Each month CodeChef runs a coding competition in which I try to solve at least two problems. Usually the easiest two are simple and can be written and verified in a few hours each.
The problem that had me a bit stuck this month was one involving probability. I haven’t done much work with probability since college (over 10 years ago), so I had to brush up on some ideas.
The outline of the problem (with an example):
1. Two teams are playing each other, team A and team B.
2. The probability that team A wins is pX. (For example, 0.51, meaning a 51% chance that team A wins.)
3. The probability that team B wins is 1 – pX.
4. We will always bet 10,000 dollars between the two teams.
5. We can bet on both teams at once.
6. If a team wins, then the money made is calculated by (2 * (1 – pX)) * x, where “x” is the amount we bet. For example, we bet 6,000 on team A, who has a 51% chance of winning … if they win, we make (2 * (1 – 0.51)) * 6000, which is 5,880 dollars.
7. If a team loses, then we lose the amount we bet. If we bet 4,000 dollars on team B and they lose, we’ve only lost 4,000 dollars.
The problem is to find the optimal amount to bet on each team so that we maximize our money.
This means that if I bet 1,000 on team A and 9,000 on team B, I make X amount of money. If I change the amount for team A from 1,000 to 2,000 (and subsequently the team B amount from 9,000 to 8,000), then I’ll make Y amount of money. We just keep changing the amounts until we find one that we know is the highest.
Whenever I read the word “optimal” I assume the problem is going to be difficult. Specifically with this problem I thought it was going to mean I would come up with a function, f(x), where I would have to find a maximum – or a place where the slope was 0. It made me think the problem was harder than it was.
I tried to do some math based on the problem details (http://www.codechef.com/JUNE13/problems/PREDICT) and could not seem to generate the example shown.
This lead me to Google around for betting formulas which lead me to the Kelly Criterion – http://en.wikipedia.org/wiki/Kelly_criterion. Aha! Complex math and formulas! I’ve done CodeChef problems before where a formula was at the heart of a problem, so I assumed this must be my answer.
Long story short, the Kelly Criterion was a dead end. I could never make it work given my variables, so I assumed it didn’t match my problem space.
The real break through came when I sat down and went slowly and thought through the problem. I had gone through the problem before, but this time I tried not to skip steps.
The example on the website had two inputs and one output. Eventually I worked through their example by hand so that I could re-produce their result. The next step was to generalize the formula. This is the result:
p = 0.51 x = 6000 total = 10000 result = ((total + ((2 * (1 - p)) * x) - (total - x)) * p) + ((total + (2 * (1 - (1 - p))) * (total - x) - (total - (total - x))) * (1 - p))
When I printed the result I got 10018, which matched perfectly.
I decided to use Python to solve this problem because it’s quick and easy to get going and several people in the comments on the site had mentioned that Python gave easy/accurate floating point accuracy which was important.
Now that I had a formula that worked in one case, it was time to graph some general results. I just ran the formula through a loop and graphed the output. This is what I got:
It’s linear. I couldn’t believe it, so I picked some other probability for team A and ran my test again. I thought I’d see a curve somehow, but got this:
Not a curve. I suppose it shouldn’t surprise me since the formula earlier shows no possibility of being a curve.
Since it was linear, this meant that based on the percentage of winning, I would either bet 0 dollars or all 10,000 on team A. Then it is simply a matter of plugging that into the formula to find the optimal amount.
This took me a lot longer than it should. Each time I struggle with a problem like this I learn to take my time and patiently work through a problem. I also learn to not make assumptions and follow rabbit trails.