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# Bayesian Nash Equilibrium

📗 Given a game \((A, R)\), \(\pi \in A\) is a Nash equilibrium if \(R\left(\pi_{i}, \pi_{-i}\right) \geq R\left(a, \pi_{-i}\right)\) for every \(a \in A_{i}\) and \(i \in \left[n\right]\).
📗 Given a Bayesian game \((A, T, R)\), \(\pi \in T \to  A\) is a Bayesian Nash equilibrium if \(E_{T_{-i}}\left[R\left(\pi_{i}\left(t\right), \pi_{-i}; t\right)\right] > E_{T_{-i}}\left[R\left(a, \pi_{-i}; t\right)\right]\) for every \(a \in A\) and \(t \in T_{i}\) and \(i \in \left[n\right]\).
➩ BNE requires common knowledge of the prior distribution of types (for expectations taken over \(T_{-i}\)).
➩ BNE mutual best responses take into consideration the existence of the other type of themselves.

# BoS Example

📗 Two types of column players: \(H, L\).
➩ For \(H\) type, the game is \(\begin{bmatrix} \left(2, 1\right) & \left(0, 0\right) \\ \left(0, 0\right) & \left(1, 2\right) \end{bmatrix}\).
➩ For \(L\) type, the game is \(\begin{bmatrix} \left(2, 0\right) & \left(0, 1\right) \\ \left(2, 0\right) & \left(1, 0\right) \end{bmatrix}\).
➩ The payoffs for the row player should be independent of the other player's type.
📗 Pure BNE.
📗 Mixed BNE.






Last Updated: November 25, 2025 at 1:46 AM