Expectancy in Wicked Environments

Expectancy is the chance of the good outcome times its size, minus the chance of the bad outcome times its size. The sign of the result says whether a bet is worth taking. The exact calculation needs published odds: a setting that states what winning and losing mean, so that chances and sizes can be measured. The choices that matter almost never publish them.

A usable estimate still exists inside almost any messy choice. The method for getting it has three steps. Remove every small outcome. Compare only the largest outcomes that remain. Rate those in bands: for size, from imperative down to insignificant; for chance, from guaranteed down to low. Combining the size band with the chance band for each remaining outcome is the whole calculation, and it is small enough for a person to work out mentally.

Core takeaways

• Expectancy is the chance of the good outcome times its size, minus the chance of the bad outcome times its size. A positive result means the bet is worth repeating, and a negative result means the bet should be refused.
• The exact calculation needs a setting that publishes what winning and losing mean. Chess and golf do this. Most real choices do not.
• For a choice without published odds, remove every small outcome, compare only the few largest positives and negatives, and rate those in broad bands of size and chance.
• Search for outcomes before removing any. The main danger is removing a major downside that was never noticed.
• To improve the biggest downside, lower its chance or lower its size. Either change alone can turn the sign from negative to positive.
• A band assigned before the information search is a guess. Where real statistics and explicit win conditions exist, use exact arithmetic there instead of bands.

The calculation when the odds are known

A game that publishes its rules allows the arithmetic to be done with known numbers. Multiply the chance of the good outcome by its size. Multiply the chance of the bad outcome by its size. Subtract the second from the first and read the sign. A positive sign means the bet is worth repeating. A negative sign means the bet should be refused.

An example with published odds: win $10 at 75 percent, lose $50 at 25 percent, over one hundred rounds. The wins add to $750. The losses add to $1,250. Expectancy is −$500. Do not play. Now shrink the loss to $20 and keep the same odds. The wins are still $750 and the losses are $500, so expectancy is +$250. The chance on each side did not change. Only the size of the loss changed, and that was enough to change the sign. The sign depends on chance and size on each side.

Why most choices do not allow the calculation

That arithmetic assumes a setting that publishes what winning and losing mean, so that chances and sizes can be measured. Chess and golf do this. Some other scored games do too. The looser claim that “most sports” do so holds less well once judging, weather, and opponents affect the result.

Outside such games, those numbers are not published. The win and the loss are both undefined. Sometimes every available outcome is a loss, and the choice is the less bad one. There are more variables than a person can multiply mentally. No one computes expectancy over a list like that, and waiting for the perfect calculation means never deciding.

Three steps for a choice without published odds

Using three steps, a person reduces a choice of that kind until the game arithmetic applies again.

First, remove the small outcomes. Any outcome that is low in chance, low in size, or both comes off the list. So does any outcome that is guaranteed but too insignificant to matter. Almost all of any long list is this kind of outcome. Small positives and small negatives roughly cancel each other. Any small outcome that does not cancel can be handled when it happens. Counting how many outcomes were removed adds nothing.

Second, compare only the largest outcomes. After the first step, judge the few most significant positives against the few most significant negatives. The choice is now small enough to fit the chance-times-size comparison.

Third, rate the remaining outcomes in broad bands. Giving a rating like 78 percent against 79.3 percent is false precision, and false precision is the main way this step fails. Size has four bands: imperative, significant, convenient, insignificant. For a risk, imperative means unacceptable. Chance has four bands: guaranteed, very high, somewhat, low. Combining the size band with the chance band for each remaining outcome is the whole calculation.

Search for outcomes before removing any

The search for outcomes comes before any removal. The danger is removing a major downside that was never noticed. A thorough information search goes first, so that no one can drop a harm they never listed. The faster that search runs, the less a good decision costs.

Lower the biggest downside

After the search, work on the biggest downside. Lower its chance or lower its size. Either change alone can change the sign. In the example above, cutting the $50 loss to $20 was a change in size. Cutting a 25 percent chance to 5 percent is a change in chance. Do the same in the other direction for the biggest upside: raise its chance or raise its size.

When a rating is a guess

A rating is only as good as the information behind it. A band assigned after an extensive search is an informed estimate. The same band assigned without a search is a guess with a rating label on it. A band assigned before the search is the signal to stop.

The estimate will sometimes be wrong. Making the estimate is still worth doing. Running a complex choice through these steps produces better decisions on average than skipping them. That claim is the reason to run the steps. It is not a measured trial of these four size labels or these four chance labels.

When to go back to exact arithmetic

Where part of a choice has real statistics and explicit win conditions, stop using bands for that part and do the exact arithmetic. Positional Decisions and Expected Value covers the position-improving move and the downside protection across repeats. That calculation stays in bands until real statistics and explicit win conditions appear.

At the end of the three steps, either the remaining outcomes fit the chance-times-size comparison, or a part with real odds appeared and the bands were dropped there. In both cases the estimate is one a person can actually run. Do not give a rating finer than the setting allows.

How to practice this

1. Before deciding, list every outcome of the choice you can find, good and bad. Run this search quickly but thoroughly. Notice whether a large downside appears that you had not thought of.
2. Remove every outcome that is low in chance, low in size, or both. Also remove guaranteed outcomes too small to matter. Notice that almost all of the list is gone.
3. Set the few largest positives against the few largest negatives. Notice that the choice is now small enough to hold in your head.
4. Rate each remaining outcome’s size as imperative, significant, convenient, or insignificant. Rate its chance as guaranteed, very high, somewhat, or low. Notice any wish to give a finer number such as 78 percent, which is false precision.
5. Take the biggest remaining downside and find a way to lower its chance or its size. Notice whether that one change flips the sign, as cutting the $50 loss to $20 did.
6. Check that every band was assigned after the search, not before. A band assigned before the search is a guess. Where real statistics and explicit win conditions exist, drop the bands there and compute the exact number.

Related pages

• Positional Decisions and Expected Value: the position-improving move and downside protection across repeats. That calculation stays in bands until real odds appear.
• Decision Making: the hub’s operating loop, with its variable-naming and downside steps. The three steps above fit into those steps.
• judging a decision by its process: process versus outcome, after the estimate is made.

Open questions

When a band stays “I don’t know” after the information search, is the next move delay or a no?

Sources

• Expected value as definitional: chance of the good times its size, minus chance of the bad times its size. Any finite-sample bet.
• Tetlock / superforecasting craft: coarse, calibrated probabilities do better than falsely fine ones for many tasks. Bound: not a trial of these four size labels or these four chance labels.
• Hogarth, Lejarraga & Soyer 2015, Current Directions in Psychological Science: a kind learning environment is one whose feedback is accurate, plentiful, and tightly linked to the action; a wicked one has missing, delayed, or misleading feedback. Bound only: that paper measures feedback quality. The wicked environment described above is one where the odds are not published and the win is not defined.