Solver algorithms

Impact Engine Allocate provides two pluggable decision rules for portfolio selection. Both share a common preprocessing pipeline (confidence filtering and return penalization) and return the same RuleResult dict.

Shared preprocessing

Three-scenario model

Each initiative provides return estimates under three scenarios:

Scenario

Field

Description

Best

return_best

Optimistic outcome

Median

return_median

Expected outcome

Worst

return_worst

Pessimistic outcome

Confidence penalty

Initiatives with low confidence have their effective returns penalized:

$$R^{\text{eff}}{i,j} = (1 - \gamma_i) \cdot R{i,j} + \gamma_i \cdot R_{i,\text{worst}}$$

where $\gamma_i = 1 - c_i$ and $c_i$ is the confidence score in $[0, 1]$.

  • High confidence ($c \to 1$): $\gamma \to 0$, returns stay as-is

  • Low confidence ($c \to 0$): $\gamma \to 1$, all returns collapse to worst-case

Both rules operate on the penalized effective returns, so better evidence enables better bets regardless of which rule is selected.

Minimax regret

The minimax regret rule selects the portfolio that minimizes the maximum regret across all scenarios. Regret is the difference between the best achievable return and the actual portfolio return under each scenario.

Step 1: Optimal scenario returns

For each scenario $j$, solve an independent binary knapsack:

$$V_j^* = \max \sum_i x_i \cdot R^{\text{eff}}_{i,j} \quad \text{s.t.} \quad \sum_i x_i \cdot \text{cost}_i \leq B$$

Step 2: Minimax regret

Minimize the maximum regret $\theta$ across all scenarios:

$$\min \theta \quad \text{s.t.} \quad \theta \geq V_j^* - \sum_i x_i \cdot R^{\text{eff}}_{i,j} \quad \forall j$$

Subject to:

  • Budget constraint: $\sum_i x_i \cdot \text{cost}_i \leq B$

  • Minimum worst return: $\sum_i x_i \cdot R_{i,\text{worst}} \geq R_{\min}$

The detail dict returns v_j_star (optimal per-scenario returns) and regrets (per-scenario regret values) for inspection.

Bayesian expected return

The Bayesian expected return rule maximizes the weighted sum of scenario returns, where weights represent the decision-maker’s prior beliefs about scenario likelihoods.

Formulation

Given scenario weights $w_j$ (non-negative, summing to 1), the weighted return for initiative $i$ is:

$$\bar{R}i = \sum_j w_j \cdot R^{\text{eff}}{i,j}$$

The solver maximizes the total weighted return:

$$\max \sum_i x_i \cdot \bar{R}_i$$

Subject to:

  • Budget constraint: $\sum_i x_i \cdot \text{cost}_i \leq B$

  • Minimum worst return: $\sum_i x_i \cdot R_{i,\text{worst}} \geq R_{\min}$

Equal weights recover the Laplace criterion (no scenario preference). Pessimistic weights (high worst weight) produce conservative portfolios similar to minimax regret.

The detail dict returns weights (the scenario weights used) and weighted_returns (per-initiative weighted returns for selected initiatives).

Implementation

Both solvers use PuLP with the CBC (COIN-OR Branch and Cut) backend. Binary decision variables $x_i \in {0, 1}$ indicate whether each initiative is selected.