31 Beyond Two Alternatives
In the previous three chapters, we focused on situations where groups must choose between exactly two alternatives. While this simplified setting helped us understand fundamental principles of group decision-making, many real-world situations are more complex. Group decisions often exhibit a combinatorial structure - they involve multiple interconnected choices.
When we move beyond two alternatives, new paradoxes emerge. In this chapter, we examine two striking examples: the multiple elections paradox and the judgment aggregation paradox.
31.1 The Multiple Elections Paradox
The first paradox, called the Multiple Elections Paradox, arises when a group makes a decision on multiple propositions, each of which can be accepted or rejected. Consider the following example from Brams, Kilgour, and Zwicker (1998):
Suppose that there are 13 voters who have to vote on three different propositions. For each proposition, each voter can vote either “Yes” (\(Y\)) or “No” (\(N\)). We describe a vote by a list of ‘\(Y\)’ and ‘\(N\)’. So, for instance, \(YNY\) means that the voter votes Yes on Proposition 1, No on Proposition 2, and Yes on Proposition 3. Suppose that the voters cast their votes as follows:
- One voter supports “Yes” to all three propositions (\(YYY\))
- One voter supports “Yes” to first two, “No” to third (\(YYN\))
- One voter supports “Yes” to first and third, “No” to second (\(YNY\))
- One voter supports “No” to first, “Yes” to others (\(NYY\))
- Three voters support “Yes” to first, “No” to others (\(YNN\))
- Three voters support “No” to first, “Yes” to second, “No” to third (\(NYN\))
- Three voters support “No” to first two, “Yes” to third (\(NNY\))
- No voters oppose all three propositions (\(NNN\))
The table below summarizes the votes:
| Voter | Proposition 1 | Proposition 2 | Proposition 3 |
|---|---|---|---|
| 1 | \(Y\) | \(Y\) | \(Y\) |
| 2 | \(Y\) | \(Y\) | \(N\) |
| 3 | \(Y\) | \(N\) | \(Y\) |
| 4 | \(N\) | \(Y\) | \(Y\) |
| 5 | \(Y\) | \(N\) | \(N\) |
| 6 | \(Y\) | \(N\) | \(N\) |
| 7 | \(Y\) | \(N\) | \(N\) |
| 8 | \(N\) | \(Y\) | \(N\) |
| 9 | \(N\) | \(Y\) | \(N\) |
| 10 | \(N\) | \(Y\) | \(N\) |
| 11 | \(N\) | \(N\) | \(Y\) |
| 12 | \(N\) | \(N\) | \(Y\) |
| 13 | \(N\) | \(N\) | \(Y\) |
Suppose that the group decides to apply majority rule to each proposition separately.
- Proposition 1: “No” wins (7 No votes vs 6 Yes votes)
- Proposition 2: “No” wins (7 No votes vs 6 Yes votes)
- Proposition 3: “No” wins (7 No votes vs 6 Yes votes)
This produces a striking result: The majority vote results in all three propositions being rejected (\(NNN\)), yet not a single voter voted for this combination! This shows that even when using majority rule - which we proved is the uniquely fair method for single yes/no decisions - applying it separately to multiple decisions can produce outcomes that no voter supports.
This paradox raises deep questions about the nature of group decision-making. As Brams, Kilgour, and Zwicker note, it “strikes at the core of social choice - both what it means and how to uncover it.” The paradox reveals a fundamental tension between two different ways of thinking about group decisions: should we consider each issue separately, or should we look at combinations of decisions as complete packages?
31.2 Judgement Aggregation
The second paradox arises when a group must make a decision on multiple interrelated propositions. This is known as the judgment aggregation paradox. Consider the following variation of an example from List and Pettit (2002):
Suppose that three people must decide whether to hire a candidate for an academic position. They evaluate two criteria:
- \(r\): the candidate has a strong research record
- \(t\): the candidate has a strong teaching record
Each person must make three judgments:
- Whether the research record is strong (Yes or No on \(r\))
- Whether the teaching record is strong (Yes or No on \(t\))
- Whether to hire the candidate (Yes or No on \(h\))
Everyone agrees that the candidate should be hired if and only if both the research and teaching records are strong. In logical notation, this is written as \((r\wedge t) \leftrightarrow h\), where “\(r\wedge t\)” means “both \(r\) and \(t\) are true” and “\(\leftrightarrow\)” means “if and only if”.
The three people make the following judgments:
| Expert | \(t\) | \(r\) | \((t\wedge r)\leftrightarrow h\) | \(h\) |
|---|---|---|---|---|
| 1 | \(Y\) | \(Y\) | \(Y\) | \(Y\) |
| 2 | \(Y\) | \(N\) | \(Y\) | \(N\) |
| 3 | \(N\) | \(Y\) | \(Y\) | \(N\) |
This creates a paradox. There are two different ways to determine whether the candidate should be hired:
Direct vote on hiring: A majority (experts 2 and 3) vote “No” on hiring, so the candidate should not be hired.
Vote on the components: A majority (experts 1 and 2) agree the teaching record is strong, and a majority (experts 1 and 3) agree the research record is strong. Since everyone agrees hiring should occur if and only if both records are strong, following this rule, the candidate should be hired.
This judgment aggregation paradox raises fundamental questions about collective decision-making. Should we aggregate people’s final judgments (their conclusions), or should we aggregate their judgments about the reasons for their final judgement? Neither approach seems obviously wrong, yet they can lead to contradictory outcomes. This paradox was first identified in the context of judicial decision-making by Kornhauser and Sager (1986), who called it the “doctrinal paradox.” The example has profound implications for group decision-making in many contexts, from corporate boards to academic committees. List and Pettit (2002) showed that this is not merely an isolated example - they proved that no voting method can guarantee consistent judgments while satisfying other reasonable requirements. For a comprehensive overview of these issues and their implications, see Cariani (2011) (available at http://cariani.org/files/JA.cariani.pdf).