# Individual Rationality Needn't Generalize to Rational Consensus

# tl;dr 

Organizations that enforce rationality at the collective level can get very different voting outcomes than organizations that enforce rationality at the individual level, per known results in social choice theory. This has implications for real-world expert panels. 

Here, "rationality" is logical consistency - it is possible for the majority of members to vote to reject a conclusion while also believing the necessary conditions to accept it hold, and vice-versa, even if they all independently evaluated the precepts and arrived at the conclusion logically. This arises because of how majority votes work.

This post summarizes [Philip Pettit's 2002 paper](https://www.princeton.edu/~ppettit/papers/Deliberative_PhilosophicalIssues_2001.pdf) outlining the issue and its implication for any deliberative democracy. It additionally summarizes Pettit and List's impossibility result on judgement aggregation rules from the Stanford Encyclopedia of Philosophy. 

# Introduction to the Discursive Dilemma

## [The Doctrinal Paradox](https://plato.stanford.edu/entries/social-choice/#DocParDisDil)

Three judges have to decide by majority whether a defendant broke a contract. 

Legal doctrine dictates that if the following two premises are true: 

(a) the defendant was contractually obliged not to do action X, and 

(b) the defendant did action X, 

then the conclusion is that the defendant broke his contract. Let's call this conclusion (c). 

The judges get to decide which of the premises, (a) and (b), are true. 

This is the simple conjunctive formula  $a \wedge b \implies c$ , and, taking one possible way the judges vote, we can construct a truth table for the same.  

|          |   A   |   B   | $A \wedge B$    |
|:--------:|:-----:|:-----:|-------|
|  Judge 1 |  True |  True | True  |
|  Judge 2 |  True | False | False |
| Judge 3  | False | True  | False |
| _Majority_ | _True_  | _True_  | _False_ |

In this permutation of votes, because the majority voted false for (c), the judges vote to pardon the defendant. 

But here's the paradox: if the majority of the judges thought (a) was true and (b) was true, then it should have implied that the majority thought (c) was true. 

You now have the unfortunate case where the majority voted true for all the precepts, but also rejected the conclusion. 

This demonstrates collective inconsistency arising from individual consistency. 

## Generalizations

A similar paradox can be constructed for disjunctive propositions $A \vee B \implies C$ . It scales to arbitrary groups of people. 

In fact, Pettit identifies the following minimal conditions:

> a. there is a conclusion to be decided among a group of people
> by reference to a conjunction (or disjunction) of independent or separable
> premises—the conclusion will be endorsed if relevant prem-
> ises are endorsed, and otherwise it will be rejected;
>
> b. each member of the group forms a judgment on each of the
> premises and a corresponding judgment on the conclusion;
>
> c. each of the premises is supported by a majority of members
> but those majorities do not coincide with one another;
>
> d. the intersection of those majorities will support the conclu-
> sion, and the others reject it, in view of a; and
>
> e. the intersection of the majorities is only a minority in the
> group as a whole.

## Implications

Although cute-sounding at first, this paradox underscores a very important issue regarding voting procedures in organizations in practice. 

First, organizations must choose to vote directly for the proposition at stake, or to vote for individual conditions and have the decision inferred from that. As the foregoing paradox demonstrates, the outcomes are very different in both - in the first procedure, the judges voted to pardon, but, under the second procedure, the judges would have had to convict. 

Second, it is not the case that organizations can always choose to vote directly and ignore preserving collective consistency, as is usually argued about electoral voting. Pettit identifies two examples of organizations where collective consistency is absolutely necessary: 

1. A committee that has been tasked to evaluate the merits of a case and arrive at a recommendation accordingly. This includes awards panels, juries, trusts acting on external instructions, and expert policy bodies. 

2. Political or activist movements that seek to hold ethically or philosophically consistent positions, where members may desert if the movement does not appear to be holistically consistent. 

Because of this generalization, the doctrinal paradox has been dubbed [the discursive dilemma](https://www.wikiwand.com/en/Discursive_dilemma), given that the issue at stake needn't have anything to do with legal doctrine at all.

Finally, it demonstrates that the root of this issue arises from the specific scheme of majority vote collection. It opens the door to thinking about alternate voting schemes that try to satisfy both individual and collective rationality.

## An Impossibility Result For Collective Rationality In the Best Case

 The more general problem of arriving at a way to collect votes on propositions is known as _judgement aggregation_. 

Typically, we might want to be able to ensure that we can have a procedure for judgement aggregation that meets a few nice properties: 

1. *Plurality*: There are no constraints on which available options people can vote for. In other words, if there are $N$ possible options, it is safe to assume that people may vote for any of the $N$ options. 

2. *Complete collective consistency*: The judgment preserves collective consistency for all propositions. In other words, independent of the proposition, the majority judgement will always faithfully evaluate the proposition consistently. 

3. *Anonymity*: If the proportion of people who voted stays the same, then the judgement stays the same, independent of *who* exactly voted. In other words, it doesn't matter which subpopulation voted, only the relative propotion. 

4. *Systematicity*: This is fairly technical, but is a slightly stronger version of [independence of irrelevant alternatives](https://www.wikiwand.com/en/Independence_of_irrelevant_alternatives), which states that a preference between two options shouldn't be changed by introducing a third option. You might prefer $A$ to $B$, but introducing $C$ shouldn't make you suddenly prefer $B$ to $A$, although you may prefer $A$ to $C$ or $B$ to $C$.  

Unfortunately, [List and Pettit](https://www.princeton.edu/~ppettit/papers/Aggregating_EconomicsandPhilosophy_2002.pdf) jointly proved in 2002 that arriving at a scheme like this is impossible. In order to arrive at a collectively consistent scheme, we must sacrifice some of the other properties. 

## An Exercise for the Interested Reader

So how *would* you preserve collective consistency and individual consistency?