**undersupervised**asks

you're in a tournament with x people (lets say x is at least 10), assume no one is colluding and/or the amount of collusion is trivial. each player is allowed to make a single bet that has a payout of y for 1, probability of winning 1/y. each player can (secretly) choose any y for their bet. the winner of the tournament is the player with the most money after the bets have been resolved.

what's the best strategy?

I wrote some code to solve discrete versions of this problem... unfortunately it doesn't scale well to large numbers of players and/or bets yet. I assumed that ties count as fractional wins for each involved player, and that ties may occur both if multiple people make the same bet, and win, or if every bets with y>1 and all lose. I also assumed that there is no collusion--- that all your opponents are playing the same mixed strategy.

Here's a result for a 5-player game with bets every half-increment from 1 to 10. (2000 fictitious play iterations, nemesis strategy earns about 0.2002 of a game, or 0.1%.)

y frequency 1.0: 0.069 1.5: 0.081 2.0: 0.088 2.5: 0.096 3.0: 0.099 3.5: 0.106 4.0: 0.107 4.5: 0.116 5.0: 0.114 5.5: 0.121 6.0: 0.000 6.5: 0.000 ... 10.0: 0.000

I didn't expect to see any bets above y=5, but y=5.5 is actually the most common. From earlier iterations I also expected to see a bimodal distribution, but that seems to have mostly smoothed out.

10 player games, integer bets 1 through 10, (1280 FP iterations, nemesis strategy earns 0.10009, again about 0.1%.)

y frequency 1: 0.075 2: 0.094 3: 0.087 4: 0.107 5: 0.091 6: 0.115 7: 0.091 8: 0.122 9: 0.093 10: 0.126

This one is a little stranger; could be the result of fewer iterations, coarser granularity, or a cap at 10 instead of 10.5 or 11.

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