$(1$pt$)$ Suppose two strangers play a single$-$shot game in which player #$1$ can get $$100$ cash on the condition that she must offer to divide it with player #$2$ and if player #$2$ rejects the offer as unfair, then neither player gets anything. There are no other payoffs and no chance to communicate anything but the offer and whether or not it is rejected.

If both players are purely selfish and rational, and they both know that they are both selfish and rationl, how much would player #$1$ get from playing this game?

a$.$ $$50$

b$.$ $$0$ because player #$1$ will probably offer a small amount to player #$2$ and player #$2$ will probably selfishly reject the offer.

c$.$ A little bit less than player #$2,$ but not a lot less.

d$.$ A little bit more than player #$2,$ but not a lot more.

e$.$ Nearly $$100,$ but not quite.