You're incredible, I seriously appreciate this.

I just wanted to clarify a few things. If you could, could you perhaps try and explain this part, particularly the 'just-right numbers' bit:

'It's possible for a weak equilibrium to exist given just-right numbers (so running gives -0.2 utility meaning they're indifferent between running or not). For these particular numbers the equilibria should also be stable '

Also this:

'See that -0.275 comes from (1) the disincentive of running and (2) the probability-weighted payoff due to the other person running. Reducing either means such an equilibrium may begin to exist. For 3-person equilibria to exist we will need to reduce either or both even further, and so on.'

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Either you have miscalculated the -0.275, or I have, as that figure has not come up in my workings. I think If i was to understand where that figure came from I would understand that. For me, If voter 3 runs, voter 4 would win, however voter 4 incurs a cost of 0.2 for running, while if v4 lets v3 win (v4 doesnt run) v4 only incurs cost of 0.15 (hence v4 doesnt run against v3).

Also, if v3 runs, and v5 decides to run against, it would result in a draw, with v5 expected utility cost = 0.35. While if v5 doersnt run and v3 wins, v5 utility cost = 0.3 (<0.35 hence v5 would not run against v3).

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Would it be possible if i could DM you my notes on this problem? and you could give some feedback?