= Solution
\b[Neither asymmetric purchase outcome is a <Nash equilibrium> for any permitted $P>0,n>0$.] At $(b,nb)$, X's chosen purchase requires $P\le2t/3$, but Y's chosen nonpurchase, against X buying, requires $P\ge t$. These requirements are incompatible because $2t/3<t$.
At $(nb,b)$, Y's chosen purchase requires $P\le t/3$, whereas X's nonpurchase against a buying Y requires $P\ge t$. Again the inequalities are incompatible. Thus every <Nash equilibrium> in <pure strategies> has both players taking the same action; the conclusion is stronger than merely observing symmetry in their available actions, since their original market shares differ.
Back to article page
