Opinion Dynamics in Social Networks through Mean-Field Games

Emulation, mimicry, and herding behaviors are phenomena that are observed when multiple social groups interact. To study such phenomena, we consider in this paper a large population of homogeneous social networks. Each such network is characterized by a vector state, a vector-valued controlled input and a vector-valued exogenous disturbance. The controlled input of each network is to align its state to the mean distribution of other networks’ states in spite of the actions of the disturbance. One of the contributions of this paper is a detailed analysis of the resulting mean field game for the cases of both polytopic and L2 bounds on controls and disturbances. A second contribution is the establishment of a robust mean-field equilibrium, that is, a solution including the worst-case value function, the state feedback best-responses for the controlled inputs and worst-case disturbances, and a density evolution. This solution is characterized by the property that no player can benefit from a unilateral deviation even in the presence of the disturbance. As a third contribution, microscopic and macroscopic analyses are carried out to show convergence properties of the population distribution using stochastic stability theory

Strategic thinking under social influence: Scalability, stability and robustness of allocations

This paper studies the strategic behavior of a large number of game designers and studies the scalability, stability and robustness of their allocations in a large number of homogeneous coalitional games with transferable utilities (TU). For each TU game, the characteristic function is a continuous-time stochastic process. In each game, a game designer allocates revenues based on the extra reward that a coalition has received up to the current time and the extra reward that the same coalition has received in the other games. The approach is based on the theory of mean-field games with heterogeneous groups in a multi-population regime

Mean-field games and two-point boundary value problems

© 2014 IEEE. A large population of agents seeking to regulate their state to values characterized by a low density is considered. The problem is posed as a mean-field game, for which solutions depend on two partial differential equations, namely the Hamilton-Jacobi-Bellman equation and the Fokker-Plank-Kolmogorov equation. The case in which the distribution of agents is a sum of polynomials and the value function is quadratic is considered. It is shown that a set of ordinary differential equations, with two-point boundary value conditions, can be solved in place of the more complicated partial differential equations associated with the problem. The theory is illustrated by a numerical example

Large-scale games in large-scale systems

Many real-world problems modeled by stochastic games have huge state and/or action spaces, leading to the well-known curse of dimensionality. The complexity of the analysis of large-scale systems is dramatically reduced by exploiting mean field limit and dynamical system viewpoints. Under regularity assumptions and specific time-scaling techniques, the evolution of the mean field limit can be expressed in terms of deterministic or stochastic equation or inclusion (difference or differential). In this paper, we overview recent advances of large-scale games in large-scale systems. We focus in particular on population games, stochastic population games and mean field stochastic games. Considering long-term payoffs, we characterize the mean field systems using Bellman and Kolmogorov forward equations.Comment: 30 pages. Notes for the tutorial course on mean field stochastic games, March 201

Large networks of dynamic agents: Consensus under adversarial disturbances

This paper studies interactions among homogeneous social groups within the framework of large population games. Each group is represented by a network and the behavior described by a two-player repeated game. The contribution is three-fold. Beyond the idea of providing a novel two-level model with repeated games at a lower level and population games at a higher level, we also establish a mean field equilibrium and study state feedback best-response strategies as well as worst-case adversarial disturbances in that context. © 2012 IEEE

Reputation and Influence in Charitable Giving: An Experiment

Abstract Previous experimental and observational work suggests that people act more generously when they are observed and observe others in social settings. But the explanation for this is unclear. An individual may want to send a signal of her generosity in order to improve her own reputation. Alternately (or additionally) she may value the public good or charity itself and, believing that contribution levels are strategic complements, give more in order to influence others to give more. We perform the first series of laboratory experiments that can separately estimate the impact of these two social effects, and test whether realized influence is consistent with the desire to influence, and whether either of these are consistent with anticipated influence. We find that �leaders� are influential only when their identities are revealed along with their donations, and female leaders are more influential then males. Identified leader�s predictions suggest that are aware of their influence. They respond to this by giving more than either the control group or the unidentified leaders. We find mixed evidence for �reputation-seeking.