A Collective Decision Problem

The first example concerns a multimember government making multiple decisions that constrain each other across time (Pettit 2001b). The propositions under consideration are as follows.

P: A new (costly) education project shall be implemented.

Q: A new (costly) health care project shall be implemented.

R: A new (costly) defense project shall be implemented.

T: Taxes shall be increased.

The pursuit of all three projects leads to a budget deficit, given current tax revenues, whereas the pursuit of two or fewer projects is possible within the existing budget. I here assume that the government members unanimously accept a balanced budget constraint: P, Q, and R can be accepted simultaneously only if T is also accepted, formally:

The propositions are considered one by one. At each point, a majority vote is taken on the proposition under consideration. If the majority opinion is consistent with all prior decisions, the government decides the new proposition according to that opinion; if it is not, the government decides the new proposition based on what those prior decisions imply. So earlier decisions constrain later ones. Consider two alternative sequences in which the propositions might come up.

Case 1. The balanced budget constraint is considered in January and accepted unanimously. The tax increase proposal is considered in February and rejected unanimously. The education and health proposals are considered in March and April and are each accepted by a two-thirds majority. When the defense proposal is considered in May, it is supported by a two-thirds majority, but the government is already committed to two costly projects, to not increasing taxes and to balancing the budget. To avoid losing credibility (and the next election), the government rejects the defense proposal, overruling its majority opinion on that proposal. Overall, the government accepts P and Q (and the budget constraint) and rejects R and T.

Case 2. Again, the balanced budget constraint is considered and accepted in January. But the education, health, and defense proposals are considered in February, March and April and are each accepted by a two-thirds majority. As a tax increase has not been ruled out yet, accepting all three proposals is still consistent with the balanced budget constraint. But in May a dilemma arises. All government members oppose a tax increase, but the prior commitments to balancing the budget and to pursuing all three costly projects necessitate accepting a tax increase. Overall, the government accepts P, Q, R, and T (and the budget constraint).

Cases 1 and 2 lead to mutually contradictory outcomes. The individual views are identical in both cases, as is the government's method of deciding each proposition by a majority vote, unless prior commitments intervene. The two cases differ only in the order in which the propositions are considered: The decision process is path-dependent. Could the government use a different decision rule to avoid such path-dependence?

An Individual Decision Problem

The second example concerns political argumentation. It is related to work by Aragones et al. (2002) on rhetoric, suggesting that one can change an agent's beliefs without giving her new information, just by organizing existing information differently. Aragones et al. analyze the effects of rhetorical analogies. The present example suggests that similar effects can be achieved by adjusting the order in which propositions are presented to the agent(s) for consideration. As this phenomenon can be strategically exploited, it is relevant to political manipulation (McLean 2002; Riker 1986). Imagine an individual agent who has a disposition to accept each of the following propositions: If she were to consider each proposition in isolation, then she would find the proposition sufficiently plausible to accept.

P: Young people are entitled to unrestricted freedom after school.

Q: Compulsory national service reduces crime among young people.

R: Compulsory national service is justifiable.

“If P, then not R”: If young people are entitled to unrestricted freedom after school, then compulsory national service is not justifiable.

“If Q, then R”: If compulsory national service reduces crime among young people, then compulsory national service is justifiable.

One may notice an inconsistency between these propositions, but I leave this point aside until later.⁶

Case 1. At times 1 and 2, the agent considers whether young people are entitled to unrestricted freedom after school (P) and whether this implies that compulsory national service is unjustifiable (“If P, then not R”). She finds each proposition plausible and accepts it. So, at time 3, when she considers whether compulsory national service is justifiable (R), she rejects that proposition. At time 4, she considers and accepts that compulsory national service would reduce crime among young people (Q). At time 5, she considers whether such crime reduction would be sufficient to justify compulsory national service (“If Q, then R”) and rejects the proposition, to respect her previous decisions. Overall, the agent accepts P, Q, and “If P, then not R” and rejects R, and “If Q, then R.”

