agent-form game

Definition

Let $\Gamma = (N, (T_i)_{i \in N}, p, S, (s_t)_{t \in T})$ be a game with incomplete information. The agent-form game $\hat{\Gamma}$ corresponding to $\Gamma$ is the following game in strategic-form:

• set of players $\cup_{i \in N} T_i$: every type of each player in $\Gamma$ is a player in $\Gamma$
• set of pure strategies of player $t_i$ in $\hat{\Gamma}$ is $A_i(t_i)$, set of available actions of that type in the game $\hat{\Gamma}$.
• payoff function $\hat{u}_{t_i}$ of player $t_i$ in $\hat{\Gamma}$ is given by$$\hat{u}_{t_i}(a) := \sum_{t_{-i} \in T_{-i}} p(t_{-i} \vert t_i) u_i(t_i, t_{-i}; (a_j(t_j))_{j \in N})$$ where $a = (a_j(t_j))_{j \in N, t_j \in T_j}$ denotes vector of actions of all players in $\hat{\Gamma}$.

See also

• Bayesian equilibrium

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 354.