Geodesics of minimal length in the set of probability measures on graphs

2019 ◽  
Vol 25 ◽  
pp. 78 ◽  
Author(s):  
Wilfrid Gangbo ◽  
Wuchen Li ◽  
Chenchen Mou
Keyword(s):  
Mean Field ◽  
Weighted Graph ◽  
Minimal Length ◽  
Connected Components ◽  
Mean Field Games ◽  
Probability Measures ◽  
Games On Graphs ◽  
Kantorovich Metric ◽  
Probability Simplex

We endow the set of probability measures on a weighted graph with a Monge–Kantorovich metric induced by a function defined on the set of edges. The graph is assumed to have n vertices and so the boundary of the probability simplex is an affine (n − 2)-chain. Characterizing the geodesics of minimal length which may intersect the boundary is a challenge we overcome even when the endpoints of the geodesics do not share the same connected components. It is our hope that this work will be a preamble to the theory of mean field games on graphs.

scholarly journals Lax connection and conserved quantities of quadratic mean field games

2021 ◽  
Vol 62 (8) ◽  
pp. 083302
Author(s):  
Thibault Bonnemain ◽  
Thierry Gobron ◽  
Denis Ullmo
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Conserved Quantities ◽  
Quadratic Mean

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.

Linear–Quadratic Time-Inconsistent Mean Field Games

2013 ◽  
Vol 3 (4) ◽  
pp. 537-552 ◽  
Author(s):  
A. Bensoussan ◽  
K. C. J. Sung ◽  
S. C. P. Yam
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Linear Quadratic ◽  
Time Inconsistent
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