Case 2. At time 1, the agent considers and accepts that, if compulsory national service were to reduce crime among young people, this would make compulsory national service justifiable (“If Q, then R”). At time 2, she considers and accepts that it would reduce crime (Q). So, at time 3, when she considers whether compulsory national service is justifiable (R), she concludes that it is. At time 4, she considers the proposition that, if young people were entitled to unrestricted freedom after school, this would make compulsory national service unjustifiable (“If P, then not R”). She accepts this proposition, having not yet considered whether young people are entitled to such freedom. At time 5, she considers this issue (P) and rejects the proposition, following her prior decisions. So she concludes that compulsory national service takes priority over young people's freedom after school. Overall, the agent accepts Q, “If Q, then R”, R, and “If P, then not R” and rejects P.

Cases 1 and 2 lead to mutually contradictory outcomes. Only a single agent is involved, who in both cases is initially disposed to accept each proposition. In each case, the agent uses only valid logical inferences. The two cases differ only in the order in which the propositions are considered: The decision process is path-dependent. What has happened?

The Propositions

I use a simple language of propositional logic, which includes both atomic and compound propositions. Atomic propositions are the most basic ones; they are represented by the (finitely many) letters P, Q, R, S, …. Compound propositions state interconnections between other propositions; they involve logical connectives, such as ¬ (not), ∧ (and), ∨ (or), → (implies), [harr ] (if and only if).

Definition

An acceptance/rejection function (“AR-function”) is a function δ: X→{1,0}.

For each proposition ϕ∈X, δ(ϕ)=1 means that the agent accepts ϕ (or has an initial disposition to accept ϕ); δ(ϕ)=0 means that the agent does not accept ϕ (or has an initial disposition not to accept ϕ). Note that δ(ϕ)=0 does not by itself mean that the agent accepts the negation of ϕ (or has such an initial disposition). The agent accepts that negation (or has such an initial disposition) if and only if δ (¬ ϕ)=1. For brevity, I usually refer to δ (ϕ)=1 as “acceptance” of ϕ and to δ (ϕ)=0 as “rejection” of ϕ, bearing in mind the more precise interpretation offered here.¹⁰

The Propositional Attitudes of a Group.If the agent is a group, ascribing propositional attitudes to that group does not presuppose any metaphysical assumptions about “group minds.” As illustrated by the government example and discussed below, the group's propositional attitude on a proposition can be its majority verdict on that proposition, or the outcome of some other aggregation over the propositional attitudes of the group members.

Definition

A decision-path on X is a one-to-one function {Ω: 1, 2,…, k} → X, where k is the number of propositions in X.

Here Ω(1), Ω(2), …, Ω(k) are the first, second, …, k th propositions to be considered.¹²

The Notion of a Sequential Decision-Process In defining a sequential decision process, it is useful to interpret the agent's propositional attitudes as initial dispositions: The agent enters the decision process without fully endorsed views on the propositions, but with initial dispositions. The agent considers the propositions in the order given by a decision-path Ω and decides at each point whether or not to accept the proposition under consideration, say, ϕ. The outcome of the decision process can then be interpreted as the set of fully endorsed views the agent has formed on the propositions after having considered them one by one along the path Ω. How does the agent decide whether or not to accept each proposition in the sequence? Suppose that proposition ϕ is under consideration. There are two cases. Either the agent's initial disposition on ϕ is consistent with the propositions (if any) the agent has accepted earlier: If so, the agent accepts or rejects ϕ according to that initial disposition. Or there is a logical inconsistency between the initial disposition on ϕ and some previously accepted propositions: If so, the agent requires a method of resolving this inconsistency.

