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Excuse the click-baity title. I notice the discounted payoff in the game theory literature usually takes the form

$$\sum_{t=1}^\infty\lambda(1-\lambda)^{t-1}R_t$$

This differs from the discounted payoff in the other dynamic optimization settings, e.g., see the Bellman equation in control theory.

Why the difference?

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In my experience, it's mainly just for cleanliness for results.

Consider an infinite horizon repeated game, with discounted payoff representation (where I use $\delta = (1-\lambda)$ in your notation) $$ (1-\delta)\sum_{t=0}^{\infty}\delta^t R_t $$ where $0 < \delta < 1$.

Suppose I play a strategy that gives me the same payoff, say $a$, for each period $t$. Then, $$ (1-\delta)\sum_{t=0}^{\infty}\delta^t a = (1-\delta)\frac{a}{1-\delta} = a $$ which is cleaner than $$ \frac{a}{1-\delta} $$ As a side note, multiplying the utility by a constant does not change preferences, so we keep the inherent preferences the same.

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    $\begingroup$ In fact, arguably utility should be considered an equivalence class, and a particular formula simply an representative of that class. Two formulas in the same equivalence class are essentially the same utility. $\endgroup$ – Acccumulation Feb 28 at 4:04
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Are you referring to repeated games, in which player $i$ obtains a payoff of the form $$(1-\delta)\sum_{t=0}^{\infty}\delta^{t}u_{i}(x_{i}^{t})$$

from the sequence of payoffs $\left\{x_{i}^{t}\right\}$? The $(1-\delta)$ is appended to the front to give the average per period payoff.

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