise from a family of partial conditional dependence assumptions based on an extended form of reciprocity, here called reciprocal path dependence. Algorithms for computing cycle statistic changescores and the cycle census are provided, as are analytical expressions for the first and approximate second moments of the cycle census under a Bernoulli null model. An illustrative application of ERG modeling using cycle statistics is also provided.

We compare replicator dynamics for some simple games with and without the addition of conformist bias. The addition of conformist bias can create equilibria, it can change the stability properties of existing equilibria, it may leave the equilibrium structure intact but change the relative size of basins of attraction, or it may do nothing at ali. Examples of each ofthe foregoing are given.

It always seems to be the case. No matter how hard you might work on a proposal, no matter how polished and complete the final product may be, when it is presented to a group for approval, there always seems to be some majority who wants to "improve It." Is this just an annoyance or is there a reason? The mathematical modeling provides an immediate explanation in terms of some interesting and unexpected mathematics. Even more; the mathematics describing this behavior underscores the reality that it can be surprising easy even for a group sincerely striving for excellence to make inferior decisions. Indeed, these difficulties are so pervasive and can arise in such unexpected ways that it is realistic to worry whether groups you belong to have been inadvertently victimized by these mathematical subtleties based on the orbits of symmetry groups. These problems can occur even if all decisions are reached by consensus during discus sions, such as a committee discussing the selection of a new calculus book. This paper addresses deliberations by discussing a branch of voting theory where Euclidean geometry models an "issue space." Then describing how it is possible to un intentionally make inferior choices, we will encounter mathematical behaviors remark- ably similar to "attractors" and "chaotic dynamics" from dynamical systems. Since the coexistence of chaotic and stable behavior is common in the Newtonian N -body problem and dynamical systems, it is interesting that this combination also coexists in the dynamics of discussions. Another connection arises when configurations central to the N -body problem play a suggestive role in the analysis; at another step we use singularity theory. What adds to the delight of this topic is that while the mathematics can be intricate, the issues can be described at a classroom level where some even lead to student level research projects. Keywords: spatial voting, paradoxes, singularity theory

Although it is well-known that there is a relationship between socio-physical dis- tance and edge probability in interpersonal networks, the predictive power of such distances for total network structure has not been established. Here, it is shown that upper bounds on the marginal edge probabilities for farflung dyads can be used to place a lower bound on the predictive power of distance, and one such bound is de- rived. Application of this bound to the special case of uniformly placed vertices on the plane suggests that only modest constraints are required for distance effects to dominate at large physical scales.