A Conflict Resolution Rule. I call a method by which the agent can resolve an inconsistency between his or her disposition on a new proposition and previously accepted propositions a conflict resolution rule. Under the priority-to-the-past rule, as in the two examples, the agent resolves the inconsistency by accepting the logical implications of previously accepted propositions and overruling the initial disposition on the new proposition. So earlier decisions constrain later ones, but not vice versa. Precedent-based decision making is a form of priority-to-the-past decision-making. For simplicity, I only consider the priority-to-the-past rule here. This is a restriction, as there are cases where other conflict resolution rules are more plausible, but the model can be generalized.¹³

When Is the Priority-to-the-Past Rule Plausible? Let me suggest an illustrative list of cases (neither exhaustive nor mutually exclusive) in which the priority-to-the-past rule is plausible. First, suppose that the decision-path is the temporal order in which the propositions are considered. Then the priority-to-the-past rule is plausible in cases where decisions, once made, are to hard to overrule. This may be because:

• Earlier decisions may have created commitments that cannot (easily) be overruled, for several reasons: (i) Legal reasons: Earlier decisions may be legally binding; they may have involved passing certain legislation, changing constitutional rules, or they may serve as precedents. (ii) Social or cognitive reasons: Earlier decisions may be so entrenched in the system of beliefs or judgments of the agent(s) that they cannot be overruled without a substantial cost. (iii) Strategic reasons: If earlier decisions are publicly known, the agent may lose credibility by not respecting those decisions; this loss of credibility may be costly; the agent might lose the next election, for example.
• Earlier decisions may have led to actions that (i) are irreversible, or (ii) are perceived by the agent(s) to be irreversible, or (iii) are too costly to reverse in practice.

Second, suppose that the decision-path is the order of importance or priority among the propositions. Then the priority-to-past rule is a natural way of making decisions based on that order: Decisions on more important propositions constrain decisions on less important ones. My model is neutral between different interpretations of the priority-to-the-past rule.

Definition

M(δ, Ω) is invariant under changes of the decision-path if, for any two decision-paths Ω₁ and Ω₂, M(δ, Ω₁)=M(δ, Ω₂).

Rationality Conditions on Propositional Attitudes

I introduce four rationality conditions that an agent's propositional attitudes may or may not satisfy. Completeness requires that the agent should accept at least one member of each proposition-negation pair. Weak consistency requires that the agent should accept at most one member of each proposition-negation pair. Strong consistency requires that the propositions accepted by the agent can be simultaneously true. Deductive closure requires that the agent should accept all implications of other propositions she accepts. These concepts provide rationality criteria over and above the standard criterion of transitivity of preferences. Let δ be the agent's AR-function:

Theorem 1

For any ϕ∈X, M (δ, Ω) is strongly path-dependent with respect to ϕ if and only if δ violates strong consistency with respect to ϕ.

Proof

The proofs of all theorems are in the Appendix.

By Theorem 1, if (and only if) the agent's propositional attitudes violate strong consistency with respect to some proposition ϕ, there exist (at least) two decision-paths such that under one path ϕ is accepted, whereas under the other ¬ϕ is accepted. An agent's propositional attitudes violate strong consistency either when they violate weak consistency or when they are weakly consistent but an inconsistency is “hidden” by a violation of deductive closure.

Proposition 3

Suppose that δ is complete and weakly consistent. For any ϕ∈X, δ violates strong consistency with respect to ϕ if and only if δ is not deductively closed with respect to one of ϕ or ¬ϕ.

The conjunction of Theorem 1 and Proposition 3 yields a necessary and sufficient condition for strong path-dependence in the case where the agent's propositional attitudes are complete and weakly consistent. In this case, there exist (at least) two decision-paths with mutually inconsistent outcomes on some proposition ϕ if and only if the agent's propositional attitudes violate deductive closure with respect to ϕ:

Theorem 2

Suppose that δ is complete and weakly consistent. For any ϕ∈X, M (δ, Ω) is strongly path-dependent with respect to ϕ if and only if δ is not deductively closed with respect to one of ϕ or ¬ϕ.

Although strong path-dependence is ruled out when the agent's propositional attitudes are strongly consistent (by Theorem 1), weak path-dependence is still possible.

Theorem 3

Suppose that δ is strongly consistent. For any ϕ∈X, M (δ, Ω) is weakly path-dependent with respect to ϕ if and only if δ is not deductively closed with respect to ϕ.

