Game theory : A and B are playing divide the dollar

Question

A and B are playing divide the dollar. Denote A’s strategy by a (the amount he claims) and B’s strategy by b. If they can agree on a division of the dollar (a + b ≤ 1) they walk away with the share they have agreed; if they cannot agree (a + b > 1) they walk away with nothing. Show that any pair of strategies {a , b} such that a + b = 1, a ≥ 0 and b ≥ 0 is an equilibrium.

Answer 1

Suppose we have the strategies $\{a,b\}$ with $a+b=1$ and $a,b>0$. If A changes strategies to $a'>a$, then he gets nothing, which is worse than $a$. If A changes strategies to $a'<a$ instead, then he gets $a'$, which is worse than $a$.