Physics News Update
The American Institute of Physics Bulletin of Physics News

Number 385 (Story #1), August 3, 1998 by Phillip F. Schewe and Ben Stein


GROUP DECISIONS CAN BE MATHEMATICALLY UNPREDICTABLE even with total knowledge of everyone's individual choices and the use of completely explicit decision making rules, a new study shows. As it turns out, the order in which the choices are presented to a group can make the course of the decision 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. Whenever a group tries to choose among three or more options by weighing two at a time, its decision can often cycle endlessly from one choice to another, especially when there is a large diversity of preferences. To address the consequences of such cycles, David Meyer (UC-San Diego, 619-534-5524) and Thad Brown (Univ. Missouri) model each possible sequence of choices as a minimum-energy configuration of a one-dimensional system, such as the string of atoms with up and down spins used in studying magnetic materials. For those sequences that cycle endlessly through different options, the system has a greater-than- zero entropy--meaning that there is an uncertainty in what the group's next choice will be. In this case the dynamics are chaotic---the next choice in the cycle is unpredictable since it depends sensitively on the order in which options are presented to the group. Some more details on the paper are given below.


Chaos in Group Decisions

Group Decisions with Three Choices

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.
 
 
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

This cycle continues endlessly:

Endless Group Decision Cycle

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.

Group DecisionIn 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.

Group Decisions with Four ChoicesIn 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.