Theorem 3 states that if the agent's propositional attitudes are strongly consistent, then there exist (at least) two decision-paths with different outcomes on some proposition ϕ if and only if these propositional attitudes violate deductive closure with respect to ϕ.²⁰

If the agent is an individual, the conditions for avoiding path-dependence are clear. If this individual's propositional attitudes are weakly consistent and deductively closed (hence strongly consistent), he or she is immune to path-dependence in a priority-to-the-past decision process. By contrast, if the agent's propositional attitudes violate strong consistency or deductive closure, he or she may be manipulable by a “heresthetician” presenting the propositions in some strategic order. Whereas it is unsurprising that irrationality can make individuals manipulable, my results imply that even a boundedly rational individual may be manipulable. As shown, a violation of deductive closure is sufficient for path-dependence, even when weak consistency is satisfied.²¹

If the agent is a group, the conditions for avoiding path-dependence are less clear. It would be desirable to find a method of aggregating the group members' views into a collective AR-function that satisfies completeness, weak consistency, and deductive closure (and thus strong consistency). By Theorems 2 and 3, this would rule out path-dependence. The multimember government example has illustrated that majority voting does not generally have these properties. I next show that this is not an accidental property of majority voting: Under certain conditions, no aggregation function with the desired properties exists.

The Setting

I now assume that the agent is a group of individuals, represented by the set N = 1, 2, …, n (n≥2). The views of each individual, i∈N, on the propositions in X are represented by an AR-function δ_i : X → 1, 0, defined as before. I here interpret the individuals' propositional attitudes as fully endorsed views and assume, as a best-case scenario, that each individual's AR-function δ_i is complete, weakly consistent, and deductively closed (and thus strongly consistent).

Definition

An aggregation function is a function δ: {0, 1}ⁿ→{0, 1}.

Examples of aggregation functions are majority voting, as used above, and a dictatorship of one individual, analogous to the definition in Arrow's (1951) preference-based model.

Definition

Majority voting is the aggregation function δ: {0, 1}ⁿ→{0, 1} defined as follows. For any 〈 d₁, d₂, …, d_n〉∈{0, 1}ⁿ,

Definition

A dictatorship of individual i∈N is the aggregation function δ : {0, 1}ⁿ→{0, 1} defined as follows. For any (d₁, d₂, …, d_n)∈{0, 1}ⁿ, δ(d₁, d₂, …, d_n)=d_i.

Majority voting has the attractive property of giving all individuals equal weight in determining the group's propositional attitude on any proposition; it satisfies anonymity. A dictatorship violates not only anonymity, but also the weaker condition of nondictatorship.

Definition

An aggregation function δ is anonymous if, for any 〈 d₁, d₂, …, d_n〉∈ {0, 1}ⁿ and any permutation σ: N→N, δ(d₁, d₂, …, d_n)=δ (d_{σ (1)}, d_{σ (2)}, …, d_{σ (n)}); δ is non-dictatorial if it is not a dictatorship of any i∈N.

How can an aggregation function generate a single collective AR-function based on a profile of individual AR-functions? For each profile 〈δ_i〉, an aggregation function δ induces a collective AR-function δ_{〈δi 〉} : X → {0, 1}, where

for each ϕ∈X, δ_{〈 δi〉} (ϕ):=δ (δ₁(ϕ), δ₂(ϕ), …, δ_n(ϕ)).

The collective AR-function δ_{〈δi 〉} represents the group's propositional attitudes, as determined by applying the aggregation function δ to the profile 〈δ_i〉.²³

A restriction is built into my definition of an aggregation function: The aggregation over the n individual views is performed on a proposition-by-proposition basis. This mirrors the restriction introduced by independence of irrelevant alternatives in Arrow's classical model. Under Arrow's independence condition, preferences are aggregated based on pairwise comparisons between alternatives, i.e., based on (what might be seen as) pairwise ranking propositions. Whether such a restriction is plausible depends on the context. In many sequential decision problems, collective decisions are taken on a proposition-by-proposition basis, by voting separately on each proposition as it arises. But, as discussed below, a possible generalization of my model is to relax propositionwise aggregation.

