A Note on Shapley's convex measure games

L. S. Shapley, in his paper 'Cores of Convex Games', introduces Convex Measure Games, those that are induced by a convex function on R, acting over a measure on the coalitions. But in a note he states that if this function is a function of several variables, then convexity for the function does not imply convexity of the game or even superadditivity. We prove that if the function is directionally convex, the game is convex, and conversely, any convex game can be induced by a directionally convex function acting over measures on the coalitions, with as many measures as players

On the nucleolus of 2 x 2 assignment games

[spa] En este artículo hallamos fórmulas para el nucleolo de juegos de asignación arbitrarios con dos compradores y dos vendedores. Se analizan cinco casos distintos, dependiendo de las entradas en la matriz de asignación. Los resultados se extienden a los casos de juegos de asignación de tipo 2 x m o m x 2.

Resistance to extreme strategies, rather than prosocial preferences, can explain human cooperation in public goods games.

The results of numerous economic games suggest that humans behave more cooperatively than would be expected if they were maximizing selfish interests. It has been argued that this is because individuals gain satisfaction from the success of others, and that such prosocial preferences require a novel evolutionary explanation. However, in previous games, imperfect behavior would automatically lead to an increase in cooperation, making it impossible to decouple any form of mistake or error from prosocial cooperative decisions. Here we empirically test between these alternatives by decoupling imperfect behavior from prosocial preferences in modified versions of the public goods game, in which individuals would maximize their selfish gain by completely (100%) cooperating. We found that, although this led to higher levels of cooperation, it did not lead to full cooperation, and individuals still perceived their group mates as competitors. This is inconsistent with either selfish or prosocial preferences, suggesting that the most parsimonious explanation is imperfect behavior triggered by psychological drives that can prevent both complete defection and complete cooperation. More generally, our results illustrate the caution that must be exercised when interpreting the evolutionary implications of economic experiments, especially the absolute level of cooperation in a particular treatment.

Evolutionary Games in Networked Populations: Models and Experiments

Cooperation and coordination are desirable behaviors that are fundamental for the harmonious development of society. People need to rely on cooperation with other individuals in many aspects of everyday life, such as teamwork and economic exchange in anonymous markets. However, cooperation may easily fall prey to exploitation by selfish individuals who only care about short- term gain. For cooperation to evolve, specific conditions and mechanisms are required, such as kinship, direct and indirect reciprocity through repeated interactions, or external interventions such as punishment. In this dissertation we investigate the effect of the network structure of the population on the evolution of cooperation and coordination. We consider several kinds of static and dynamical network topologies, such as Baraba´si-Albert, social network models and spatial networks. We perform numerical simulations and laboratory experiments using the Prisoner's Dilemma and co- ordination games in order to contrast human behavior with theoretical results. We show by numerical simulations that even a moderate amount of random noise on the Baraba´si-Albert scale-free network links causes a significant loss of cooperation, to the point that cooperation almost vanishes altogether in the Prisoner's Dilemma when the noise rate is high enough. Moreover, when we consider fixed social-like networks we find that current models of social networks may allow cooperation to emerge and to be robust at least as much as in scale-free networks. In the framework of spatial networks, we investigate whether cooperation can evolve and be stable when agents move randomly or performing Le´vy flights in a continuous space. We also consider discrete space adopting purposeful mobility and binary birth-death process to dis- cover emergent cooperative patterns. The fundamental result is that cooperation may be enhanced when this migration is opportunistic or even when agents follow very simple heuristics. In the experimental laboratory, we investigate the issue of social coordination between indi- viduals located on networks of contacts. In contrast to simulations, we find that human players dynamics do not converge to the efficient outcome more often in a social-like network than in a random network. In another experiment, we study the behavior of people who play a pure co- ordination game in a spatial environment in which they can move around and when changing convention is costly. We find that each convention forms homogeneous clusters and is adopted by approximately half of the individuals. When we provide them with global information, i.e., the number of subjects currently adopting one of the conventions, global consensus is reached in most, but not all, cases. Our results allow us to extract the heuristics used by the participants and to build a numerical simulation model that agrees very well with the experiments. Our findings have important implications for policymakers intending to promote specific, desired behaviors in a mobile population. Furthermore, we carry out an experiment with human subjects playing the Prisoner's Dilemma game in a diluted grid where people are able to move around. In contrast to previous results on purposeful rewiring in relational networks, we find no noticeable effect of mobility in space on the level of cooperation. Clusters of cooperators form momentarily but in a few rounds they dissolve as cooperators at the boundaries stop tolerating being cheated upon. Our results highlight the difficulties that mobile agents have to establish a cooperative environment in a spatial setting without a device such as reputation or the possibility of retaliation. i.e. punishment. Finally, we test experimentally the evolution of cooperation in social networks taking into ac- count a setting where we allow people to make or break links at their will. In this work we give particular attention to whether information on an individual's actions is freely available to poten- tial partners or not. Studying the role of information is relevant as information on other people's actions is often not available for free: a recruiting firm may need to call a job candidate's refer- ences, a bank may need to find out about the credit history of a new client, etc. We find that people cooperate almost fully when information on their actions is freely available to their potential part- ners. Cooperation is less likely, however, if people have to pay about half of what they gain from cooperating with a cooperator. Cooperation declines even further if people have to pay a cost that is almost equivalent to the gain from cooperating with a cooperator. Thus, costly information on potential neighbors' actions can undermine the incentive to cooperate in dynamical networks.

