With the "right" scoring function, it is possible that $best(\alpha_{i}) = p-1$: Suppose our winner is next-to-last in every race, that each of the other racers is last in at least one race, and the scoring function awards $100^{p}-k+1$ (or some other large enough number) points for position $k=1,\ldots, p-1$ and $0$ points for position $p$. (EDIT: Also suppose $n \geq p-1$.) The scoring function gives roughly equal awards to the racers who don't come in last and severely penalizes a racer who comes in last. Our consistently next-to-last racer is the overall winner since he or she is the only one who never comes in last.
With the "right" scoring function, it is possible that $best(\alpha_{i}) = p-1$: Suppose our winner is next-to-last in every race, that each of the other racers is last in at least one race, and the scoring function awards $100^{p}-k+1$ (or some other large enough number) points for position $k=1,\ldots, p-1$ and $0$ points for position $p$. The scoring function gives roughly equal awards to the racers who don't come in last and severely penalizes a racer who comes in last. Our consistently next-to-last racer is the overall winner since he or she is the only one who never comes in last.