= Solution
At joint purchase, each player's gain from continuing to buy rather than deviating to $nb$ is $(t-P)/2$. Both actions are <best responses> exactly when this is nonnegative. Therefore
$$
\boxed{(b,b)\text{ is a Nash equilibrium iff }0<P\le\frac n{n+3}.}
$$
At equality both players are indifferent to their individual <unilateral deviations>, which is permitted in a <Nash equilibrium>. The condition is less restrictive than requiring either player's action to be a <weakly dominant strategy>: only the opponent's actual action needs to be considered.
Back to article page
