 Solution

Write $P=999/1000$. Ifthen both players prefer buying against a buyer, since $(t-P)/2>0$. Both prefer not buying against a nonbuyer: $2t/3<2/3<P$ and $t/3<1/3<P$. These strict opposite best responses give the interior crossing of a coordination game.

An explicit fully mixed Nash equilibrium follows from indifference. Let $p$ be X's probability of buying and $q$ Y's. X is indifferent whenand Y is indifferent whenSolving givesBoth numerators and denominators are positive. For $p<1$ the required inequality is $2P-2t/3<P+t/3$, equivalent to $P<t$; the same comparison for $q<1$ again reduces to $P<t$. Thus $p,q\in (0,1)$, and the equations make both actions best responses for each player. By linearity of expected payoff, neither player can profit by any other mixed strategy. At $n=2997$ these formulas reach $p=q=1$ and are not fully mixed; the strict threshold matters.