AD/BC problem

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The chapter "Risk Policies", in Kahneman's "Thinking, Fast and Slow", opens with this example, which makes vivid the pitfalls of relying on our intuitions in choosing between bets:

Imagine that you face the following pair of concurrent decisions. First examine both decisions, then make your choices.

  • Decision (i): Choose between
    • A. sure gain of $240
    • B. 25% chance to gain $1,000 and 75% chance to gain nothing
  • Decision (ii): Choose between
    • C. sure loss of $750
    • D. 75% chance to lose $1,000 and 25% chance to lose nothing

Most people, looking at both concurrently, choose A and D. Now consider this second choice:

  • AD. 25% chance to win $240 and 75% chance to lose $760
  • BC. 25% chance to win $250 and 75% chance to lose $750

Clearly, any sane person will choose BC here; it dominates option AD. However, AD is exactly the combination of options A and D, while BC is the combination of B and C.