Choose by Probability, not by Expectation

A lot of financial thinking and intuition attempts to phrase decisions and choices in terms of claimed returns or in terms of expected values. For example:

• An ultra greedy gambler may pick the highest claimed winning payoff, independent of win or success probabilities.
• A greedy gambler may pick the highest expected value, independent of variance.
• The famous Sharpe ratio suggests allocations in terms of expected values and variances alone (seemingly without any direct references to underlying probabilities).

The idea is: such reasoning is relatively simple and aggressive. I’d like to argue that it is cleaner to work more over probabilities.

To make things more specific, consider the following simplified investment or gambling situation. We have:

• k possible bets, of which at most one of will pay off.
• The ith outcome “succeeds” with probability pi. Note this implies: Σi pi ≤ 1.
• A bet of w dollars on ith outcome pays a total of w vi dollars on success, and zero otherwise.
• The expected return rate of the ith bet is then ei = pi vi per dollar bet.
• We have filtered the opportunities to consider only expectation favorable opportunities where ei > 1.

Our task is to well-allocate our total holdings between all the possible bets and a hold-out reserve fund.

Let * denote the index of a maximal ei. If we try to maximize expected wealth (which typically is an unsafe practice) we would put all of our money on the *-outcome. Then our expected wealth is e* W (where W is our initial wealth). However with probability 1 - p* we lose everything. If p* is small we are almost certainly ruined.

If we instead try to maximize expected log-wealth (a not completely unnatural set up when we assume we are going to have to bet avoiding ruin many times in our lifetime) the solution is:

• Hold out a h = (1 - Σj pj) / (1 - Σj 1 / vj) faction of your wealth from betting.
• For the ith opportunity bet a pi - h / vi fraction of your wealth.

This is a variation of the Kelly criteria.

Notice the new claimed optimal betting strategy uses the probability of an event happening (the pi) very differently than how it uses the value of the events (the vi). In fact we can not write down the betting allocation in terms of only expected values ei alone.

In fact if the vi are all large we are allocating our wealth not too differently than the probabilities pi. For large claimed payoffs the payoff amounts are almost ignored. Likely profitable events turn out to be much more important than unlikely profitable events. In a sense claimed payoffs and expected values are distractions. Our intended lesson is: sell on expected values and buy on probabilities.

(Note: For our win/lose binomial events the Sharpe ratio is close to the square-root of the odds, so it also becomes a nearly probability alone allocation.)

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Tagged as: decision analysis Kelly Criterion