Impossibility Results on Path-(In)Dependence

Consider the implications of Propositions 4 and 5. Any anonymous or just nondictatorial (propositionwise) aggregation function generating complete and weakly consistent collective AR-functions necessarily produces deductive closure violations for some profiles in the universal domain. By Theorem 2, for these profiles, there exist different decision-paths with mutually inconsistent outcomes. This mechanism underlies the following result.

Theorem 5

Let δ be an aggregation function. For every 〈δ_i〉∈U, M (δ_〈δi〉, Ω) is decisive and invariant under changes of the decision-path if and only if δ is a dictatorship of some individual i∈N.

In summary, suppose that a priority-to-the-past decision process satisfies two conditions. Universal domain: Any logically possible profile of (rational) individual views on the propositions is acceptable as input. Decisiveness: The decision process produces a determinate decision on every proposition. Then the unique (propositionwise) aggregation function for which the decision process is generally path-in dependent is a dictatorship of one individual. Add a further condition. Anonymity: All individuals have equal weight in the decision on each proposition. Or, even more weakly, nondictatorship: There does not exist a dictatorial individual. Then there exists no (propositionwise) aggregation function for which the decision process is generally path-independent. This does not mean that the decision process is path-dependent for every profile. But no anonymous or nondictatorial aggregation function guarantees path-independence. Under any such aggregation function, there exist some profiles for which the decision process is (strongly) path-dependent.²⁵

Does this imply that, at a collective level, path-dependence is in principle unavoidable? And, if so, what is the cost of this? I first address the second question, and show that path-dependence opens up two types of strategic manipulation. I then turn to the first question, and explore some escape-routes from path-dependence.

Manipulation by Agenda Setting

Whenever the agent's propositional attitudes violate strong consistency with respect to ϕ, an agenda-setter—whoever chooses the decision-path—may have power to determine whether the outcome of the decision process will be ϕ or ¬ϕ. By Theorem 1, given a violation of strong consistency with respect to ϕ, there exists one decision-path leading to ϕ and another leading to ¬ϕ. Given sufficient information and computational power, an agenda-setter can determine the decision-path required to bring about the preferred outcome. In the multimember government example, an agenda-setter who cares strongly about the defense proposal (proposition R) might advocate the decision-path of case 2, which results in the acceptance of R. An agenda-setter who opposes the defense proposal might advocate the decision-path of case 1, which results in the rejection of R. This parallels the problem of agenda-dependence in preference-based models, albeit within the new domain of decisions on multiple interconnected propositions (e.g., Plott and Levine 1978; Riker 1982, chap. 7).

Manipulation by Expression of Untruthful Views

Suppose that a decision process is (strongly) path-dependent but a particular decision-path has been fixed. If some individual (or group) cares strongly about certain later propositions in the decision-path, they might strategically express untruthful views on certain earlier ones, as decisions on the later propositions will be affected by decisions on those earlier ones. In the government example, suppose that individual 3 cares most about the defense proposal (proposition R) and is willing to sacrifice his or her support for the health proposal (proposition Q), to get his or her way on the defense issue. Suppose that the decision-path is as in case 1. In April, when Q is considered, individual 3 might untruthfully vote against Q, bringing about a majority rejection of Q. Then, in May, when R is considered, there would no longer be a conflict between prior commitments and the majority verdict on R; the government would be able to follow that majority verdict and accept R. Without individual 3's strategic intervention, the outcome set of the decision process would have been {((P∧Q∧R)→T), P, Q,¬ R,¬ T. With that intervention, the outcome set is {((P∧Q∧R)→T), P,¬ Q, R,¬ T, an outcome individual 3 prefers: Individual 3 has an incentive to express an untruthful view on Q.²⁶

