Mean-field Equilibrium Research Articles

Dynamic Demand and Mean-Field Games

Within the realm of smart buildings and smart cities , dynamic response management is playing an ever-increasing role, thus attracting the attention of scientists from different disciplines. Dynamic demand response management involves a set of operations aiming at decentralizing the control of loads in large and complex power networks. Each single appliance is fully responsive and readjusts its energy demand to the overall network load. A main issue is related to mains frequency oscillations resulting from an unbalance between supply and demand. In a nutshell, this paper contributes to the topic by equipping each consumer with strategic insight. In particular, we highlight three main contributions and a few other minor contributions. First, we design a mean-field game for a population of thermostatically controlled loads, study the mean-field equilibrium for the deterministic mean-field game, and investigate on asymptotic stability for the microscopic dynamics. Second, we extend the analysis and design to uncertain models, which involve both stochastic or deterministic disturbances. This leads to robust mean-field equilibrium strategies guaranteeing stochastic and worst-case stability, respectively. Minor contributions involve the use of stochastic control strategies rather than deterministic and some numerical studies illustrating the efficacy of the proposed strategies.

Phase structural ordering kinetics of second-phase formation in the vicinity of a crack

The formation of a second phase in presence of a crack in a crystalline material is modelled and studied for different prevailing conditions in order to predict and a posteriori prevent failure, e.g. by delayed hydride cracking. To this end, the phase field formulation of Ginzburg–Landau is selected to describe the phase transformation, and simulations using the finite volume method are performed for a wide range of material properties. A sixth order Landau potential for a single structural order parameter is adopted because it allows the modeling of both first and second order transitions and its corresponding phase diagram can be outlined analytically. The elastic stress field induced by the crack is found to cause a space-dependent shift in the transition temperature, which promotes a second-phase precipitation in vicinity of the crack tip. The spatio-temporal evolution during nucleation and growth is successfully monitored for different combinations of material properties and applied loads. Results for the second-phase shape and size evolution are presented and discussed for a number of selected characteristic cases. The numerical results at steady state are compared to mean-field equilibrium solutions and a good agreement is achieved. For materials applicable to the model, the results can be used to predict the evolution of an eventual second-phase formation through a dimensionless phase transformation in the crack-tip vicinity for given conditions.

Iterative control strategies for nonlinear systems.

In this paper, we focus on the control of the mean-field equilibrium of nonlinear networks of the Langevin type in the limit of small noise. Using iterative linear approximations, we derive a formula that prescribes a control strategy in order to displace the equilibrium state of a given system and remarkably find that the control function has a "universal" form under certain physical conditions. This result can be employed to define universal protocols useful, for example, in the optimal work extraction from a given reservoir. Generalizations and limits of application of the method are discussed.

Discrete-Time Mean Field Partially Observable Controlled Systems Subject to Common Noise

In this article, we provide the first systemic study on discrete time partially observable mean field systems in the presence of a common noise. Each player makes decision solely based on the observable process. Both the mean field games and the related tractable mean field type stochastic control problem are studied. We first solve the mean field type control problem using classical discrete time Kalman filter with notable modifications. The unique existence of the resulted forward backward stochastic difference system is then established by separation principle. The mean field game problem is also solved via an application of stochastic maximum principle, while the existence of the mean field equilibrium is shown by the Schauder’s fixed point theorem.

