trembling hand for more-than-2-players might imply converging on blender depending on how much you value your friends' lives and don't want them to slip into the blender (loving people is part of the reality (or at least hypergame) that these boxes don't track!)
Trembling hand equilibria and friends' lives in the multiplayer pill game
13 tweets · August 2023 · 42 likes · 0 retweets · read on Twitter
suppose you play this with 10 ppl & a 1% trembling hand rate if everyone tries to play red, there's a 9.1% chance that one of you gets blenderized (0.91% chance it's you). ~0.44% that 2+ of you get blenderized if 100% selfish, unilateral move to blue still sucks (~99% you die)
but unless you value your friends' lives more than your own, then no reason to switch ...unless you factor not just the lives but living with the guilt of having chosen selfishly and survived, which is something most non-psychopaths would have might not outweigh, but it's a lot
anyway, so depending on how to weight that, which is impossible to numericize and will vary dramatically by person, and so on... ...the red option is still probably a trembling hand equilibrium, assuming zero communication
but now I'm pretty sure "everyone chooses blue" is ALSO an equilibrium, and a better one (tho unfortunately it's hard to draw those little toy payoff matrices when there are 10 dimensions) ...hours later, I come up for air to realize I accepted that challenge HARD.
well 10 was a bit too hard but I managed to get 5 going I valued "own life" at 100 and "each friends' life" at 1, mostly so it was easy to see where the numbers went and there's now an equilibrium at 100% blue. it's small because small n.
I did another one with the error rate set to 10%, and interestingly this makes the one where half of the other people pick red/blue switch in valence this is partially a function of the ratio between own & friends' lives
anyway, I have spent way too long on this it was kinda nice to brush up on binomial theorem and game theory though, both of which I studied but hadn't formally worked with in years if you want to try to extend my sheet, it's here: docs.google.com/spreadsheets/d…
but anyway, the case is honestly a lot messier at 5 but again, with an all-red strategy 🔴 - with 10 people there's a ~10% chance of someone dying - with 100 people it's more likely than not that 1-2 die - by 1000, it becomes vanishingly unlikely that all live
ANYWAY going back to feelings, imagine how you'd feel in the scenario where you all pick red and you get lucky and nobody slips, and you all survive (90% likely) like WHEW OKAY WE GOT LUCKY even if it feels like yall made the right choice, it's still a crazy risk to take
then imagine how you'd feel in the scenario where you all pick blue and you all survive (99.9999%+ likely) it'd be like "nice! we did it! of course! we know how to handle this!" everybody knew that if everyone else made the right choice, it would *definitely* work out fine
like once there's a tiny bit of noise, obviously converging on blue is the thing to do
here's another great analysis, which does the math at the limit as N increases, and points out some of the same things I did but more clearly, plus other things
