Strategy question for radio game

[bozlax]

Ok, local radio station is offering the following game: they have an unknown amount of money in an "account" for you. The first person to correctly guess how much money (dollars and cents) is in their account gets the money. Everybody's amount is theoretically different. They have throttled it so you can only guess a certain number of times per day. Each time you guess incorrectly, they tell you if your amount is higher or lower than your guess.

If I start out by guessing $10,000.00 and am told my number is lower, what should my next guess be?

The most certain way to do it is to just cut it in half each time, i.e. my next guess is $5,000.00, the next one after that is $5,000.00 plus or minus $2,500.00, next is the result plus or minus $1,250.00, etc.

However, if you guess, say top 20% (so your next guess would be $8,000.00), as long as your guess lower than the actual number, you've reduced the number of guesses necessary to find the actual number, and you're ahead in the race with the rest of the contestants. By the bye, you have no way of knowing where in the range your actual number falls, in theory; in reality, I doubt they'd put it below, say, $2,000, for fear of irritating the eventual winner.

What do people think the best strategy is, and how do you quantify it?

[maddog2030]

My friend was playing this game and we tried to figure out what would be the best strategy. The number I think tends to be around $1,000-$1,500 +/- $500. She played for a few weeks and then gave up I think, but the winner during the time was $1400.

You can take chances but your smallest average number of guesses will always be cutting it in half. But perhaps you want to take a few chances early and not waste your time if you were off with those (since I'm sure there will be others employing the "correct" strategy, and you need to get a heads up on them). It's better to take chances early rather than later. If you get lucky early, then go back to the proper strategy. If you get unlucky, you can probably quit because there will be those players who are way ahead of you.

I think the "correct" play depends on the strategies of the other players. I'd imagine a "75/25 split early and then graduating to 50/50 guessing as you get closer" would be decent strategy against a mix of people who are doing 50/50 everytime and those who are randomly guessing. It's just a hunch though.

Edit: The more I think about this, the more I think I'm wrong.

[PairTheBoard]

I have no idea. But I can't help making the observation that if you make the $8,000 guess and get lucky you are getting lucky on a higher amount possibly making up in EV what you're giving away in effeciency. Also, you may have little chance to win with the slow but sure 50% method but have a good chance to win with a 10% or 20% method that gives you a chance to get lucky and narrow it down fast.

It might make a difference knowing how many people are playing. It also seems that to formally derive a "best strategy" you would have to make some kind of assumptions about the probabilty distribution for the account values.

PairTheBoard

[AaronBrown]

maddog2030 and PairTheBoard have it right. To get a precise solution, as PairTheBoard says, we need a probability distribution of the amounts and some assumption about the number and strategies of the other players.

Let's say, for example, that the amounts are equally likely to be any number from $0.01 to $10,000.00. That means it takes 20 guesses to guarantee the hit. However, someone is going to get lucky and guess the number earlier. Let's say you have an expected 10 guesses. Rather than trying to model the exact strategy of all the players, and figuring out that you have 90% chance of winning if you get it in 8 guesses, 75% if you get it in 9 and so on; let's just assume you win if you get it in 10 guesses or less, and not otherwise. We can make it more complicated later.

With 10 guesses to work with, you can pick 1,023 numbers out of the 1,000,000 possible amounts. If one of these is your number, you win, otherwise you lose. It obviously makes sense to pick the 1,023 highest amounts, since these give you the most if you do win. So your first guess would be $9,994.89. Your next guess would be higher or lower by $5.11, then $2.55 and so on.

I know this seems crazy, it only wins if your number is $9,989.78 to $10,000.00. But any strategy only works for 1,023 amounts. 1-2-3-4-5-6 is as likely to win the lottery as 7-19-24-31-41-47.

Now back to the assumption that you must win in 10 guesses or lose. Obviously, you don't know the exact number of guesses you'll have before someone wins. If it were a higher number, you'd try for a larger range. If it were a lower number, you'd shrink your range.

The problem is not symmetrical. If you think you have 10 guesses and you really have 11, you get essentially no advantage from your last guess, it's just a random shot, with no information other than the amount is less that $9,989.78. But if you think you have 10 and you really have 9, you have the same chance of winning as if you picked the optimal solution for 9, you just win an average of $5.11 less if you do win.

Therefore, even if the expected number of guesses is 10, you might be wise to assume 13 or more. Underestimating is more costly than overestimating.

The other unrealistic assumption is that the amounts are uniformly distributed. More likely, there's an average amount, say $1,500.00 and amounts get less likely on either side of it. Moreover, it's a good guess it's not a round amount, radio stations love to say "one thousand six hundred fifty eight dollars and thirty two cents!" It makes it all sound harder, and more fun.

What that means is you want to pick a range near the most likely numbers, but on the high side. If you think it's $1,500 plus or minus $500, maybe concentrate your efforts from $1,800 to $1,850. I'd rule out the round numbers and just go for the uneven ones.

I do think it makes sense to pick a fairly small range and just guess that your number is in it; then systematically search within the range. It doesn't reduce your odds versus a broader but coarser search, and if you do win, you'll win more money. It does make sense to search in the region with the highest product of probability and amount.

[maddog2030]

[ QUOTE ]
My friend was playing this game and we tried to figure out what would be the best strategy. The number I think tends to be around $1,000-$1,500 +/- $500. She played for a few weeks and then gave up I think, but the winner during the time was $1400.

[/ QUOTE ]

I'd definitely start out around this range, or try to find out what the previous number range has been. I forget how many guesses it took for that 1 person I heard about to win, but I remember it was like 3 guesses or so under the "divide by 2" approach given you start off somewhere in the ballpark.

Note: when I said the prize was $1400 that was a round number. It was like $1413.83 or something random. They gotta make this thing last as long as possible.

I vaguely remember ~$800 also being won, but don't hold me to it. If you google I'm sure you'll find something that should be useful.