Two players, player 1 and player 2 simultaneously choose whether to be offensive (A) or passive (P) in a pizza sharing game. If they fight, they both suffer injuries that cost player 1 c1 and player 2 c2. There is only one fight if and only if the two decide to be aggressive. If both are aggressive or passive, they share the pizza. If one is aggressive, this one stays with the pizza. Everyone values pizza in Z Therefore this game can be represented by the

following matrix:

Player 2

A P

Player 1 A (Z/2 c1, Z/2 c2) (Z, 0)

P (0, Z) (Z/2, Z/2)

It is common knowledge that c1 Z / 2 but that c2 is 0 (player 2 is very Rough) strictly greater than Z / 2 (player 2 is Soft). Player 2 knows c2 while Player 1 believes that c2> Z / 2 with probability q (0, 1) and that c2 = 0 with probability 1 q. All of this is common knowledge