Interesting HI/LO game dilemma (not poker)

So, the local radio station has a game called HI LOW. Basically, they pick a random (or is it really random?) number between 100 and 1000, then 15 callers get to call in. Each one guesses the number, and if they get it right, they win that ammount in dollars. If they get it wrong (as is likely the case), they are told that the true number is either HIGHER or LOWER than the guess.

This problem occurred to me when I heard the first caller guess 550. At first, this seems like a reasonable guess --but only if this caller were allowed to make all 15 guesses himself. Knowing that each caller only gets one guess, then mathematically it would be correct for the first person to guess 1000, the 2nd person to guess 999, etc. However, although this is the optimal strategy for each individual caller, it means that as a whole the callers will only win money 15/900 of the time. If they used correct strategy as a whole, they could beat the system and win money every time (by guessing the middle of the two known numbers that it is below and higher than). So, either the first caller is an altruist or doesn't understand game theory. Or perhaps I'm missing something. Either way, I thought it was interesting to see that the optimal strategy for each individual caller ended up being the worst strategy cumulatively (or exactly what the radio station would want you to do).

Discuss

-Grant

[Tom1975]

If the number is truly random, and you only get one guess, why do you say it's mathematically correct for the first caller to guess 1000? All 901 numbers are equally likely. Guessing 1000 does make it harder for someone else to win, but if you only get one guess, making it harder for someone else to win does not increase your odds of winning since you've already had your guess. Therefore, why not help your fellow man and go with 550?

[TomCollins]

Because the EV of it is higher, since $1000 better than $502.

[Tom1975]

Duh, I overlooked the part about the amount they win being the actual number. Now it makes sense.

lets say you are the first person to guess. if you guess 1000, your EV is (1/900)*(1000). if you guess 550, your EV drops too: (1/900)*(550). Basically,do you want a 1 in 900 chance at 1000 or a 1 in 900 chance at 550?

i do however like your instinct to help your fellow man /images/graemlins/smile.gif

-grant

[DavidC]

[ QUOTE ]
Because the EV of it is higher, since $1000 better than $502.

[/ QUOTE ]

Brilliant!

Absolutely brilliant.

I love it! /images/graemlins/smile.gif

[AaronBrown]

You're right of course, unless you think it's possible for me or a friend to get one of the subsequent guesses. That's not improbable, my guess is the number of people who try hard to get into these contests is pretty small.

But if I had the first guess, and only one guess, I wouldn't guess $1,000. I don't think the number selection is random, unless they play this game a lot and reveal the number afterwards. My guess is they avoid round numbers, and tend to keep it between $600 and $850 to get a reasonable number of Highs and Lows, but also avoid the loss of suspense that occurs if people find out the answer is under $200. So I'd go with $783 or something like that.