Let’s generalize from last time to betting on a subset of mutually exclusive events `E_1, ... E_n`

. Event `E_i`

occurs with (true) probability `p_i`

, and the market thinks that it occurs with (possibly untrue) probability `l_i`

, and thus a long contract costs `l_i`

. If we invest `d_i`

dollars in `E_i`

occurring, we get `d_i/l_i`

shares. These shares resolve to being worth `$1`

if `E_i`

occurs, and thus our `d_i/l_i`

shares resolve to be worth `d_i/l_i`

dollars.

Our total cost is `d = d_1 + d_2 + ... + d_n`

, so in net, if `E_i`

occurs, our bankroll changes from (wlog) `$1`

to `$1 + d_i/l_i - d`

. Knowing this, we can write out the expected log value:

The Kelly-Optimal bet is the set of values that maximize this expression, subject to the constraint that .

Let’s try this out with a numerical solver in Excel:

Here we’re using solver to find the value of the yellow cells that maximizes `E(log(X))`

. Excel is telling us to put 22% of our bankroll into Event 2 and 38% of our bankroll into Event 3. Notice that Event 3 has a 50% chance of occurring, though the market believes the chance is only 30%. Using the formula from our last post, we know that if we could bet in this contract alone, Kelly would have us wager (50% - 30%)/70% = 28.6 of our payroll. In this case, having more contracts allows us to increase our return.