Individual upper semicontinuity and subgame perfect ϵ-equilibria in games with almost perfect information

We study games with almost perfect information and an infinite time horizon. In such games, at each stage, the players simultaneously choose actions from finite action sets, knowing the actions chosen at all previous stages. The payoff of each player is a function of all actions chosen during the game. We define and examine the new condition of individual upper semicontinuity on the payoff functions, which is weaker than upper semicontinuity. We prove that a game with individual upper semicontinuous payoff functions admits a subgame perfect \(\epsilon \)-equilibrium for every \(\epsilon >0\), in eventually pure strategy profiles.