Harsanyi game with incomplete information

Definition (Harsanyi game with incomplete information)

A Harsanyi game with incomplete information is a vector $(N, (T_i)_{i \in N}, p, S, (s_t)_{t \in \times_{i \in N}T_i})$ where:

• $N$ is a finite set of players
• $T_i$ is a finite set of types for player $i$, for each $i \in N$.
  • $T = \times_{i \in N} T_i$, set of type vectors
• $p \in \Delta(T)$, probability distribution over set of type vectors that satisfy $p(t_i) := \sum_{t_{-i} \in T_{-i}} p(t_i, t_{-i}) > 0$ for every player $i \in N$ and every type $t_i \in T_i$
• $S$, set of states of nature, called state games.
  • Every state of nature $s \in S$ is a vector $s = (N, (A_i)_{i \in N})$, where $A_i$ is a nonempty set of actions of player $i$ and $u_i : \times_{i \in N} A_i \to \mathbb{R}$ is the payoff function of player $i$.
• $s_t = (N, (A_i(t_i))_{i \in N}, (u_i(t))_{i \in N}) \in S$ is the state game for the type vector $t$, for every $t \in T$.
  • Thus, player $i$'s action set in state game $s_t$ depends on his type $t_i$ only, and is independent of the types of the other players.

Definition (Harsanyi model of incomplete information)

Refer to a Harsanyi game as a Harsanyi model of incomplete information when it is analyzed without specifiying the state game.

Notes

• Harsanyi games with incomplete information may be analyzed at the ex ante stage, before players know their types, and at the interim stage, after they have learned what their types are.
• Two types of equilibria can therefore be defined:
  • Nash equilibrium in Harsanyi games, where no player can profit by unilateral deviation before knowing his type, and
  • Bayesian equilibrium, where no player $i$ can profit by deviating at the interim stage, after learning his type $t_i$.

See also

• Aumann model of incomplete information
• Aumann model of incomplete information with beliefs

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 347-349.