[I’m not completely sure EDT can’t do better than this, so corrections with even more elaborate schemes encouraged]
I blindfold myself, weigh two random boxes, then weigh the other two random boxes. I pick the box pair which weighs the least then randomly select between those two. If no weight difference then select randomly. This should net you the maximum amount of $301 if the hosts naively compete against each other as you describe in your scenario (i.e. competing against each other by putting more money in boxes just to arrive at the same 25% equilibrium without any sort of cooperation between them).
Hosts are incentivized to put the maximum amount of money in each other box because if only one Host is putting money in the other boxes they guarantee themselves to be in that least heavy pair (total weight of $202 in pairs without their box and $102 in the pair with their box). If 3 of the Hosts are putting money in the other boxes but 1 Host isn’t, he’s screwing himself because his box will never be the least heavy pair (total weight of $502 in the pair with their box and only $402 in the pair with the other two boxes).
Are you sure at the critical point in the plan EDT really would choose to take randomly from the lighter pair than the heavier pair? She’s already updated from knowing the weights of the pairs, and surely a random box from the more heavy pair has more money in expectation than a random box from the less heavy pair, the expected value of it is just half the total weight? If it was a tie (as it certainly will be) it wouldn’t matter. If there’s not a tie somehow one Host made an impossible mistake: if she chooses from the lighter she can expect the Hosts mistake was not putting money in since that would have been optimal (so the boxes have 301, 301, 301, 201, and choosing from the lighter has expected value 251), but if she chooses from the heavier the Hosts mistake was putting money in when it shouldn’t have (so the boxes weigh 101, 101, 101, 1), and choosing from the heavier guarantees 101, which would be less? Actually okay yeah I’m persuaded that this works. I imagined when I first wrote this that weighing a group of boxes lets you infer the total value, so she’d defect on the plan and choose from the heavier pair expecting more returns that way, but so long as she only knows which pair of boxes is heavier (a comparative weighing) instead of how much each pair of boxes actually weighs exactly (from which she would infer the amount on money in each pair total) she can justify choosing the lighter and get 301, I think?
I blindfold myself, weigh two random boxes, then weigh the other two random boxes. I pick the box pair which weighs the least then randomly select between those two. If no weight difference then select randomly. This should net you the maximum amount of $301 if the hosts naively compete against each other as you describe in your scenario (i.e. competing against each other by putting more money in boxes just to arrive at the same 25% equilibrium without any sort of cooperation between them).
Hosts are incentivized to put the maximum amount of money in each other box because if only one Host is putting money in the other boxes they guarantee themselves to be in that least heavy pair (total weight of $202 in pairs without their box and $102 in the pair with their box). If 3 of the Hosts are putting money in the other boxes but 1 Host isn’t, he’s screwing himself because his box will never be the least heavy pair (total weight of $502 in the pair with their box and only $402 in the pair with the other two boxes).
Are you sure at the critical point in the plan EDT really would choose to take randomly from the lighter pair than the heavier pair? She’s already updated from knowing the weights of the pairs, and surely a random box from the more heavy pair has more money in expectation than a random box from the less heavy pair, the expected value of it is just half the total weight?
If it was a tie (as it certainly will be) it wouldn’t matter. If there’s not a tie somehow one Host made an impossible mistake: if she chooses from the lighter she can expect the Hosts mistake was not putting money in since that would have been optimal (so the boxes have 301, 301, 301, 201, and choosing from the lighter has expected value 251), but if she chooses from the heavier the Hosts mistake was putting money in when it shouldn’t have (so the boxes weigh 101, 101, 101, 1), and choosing from the heavier guarantees 101, which would be less?
Actually okay yeah I’m persuaded that this works. I imagined when I first wrote this that weighing a group of boxes lets you infer the total value, so she’d defect on the plan and choose from the heavier pair expecting more returns that way, but so long as she only knows which pair of boxes is heavier (a comparative weighing) instead of how much each pair of boxes actually weighs exactly (from which she would infer the amount on money in each pair total) she can justify choosing the lighter and get 301, I think?