Remarks on Nash equilibria in mean field game models with a major player

For a mean field game model with a major and infinite minor players, we characterize a notion of Nash equilibrium via a system of so-called master equations, namely a system of nonlinear transport equations in the space of measures. Then, for games with a finite number N of minor players and a major player, we prove that the solution of the corresponding Nash system converges to the solution of the system of master equations as N tends to infinity

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure

Coordination after gains and losses: Is prospect theory’s value function predictive for games?

We analyze the effects of prior gain and loss experiences on individuals’ behavior in two coordination games: battle of the sexes and simultaneous market entry. We propose subjectively transformed games that integrate elements of prospect theory, aggregation of prior and subsequent payoffs, and social projection. Mathematical predictions of behavior are derived based on equilibrium selection concepts. Males’ behavior in our experimental studies is largely consistent with our predictions. However, the behavior of many female respondents appears to be rather consistent with interpreting the initial random lottery outcomes used to manipulate prior experiences as a signal for the players’ abilities to compete. This could be related to females’ known uneasiness of competing against counterparts that might be male and thus, a generally higher salience of rivalry in our incentivized experiments. Females also chose to play far more mixed strategies than males indicating some uncertainty about what type of behavior is appropriate.Prospect Game Theory, Prior Outcomes, Coordination, Equilibrium Selection, Economic Experiment, Agribusiness, Agricultural and Food Policy, Financial Economics, Institutional and Behavioral Economics, Research Methods/ Statistical Methods, Risk and Uncertainty,

An Extended Mean Field Game for Storage in Smart Grids

We consider a stylized model for a power network with distributed local power generation and storage. This system is modeled as network connection a large number of nodes, where each node is characterized by a local electricity consumption, has a local electricity production (e.g. photovoltaic panels), and manages a local storage device. Depending on its instantaneous consumption and production rates as well as its storage management decision, each node may either buy or sell electricity, impacting the electricity spot price. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. In a non-cooperative game setting, we are led to the analysis of a non-zero sum stochastic game with $N$ players where the interaction takes place through the spot price mechanism. For an infinite number of agents, our model corresponds to an Extended Mean-Field Game (EMFG). In a linear quadratic setting, we obtain and explicit solution to the EMFG, we show that it provides an approximate Nash-equilibrium for $N$-player game, and we compare this solution to the optimal strategy of a central planner.Comment: 27 pages, 5 figures. arXiv admin note: text overlap with arXiv:1607.02130 by other author