Consider the following multistage game. Player 1 first has to choose how to divide $2 between himself and player 2 (with only non-negative integer divisions being possible). Both players observe the division, and then they play the simultaneous-move game with the dollar payoffs shown below.
\begin{array} x & A & B & C\\ U& x,x & 0,0 & -2,-2\\ D&0,0 & 1,1 & -2,-2\\ \end{array}
Assume that each player has utility equal to the sum of the expected number of dollars he receives in the divide-the-two-dollars game and the expected dollar payoff he receives in the second-stage game.
(a) Show that for any x the game has a Nash equilibrium in which player 1 chooses to give both dollars to player 2 in the initial divide-the-two-dollars game. (b) For what values of x will the game have a unique subgame-perfect equilibrium? (c) For what values of x is there a subgame-perfect equilibrium in which player 1 gives both dollars to player 2 in the initial divide-the-two-dollars game?