We can make choices based on expected values and standard deviations.

Suppose that you have two urns (jars) containing colored balls and that certain payoffs are associated with the colors.

Suppose that yellow balls are worth $100,

blue balls are worth $ 20,

and red balls are worth -$ 50.

Designed by Gary Simon, September 2007.

Y = $100, B = $20, R = -$50

If the urn has

40 yellow balls

30 blue balls

30 red balls

the expected value is $31.

You might be willing to spend $15 in development costs to get to select one ball from this urn.

It is possible to compute a standard deviation around the expected value of $31. Here’s how:

Variance = E[ squared difference from $31 ]

= ($100 - $31)2 × 0.40

+ ($ 20 - $31)2 × 0.30

+ ( (-$50) - $31)2 × 0.30

= $2 3,909

Standard deviation = $ $62.523,909

Would you regard this as risky?

You can also compare two jars.

Jar A has

Jar B has

20 yellow balls ($100)

70 blue balls ($ 20)

10 red balls (-$50)

13 yellow balls ($100)

85 blue balls ($ 20)

2 red balls (-$50)

For both jars, the expected value is $29.

…but SD(Jar A) = $41.10 > SD(Jar B) = $29.14.

At times you might be willing to take the larger standard deviation, provided that you also get the larger mean!

Jar C has

expected value = $37, standard deviation = $46.05

30 yellow balls ($100)

60 blue balls ($ 20)

10 red balls (-$50)

Jar D has

expected value = $41, standard deviation = $47.84

35 yellow balls ($100)

55 blue balls ($ 20)

10 red balls (-$50)

In some decisions, you just don’t know enough!

Consider a new urn, with yellow, green, and purple balls.

You know that there are 900 balls in all:

300 balls are yellow

an unknown number are green

an unknown number are purple

(green + purple = 600)

300 yellow ; green + purple = 600

Now select one ball. Which payout scheme would you prefer?

Scheme Yellow Green Purple

A $60 0 0

B 0 $60 0

Write down your answer before proceeding. . . .

Let’s play the game again. Return the ball and mix up everything again.

You may have an opinion about the number of green balls. After seeing the first draw, you should do a Bayesian updating on that opinion. Since the urn is large (900 balls) and since the information is minimal (one draw), we will skip this step just for simplicity.

Now, which of C and D do you prefer?

Scheme Yellow Green Purple

C $60 0 $60

D 0 $60 $60

Write down your answer before proceeding. . . .

Most people choose A over B and then choose D over C.

Why?

Let’s think what this means.

Scheme Yellow Green Purple

A $60 0 0

B 0 $60 0

300 yellow ; green + purple = 600

Choosing A over B seems to be invoking a belief that there are more yellow balls than green balls.

Scheme Yellow Green Purple

C $60 0 $60

D 0 $60 $60

300 yellow ; green + purple = 600

Choosing D over C seems to be invoking a belief that there are more green balls than yellow balls.

It is logically inconsistent to choose A over B and then choose D over C.

A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines. Ralph Waldo Emerson

Many explanations have been proposed as to why this happens.

The choices of A and D are those that eliminate the uncertainty by not knowing the number of green balls.

Scheme Yellow Green Purple

A $60 0 0

B 0 $60 0

Scheme Yellow Green Purple

C $60 0 $60

D 0 $60 $60

The choices of A and D can be thought of as “maximin,” in that they guard against the worst thing that can happen.

You cannot describe these choices as “eliminating uncertainty.” There are two sources of variation here:

The variability regarding the number of green balls.

The variability associated with random sampling.

There is an additional perspective that appears when the choices are displayed next to each other.

Scheme Yellow Green Purple

A $60 0 0

B 0 $60 0

Scheme Yellow Green Purple

C $60 0 $60

D 0 $60 $60

The games are identical, except that the second has been enriched by paying out $60 for the purple balls!

If you prefer A over B, logic demands that you prefer C over D.

This is known now as Ellsberg’s paradox, 1962. There are many variations of it, and it has ignited many arguments.

Daniel Ellsberg is better known for his tangential role in the Watergate crisis of the 1970s than for this wonderful intellectual gem.

Ellsberg’s psychiatrist was the victim of an office break-in. The intruders were suspected of looking for embarrassing information on Ellsberg.