Answer Happy

Player 1 and Player 2 are in a contest about who gets an object of value. Each can play "hawk” (i.e. aggressively) or "dove" (i.e. giving up). If one plays hawk while the other plays dove, the player who played hawk gets the object and the other player gets nothing. If both play dove, they split the object equally. If both play hawk, they split the object equally, but each also incurs a cost of fighting equal to 1. Player 1 hawk dove hawk 0.5V-1, 0.5V-1 Player 2 0, V dove V, 0 0.5V, 0.5V
Now suppose that we make this a game of incomplete information: Player 2 knows whether V is 1 or 4, but Player 1 only knows that V is either 1 or 4, with each occurring with 50% chance. What is the unique pure strategy BNE of the game? (hint: you can think of there being two types of Player 2 – one that knows that V=1 and one that knows that V=4. Remember that to describe a BNE you need a strategy for each player-type, e.g. "Player 1 plays hawk, Player 2 plays hawk when V=1 and dove when V=4". To get started with this question, ask yourself what Player 2 would do if he knew that V=4...)