A Two-Sided Market Analysis of Behaviorally Targeted Display Advertising

Targeting breadth is defined as the number of consumer interests that an ad network uses to form the set of potential advertisers who compete for an ad impression to a consumer in the context of display advertising. Despite its relevance and importance to display ad platforms, the impact and optimal design of targeting breadth remain largely unexplored. We develop an integrated model of decisions made by consumers, advertisers, and the platform, where advertisers’ bids are characterized by the mean field equilibrium. Using a proprietary dataset obtained from a large ad platform that comprises of 1.26 million ad impressions over 56,000 unique ads from 4,400 consumers over six weeks, we estimate model parameters, draw insights, and configure the optimal targeting breadth to improve the platform’s ad revenue. Counterfactual experiments indicate that both click-through rate (CTR) and cost-per-click (CPC) first increase and then decrease, when a broader targeting is used in ad serving. The inverted-U shaped effect of targeting breadth on CTR is because a broader targeting tends to serve more ads that are less relevant to consumers (which decreases CTR), but also from different product categories and therefore alleviate the negative spillovers from the highly similar ads delivered simultaneously (which increases CTR). For CPC, the non-monotonic effect is driven by the increased competition (which increases CPC) and the lower match value of targeted audience (which decreases CPC) under a broader targeting. Moreover, the platform’s revenue can be improved by 9.2% if optimizing the targeting breadth uniformly and by another increase of 6.8% if setting the targeting breadth individually.

Are Mean-field Games the Limits of Finite Stochastic Games?

Mean-field games model the rational behavior of an infinite number of indistinguishable players in interaction. An important assumption of mean-field games is that, as the number of player is infinite, the decisions of an individual player do not affect the dynamics of the mass. Each player plays against the mass. A mean-field equilibrium corresponds to the case when the optimal decisions of a player coincide with the decisions of the mass. Many authors argue that mean-field games are a good approximation of symmetric stochastic games with a large number of players, the rationale behind this being that the impact of one player becomes negligible when the number of players goes to infinity. In this paper, we question this assertion. We show that, in general, this convergence does not hold. In fact, the “tit for tat” principle allows one to define many equilibria in repeated or stochastic games with N players. However, in mean-field games, the deviation of a single player is not visi- ble by the population and therefore the “tit for tat” principle cannot be applied. The conclusion is that, even if N-player games have many equilibria with a good social cost, this may not be the case for the limit game.

Mean Field Equilibria of Pricing Games in Internet Marketplaces

We model an Internet marketplace using a set of servers that choose prices for performing jobs. Each server has a queue of unfinished jobs, and is penalized for delay by the market maker via a holding cost. A server completes jobs with a low or high "quality", and jobs truthfully report the quality with which they were completed. The best estimate of quality based on these reports is the "reputation" of the server. A server bases its pricing decision on the distribution of its competitors offered prices and reputations. An entering job is given a random sample of servers, and chooses the best one based on a linear combination of price and reputation. We seek to understand how prices would be determined in such a marketplace using the theory of Mean Field Games. We show the existence of a Mean Field Equilibrium and show how reputation plays a role in allowing servers to declare larger prices than their competitors. We illustrate our results by a numerical study of the system via simulation with parameters chosen from data gathered from existing Internet marketplaces.

Downlink Power Control in Self-Organizing Dense Small Cells Underlaying Macrocells: A Mean Field Game

A novel distributed power control paradigm is proposed for dense small cell networks co-existing with a traditional macrocellular network. The power control problem is first modeled as a stochastic game and the existence of the Nash Equilibrium is proven. Then, we extend the formulated stochastic game to a mean field game (MFG) considering a highly dense network. An MFG is a special type of differential game which is ideal for modeling the interactions among a large number of entities. We analyze the performance of two different cost functions for the mean field game formulation. Both of these cost functions are designed using stochastic geometry analysis in such a way that the cost functions are valid for the MFG setting. A finite difference algorithm is then developed based on the Lax-Friedrichs scheme and Lagrange relaxation to solve the corresponding MFG. Each small cell base station can independently execute the proposed algorithm offline, i.e., prior to data transmission. The output of the algorithm shows how each small cell base station should adjust its transmit power in order to minimize the cost over a predefined period of time. Moreover, sufficient conditions for the uniqueness of the mean field equilibrium for a generic cost function are also given. The effectiveness of the proposed algorithm is demonstrated via numerical results.

