it's been a while since i watched the open yale Game Theory lectures and so the terminology isn't fresh in my head but I think you'll understand if I describe what I'm thinking?
On the topic of recursion, one of the games that was discussed was the case of two passengers who both lost their luggage at the same time with the same luggage contents, say they both had an identical vase or something. The airline asks them each to write down separately the value of the vase not exceeding $100. The lowest number will be assumed to be the true value of the vase, and the person who wrote it will be given an extra $2 for their honesty and the person who didn't will be penalised $2 for their dishonesty.
If they both write $100 they both get $100. But if one of them writes $99 and the other writes $100 then the first one will get $101 and the second will get $98, which seems better, but the second one, knowing the other guy will write $99 then has an incentive to write $98, and by recursion it goes all the way down to a nash equilibrium of them both writing $1. Which is terrible. Obviously $100 is better than $1 and they should prefer to write $100 because who cares about the extra $1. I would rather receive money in the range $98-$101 depending on what the other guy does than to receive $1 because logic. $1 is the right answer but it's the dumb answer.
We're not exactly in that situation because we can consult with each other beforehand. My girlfriend and I have agreed to co-operate if we're ever in a prisoner's dilemma which breaks it.
So maybe the nash equilibrium if we all choose our best strategy is coinflip. But maybe there's a broader a solution that, while still not our preferred strategy, if adopted is overall more preferable to each of us than the coinflip solution? If everybody's second best option leads to a higher winchance for the group.
I mean, from where I sit I would rather convince you both to vote michael than coinflip but will accept coinflip if it comes down to it.
