4.3a Payoff Discounting

Discounting

Discounting is the process of reducing the value of payoffs in future stages of an infinitely repeated game to account for the preference for immediate rewards over delayed ones. This is represented by a discount factor $\delta \in (0,1)$, applied to future payoffs to compute their present value. A player $i$‘s discounted payoff, provided that $i$ plays a strategy $s_i\in S_i$, is then given as
${\sum }_{i=1}^{}{\delta }^{i-1}{u}_i(s_i)=\frac{u_i(s_i)}{1-\delta },$
where $u_i$ is the payoff function of player $i$.

Proof