Paper: Bar-Hillel, "Cycles of Intransitive Choice"

Transitivity is a property of preferences where if A is preferred to B (denoted A > B) and B preferred to C (B > C) then A must be preferred to C (A > C). The paper discussions various examples where this property doesn’t hold.

The most famous “intransitivity” is in group decision making by majority rule. If person one has preferences A > B > C, person two has preferences B > C > A and person 3 has preferences C > A > B. A majority of people prefer A to B and B to C, but a majority also prefers C to A. Thus, for the group A > B > C > A > B and so on.

Group decision making is analogous to multi-dimensional decision making. If you’re buying a new computer, you may think CPU speed, amount of memory and hard drive space are the important attributes. Let’s say model A is fastest, B second fastest and C is the slowest. Likewise, B has the most memory, C the second most and A the least and C has the biggest hard drive, A the second biggest and B the smallest. You can see that this situation is very similar to the above and the decision rule “pick the model that has most preferred categories” gets you no where.

The paper also talks about “inherently comparative choice”. The idea is that in the case where the satisfaction (or dissatisfaction) derived from a choose is dependent on the thing chosen and the things not chosen (e.g. regret), transitivity can be violated. A good example compares the difference in regard to stock market returns in the case where you bought a share of stock yesterday that doubled today versus the case where you’ve owned 2 shares of the same stock for a long while and it doubled. You may find yourself more pleased with the first situation because you where able to “time the market” even though the outcomes in the second case is better in a money-in-your-pocket sense. Comparing these situations to the situation where you had 4 shares of the stock yesterday but then sold one share, you may find yourself regretting that choice today when the stock price doubled. Even though you have higher return in this case than in the second case, the feeling of regret may make you to prefer the second case to the third. However, despite the regret in the third case, the money gain is so high that you still prefer the third to the first case. Thus, we have a violation of transitivity… timing the market with a small return is preferred to buy and hold with a medium return is preferred to regretting a sale with a large return is preferred timing the market with a small return, etc.

The paper concludes by remarking that it is implicit in revealed preference that choice is driven by preferences, but that perhaps preferences are “constructed” by choice and thus “choice procedures are geared towards justification and conflict resolution rather than towards optimization.”


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