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A model problem
for first-order mean field games with discrete initial
data
Philip Jameson Graber and Brady Zimmerman |
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Vol. 19 (2026), No. 1, 121–143
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Abstract |
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We study a simplified version of a density-dependent first-order mean field game, in which the players face a penalization equal to the population density at their final position. We consider the problem of finding an equilibrium when the initial distribution is a discrete measure. We show that the problem becomes finite-dimensional: the final piecewise smooth density is completely determined by the weights and positions of the initial measure. We establish existence and uniqueness of a solution using classical fixed point theorems. Finally, we show that Newton’s method provides an effective way to compute the solution. Our numerical simulations provide an illustration of how density penalization in a mean field game tends to the smoothen the initial distribution. |
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Keywords
mean field games, optimal control, Nash equilibrium,
congestion games
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Mathematical Subject Classification
Primary: 35Q89, 49N80
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Milestones
Received: 1 May 2024
Revised: 24 July 2024
Accepted: 14 August 2024
Published: 25 January 2026
Communicated by Martin Bohner
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Authors |
| Philip Jameson Graber | |
| Department of Mathematics Baylor University Waco, TX United States |
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| Brady Zimmerman | |
| Department of Mathematics Baylor University Waco, TX United States |
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| © 2026 MSP (Mathematical Sciences Publishers). |
