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I don't understand why I would care about the median national voter. Shouldn't political parties focus on winning specific state-level races, both for congress and president? The national popular vote is meaningless.

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Sure, but politicians running for national level positions (i.e. President) run a single campaign, and they have to select a single position set for that campaign (1).

You're right that the game theory is more complicated than it would be in a national popular vote contest, but I suspect that the optimal position isn't all that far off from where it would be in a single nationwide Hotelling game.

(1) Mostly, at least, I know they can of course run different ads and given different speeches in different states, flip flop, etc, but their reputation and messaging is inevitably going to bleed across state lines / time quite a bit.

I mean, it's possible that the optimal position isn't all that far off, but I would need to see that actually demonstrated before I start caring about the national median. It's not like the relevant demographics aren't available for swing voters in swing states - the campaigns already look at those.

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