Ponder the following:
Larry, Moe, and Curly are on Jeopardy.
Going into Final Jeopardy:
Larry has $50,000, Moe has $10,000, Curly has $10,000
Larry bets $29,999, Moe bets $10,000, Curly bets $10,000
These bets are ALL RATIONAL and ALL MATTER independent of what the category is. For example, these bets make sense whether the category is THE THREE STOOGES or CIRCUIT LOWER BOUNDS.
Explain why this is.
I'll answer in my next post or in the comments of this one
depending on... not sure what it depends on.
"Contestants who provide the correct response receive the values of their wagers, while contestants who fail to provide the correct response, or to phrase their response in the form of a question (even if correct), have that amount subtracted from their scores."ReplyDelete
"Non-winners receive consolation prizes. Since May 16, 2002, consolation prizes have been $2,000 for the second-place contestant(s) and $1,000 for the third-place contestant."
So basically it's come first or get a fixed consolation prize, meaning Moe/Curly may as well go all in because it doesn't really matter if they lose everything.
Larry on the other hand has a lot to lose, he's all but guaranteed to come back the next day since neither of the other 2 can reach 50k. If Larry doesn't get the question right he'll end on 20,001, which is higher than the other 2 even if they get the answer correct. As he's doing quite well he's confident his chance of getting the answer correct is high enough that it's worth gambling the ~30k
What I don't get is why Moe or Curly don't bet $1 less in the hope that their counterpart is still betting everything. They're unlikely to come first as surely Larry isn't silly enough to bet more than 29,999, so why not put $1 away to be safe and hope that they get second place instead of 3rd.
It's not unknown for people to make mistakes when writing down the amount they wagered, and there's no reason for Moe and Curly not to bet everything in case Larry accidentally bets $30k or more.ReplyDelete
Sam is correct in pointing out WHY this scenario is puzzling.ReplyDelete
Anon 11:49--- I am assuming everyone plays completely rationally.
I"ll post answer tommorow (Wed July 29)
Larry has a forced win. Judging by Sam's notes above, Moe and Curly are individually incentivized to win second place (or tie at second place). Regardless of the probability distribution of whether Moe and Curly answer correctly, Moe is most likely to win/tie second place if he matches Curly's bet. By symmetry, Curly wants to match Moe's bet. As such, Moe and Curly's bets appear to be rational.ReplyDelete
Is a $10K bet by Moe/Curly somehow more rational than a $0 bet? Also, it seems that Larry's bet should depend on the category, unless his personal objective is to maximize his possible winnings subject to wining first place (instead of maximizing expected winnings).