D: The average outcome over a large number of trials.

Suppose you were offered a game of triple or nothing. In this game you get a choice of

  1. You get twenty dollars; or
  2. You throw a coin, and if it is heads you get sixty dollars, and if it is tails you get nothing.
Which would you choose?

Common responses to this question include

  1. Take the twenty dollars because it is a certainty, and a certainty is always greater that a probability, therefore it is the better choice.
  2. Take the twenty dollars, because the other option is gambling, which is bad.
  3. The question can't be answered because you can't equate the two.
  4. Take the toss because on average it will give you a return of $30.
  5. Take option 2, option 1 has an expectation value of $20, and option 2 has an expectation value of $30.

Argument 1 is an odd argument, and reflects an inability to deal with fuzziness, and need to classify everything as true or false and work from there. People who use argument 1 normally have serious personal issues to resolve.

Argument 2 reflects a totally risk averse attitude, and usually comes with a denial that that same person takes risks in everyday life.
A: 'Every time you drive through an intersection, every time you board an airplane you take risks. The risks are small, but still significant.'
B: 'That is true, but I always minimize any risk'.
A: 'No you don't - your risk will be minimized by choosing not to fly, but to stay home; by choosing not to go out visiting at night, but staying home, you choose not to do this, you choose to take those risks because you regard a certain level of risk as acceptable.'
B: 'But that's just impractical. You just can't live like that'.
A: 'No - you can live like that. You won't be very happy doing it, but you can. You choose not to. You weigh your happiness against your risk, and choose the risk because it is small'.

These people also normally have serious personal issues to resolve.

Argument 3 reflects denial of the practicalities of everyday life. The simple fact is that, in life we have to make decisions. We can refuse to play the game 'what would you do if ..', but we can't avoid these decisions in real life. In real life we have to make a decision - do we play the game or not.

At some point all of us have to make some (or all) of these decisions. When they come up conversation and thought-experiments, they can be avoided, in real life they cannot. At every point in our lives there are many possible choices, and we have to choose between them.

While in everyday conversation we may say 'I had no choice', this is in fact rarely true. If someone holds a gun to your head and says 'give me your money', you in fact do have a choice - give them the money or be shot. Most of us would conclude immediately that giving up the money is the preferable choice. When we say 'I had no choice' what we really mean is that the decision was easy - that was so much better than the other it involved very little thought.

Argument 4 and 5 are variations of the same argument (informal and formal). The reasoning is: Expectation value is defined as the sum of the products of the outcomes and their probabilities.

In choice 1, there is only one possible outcome, hence it has a probability of 1.0, and the amount you get in that case is $20, hence the expectation value is 

EV(1) = 1.0 * $20.00
      = $20.00
which is saying that over a large number of trials you can expect to average getting $20.00 per time.

In choices 2, there are two outcomes, each with a probability of 0.5 (or 50%), one of which returns zero, and the other which returns $50.00. 

EV(2) = 0.5 * 0.00 + 0.5 * $60.00
      = $30.00
which is saying that over a large number of trials you can expect to average getting $30.00 per time.

The philosophy of expectation value is that you always choose the option with the highest expectation value, in this case option 2. However it is useful that utility, not necessarily money is the good which most people are trying to maximise, and the relationship is usually not linear.

See