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Wednesday, August 12, 2026
☀ Light
Minimax
max
x
∈
Δ
min
y
∈
Δ
x
⊤
A
y
=
v
∗
\max_{x \in \Delta} \min_{y \in \Delta} \, x^\top\! A \, y = v^*
max
x
∈
Δ
min
y
∈
Δ
x
⊤
A
y
=
v
∗
§
Zero-Sum
∑
i
=
1
n
u
i
(
s
)
=
0
∀
s
∈
S
\textstyle\sum_{i=1}^{n} u_i(s) = 0 \;\; \forall \, s \in S
∑
i
=
1
n
u
i
(
s
)
=
0
∀
s
∈
S
§
Nash Eq.
x
∗
⊤
A
y
≥
x
∗
⊤
A
y
∗
≥
x
⊤
A
y
∗
x^{*\top}\! A y \geq x^{*\top}\! A y^* \geq x^\top\! A y^*
x
∗
⊤
A
y
≥
x
∗
⊤
A
y
∗
≥
x
⊤
A
y
∗
§
Expected
E
[
u
1
]
=
x
⊤
A
y
\mathbb{E}[u_1] = x^\top\! A \, y
E
[
u
1
]
=
x
⊤
A
y
§
Regret
R
T
=
max
i
∑
t
=
1
T
u
i
t
−
∑
t
=
1
T
u
a
t
t
R_T = \max_i \sum_{t=1}^{T} u_i^t - \sum_{t=1}^{T} u_{a_t}^t
R
T
=
max
i
∑
t
=
1
T
u
i
t
−
∑
t
=
1
T
u
a
t
t
§
LP Dual
min
y
,
w
w
s.t.
A
y
≤
w
⋅
1
\min_{y,w} w \;\; \text{s.t.} \;\; Ay \leq w \cdot \mathbf{1}
min
y
,
w
w
s.t.
A
y
≤
w
⋅
1
§
Payoff
u
1
(
i
,
j
)
=
a
i
j
,
u
2
(
i
,
j
)
=
−
a
i
j
u_1(i,j) = a_{ij}, \;\; u_2(i,j) = -a_{ij}
u
1
(
i
,
j
)
=
a
ij
,
u
2
(
i
,
j
)
=
−
a
ij
§
Minimax
max
x
∈
Δ
min
y
∈
Δ
x
⊤
A
y
=
v
∗
\max_{x \in \Delta} \min_{y \in \Delta} \, x^\top\! A \, y = v^*
max
x
∈
Δ
min
y
∈
Δ
x
⊤
A
y
=
v
∗
§
Zero-Sum
∑
i
=
1
n
u
i
(
s
)
=
0
∀
s
∈
S
\textstyle\sum_{i=1}^{n} u_i(s) = 0 \;\; \forall \, s \in S
∑
i
=
1
n
u
i
(
s
)
=
0
∀
s
∈
S
§
Nash Eq.
x
∗
⊤
A
y
≥
x
∗
⊤
A
y
∗
≥
x
⊤
A
y
∗
x^{*\top}\! A y \geq x^{*\top}\! A y^* \geq x^\top\! A y^*
x
∗
⊤
A
y
≥
x
∗
⊤
A
y
∗
≥
x
⊤
A
y
∗
§
Expected
E
[
u
1
]
=
x
⊤
A
y
\mathbb{E}[u_1] = x^\top\! A \, y
E
[
u
1
]
=
x
⊤
A
y
0∑×
$0 sometimes
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