Bayesian Nash Equilibrium -  Normal-Form Representation

Let player  i's  possible payoff functions  be  represented  byui(a₁,  ..  .,  a_n;  t₁)  , where t, is called player i's  type and belongs to a set of possible types (or type space) T_i.  Each  type t_i, corresponds to a different payoff function that player  i might have. Given  the definition  of  player's type, saying that player  i  know her  own payoff function is equivalent to saying that player i knows her type. Likewise, saying that player i may be uncertain about the other players' payoff functions is equivalent to saying that player i may be  uncertain about the types of other players, denoted by t-i = (t₁,  ...,  t_{i - 1}, t_{i + 1},  ...,  t_n). We use  t_i,  to denote the set of all possible values of t_i,  and we use the probability distribution  p_i(t - i | t_i) to denote player  i's  belief  about the other players'  types t_i, given player  1's knowledge of her own type, t,.

Joining the new concepts of  types and beliefs with the familiar elements of the normal-form  representation  of a  static game of complete information yields the normal-form representation of a static Bayesian game.  

Definition: The  normal-form representation  of  an  n-player static Bayesian game  specifies the players' action spaces  A₁,  ..., A_n,  their  type  spaces T_I  ,...,  T_n,  their beliefs p₁,  ...,  p_n,  and their payoff functions u₁,  ...,  u_n.  Player  i's .type,  t_i,  is  privately known  by  player  i,  determines player i's payoff function, u_i(a₁,  .  . .  ,  a_n;  t_i)  ,  and  is a member  of  the set  of  possible types, T,. Player  i's  belief  p_i(t  - i |t_i) describes  i's'uncertainty  about  the  n-1  other players'  possible types,  t.,,  given  i's  own  type,  ti. We  denote  this  game  by G = {A₁,  ...,  A_n; T₁,  ...,  T_n; P₁,  ...,  p_n; U₁,  ...,  u_n) . Following Harsanyi  (1967),  we  will assumes that the timing  of a static Bayesian game is as follows:

1)  Nature draws a type vectort = (t_i,  ...,  t_n)  ,  where t, is drawn form  the set of possible type vector T_i.
2) Nature reveals ti  to player i  but not to any other player.

3)  The  players simultaneously choose actions, player  i choosing  a, from  the feasible set Ai.
4)  Payoffs  u_i(a_i,  . . .  ,  a_n;  t_i)  are received.