Extensive-form game with imperfect information

MAS

It’s a

1. The mediator uses a probability distribution $p$ over $S=S_1\times \cdot \cdot \cdot \times S_n$ to generate randomly an action profile $s=(s_1,...,s_n)\in S$
2. The mediator tells each player i only $s_i$
3. Each player i is free to choose any action $s_i^{\prime }\in S_i$
4. The resulting utility is $u_i(s_1^{\prime },\dots ,s_n^{\prime })$

Strategy is a mapping from private recommendations to individual actions. ${\sigma }_i^{\ast }(s_i)=s_j$ Expected utility of player $i$ under $({\sigma }_1^{\ast },\dots ,{\sigma }_n^{\ast })$ is ${\sum }_{{\mathbf{s}}_{-i}\in {\mathbf{S}}_{-i}}p({\mathbf{s}}_{-i}|s_i)\cdot u_i(s_i,{\mathbf{s}}_{-i})\forall s_i\in S_i$

• conditional probability that others got recommendations ${\mathbf{s}}_{-i}$, given that you got $s_i$​