In Back to the Future Part II, Biff Tannen made his fortune betting on outcomes of sporting events. But what if he only knew a team’s final record without knowing the scores of individual matches?
I like the problem I originally posed because, whenever your team wins one more game than it loses (and you're only allowed to bet on your team), you can exactly double your original bankroll. I've never seen an intuitive explanation for why this happens.
For the main problem, after the first game the result of the second game is known, so you would reasonably bet all or nothing. For the extra credit, on the first game you bet x for some x. But then what can you bet on the second game? Only x again or all or nothing? Or an arbitrary amount y?
You can bet arbitrary values (up to how much money you currently have) after each game. These values don't have to be the same from one game to the next.
Do we need to define a specific amount for each game or just outline the strategy? In other words, can the amount for bet 2 be determined or adjusted after game 1 is over?
Can we get clarification on "guarantees you’ll have as much money as possible after both games"? Highest minimum value? The approach for that seems trivial, but that seems like a more natural way to interpret "guarantee" than something like highest expected value.
Aug 25, 2023·edited Aug 25, 2023Liked by Zach Wissner-Gross
I failed to note that you can only bet on Fiddly to win, not on the other team. Less trivial. Although, the problem states that if they win the first game you'll have $100 + x to bet on the second, which you know they'll lose, so you don't bet....
This is a variation of a problem I posed in the _American Mathematical Monthly_: https://www.jstor.org/stable/2589190
Robb T.Koether and John K. Osoinach, Jr. explored this further in "Outwitting the Lying Oracle" (_Mathematics Magazine_, 2005): https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/koether98.pdf
I like the problem I originally posed because, whenever your team wins one more game than it loses (and you're only allowed to bet on your team), you can exactly double your original bankroll. I've never seen an intuitive explanation for why this happens.
I'm horribly amused that JSTOR thinks "Glen Ellyn" is a co-author.
For the main problem, after the first game the result of the second game is known, so you would reasonably bet all or nothing. For the extra credit, on the first game you bet x for some x. But then what can you bet on the second game? Only x again or all or nothing? Or an arbitrary amount y?
You can bet arbitrary values (up to how much money you currently have) after each game. These values don't have to be the same from one game to the next.
This would make sense if Biff used the time travel technology from Twelve Monkeys, where the by-product of time travel is some degree of confusion
Do we need to define a specific amount for each game or just outline the strategy? In other words, can the amount for bet 2 be determined or adjusted after game 1 is over?
In making your wager for Game 2, you know the outcome of Game 1. Your solution should be how much money you have after both games are played.
Can we get clarification on "guarantees you’ll have as much money as possible after both games"? Highest minimum value? The approach for that seems trivial, but that seems like a more natural way to interpret "guarantee" than something like highest expected value.
Highest minimum value is correct. If you find it trivial, then you're ready to move on to the Extra Credit for sure.
I failed to note that you can only bet on Fiddly to win, not on the other team. Less trivial. Although, the problem states that if they win the first game you'll have $100 + x to bet on the second, which you know they'll lose, so you don't bet....
Correct. In that instance, you'd wager nothing.
Is it correct to assume even money odds for each game in the extra credit?
That is correct -- even money odds. I'll clarify!