I now define strategic incentives formally. Let M be the set of all possible outcome sets M(δ, Ω) (⊆ X) of a priority-to-the-past decision process. I assume that each individual has certain preferences over these possible outcome sets. Individual i's most preferred outcome set—for brevity, denoted δ_i like the individual's AR-function—is the one that includes precisely those propositions that she individually accepts, ϕ ∈ X : δ_i (ϕ)=1}. The closer an outcome set is to this most preferred one—in a sense defined below—the more he or she prefers that outcome set. The individual's preferences over the outcome sets are represented by an ordering R_i,δi on M (reflexive, transitive and, connected). The subscripts i and δ_i indicate that the ordering depends not only on individual i, but also on his or her AR-function δ_i. For any two outcome sets Φ₁, Φ₂, Φ₁R_i,δiΦ₂ means that individual i weakly prefers Φ₁ to Φ₂. I write Φ₁P_i,δiΦ₂ to mean that individual i strictly prefers Φ₁ to Φ₂, formally (Φ₁R_i,δiΦ₂ and not {Φ₂R_i,δiΦ₁). I now formalize the assumption that the closer an outcome set is to individual i's most preferred one, the more he or she will prefer that outcome set.

Definition

An outcome set Φ₁ is at least as close to δ_i as an outcome set Φ₂ if, for every ϕ∈X, δ_i(ϕ)=Δ₂(ϕ) implies δ_i(ϕ)=Δ₁(ϕ), where, for j=1, 2, Δ_j(ϕ)=1 if ϕ∈Φ_j and Δ_j(ϕ)=0 if ϕ ∉Φ_j.

Informally, one outcome set is at least as close to an individual's most preferred outcome set as another if there is no proposition that the individual accepts and that is included in the second outcome set but not in the first, and no proposition that the individual rejects and that is excluded from the second outcome set but included in the first.

Assumption 1

If an outcome set Φ₁ is at least as close to δ_i as another outcome set Φ₂, then individual i weakly prefers Φ₁ to Φ₂, i.e., Φ₁R_i,δ_iΦ₂.

As the “at least as close” relation is only a partial, not generally complete, ordering over the possible outcome sets, this assumption is typically consistent with more than one preference ordering; so it is not maximally restrictive. My definition of strategic incentives is a translation of the classical definition into the framework of decisions on multiple propositions (compare Gibbard 1973 and Satterthwaite 1975).

Definition

In the decision process M(δ, Ω) defined on the domain D, an individual i∈N has an incentive to express an untruthful AR-function at the profile 〈δ_i〉∈D if there exists an AR-function δ*_i(≠δ_i) (where 〈δ₁,…, δ*_i,…,δ_n〉∈D) such that Φ*P_i,δ_iΦ, where Φ*=M(δ_〈δ1, …, δ*_i, …, δ_n〉, Ω) and Φ=M(δ_〈δi〉, Ω).²⁷

Informally, individual i has an incentive to express an untruthful AR-function at the profile 〈δ_i〉 if three conditions hold. (i) If individual i expresses her truthful AR-function δ_i (holding the other individuals' AR-functions fixed), the decision process leads to the outcome set Φ. (ii) If individual i expresses the “untruthful” AR-function δ*_i (holding the other individuals' AR-functions fixed), the decision process leads to the outcome set Φ*. (iii) Individual i strictly prefers Φ* to Φ.

The government example shows that path-dependent decision processes may give individuals incentives to express untruthful AR-functions. Whether or not an individual has such an incentive in a given case depends on several factors: the individual's preference ordering, the decision-path, and whether the individual's views are pivotal for the collective outcome on some propositions. Below I prove that (weak) path-dependence is a necessary condition for the existence of individuals with incentives to express untruthful views. It is not a sufficient condition. Even in cases of (strong) path-dependence there may not exist a single individual who is pivotal for the outcome on a relevant proposition; so there may not exist an individual who can single-handedly manipulate the outcome. A more technical analysis might be used to show that, under some conditions, (strong) path-dependence implies that there exists a coalition of individuals with an incentive to express untruthful views.

Relaxing the Universal Domain: The break Domain Restriction Approach

This escape route relaxes the requirement that the decision process should accept as admissible input any logically possible profile of individual AR-functions. Suppose that not all such profiles will occur in collective decisions, for one of the following reasons. (i) Coercive reasons: Certain profiles are explicitly ruled out by restrictions on the views individuals are permitted to express. (ii) Empirical reasons: Certain profiles do not occur in practice, as a matter of contingent fact. (iii) Deliberative reasons: Although all logically possible profiles can in principle occur, the individuals engage in group deliberation prior to a collective decision, where such group deliberation transforms the individual views and thereby restricts the domain of postdeliberation profiles.

