Expected Value of Perfect Information (EVPI)

The expected value of perfect information (EVPI) quantifies how much a decision-maker would gain, on average, from knowing the realized scenario before committing to a decision, rather than choosing under uncertainty as required by the non-anticipativity constraint. EVPI is computed as the gap between the wait-and-see value WS (the expected best-per-scenario objective) and the recourse-problem value RP (the optimal here-and-now expected objective).

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The Wait-and-See Value

In a stochastic optimization problem, the non-anticipativity constraint forces a single decision to be made before the uncertain scenario is revealed. The expected value of perfect information (EVPI) quantifies how much we would gain, in expectation, if that constraint were lifted -- that is, if we could observe the realized scenario before deciding.

The first ingredient is the wait-and-see (WS) value. Suppose the uncertainty is modeled by scenarios $s = 1, \ldots, S$ with probabilities $p_1, \ldots, p_S$ (so $\sum_s p_s = 1$ ). For each scenario, let

z_s^{*} \,=\, \max_{x \in X} \, f(x, s)

be the optimal objective value of the deterministic problem solved as if scenario $s$ were known to occur. (For a cost-minimization problem, replace $\max$ by $\min$ .) The wait-and-see value is the probability-weighted average of these scenario-by-scenario optima:

WS \,=\, \sum_{s=1}^{S} p_s \, z_s^{*}.

This is the expected objective achievable if the decision-maker is allowed to react to the scenario after observing it -- i.e., to tailor a separate optimal decision to each realization.