Trader Probabilities Derivation And K-L Divergence (Part 1)

For any sports event, we need to come up with a set of probabilities against which to match the current market prices. We do this in order to identify whether or not we have at least one positive expectation in that event. Obviously, the market itself reflects in its prices (and implied probabilities) the combined insights of all those who wish to trade on that specific event. Naturally, we have our own ideas as to the likely contenders but these views may be quite vague.

So, how do we create a coherent set of probabilities that does not stray too far from the implied market probabilities while taking into account our own insights?

To illustrate our approach, let us assume that we have identified a horse-race with five runners that meets our WCMI trading threshold.

In reality, we never know the true win probabilities of individual horses. However, we almost always have some opinions to work with. For example:
* 70% chance that winner will come from one of three horses - Alpha, Bravo, or Charlie;
* 5% edge on implied market probability for Charlie, and
* All horses have at least 7% chance of winning.

First, anchoring our initial range [$H8:$H12] (set as starting values) to the set of implied market probabilities, we can use Excel Solver and Kullback-Leibler Divergence calculations to search for the set of coherent win probabilities that minimizes the distance from the set of implied market probabilities while simultaneously satisfying our constraints.

As a sanity check. we can test the solved set of probabilities against our constraints:
* (28% + 23% + 19%) <= 70%;
* ((5.500 * 19%) - 1.000) >= 0.050 (rounding up);
* MIN(28%, 23%, 19%, 23%, 7%) >= 7%.

In sum, we have expertly combined the wisdom of the crowds with our own limited insights to produce a coherent and valid set of win probabilities!

Note that the market odds, trader probabilities, and constraints are not necessarily realistic!