Long-run average payoff (${\lim }_{T\to \infty }{T}^{-1}{\sum }_{t=0}^{T-1}{u}_i^t$)

A repeated game criterion evaluating the average stage payoff over increasingly long horizons. State how paths without a limit are treated, for example by using a lower limit. Changing a finite prefix does not change a convergent long-run average payoff. Thus a one-time gain can be outweighed by permanent future punishment without specifying a discount factor.