A model for the evolution of reinforcement learning in fluctuating games

Many species are able to learn to associate behaviours with rewards as this gives fitness advantages in changing environments. Social interactions between population members may, however, require more cognitive abilities than simple trial-and-error learning, in particular the capacity to make accurate hypotheses about the material payoff consequences of alternative action combinations. It is unclear in this context whether natural selection necessarily favours individuals to use information about payoffs associated with nontried actions (hypothetical payoffs), as opposed to simple reinforcement of realized payoff. Here, we develop an evolutionary model in which individuals are genetically determined to use either trial-and-error learning or learning based on hypothetical reinforcements, and ask what is the evolutionarily stable learning rule under pairwise symmetric two-action stochastic repeated games played over the individual's lifetime. We analyse through stochastic approximation theory and simulations the learning dynamics on the behavioural timescale, and derive conditions where trial-and-error learning outcompetes hypothetical reinforcement learning on the evolutionary timescale. This occurs in particular under repeated cooperative interactions with the same partner. By contrast, we find that hypothetical reinforcement learners tend to be favoured under random interactions, but stable polymorphisms can also obtain where trial-and-error learners are maintained at a low frequency. We conclude that specific game structures can select for trial-and-error learning even in the absence of costs of cognition, which illustrates that cost-free increased cognition can be counterselected under social interactions.

On the structure of cooperative and competitive solutions for a generalized assignment game

We study cooperative and competitive solutions for a many- to-many generalization of Shapley and Shubik (1972)'s assignment game. We consider the Core, three other notions of group stability and two al- ternative definitions of competitive equilibrium. We show that (i) each group stable set is closely related with the Core of certain games defined using a proper notion of blocking and (ii) each group stable set contains the set of payoff vectors associated to the two definitions of competitive equilibrium. We also show that all six solutions maintain a strictly nested structure. Moreover, each solution can be identified with a set of ma- trices of (discriminated) prices which indicate how gains from trade are distributed among buyers and sellers. In all cases such matrices arise as solutions of a system of linear inequalities. Hence, all six solutions have the same properties from a structural and computational point of view.

Mathematics inspired by Darwin. Adaptive dynamics of dispersal and cooperation

In 1859, Charles Darwin published his theory of evolution by natural selection, the process occurring based on fitness benefits and fitness costs at the individual level. Traditionally, evolution has been investigated by biologists, but it has induced mathematical approaches, too. For example, adaptive dynamics has proven to be a very applicable framework to the purpose. Its core concept is the invasion fitness, the sign of which tells whether a mutant phenotype can invade the prevalent phenotype. In this thesis, four real-world applications on evolutionary questions are provided. Inspiration for the first two studies arose from a cold-adapted species, American pika. First, it is studied how the global climate change may affect the evolution of dispersal and viability of pika metapopulations. Based on the results gained here, it is shown that the evolution of dispersal can result in extinction and indeed, evolution of dispersalshould be incorporated into the viability analysis of species living in fragmented habitats. The second study is focused on the evolution of densitydependent dispersal in metapopulations with small habitat patches. It resulted a very surprising unintuitive evolutionary phenomenon, how a non-monotone density-dependent dispersal may evolve. Cooperation is surprisingly common in many levels of life, despite of its obvious vulnerability to selfish cheating. This motivated two applications. First, it is shown that density-dependent cooperative investment can evolve to have a qualitatively different, monotone or non-monotone, form depending on modelling details. The last study investigates the evolution of investing into two public-goods resources. The results suggest one general path by which labour division can arise via evolutionary branching. In addition to applications, two novel methodological derivations of fitness measures in structured metapopulations are given.

Cooperative of Noncooperative Behavior?

In an abstract two-agent model, we show that every deterministic joint choice function compatible with the hypothesis that agents act noncooperatively is also compatible with the hypothesis that they act cooperatively. the converse is false.

Fuzzy Topological Games and Related Topics

The main purpose of study is to extend the concept of the topological game G(K, X) and some other kinds of games into fuzzy topological games and to obtain some results regarding them. Owing to the fact that topological games have plenty of applications in covering properties, it made an attempt to explore some inter relations of games and covering properties in fuzzy topological spaces. Even though the main focus is on fuzzy para-meta compact spaces and closure preserving shading families, some brief sketches regarding fuzzy P-spaces and Shading Dimension is also provided. In a topological game players choose some objects related to the topological structure of a space such as points, closed subsets, open covers etc. More over the condition on a play to be winning for a player may also include topological notions such as closure, convergence, etc. It turns out that topological games are related to the Baire property, Baire spaces, Completeness properties, Convergence properties, Separation properties, Covering and Base properties, Continuous images, Suslin sets, Singular spaces etc.

