Agnieszka Rusinowska

We take a novel approach based on differential games to the study of criminal networks. We extend the static crime network game (Ballester et al., 2006, 2010) to a dynamic setting where criminal activities negatively impact the accumulation of total wealth in the economy. We derive a Markov Feedback Equilibrium and show that, unlike in the static crime network game, the vector of equilibrium crime rates is not necessarily proportional to the vector of Bonacich centralities. Next, we conduct a comparative dynamic analysis with respect to the network size, the network density, and the marginal expected punishment, finding results in contrast with those arising in the static crime network game. We also shed light on a novel issue in the network theory literature, i.e., the existence of a voracity effect. Finally, we study the problem of identifying the optimal target in the population of criminals when the planner’s objective is to minimize aggregate crime at each point in time. Our analysis shows that the key player in the dynamic and the static setting may differ, and that the key player in the dynamic setting may change over time.

<div><p>This paper aims to connect the social network literature on centrality measures with the economic literature on von Neumann-Morgenstern expected utility functions using cooperative game theory. The social network literature studies various concepts of network centrality, such as degree, betweenness, connectedness, and so on. This resulted in a great number of network centrality measures, each measuring centrality in a different way. In this paper, we aim to explore which centrality measures can be supported as von Neumann-Morgenstern expected utility functions, reflecting preferences over different network positions in different networks. Besides standard axioms on lotteries and preference relations, we consider neutrality to ordinary risk . We show that this leads to a class of centrality measures that is fully determined by the degrees (i.e. the numbers of neighbours) of the positions in a network. Although this allows for externalities, in the sense that the preferences of a position might depend on the way how other positions are connected, these externalities can be taken into account only by considering the degrees of the network positions. Besides bilateral networks, we extend our result to general cooperative TU-games to give a utility foundation of a class of TU-game solutions containing the Shapley value.</p></div>

We propose a model of the joint evolution of opinions and social relationships in a setting where social influence decays over time. The dynamics are based on bounded confidence: social connections between individuals with distant opinions are severed while new connections are formed between individuals with similar opinions. Our model naturally gives raise to strong diversity, i.e., the persistence of heterogeneous opinions in connected societies, a phenomenon that most existing models fail to capture. The intensity of social interactions is the key parameter that governs the dynamics. First, it determines the asymptotic distribution of opinions. In particular, increasing the intensity of social interactions brings society closer to consensus. Second, it determines the risk of polarization, which is shown to increase with the intensity of social interactions. Our results allow to frame the problem of the design of public debates in a formal setting. We hence characterize the optimal strategy for a social planner who controls the intensity of the public debate and thus faces a trade-off between the pursuit of social consensus and the risk of polarization. We also consider applications to political campaigning and show that both minority and majority candidates can have incentives to lead society towards polarization.

We explore the possibility to compare positions in different directed and undirected graphs. We assume an agent to have a preference relation over positions in different weighted (directed and undirected) graphs, stating pairwise comparisons between these positions. Ideally, such a preference relation can be expressed by a utility function, where positions are evaluated by their assigned ‘utility’. Extending preference relations over the mixture set containing all lotteries over graph positions, we specify axioms on preferences that allow them to be represented by von Neumann–Morgenstern expected utility functions. For directed graphs, we show that the only vNM expected utility function that satisfies a certain risk neutrality, is the function that assigns to every position in a weighted directed graph the same linear combination of its outdegree and indegree. For undirected graphs, we show that the only vNM expected utility function that satisfies this risk neutrality, is the degree measure that assigns to every position in a weighted graph its degree. In this way, our results provide a utility foundation for degree centrality as a vNM expected utility function. We obtain the results following the utility approach to the Shapley value for cooperative transferable utility games of Roth (1977b), noticing that undirected graphs form a subclass of cooperative games as expressed by Deng and Papadimitriou (1994). For directed graphs, we extend this result to a class of generalized games. Using the relation between cooperative games and networks, we apply our results to some applications in Economics and Operations Research.

We investigate the phenomenon of diffusion in a countably infinite society of individuals interacting with their neighbors in a network. At a given time, each individual is either active or inactive. The diffusion is driven by two characteristics: the network structure and the diffusion mechanism represented by an aggregation function. We distinguish between two diffusion mechanisms (probabilistic, deterministic) and focus on two types of aggregation functions (strict, Boolean). Under strict aggregation functions, polarization of the society cannot happen, and its state evolves towards a mixture of infinitely many active and infinitely many inactive agents, or towards a homogeneous society. Under Boolean aggregation functions, the diffusion process becomes deterministic and the contagion model of Morris (2000) becomes a particular case of our framework. Polarization can then happen. Our dynamics also allows for cycles in both cases. The network structure is not relevant for these questions, but is important for establishing irreducibility, at the price of a richness assumption: the network should contain at least one complex star and have enough space for storing local configurations. Our model can be given a game-theoretic interpretation via a local coordination game, where each player would apply a best-response strategy in a random neighborhood.

We consider plurality voting games being simple games in partition function form such that in every partition there is at least one winning coalition. Such a game is said to be weighted if it is possible to assign weights to the players in such a way that a winning coalition in a partition is always one for which the sum of the weights of its members is maximal over all coalitions in the partition. A plurality game is called decisive if in every partition there is exactly one winning coalition. We show that in general, plurality games need not be weighted, even not when they are decisive. After that, we prove that (i) decisive plurality games with at most four players, (ii) majority games with an arbitrary number of players, and (iii) decisive plurality games that exhibit some kind of symmetry, are weighted. Complete characterizations of the winning coalitions in the corresponding partitions are provided as well.

In this note we present a short overview of different ranking methods. We recall the ranking methods for directed graphs and focus on axiomatic characterizations of the ranking methods by outdegree, Copeland score, and the β-measure.

We study within one theoretical framework two related phenomena – ingratiation by subordinates and favoritism of superiors towards their employees. We express ingratiation by opinion conformity of the worker when reporting his opinion to the manager. Favoritism of the manager is inferred from a bias when reporting toa firm her observation of the worker’s performance. We show interdependences of favoritism and ingratiation by investigating their influence on wages and profit. We study the more sophisticated manager and firm that try to infer the worker’s opinion and the manager’s observation. Such higher degrees of sophistication can mitigate the consequences of ingratiation and favoritism.