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Abstract
Here, we consider one-dimensional forward-forward mean-field games (MFGs) with congestion, which were introduced to approximate stationary MFGs. We use methods from the theory of conservation laws to examine the qualitative properties of these games. First, by computing Riemann invariants and corresponding invariant regions, we develop a method to prove lower bounds for the density. Next, by combining the lower bound with an entropy function, we prove the existence of global solutions for parabolic forward-forward MFGs. Finally, we construct traveling-wave solutions, which settles in a negative way the convergence problem for forward-forward MFGs. A similar technique gives the existence of time-periodic solutions for non-monotonic MFGs.
Highlights
Mean-field games (MFGs) model competitive interactions in large populations in which the agents’ actions depend on statistical information about the population. These games are modeled by a Hamilton-Jacobi equation coupled with a FokkerPlanck equation
Together with terminal-initial conditions; we prescribe the initial value of m at t = 0 and the terminal value of u, at t = T : u(x, T ) = uT (x) m(x, 0) = m0(x)
We investigate the existence of entropies and determine conditions under which these entropies are convex
Summary
Mean-field games (MFGs) model competitive interactions in large populations in which the agents’ actions depend on statistical information about the population. In [1], the authors propose the forward-forward MFG model and study numerically its convergence to a stationary MFG This scheme relies on the parabolicity in (1.1) to force the long-time convergence to a stationary solution. In [13], the authors studied parabolic forward-forward MFGs and proved the existence of a solution. For quadratic Hamiltonians, we convert the forward-forward problem into a system of conservation laws and compute new convex entropies (Lemma 2.1) and Riemann invariants (Lemmas 2.5 and 2.6). These Riemann invariants give lower bounds for the density, m, and, for parabolic MFGs, these bounds combined with an entropy estimate gives the existence of a classical global solution (Theorem 3.6). We see that η as defined in (2.5) satisfies (2.4) if and only if (2.6) holds
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