有笔误，下面已改正

Consider the following normal-form game between two players, P1 and P2, who try to share $1 between them: P1 chooses x 属于 [0.1, 0.9] and P2 chooses either A (accept) or R (reject). Note that they choose simultaneously. The payoff/utility of players are as follows: If P1 chooses x and P2 chooses A, P1’s payoff is (1 − x) and P2’s payoff is x; if P1 chooses x and P2 chooses R, both players get a payoff of 0.
(Hint: x may be interpreted as the share of $1 that P1 proposes for P2. Note that P1 is not allowed to propose x below 0.1 or above 0.9, and that P2 decides to accept (A) or reject (R) without observing the actual proposal x, although P2 understands P1’s strategy correctly in equilibrium. Also note that we omit $ sign.)
Part A.
(1) Find and describe a Nash equilibrium of the game. If there are more than one, find all of them.
(2) Find the reservation utility levels (minimax values) of each player. Draw a digram of the set V of feasible payoff vectors. Describe the set of payoff vectors in V that can be generated as equilibrium payoff vector of a Nash equilibrium of the supergame for sufficiently large δ< 1.
(3) Show that the following strategy profile is a subgame-perfect equilibrium (SPE) of the supergame for all δ < 1: P1 chooses x = 0.1 and P2 chooses A in every period regardless of what happened in previous periods.
(4)Consider v* = (1 − x, x) for an arbitrary x 属于 [0.1, 0.9]. Find and describe a Nash equilibrium (NE) of the supergame that generates the payoff vector v* for sufficiently large δ→ 1. (Your description of NE needs to specify what the players do after every possible history.) For which values of δ, is the NE you describe indeed an equilibrium? Is it a subgame-perfect equilibrium (SPE)?