Formulating generalised 'goal games' against nature: An illustration from decision-making under uncertainty in agriculture

The games-against-nature approach to the analysis of uncertainty in decision-making relies on the assumption that the behaviour of a decision-maker can be explained by concepts such as maximin, minimax regret, or a similarly defined criterion. In reality, however, these criteria represent a spectrum and, the actual behaviour of a decision-maker is most likely to embody a mixture of such idealisations. This paper proposes that in game-theoretic approach to decision-making under uncertainty, a more realistic representation of a decision-maker's behaviour can be achieved by synthesising games-against-nature with goal programming into a single framework. The proposed formulation is illustrated by using a well-known example from the literature on mathematical programming models for agricultural-decision-making. (c) 2005 Elsevier Inc. All rights reserved.

Discourse and Cooperative Learning in the Math Classroom

In this action research study of my 6th grade math classroom I investigated the effects of increased student discourse and cooperative learning on the students’ ability to explain and understand math concepts and problem solving, as well as its effects on their use of vocabulary and written explanations. I also investigated how it affected students’ attitudes. I discovered that increased student discourse and cooperative learning resulted in positive changes in students’ attitudes about their ability to explain and understand math, as well as their actual ability to explain and understand math concepts. Evidence in regard to use of vocabulary and written explanations generally showed little change, but this may have been related to insufficient data. As a result of this research, I plan to continue to use cooperative learning groups and increased student discourse as a teaching practice in all of my math classes. I also plan to include training on cooperative learning strategies as well as more emphasis on vocabulary and writing in my math classroom.

Writing in the Mathematics Classroom: Does It Have an Effect on Students’ Mathematical Reasoning?

In this action research study of my classroom of 5th grade mathematics, I investigate how to improve students’ written explanations to and reasoning of math problems. For this, I look at journal writing, dialogue, and collaborative grouping and its effects on students’ conceptual understanding of the mathematics. In particular, I look at its effects on students’ written explanations to various math problems throughout the semester. Throughout the study students worked on math problems in cooperative groups and then shared their solutions with classmates. Along with this I focus on the dialogue that occurred during these interactions and whether and how it moved students to a deeper level of conceptual understanding. Students also wrote responses about their learning in a weekly math journal. The purpose of this journal is two-fold. One is to have students write out their ideas. Second, is for me to provide the students with feedback on their responses. My research reveals that the integration of collaborative grouping, journaling, and active dialogue between students and teacher helps students develop a deeper understanding of mathematics concepts as well as an increase in their confidence as problem solvers. The use of journaling, dialogue, and collaborative grouping reveals themselves as promising learning tasks that can be integrated in a mathematics curriculum that seeks to cultivate students’ thinking and reasoning.

Cooperative Learning Groups in the Eighth Grade Math Classroom

In this action research study of my classroom of 8th grade mathematics students, I investigated whether cooperative learning would lead to a better understanding of the mathematical concepts and thus more success for the students. I used my three eighth grade classes with two using cooperative groups and the third not. I discovered that the students who wanted to work in cooperative groups were more successful than they had been. I also discovered that the grouping itself has a great effect on how the group works together. The wrong grouping of students can lead to disaster and many headaches for the teacher. Overall the two classes that used cooperative groups did better grade wise than the one class that was taught using the traditional way of not using cooperative groups. As a result of this research, I plan to continue using cooperative groups but will be more aware of the students who are grouped together.

An Uphill Battle: Incorporating cooperative learning using a largely individualized curriculum

In this action research study of my classroom of 8th grade mathematics, I investigated if cooperative learning could be an effective teaching method with the Saxon curriculum. Saxon curriculum is largely individualized in that most lessons could be completed without much group interaction. I discovered that cooperative learning was very successful with the curriculum as long as it was structured. Ninety-five percent of the students in the study preferred to work in groups, and I observed mathematical communication grow with most of the students. As a result of this research, I plan to continue to incorporate cooperative learning into my mathematics classroom. I will use cooperative learning with all of my mathematics classes, even the ones that do not use the Saxon curriculum. I believe in the power of working together.

Using Cooperative Learning In A Sixth Grade Math Classroom

In this action research study of my classroom of sixth grade mathematics, I investigated the impact of cooperative learning on the engagement, participation, and attitudes of my students. I also investigated the impact of cooperative learning upon my own teaching. I discovered that my students not only preferred to learn in cooperative groups, but that their levels of engagement and participation, their attitudes toward math, and their quality of work all improved greatly. My teaching also changed, and I found that I began to enjoy teaching more. As a result of this research, I plan to continue and expand the amount of cooperative group work that happens in my classroom.