Modeling and computation of mean field equilibria in producers’ game with emission permits trading

In this paper, we present a mean field game to model the production behaviors of a very large number of producers, whose carbon emissions are regulated by government. Especially, an emission permits trading scheme is considered in our model, in which each enterprise can trade its own permits flexibly. By means of the mean field equilibrium, we obtain a Hamilton–Jacobi–Bellman (HJB) equation coupled with a Kolmogorov equation, which are satisfied by the adjoint state and the density of producers (agents), respectively. Then, we propose a so-called fitted finite volume method to solve the HJB equation and the Kolmogorov equation. The efficiency and the usefulness of this method are illustrated by the numerical experiments. Under different conditions, the equilibrium states as well as the effects of the emission permits price are examined, which demonstrates that the emission permits trading scheme influences the producers’ behaviors, that is, more populations would like to choose a lower rather than a higher emission level when the emission permits are expensive.

Mean-Field Games for Resource Sharing in Cloud-Based Networks

In this paper, we consider last level cache (LLC) sharing problems in large-scale cloud networks with a fair payoff function. We formulate the problem as a strategic decision-making problem (i.e., a game). We examine the resource-sharing game with finite and infinite number of players. Exploiting the aggregate structure of the payoff functions, we show that the resource-sharing game has a Nash equilibrium in a wide range of return index. We show that the Nash equilibrium is not an evolutionarily stable strategy in the finite regime. Then, we introduce a myopic mean-field response where each player implements a mean-field-taking strategy. We show that such a mean-field-taking strategy is an evolutionarily stable strategy in both finite and infinite regime. We provide closed-form expression of the optimal pricing that gives an efficient resource-sharing policy. As the number of active players grows without bound, we show that the equilibrium strategy converges to a mean-field equilibrium, and the optimal prices for resources converge to the optimal price of the mean-field game. Then, we address the demand satisfaction problem for which a necessary and sufficient condition for satisfactory solutions is provided. In addition, a very fast mean-field learning algorithm is provided.

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 aims 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 $\mathcal L_2$ 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.

Mean-Field Game Modeling the Bandwagon Effect with Activation Costs

This paper provides a mean-field game theoretic model of the bandwagon effect in social networks. This effect can be observed whenever individuals tend to align their own opinions to a mainstream opinion. The contribution is threefold. First, we describe the opinion propagation as a mean-field game with local interactions. Second, we establish mean-field equilibrium strategies in the case where the mainstream opinion is constant. Such strategies are shown to have a threshold structure. Third, we extend the use of threshold strategies to the case of time-varying mainstream opinion and study the evolution of the macroscopic system.

Chemical Equilibrium on Low Dimensional Supports: Connecting the Microscopic Mechanism to the Macroscopic Observations

Classical chemical thermodynamics predicts that the equilibrium composition of a reactive system is entirely defined by the equilibrium constants of the different reactions involved. In this paper we show that for nonlinear reactions taking place on a low-dimensional support this is not true anymore: the equilibrium state depends on the mechanistic details of the chemical processes, so that even two reactions having the same mean field kinetics and equilibrium constants can reach a different equilibrium composition, depending on the microscopic mechanism. We illustrate this point by simulations and mathematical analyses of a simple autocatalytic scheme, and we propose a theoretical route to discriminate between the different cases.

Mean Field Equilibria of Dynamic Auctions with Learning

We study learning in a dynamic setting where identical copies of a good are sold over time through a sequence of second-price auctions. Each agent in the market has an unknown independent private valuation that determines the distribution of the reward she obtains from the good; for example, in sponsored search settings, advertisers may initially be unsure of the value of a click. Though the induced dynamic game is complex, we simplify analysis of the market using an approximation methodology known as mean field equilibrium (MFE). The methodology assumes that agents optimize only with respect to long-run average estimates of the distribution of other players' bids. We show a remarkable fact: In a mean field equilibrium, the agent has an optimal strategy where she bids truthfully according to a conjoint valuation. The conjoint valuation is the sum of her current expected valuation, together with an overbid amount that is exactly the expected marginal benefit of one additional observation about her true private valuation. Under mild conditions on the model, we show that an MFE exists, and that it is a good approximation to a rational agent’s behavior as the number of agents increases. Formally, if every agent except one follows the MFE strategy, then the remaining agent’s loss on playing the MFE strategy converges to zero as the number of agents in the market increases. We conclude by discussing the implications of the auction format and design on the auctioneer’s revenue. In particular, we establish the revenue equivalence of standard auctions in dynamic mean field settings, and discuss optimal selection of reserve prices in dynamic auctions. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

Mean-field games for marriage.

