Average discounted payoffs after T periods formula.

Question

I am being told that;

The average discounted payoffs after T periods is given by;

$$\pi_i = \frac{1 - \delta }{1 - \delta^T}\sum_{t=0}^{T-1} \delta^tg_i(a^t) $$

$\delta$ is the discount rate

$g_i(a^t)$ is the payoff given to player i for taking action a in period t.

I am being told that I should know this formula as it should have been covered in previous game theory and microeconomics courses therefore no derivation is given. However it is the first time I have seen it.

Can someone kindly explain how we get this formula.

Thank you.

Answer 1

The question is equivalent to asking why a payoff at time $t$ should be weighted by

$$ \frac{1-\delta}{1-\delta^T}\delta^t\;. $$

Discounting with a discount rate $\delta$ yields a weight factor $\delta^t$. Normalising these factors by their sum,

$$ \sum_{t=0}^{T-1}\delta^t=\frac{1-\delta^T}{1-\delta}\;, $$

leads to the above weights. These are the weights of an "average discounted payoff" in the sense that they discount the payoffs at rate $\delta$ and are normalised such that getting the same payoff $p$ in every period will yield an average of $p$.