In the Appendix to Chapter 17, we introduced the Allais Paradox. It went as follows: Suppose there are three closed doors — with $5 million, $1 million and $0 behind them. You are first offered a choice between Gamble 1 (G1) that will reveal the $1 million door with certainty and Gamble 2 (G2) that will open the $5 million door with probability 0.1, the $1 million door with probability 0.89 and the $0 door with probability 0.01. You get to keep whatever is behind the door that is revealed. Then, you are offered the following choice instead: either Gamble 3 (G3) that reveals the $1 million door with probability 0.11 and the $0 door with probability 0.89, or Gamble 4 (G4) that opens the $5 million door with probability 0.1 and the $0 door with probability 0.9.

A: It turns out that most people will pick G1 over G2 and G4 over G3.

(a) Why is this set of choices inconsistent with standard expected utility theory?

(b) Suppose that people use reference-based preferences to evaluate out comes when making their choice between gambles. Why might the most reasonable reference point in the choice between G₁ and G₂ be $1 million while the most reasonable reference point in the choice between G₃ and G₄ is $0?

(c) Can you explain how such reference-based preferences might explain the Allais paradox?

B: Suppose that individuals’ reference-based tastes can be described by u(x, r) = (x−r) _0.5 when x ≥ r and by v(x, r) = (r − x) _0.75 when x < r (where x is the dollar value of the outcome and r is the reference point.)

(a) Consider the case where the reference points are as described in A (b). What are the utility values associated with the three outcomes when the choice is between G₁ and G₂? What are they when the choice is between G₃ and G₄?

(b) Which gamble would be chosen by someone with such preferences in each of the two choices? How does this compare to the choices people actually make?

(c) Show that the Allais paradox would arise if the reference point were always $0 rather than what you assumed in your resolution to the Allais paradox.

(d) We mentioned in the text that prospect theory also allows for the possibility of probability weighting. If people overestimate what low probabilities mean, could this also help explain the Allais paradox?