= Final entrant payoff in a discrete location game
For $k\geq3$, suppose two distinct occupied sites are $\ell<r$, a final firm must choose a different site, customers have value two per site and nearest-firm ties are split equally. A final firm left of both incumbents maximizes its <payoff> uniquely at $\ell-1$, when that site exists, obtaining $2(\ell-1)$. To the right it maximizes uniquely at $r+1$, obtaining $2(k-r)$. At every available interior site it obtains the same <payoff> $r-\ell$. Thus all <best responses> are obtained by comparing these three values, omitting regions with no available site and retaining every tied maximizing choice.
More generally, three distinct locations $a<b<c$ give their left, middle and right firms <payoffs> $a+b-1$, $c-a$ and $2k-b-c+1$. Counting the two midpoint boundaries proves these formulas; the middle firm's demand is the difference between the cumulative demands at the two boundaries. The formulas retain all even/odd midpoint ties automatically.
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