This article examines mean-field games for marriage. The results support the argument that optimizing the long-term well-being through effort and social feeling state distribution (mean-field) will help to stabilize marriage. However, if the cost of effort is very high, the couple fluctuates in a bad feeling state or the marriage breaks down. We then examine the influence of society on a couple using mean-field sentimental games. We show that, in mean-field equilibrium, the optimal effort is always higher than the one-shot optimal effort. We illustrate numerically the influence of the couple’s network on their feeling states and their well-being.

Risk-Sensitive Mean-Field Games

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton -Jacobi- Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics.

On the Efficiency of Equilibria in Mean-Field Oscillator Games

A key question in the design of engineered competitive systems has been that of the efficiency of the associated equilibria. Yet, there is little known in this regard in the context of stochastic dynamic games in a large population regime. In this paper, we revisit a class of noncooperative games, arising from the synchronization of a large collection of homogeneous oscillators. In [1], we derived a PDE model for analyzing the associated equilibria in large population regimes through a mean field approximation. Here, we examine the efficiency of the associated mean-field equilibria with respect to a related welfare optimization problem. We construct variational problems both for the noncooperative game and its centralized counterpart and employ these problems as a vehicle for conducting this analysis. Using a bifurcation analysis, we analyze the variational solutions and the associated efficiency loss. An expression for the efficiency loss is obtained. Finally, our results are validated through detailed numerics.

Mean-Field Games and Dynamic Demand Management in Power Grids

This paper applies mean-field game theory to dynamic demand management. For a large population of electrical heating or cooling appliances (called agents), we provide a mean-field game that guarantees desynchronization of the agents thus improving the power network resilience. Second, for the game at hand, we exhibit a mean-field equilibrium, where each agent adopts a bang-bang switching control with threshold placed at a nominal temperature. At equilibrium, through an opportune design of the terminal penalty, the switching control regulates the mean temperature (computed over the population) and the mains frequency around the nominal value. To overcome Zeno phenomena we also adjust the bang-bang control by introducing a thermostat. Third, we show that the equilibrium is stable in the sense that all agents’ states, initially at different values, converge to the equilibrium value or remain confined within a given interval for an opportune initial distribution.

Mean Field Equilibrium in Dynamic Games with Strategic Complementarities

We study a class of stochastic dynamic games that exhibit strategic complementarities between players; formally, in the games we consider, the payoff of a player has increasing differences between her own state and the empirical distribution of the states of other players. Such games can be used to model a diverse set of applications, including network security models, recommender systems, and dynamic search in markets. Stochastic games are generally difficult to analyze, and these difficulties are only exacerbated when the number of players is large (as might be the case in the preceding examples). We consider an approximation methodology called mean field equilibrium to study these games. In such an equilibrium, each player reacts to only the long-run average state of other players. We find necessary conditions for the existence of a mean field equilibrium in such games. Furthermore, as a simple consequence of this existence theorem, we obtain several natural monotonicity properties. We show that there exist a “largest” and a “smallest” equilibrium among all those where the equilibrium strategy used by a player is nondecreasing, and we also show that players converge to each of these equilibria via natural myopic learning dynamics; as we argue, these dynamics are more reasonable than the standard best-response dynamics. We also provide sensitivity results, where we quantify how the equilibria of such games move in response to changes in parameters of the game (for example, the introduction of incentives to players).

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