Mixed strategy

MAS

Mixed strategy of a player $i$ is a probability distribution $p_{i}$ on $S_{i}$ . Expected utility of a player $i$ is $U_{i} (p_{1}, \dots, p_{n}) = \sum_{s \in S} u_{i} (s) \prod_{j \in N} p_{j} (s_{j})$

Utility is a multilinear function (special kind if polynomial)

Example

	x	y	z
a	1,7	1,5	3,4
b	2,3	0,4	0,6
p := $p_{1} (a)$
q := $p_{2} (x)$ , r := $p_{2} (y)$

Derive $p_{1} (b) = 1 - p$ ,

$U_{1} (p, q, r) = 1 \cdot P r (a, x) + 1 \cdot P r (a y) + 3 \cdot P r (a z) + 2 \cdot P r (b x) + 0 \cdot P r (b y) + 0 \cdot P r (b z)$ $= pq + p r + 3 p (1 - q - r) + 2 (1 - p) q$

$U_{2} (p, q, r) = 7 \cdot P r (a, x) + 5 \cdot P r (a y) + 4 \cdot P r (a z) + 3 \cdot P r (b x) + 4 \cdot P r (b y) + 6 \cdot P r (b z)$

Vojtěch Tóth

Explorer

Mixed strategy

Example

Graph View