The Informal Independence Axiom
The observer prefers outcome ‘X’ over outcome ‘Y’. Adding some other outcome (‘Z’) equally to both ‘X’ and ‘Y’ does not change the preference for ‘X’ over ‘Y’.
Here is a simple example. Lottery ‘A1’ pays $6000 on drawing numbers #1-33, but nothing for #34-100. The probability of winning is 33%, and the expected value is about $2000.
On the other hand, lottery ‘B1’ pays $1000 for #1-33, also pays $1000 for 34, and still pays nothing for #35-100. The probability of a win is 34% (versus 33% for lottery ‘A’), but the expected value dropped to about $334.
A rational observer would prefer ‘A1’ over ‘B1’, based on the expected value. In the left-hand side of the Independence Axiom, “A1 > B1”.
Let’s add the “other outcome”. Lottery ‘A2’ pays exactly as ‘A1’, except that #100 pays a princely $10. Likewise, ‘B2’ changes the ‘B1’ payout to include $10 for #100.
A rational observer applies the Independence Axiom and would prefer “A2 > B2”, just as “A1 > B1”.
Introducing the Allais Paradox
Suppose lottery ‘A3’ pays $6000 for #1-33, zero for #34, and $1000 for #35-100; and lottery ‘B3’ pays $1000 for all #1-100. The expected value of ‘A3’ is still much larger than that of ‘B3’. As well, the payout for drawing #35-100 is the same between the two lotteries.
(The expected value of ‘A3’ is approximately $2,666; ‘B3’ is worth exactly $1000).
A rational observer who applies the Independence Axiom would prefer “A3 > B3”, just as “A1 > B1” and “A2 > B2”.
However, noted French mathematician, economist and Nobel Prize winner Maurice Allais found that people may well choose “B3 > A3”. (These are not the exact lotteries Allais studied; these are simply samples to illustrate this article).
This human preference may depend on the relative probabilities and expected values among the various lotteries, rather than strictly following the Independence Axiom.
Does the Allais Paradox Demonstrate Fear and Greed Win over Mathematics?
For example, a person may prefer “B3 > A3” because they fear the 1% chance of missing out on any payment if #34 is drawn in the lottery. That fear may outweigh the greed for an extra $1666
However, in the case of “A1 > B1”, changing the payout odds from 33% to 34% may not seem nearly as significant as the lost expected value of the overall lottery.
Clearly, no-one would choose to pay $1,100 for lottery ‘B3’ with a guaranteed payout of $1,000; this is simply a means to waste $100. Yet it would be a mathematically fair price for any of the ‘A#’ lotteries.
Finally, let’s consider changing “A1 > B1”. Suppose the payout for #34 were $1,000,000 rather than $10. Adding an extra expected overall value of $1,000 still leaves “A1 > B1”, with $2000 versus $1,334. Yet we greedy people might be happy to wish for the 1% probability outcome leading to early retirement.
Mathematicians Must Reflect
As in any branch of mathematics, axioms are chosen to create an interesting and self-consistent topic of study. Often the resulting mathematical model accurately reflects the world which we experience. Some mathematicians are concerned that the Independence Axiom is valuable as a theoretical axiom but, since it does not reflect the way people make decisions, it may need to be replaced with something else.
Our Readers Reflect
Even if you appreciate the Independence Axiom as a theory, would you agree with “B3 > A3” to avoid the possibility of missing a payout?
At what point would you be indifferent between “A3” and “B3”? Specifically, how much should “A3” pay for #1-33 to make it equivalent to “B3”, if you were to pay $10 to play?
Allais, Maurice. Autobiography. Nobel Prize.org. (1988). Accessed Dec. 8, 2011.
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