Iterated Weighted Sums Commute

For finite indices I and J, simplex elements x ∈ stdSimplex 𝕜 I and y ∈ stdSimplex 𝕜 J, and a payoff matrix A : I → J → 𝕜, $$ \sum_{i \in I} x_i \!\left( \sum_{j \in J} y_j \, A_{ij} \right) \;=\; \sum_{j \in J} y_j \!\left( \sum_{i \in I} x_i \, A_{ij} \right). $$

In wsum-notation: $$ \operatorname{wsum} x \, (\lambda i.\; \operatorname{wsum} y\, A_{i\,\cdot}) \;=\; \operatorname{wsum} y \, (\lambda j.\; \operatorname{wsum} x\, A_{\cdot\, j}). $$

This is the finite-sum Fubini fact that lets matrix-game expected payoffs be computed in either order. It is the algebraic backbone of the minimax-style manipulations (Loomis) on the value of a mixed extension.

References

• [MSZ, Chapter 5] Maschler, Solan, and Zamir, Game Theory. Expected payoff of a matrix game evaluated as a bilinear form in the two players' mixed strategies.