Consider a modified version of the ultimatum game, with envy. There is a pie of size40to be split between Proposer (P) and Responder (R). In the first stage, P offers a share0x1of the pie to R. In the second stage, R accepts or rejects P's offer. If R accepts, the pie is split according to the proposed division, with P receiving(1x)40and R receivingx40. If R rejects, both players receive none of the pie.

But players'payoffsare not equal to how much of the pie they receive. In particular, payoffs in the event of an accepted offer are:

UP=(1x)40sx40

UR=x40s(1x)40

(Payoffs in the event of a rejected offer are 0 for both players.)

Here, the parameters0measures how envious players are, i.e. how much "unhappier" a player is when his opponent receives more pie.

For each field below, please input only a numerical answer in decimal form, rounded to at least 3 decimal places if rounding is needed. Do not input words or symbols (like $ or %) in your answer.

a) Supposes=1.6. In the subgame perfect Nash equilibrium of the game, R's strategy is to accept all offers greater than or equal to ________ (please give the amount of pie, not the share).

b) Supposes=1.6. What is the highest payoff P can attain in any subgame perfect Nash equilibrium of the game?

c) Supposes=0.3What is the highest payoff P can attain in any subgame perfect Nash equilibrium of the game?