Parallel mechanisms of domain restriction have been discussed in relation to classical models of social choice (Dryzek and List 2003; Knight and Johnson 1994; D. Miller 1992). I here remain neutral on how feasible such mechanisms are. I identify a structure condition with the property that, if the domain of admissible profiles includes only ones satisfying that condition, then a priority-to-the-past decision process is path-independent. The condition—called unidimensional alignment—is similar in spirit, but not in form, to Black's (1948) condition of single-peakedness in the context of preferences. A profile satisfies unidimensional alignment if there exists a single ordering of all individuals from left to right—a structuring ordering—such that, for every proposition in X, the individuals accepting the proposition are either all to the left or all to the right of those rejecting it. The profile shown in Table 2 satisfies unidimensional alignment. The corresponding structuring ordering is 4, 1, 5, 2, 3. Individual 5 is the median individual, as defined below.

For a formal definition, fix a profile 〈δ_i〉. For each ϕ∈X, let N_accept-ϕ be the set of individuals accepting ϕ, and N_reject-ϕ the set of individuals not accepting ϕ, i.e.,

and

. A structuring orderingis a linear ordering ω on the set of individuals N, where iωj means that individual i is to the left of individual j. For any two subsets N₁, N₂ ⊆ N, I write N₁ ω N₂ as an abbreviation for [for all i ∈ N₁ and all j∈ N₂ i ωj], i.e., the individuals in N₁ are all to the left of those in N₂.

Definition (List 2003).

A profile 〈δ_i〉 satisfies unidimensional alignment if there exists a structuring ordering ω such that, for every ϕ∈X, either N_accept-ϕ ω N_reject-ϕ or N_reject-ϕ ω N_accept-ϕ.³²

Definition.

The unidimensional alignment domain, R, is the set of all logically possible profiles of individual AR-functions satisfying completeness, weak consistency, deductive closure, and unidimensional alignment.

Suppose that a profile 〈δ_i〉 satisfies unidimensional alignment. Order the n individuals along a corresponding structuring ordering, say ω. There are two cases.

n is odd. Individual m is the median individual with respect to ω if there are as many individuals to m's left as to m's right, formally,

. By unidimensional alignment, the median individual shares the majority view on every proposition. Hence the collective AR-function induced by majority voting is the AR-function of the median individual, δ_m. Provided that the median individual's AR-function is complete, weakly consistent, and deductively closed, so is the collective AR-function.

n is even. Here there exists no single median individual, but a median pair. Individuals m₁ and m₂ are the median pair with respect to ω if they are adjacent (with m₁ left of m₂) and there are as many individuals to m₁'s left as to m₂'s right, formally, (i) m₁ ω m₂, (ii) there is no i∈$N such that m₁ωi and iωm₂, and (iii)

. By unidimensional alignment, if (and only if) the median pair of individuals agrees on a proposition, then their view is also the majority view on that proposition. So the collective AR-function induced by majority voting is the product (the “intersection”) of the AR-functions of the median pair of individuals, δ_m1 δ_m2.³³

. Thus the collective AR-function is “almost” complete.

Definition.

Given a profile 〈δ_i〉,

is almost complete if, for any

implies that

or

.

Proposition 6 (List 2003).

Let δ be propositionwise majority voting. Then δ satisfies anonymity and induces, for every

(with structuring ordering ω), an almost-complete, weakly consistent, and deductively closed collective AR-function

where:

This result mirrors Black's (1948) median voter theorem, albeit in a new context; it is not a corollary of Black's result, as unidimensional alignment and single-peakedness are formally different (for a more general domain restriction result in a preference context, see Sen 1966). By Proposition 6, for any profile satisfying unidimensional alignment, the group's propositional attitudes, determined by majority voting, are almost complete, weakly consistent, and deductively closed. I next show that the resulting priority-to-the-past decision process is path-independent. It is not generally decisive if n is even, but “almost” decisive.

