When a group of individuals tries to choose between three or more options, their decision can often be mathematically unpredictable, research has shown, even when everyone's individual preferences are known and completely explicit decision-making rules are used.
In the journal Physical Review Letters, David A. Meyer of UC-San Diego and Thad A. Brown of the University of Missouri-Columbia have shown 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 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.
| Person's Name | Alicia | Brandon | Claire |
| First Choice | a | b | c |
| Second Choice | b | c | a |
| Third Choice | c | a | 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 the above 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 the above 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.