Consensus as a Nash Equilibrium of a Dynamic Game

Consensus formation in a social network is modeled by a dynamic game of a prescribed duration played by members of the network. Each member independently minimizes a cost function that represents his/her motive. An integral cost function penalizes a member's differences of opinion from the others as well as from his/her own initial opinion, weighted by influence and stubbornness parameters. Each member uses its rate of change of opinion as a control input. This defines a dynamic non-cooperative game that turns out to have a unique Nash equilibrium. Analytic explicit expressions are derived for the opinion trajectory of each member for two representative cases obtained by suitable assumptions on the graph topology of the network. These trajectories are then examined under different assumptions on the relative sizes of the influence and stubbornness parameters that appear in the cost functions.Comment: 7 pages, 9 figure, Pre-print from the Proceedings of the 12th International Conference on Signal Image Technology and Internet-based Systems (SITIS), 201

Generalized Opinion Dynamics from Local Optimization Rules

We study generalizations of the Hegselmann-Krause (HK) model for opinion dynamics, incorporating features and parameters that are natural components of observed social systems. The first generalization is one where the strength of influence depends on the distance of the agents' opinions. Under this setup, we identify conditions under which the opinions converge in finite time, and provide a qualitative characterization of the equilibrium. We interpret the HK model opinion update rule as a quadratic cost-minimization rule. This enables a second generalization: a family of update rules which possess different equilibrium properties. Subsequently, we investigate models in which a external force can behave strategically to modulate/influence user updates. We consider cases where this external force can introduce additional agents and cases where they can modify the cost structures for other agents. We describe and analyze some strategies through which such modulation may be possible in an order-optimal manner. Our simulations demonstrate that generalized dynamics differ qualitatively and quantitatively from traditional HK dynamics.Comment: 20 pages, under revie

Collective states in social systems with interacting learning agents

We consider a social system of interacting heterogeneous agents with learning abilities, a model close to Random Field Ising Models, where the random field corresponds to the idiosyncratic willingness to pay. Given a fixed price, agents decide repeatedly whether to buy or not a unit of a good, so as to maximize their expected utilities. We show that the equilibrium reached by the system depends on the nature of the information agents use to estimate their expected utilities.Comment: 18 pages, 26 figure

Crises and collective socio-economic phenomena: simple models and challenges

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the Random Field Ising model (RFIM) indeed provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilising self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of RFIM-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith's invisible hand can badly fail at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria to be reached. As a theoretical challenge, the study of so-called "detailed-balance" violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.Comment: Review paper accepted for a special issue of J Stat Phys; several minor improvements along reviewers' comment

Mean-Field-Type Games in Engineering

A mean-field-type game is a game in which the instantaneous payoffs and/or the state dynamics functions involve not only the state and the action profile but also the joint distributions of state-action pairs. This article presents some engineering applications of mean-field-type games including road traffic networks, multi-level building evacuation, millimeter wave wireless communications, distributed power networks, virus spread over networks, virtual machine resource management in cloud networks, synchronization of oscillators, energy-efficient buildings, online meeting and mobile crowdsensing.Comment: 84 pages, 24 figures, 183 references. to appear in AIMS 201