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Unique Ergodicity of Deterministic Zero-Sum Differential Games

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Abstract

We study the ergodicity of deterministic two-person zero-sum differential games. This property is defined by the uniform convergence to a constant of either the infinite-horizon discounted value as the discount factor tends to zero, or equivalently, the averaged finite-horizon value as the time goes to infinity. We provide necessary and sufficient conditions for the unique ergodicity of a game. This notion extends the classical one for dynamical systems, namely when ergodicity holds with any (suitable) perturbation of the running payoff function. Our main condition is symmetric between the two players and involve dominions, i.e., subsets of states that one player can make approximately invariant.

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Notes

  1. Note that the counterexample to Hamilton–Jacobi homogenization given in [22] has been preceded by counterexamples for the convergence of the value of repeated games given by Vigeral [20] and Ziliotto [21].

  2. We refer the reader to [2, Sec. 6.1] for the connections between classical ergodic theory and ergodicity of games or Hamiltonians.

  3. In order to simplify the notation, we shall equally denote by a and b single elements of A and B, respectively, and controls of player 1 and player 2, i.e., elements of \({\mathscr {A}}\) and \({\mathscr {B}}\), respectively. The distinction should be clear from the context.

  4. We recall that a modulus of continuity is a nondecreasing function \(\omega : [0,+\infty ) \rightarrow [0,+\infty )\), vanishing and continuous at 0, that is, such that \(\lim _{r \rightarrow 0} \omega (r) = \omega (0) = 0\).

  5. In this paper, the solutions of PDEs will always be in the continuous viscosity sense.

  6. We mention that the notion of discriminating/leadership domain, hence of dominion, relates with the ones of B-set and approachability in repeated games with vector payoffs. Indeed, In [7], As Soulaimani, Quincampoix and Sorin proved that the B-sets for one player (which provide a sufficient condition for approachability) coincide with the discriminating domains for that player in an associated differential game.

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  1. Facultad de Ingeniería y Ciencia, Universidad Adolfo Ibáñez, Diagonal Las Torres 2640, Santiago, Chile

    Antoine Hochart

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The author is supported by FONDECYT Grant 3180662.

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Hochart, A. Unique Ergodicity of Deterministic Zero-Sum Differential Games. Dyn Games Appl 11, 109–136 (2021). https://doi.org/10.1007/s13235-020-00355-y

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