The Kelly Criterion and a Sure Thing

The Kelly Criterion is an alternative to standard utility theory, which seeks to maximize expected utility. Instead, the Kelly Criterion seeks to maximize expected growth. That is, if we start out with an initial bankroll $B_0$, we seek to maximize $\mathrm{E}[g(t)]$, where $B_t = B_0\cdot e^{g(t)}$.
As a simple example, consider the following choice. We can have a sure $3000, or we can take the gamble of a $\frac{4}{5}$ chance of $4000 and a $\frac{1}{5}$ chance of $0. What does Kelly say?
Assume we have a current bankroll of $B_0$. After the first choice we have $B_1 = B_0+3000$, which we can write as \[\mathrm{E}[g(1)] = \log\left(\frac{B_0+3000}{B_0}\right);\]for the second choice we have \[\mathrm{E}[g(1)] = \frac{4}{5} \log\left(\frac{B_0+4000}{B_0}\right).\]And so we want to compare $\log\left(\frac{B_0+3000}{B_0}\right)$ and $\frac{4}{5} \log\left(\frac{B_0+4000}{B_0}\right)$.
Exponentiating, we're looking for the positive root of \[{\left({B_0+3000}\right)}^5 - {B_0}\cdot {\left({B_0+4000}\right)}^4=0.\]Wolfram Alpha now tells us that we should go with the sure thing if $B_0 < $4942.92$, and take the gamble otherwise.

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Mining the First 3.5 Million California Unclaimed Property Records

As I mentioned in my previous article the state of California has over $6 billion in assets listed in its unclaimed property database . The search interface that California provides is really too simplistic for this type of search, as misspelled names and addresses are both common and no doubt responsible for some of these assets going unclaimed. There is an alternative, however - scrape the entire database and mine it at your leisure using any tools you want. Here's a basic little scraper written in Ruby . It's a slow process, but I've managed to pull about 10% of the full database in the past 24 hours ( 3.5 million out of about 36 million). What does the distribution of these unclaimed assets look like? Among those with non-zero cash reported amounts: Total value - $511 million Median value - $15 Mean value - $157 90th percentile - $182 95th percentile - $398 98th percentile - $1,000 99th percentile - $1,937 99.9th percentile - $14,203 99.99th perc...

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A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem . Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way. Standard notation: probability of A given B is written $ \Pr(A \mid B) $ probability of B is written $ \Pr(B) $ Bayes' Theorem: Using the notation above, Bayes' Theorem can be written: \[ \Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)} \] Let's apply Bayes' Theorem to the Monty Hall problem . If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random. Assume we pick Door 1 an...

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