scholarly journals Quantum Mean-Field Games with the Observations of Counting Type

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 7
Author(s):  
Vassili N. Kolokoltsov
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Open Problem ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Mean Field Games ◽  
Quantum Game ◽  
Quantum Games ◽  
Interesting Open Problem

Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.

scholarly journals Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 133
Author(s):  
Olivier Féron ◽  
Peter Tankov ◽  
Laura Tinsi
Keyword(s):  
Nash Equilibrium ◽  
Electricity Markets ◽  
Mean Field ◽  
Optimal Strategies ◽  
Price Formation ◽  
Mean Field Games ◽  
Optimal Trading ◽  
Major Player ◽  
Major Producer ◽  
Small Producers

We study price formation in intraday electricity markets in the presence of intermittent renewable generation. We consider the setting where a major producer may interact strategically with a large number of small producers. Using stochastic control theory, we identify the optimal strategies of agents with market impact and exhibit the Nash equilibrium in a closed form in the asymptotic framework of mean field games with a major player.

scholarly journals Mean field games models of segregation

2017 ◽  
Vol 27 (01) ◽  
pp. 75-113 ◽  
Author(s):  
Yves Achdou ◽  
Martino Bardi ◽  
Marco Cirant
Keyword(s):  
Numerical Methods ◽  
Existence Of Solutions ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Games ◽  
Residential Choice ◽  
Urban Settlements ◽  
Thomas Schelling ◽  
Two Populations ◽  
Large Population Limit

This paper introduces and analyzes some models in the framework of mean field games (MFGs) describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of MFG theory, in the stationary and in the evolutive case. Numerical methods are proposed with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

scholarly journals A Novel Mean Field Game-Based Strategy for Charging Electric Vehicles in Solar Powered Parking Lots

Energies ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 8517
Author(s):  
Samuel M. Muhindo ◽  
Roland P. Malhamé ◽  
Geza Joos
Keyword(s):  
Nash Equilibrium ◽  
Solar Energy ◽  
Electric Vehicles ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Games ◽  
Parking Lot ◽  
Equal Sharing ◽  
Solar Powered ◽  
Energy Forecast

We develop a strategy, with concepts from Mean Field Games (MFG), to coordinate the charging of a large population of battery electric vehicles (BEVs) in a parking lot powered by solar energy and managed by an aggregator. A yearly parking fee is charged for each BEV irrespective of the amount of energy extracted. The goal is to share the energy available so as to minimize the standard deviation (STD) of the state of charge (SOC) of batteries when the BEVs are leaving the parking lot, while maintaining some fairness and decentralization criteria. The MFG charging laws correspond to the Nash equilibrium induced by quadratic cost functions based on an inverse Nash equilibrium concept and designed to favor the batteries with the lower SOCs upon arrival. While the MFG charging laws are strictly decentralized, they guarantee that a mean of instantaneous charging powers to the BEVs follows a trajectory based on the solar energy forecast for the day. That day ahead forecast is broadcasted to the BEVs which then gauge the necessary SOC upon leaving their home. We illustrate the advantages of the MFG strategy for the case of a typical sunny day and a typical cloudy day when compared to more straightforward strategies: first come first full/serve and equal sharing. The behavior of the charging strategies is contrasted under conditions of random arrivals and random departures of the BEVs in the parking lot.

scholarly journals Mean Field Games Flock! The Reinforcement Learning Way

2021 ◽  
Author(s):  
Sarah Perrin ◽  
Mathieu Laurière ◽  
Julien Pérolat ◽  
Matthieu Geist ◽  
Romuald Élie ◽  
...  
Keyword(s):  
Reinforcement Learning ◽  
Nash Equilibrium ◽  
Population Distribution ◽  
Mean Field ◽  
High Dimensional ◽  
Mean Field Games ◽  
Fictitious Play ◽  
Mean Field Game ◽  
Best Response

We present a method enabling a large number of agents to learn how to flock. This problem has drawn a lot of interest but requires many structural assumptions and is tractable only in small dimensions. We phrase this problem as a Mean Field Game (MFG), where each individual chooses its own acceleration depending on the population behavior. Combining Deep Reinforcement Learning (RL) and Normalizing Flows (NF), we obtain a tractable solution requiring only very weak assumptions. Our algorithm finds a Nash Equilibrium and the agents adapt their velocity to match the neighboring flock’s average one. We use Fictitious Play and alternate: (1) computing an approximate best response with Deep RL, and (2) estimating the next population distribution with NF. We show numerically that our algorithm can learn multi-group or high-dimensional flocking with obstacles.

scholarly journals ANALYSIS OF TREMBLING HAND PERFECT EQUILIBRIA IN QUANTUM GAMES

2008 ◽  
Vol 08 (01) ◽  
pp. L23-L30 ◽  
Author(s):  
IRENEUSZ PAKUŁA
Keyword(s):  
Game Theory ◽  
Probability Distribution ◽  
Distribution Functions ◽  
Quantum Game ◽  
Quantum Games ◽  
Probability Distribution Functions ◽  
Perfect Equilibria ◽  
Quantum Strategies

We analyse Selten' concept of trembling hand perfect equilibria in the context of quantum game theory. We define trembles as mixed quantum strategies by replacing discrete probabilities with probability distribution functions. Explicit examples of analysis are given.

scholarly journals Existence and non-existence for time-dependent mean field games with strong aggregation

2021 ◽  
Author(s):  
Marco Cirant ◽  
Daria Ghilli
Keyword(s):  
Population Density ◽  
State Space ◽  
Time Horizon ◽  
Existence Of Solutions ◽  
Mean Field ◽  
The State ◽  
Time Dependent ◽  
Classical Solutions ◽  
Mean Field Games ◽  
Quadratic Mean

AbstractWe investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $$-\sigma m^\alpha $$ - σ m α ,$$\alpha \ge 2/N$$ α ≥ 2 / N , where m is the population density and N is the dimension of the state space. We prove the existence of solutions under the assumption that $$\sigma $$ σ is small enough. For large $$\sigma $$ σ , we show that existence may fail whenever the time horizon T is large.

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