# Binary Kelly Trainer: Ship Investor Game

A few years ago I made a simple web game to practise applying the Kelly criterion to binary bets. Play Ship Investor here if you want.

You will be presented with a situation where you have to invest in an opportunity that has some probability of suceeeding, and some return-on-investment if it does. Your job is to decide how much you invest in it. Invest too little, and you won’t benefit from the big payout. Invest too much, and you risk going broke quickly if Lady fortune does not smile in your direction.

When this is repeated multiple times, *there is actually a right answer* to each
situation, and you get there by applying the Kelly criterion. The right answer
is that which causes your wealth to grow faster than by any other method.

If you want to compete with a friend, click “link to this scenario” and send them to that page also. This gives you both the same random seed, i.e. the only difference between your runs will be how well you size your bets. Then you can play together and see who is richest after two minutes.

Applying the full Kelly criterion to these situations is cumbersome, but there is a convenient shortcut for binary bets. The fraction of your wealth to invest is “the edge over the odds”, i.e. the expected value of the bet divided by the Hong Kong odds.

Broken down further into \(p\), the probability of success; \(q\), the probability
of failure; and \(B\), the Hong Kong odds corresponding to the
return-on-investment, the fraction to invest is^{1} Many people think this *is*
the Kelly criterion, but as readers of the previous article know, this is just
a simplification for the case of binary bets. The Kelly criterion is more
general than that.

\[ f^\ast = \frac{Bp - q}{B} \]

As a concrete example if I open the Ship Investor, it asks me to invest in a shipment across the Bering strait. This seems to have a 45 % probability of success, and I would get 135 ducats if I invest 52. The profit, then, is 83 ducats, giving Hong Kong odds of \(83/52 \approx 1.6\). According to the equation above, I should invest

\[ f^\ast = \frac{1.6 × 0.45 - 0.55}{1.6} \approx 0.11 \]

i.e. 10 % of my wealth in the shipment.