Bayesian game

For each player $i \in N$ , there are
- Type set $T_{i}$
- set of actions $S_{i}$
- Utility function $u_{i} : S \times T \to R$
A common prior distribution $ρ$ on $T$

Players know all elements of the game, values of utility, but they might not know the actual type $t \in T$ . They only have their believes about it.

Strategies

Pure strategy of player i is mapping $s_{i}$ from types to actions, $s_{i} : T_{i} \to S_{i}$ Behavioral strategy of player i is mapping $p_{i}$ from types to mixed actions, $p_{i} : T_{i} \to Δ_{i}$

Utilities by types

Ex post

Player i knows types of all players $t$ . $u_{i} (s ∣ t) = u_{i} (s_{1} (t_{1}), s_{2} (t_{2}), \dots, s_{n} (t_{n}) ∣ t)$

Interim

Player i knows only $t_{i}$ , not $t_{- i}$ . $u_{i} (s ∣ t_{i}) = \sum_{t_{i} \in T_{i}} ρ (t_{- i} ∣ t_{i}) \cdot u_{i} (s ∣ t)$

Ex ante

Player doesn’t know any type

Vojtěch Tóth

Explorer