When a group of individuals tries to choose between three or more options, their decision can often be mathematically unpredictable, a new paper has shown, even when everyone's individual preferences are known and completely explicit decision-making rules are used.
In an upcoming paper to appear in the journal Physical Review Letters, David A. Meyer of UC-San Diego and Thad A. Brown of the University of Missouri-Columbia show that the order in which different choices are presented to a group can make the decision-making process impossible to anticipate. Not only does this new result address the murky human process of amending Congressional bills, but it also confirms the idea that groups of computerized "intelligent agents" each with its own rules for buying and selling commodities can cause prices to fluctuate in a mathematically chaotic fashion.
Consider a group of three people--Alicia, Brandon, and Claire--trying to choose between options a, b, and c. Let's assume that they make their decision by weighing two options at a time and then selecting the majority choice. They then compare the majority choice to the third option.
In this example, each person ranks their preferences differently. For
example, Alicia prefers option a the most, followed by b and c. Brandon
has option b as the first choice, followed by c and a. Claire likes c,
followed by a and b.
Let's assume now that the group compares options a and b first.
|Alicia --> Prefers a to b|
|Brandon--> Prefers b to a|
|Claire --> Prefers a to b|
|GROUP CHOICE: a|
Two out of the three people prefer a to b, so the group decides on a.
They then compare their preferred option a to option c.
|Alicia --> Prefers a to c|
|Brandon --> Prefers c to a|
|Claire -->Prefers c to a|
|GROUP CHOICE SHIFTS TO c|
Comparing c to b changes the group decision again!
|Alicia --> Prefers b to c|
|Brandon --> Prefers b to c|
|Claire --> Prefers c to b|
|GROUP DECISION SHIFTS TO b|
A French mathematician known as the Marquis de Condorcet observed this phenomenon more than 200 years ago: namely, the preferences of a group may cycle through several options when the options are compared pairwise. Kenneth Arrow, the 1972 Nobel Prize winner in economics, demonstrated the inevitability of such cycles for some set of individual preferences in any group of at least 3 people choosing among 3 or more options according to any voting rule satisfying certain reasonable conditions.
In this case,
the group decision-making process is not endless.
This situation can occur when Alicia, Brandon, and Claire have a different
set of preferences such that when the group settles upon option c, it never
deviates from the decision.
In this diagram, the group now considers four options: a,b,c,d. In the right-hand diagram, the group cycles endlessly through all four options, whereas in the left hand diagram, they only cycle through three of the four (no arrows point to a). Physicists would say that the situation on the right has greater entropy (disorder) because there is a greater amount of uncertainty in what the group's next choice might be. By comparing the mathematical logic of the decision making process to the behavior of a string of magnetic atoms, Meyer and Brown demonstrate that the decision process can often be unpredictable: namely the sequence of group choices can be impossible to determine under certain situations.