# 11.5 Group Decision Making

Often groups of people have to make decisions about what the group will do. It may seem that voting is a good way to determine what a group wants, and when there is a clear most-preferred choice, it is. However, there are major problems with voting when there is not a clear preferred choice, as shown in the following example.

###### Example 11.17.

Consider a purchasing agent that has to decide on a holiday destination for a group of people, based on their preference. Suppose there are three people, Alice, Bob and Cory, and three destinations, $X$, $Y$, and $Z$. Suppose the agents have the following preferences, where $\succ$ means strictly prefers:

• Alice: $X\succ Y\succ Z$

• Bob: $Y\succ Z\succ X$

• Cory: $Z\succ X\succ Y$

Given these preferences, in a pairwise vote, $X\succ Y$ because two out of the three prefer $X$ to $Y$. Similarly, in the voting, $Y\succ Z$ and $Z\succ X$. Thus, the preferences obtained by voting are not transitive. This example is known as the Condorcet paradox. Indeed, it is not clear what a group outcome should be in this case, because it is symmetric between the outcomes.

A social preference function gives a preference relation for a group. We would like a social preference function to depend on the preferences of the individuals in the group. It may seem that the Condorcet paradox is a problem unique to pairwise voting; however, the following result due to Arrow [1963] shows that such paradoxes occur with any social preference function.

###### Proposition 11.1 (Arrow’s impossibility theorem).

If there are three or more outcomes, the following properties cannot simultaneously hold for any social preference function:

• The social preference function is complete and transitive.

• Every individual preference that is complete and transitive is allowed.

• If every individual prefers outcome $o_{1}$ to $o_{2}$, the group prefers $o_{1}$ to $o_{2}$.

• The group preference between outcomes $o_{1}$ and $o_{2}$ depends only on the individual preferences on $o_{1}$ and $o_{2}$ and not on the individual preferences on other outcomes.

• No individual gets to unilaterally decide the outcome (nondictatorship).

When building an agent that takes the individual preferences and gives a social preference, you should be aware that you cannot have all of these intuitive and desirable properties. Rather than giving a group preference that has undesirable properties, it may be better to point out to the individuals how their preferences cannot be reconciled.