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Decision Strategies: Beyond Expected Value


Oftentimes when making some kind of uncertain decision, the decision maker will use a measure such as expected value to make that decision.  Imagine the case of a single coin flip where the better pays 5 dollars to play, and gets 2 dollars for heads and 10 dollars for tails.  The expected value of this game is to pay 5 dollars to enter and make 6 dollars, giving an expected one dollar profit.

But this neglects to take into account two core concepts: ruin and windfall.

Ruin and windfall are two sides of the same coin, and represent the expected maximum loss and gain respectively.  As Nassim Taleb puts it often, there is a path dependence to games: losing a million then making making a million is far worse than making a million before losing a million.  There is some amount of loss that is so great that the game is forced to end, and if that amount of loss is expected, then expected value loses is meaningfulness.  Likewise, there is some amount of gain, windfall, that is so large that any rational player would exit the game.

The windfall concept is well captured in the Kelly Criterion:

\(f^{*} = \frac{bp – q}{b} = \frac{p(b + 1) – 1}{b} \)

Where the portion of the bettors bankroll to bet is seen to be proportionate to the magnitude of payoff and the likelihood of the payoff, so even something very unlikely to happen, if the payoff is sufficiently large, is seen as rational to play.  An interesting use case of this is that for the very wealthy, the lottery becomes rational.

In the recent 1.5 billion jackpot powerball, the odds were roughly 1 in 292 million or 0.000000342% chance.  Using the Kelly Criterion that works out to allocation 0.000000208% of the bettors bankroll towards powerball tickets.   So anyone with a bankroll of more than about 9.6 billion would have been prudent to purchase a single $2 powerball ticket, strictly speaking.

Using an expected value only approach, one sees that the expected value of a ticket is about $0.0051, which is far below the $2 cost, making it irrational.

We see that there are three metrics by which to pick strategies, each of which have their own time and place.

Maximize Expected Value

The classical approach (Pascal’s at least), is to maximize the expected value. So given something like a petersburg graph, the aim is to select the subgraph with the highest mean outcome from a large number of game simulations.

Note that a game can take two primary forms: cumulative or single shot. In a cumulative game, the bettor starts with a bankroll and plays over and over until ruin, some winning threshold, or a maximum number of iterations (maybe infinite).  In a single shot game the player can only play the game one time.

Minimize Expected Ruin

The minimal expected ruin strategy (or maxmin) is to maximize the minimum simulated outcome of a game.  So the strategy with the highest minimum outcome across a large number of simulations is determined best.  This is a risk-averse approach.

Maximize Expected Windfall

The maximum expected windfall approach is to select the strategy which has the maximum maximum across a large number of simulations.  This is a particularly useful strategy when the cost to play is low, the chance of ruin for all options is low, and better’s need for smaller more likely reward is low.


Originally posted at willmcginnis.com