Asymptotic value in frequency-dependent games: A differential approach

Joseph M. Abdou (1, 2) , Nikolaos Pnevmatikos (3, 4)

1 CES - Centre d'économie de la Sorbonne
2 PSE - Paris School of Economics
3 UP2 - Université Panthéon-Assas
4 LEMMA - Laboratoire d'économie mathématique et de microéconomie appliquée

We study the asymptotic value of a frequency-dependent zero-sum game following a differential approach. In such a game the stage payoffs depend on the current action and on the frequency of actions played so far. We associate in a natural way a differential game to the original game and although it presents an irregularity at the origin, we prove existence of the value on the time interval [0,1]. We conclude, using appropriate approximations, that the limit of Vn, as n tends to infinity exists and coincides with the value of the associated continuous time game. We extend the existence of the asymptotic value to discounted payoffs and we show that Vλ as λ tends 0, converges to the same limit.

Mots clés: Hamilton-Jacobi-Bellman-Isaacs equation; frequency dependent payoffs; stochastic game; continuous-time game

Domaines: Economies et finances; Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Joseph M. Abdou, Nikolaos Pnevmatikos. Asymptotic value in frequency-dependent games: A differential approach. 2018. ⟨halshs-01400267v2⟩