“”Wednesday, August 12, 2026

Minimax

\max_{x \in \Delta} \min_{y \in \Delta} \, x^\top\! A \, y = v^*

§Zero-Sum

\textstyle\sum_{i=1}^{n} u_i(s) = 0 \;\; \forall \, s \in S

§Nash Eq.

x^{*\top}\! A y \geq x^{*\top}\! A y^* \geq x^\top\! A y^*

§Expected

\mathbb{E}[u_1] = x^\top\! A \, y

§Regret

R_T = \max_i \sum_{t=1}^{T} u_i^t - \sum_{t=1}^{T} u_{a_t}^t

§LP Dual

\min_{y,w} w \;\; \text{s.t.} \;\; Ay \leq w \cdot \mathbf{1}

§Payoff

u_1(i,j) = a_{ij}, \;\; u_2(i,j) = -a_{ij}

§Minimax

\max_{x \in \Delta} \min_{y \in \Delta} \, x^\top\! A \, y = v^*

§Zero-Sum

\textstyle\sum_{i=1}^{n} u_i(s) = 0 \;\; \forall \, s \in S

§Nash Eq.

x^{*\top}\! A y \geq x^{*\top}\! A y^* \geq x^\top\! A y^*

§Expected

\mathbb{E}[u_1] = x^\top\! A \, y

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