Average Payoff Criterion

We are now interested in analyzing the average payoff game, that is, the game in ( 1.3) with the long-run expected average payoff in (1.13): $$\displaystyle J(x,\pi ^{1},\pi ^{2}):=\liminf \limits _{n\rightarrow \infty }\frac {1}{n} E_{x}^{\pi ^{1},\pi ^{2}}\sum _{t=0}^{n-1}r(x_{t},a_{t},b_{t}) {} $$