Positive Aggregates xB and By
A bookkeeping lemma used throughout the direct Loomis induction proof: an entrywise-positive matrix $B$ stays positive after weighting by a probability vector on either side.
Statement
Let $I$ and $J$ be finite nonempty index types, and let $B \colon I \times J \to \mathbb{R}$ satisfy $B_{ij} > 0$ for every $(i, j) \in I \times J$. Then:
- For every $x \in \Delta(I)$ and every $j \in J$, $(xB)_j = \sum_i x_i B_{ij} > 0$.
- For every $y \in \Delta(J)$ and every $i \in I$, $(By)_i = \sum_j B_{ij}\, y_j > 0$.
- For every $x \in \Delta(I)$ and every $y \in \Delta(J)$, $x B y = \sum_{i, j} x_i B_{ij} y_j > 0$.
Proof
(i) Fix $x \in \Delta(I)$ and $j \in J$. Each summand $x_i B_{ij}$ is nonnegative because $x_i \ge 0$ and $B_{ij} > 0$. Since $\sum_i x_i = 1$, there is some $i^* \in I$ with $x_{i^*} > 0$; for that index $x_{i^*} B_{i^* j} > 0$ strictly. Adding the nonnegative remaining terms keeps the sum strictly positive.
(ii) Symmetric in $y$ and the column variable.
(iii) Weight (ii) by $x \ge 0$ with $\sum_i x_i = 1$. The argument of (i) applied to the strictly positive function $i \mapsto (By)_i$ gives $\sum_i x_i (By)_i > 0$, which is exactly $x B y$. $\square$
Use
Used wherever the Loomis ratios $\frac{(xA)_j}{(xB)_j}$ and $\frac{(Ay)_i}{(By)_i}$ appear: the denominators are strictly positive, so the ratios are well-defined real numbers and continuous in their simplex arguments. In particular it underwrites the value-existence and weak-duality lemmas of the direct Loomis induction proof.
References
- [MFoGT, Section 2.5] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Aggregate positivity used to define the Loomis ratios.