Definition

Given a profile 〈δ_i〉,

) is almost decisive if, for every ϕ∈X,

implies that

) or

).

Theorem 8

Let δ be propositionwise majority voting. Then δ atisfies anonymity and, for any

(with structuring ordering

), M (

) is almost decisive and invariant under changes of the decision-path, where:

So if the group uses majority voting as its aggregation function, then the resulting priority-to-the-past decision process is almost decisive and path-independent for every profile in the restricted domain R. The outcome set of the process is the set of propositions accepted by the median individual (if n is odd) or the intersection of the sets of propositions accepted by the median pair of individuals (if n is even).

Is unidimensional alignment just an artificial condition, or can it be met in plausible situations? Suppose, first, that different individuals disagree on the propositions, but they reach some “meta”-agreement on what their disagreement is about: They agree on a single dimension (e.g., from “most liberal” to “most conservative”) on which their positions can all be placed; each individual takes a position on that dimension. I call it a left–right dimension, but different interpretations are possible. Suppose further that, for each proposition, the extreme positions on the left–right dimension correspond to either clear acceptance or clear rejection of the proposition, and there exists an “acceptance threshold” on the dimension (possibly different for different propositions) such that all individuals to the threshold's left accept the proposition and all individuals to its right reject it (or vice versa). These conditions entail unidimensional alignment; i.e., meta-agreement of the form described can induce unidimensional alignment. This suggests that, to the extent that such meta-agreement is feasible, the escape route from path-dependence opened up by unidimensional alignment may have some promise. But, for my purposes here, the specific interpretation of unidimensional alignment is less relevant than the general insight that path-dependence can be avoided if there is a sufficient level of structure among the individuals' views.

Relaxing Propositionwise Aggregation: The Generalized Aggregation Approach

In my model the group's initial dispositions are defined by propositionwise aggregation, in analogy to Arrow's independence requirement. The aggregation function maps each vector of zeros and ones across individuals to a corresponding single collective disposition of either zero or one. Although plausible in sequential decisions where separate votes are taken on each proposition, this is a restrictive definition of aggregation.³⁵

My assumption of propositionwise aggregation is demanding, requiring that (i) the group's propositional attitude on each proposition be determined by aggregation over the individual views on that proposition (independent of other propositions), and (ii) the same aggregation function be used for all propositions. Pauly and van Hees (2003) and Dietrich (2003) have recently investigated weaker assumptions of propositionwise aggregation—e.g., keeping (i) but dropping (ii)—and obtained results broadly similar to Propositions 4 and 5, suggesting that results similar to Theorems 4 and 5 continue to hold. So even a weaker assumption of propositionwise aggregation may not provide an escape route from path-dependence.

A generalized aggregation function, F, maps each profile 〈δ_i〉 to a collective AR-function δ: X → {0, 1}. Such a generalized aggregation function F provides an escape-route from path-dependence if and only if it generates a weakly consistent and deductively closed AR-function δ.

A function F with this property can be defined by “feeding” each profile 〈δ_i〉 into an appropriate sequential decision process. For each profile 〈δ_i〉, define F(〈δ_i〉) :=δ_F by:

where δ_〈δi〉 is the result of applying propositionwise majority voting to 〈δ_i〉 and Ω_F is some fixed decision-path. By definition, δ_F is (almost) complete, weakly consistent, and deductively closed for any profile

U. Suppose now that δ_F is itself used as an AR-function in a sequential decision process. The weak consistency and deductive closure of δ_F imply that M(δ_F, Ω) is path-independent. In this technical sense, the generalized aggregation function F provides an escape route from path-dependence.

Some aggregation functions that have been proposed for solving the “doctrinal paradox” are of this form. The “premise-based procedure,” for example, applies majority voting to the premises and decides other propositions based on what the votes on the premises imply (Chapman 2002; Pettit 2001a). This can be represented as a generalized aggregation function F, defined in terms of a decision-path Ω_F where the premises occur before the conclusion.

Now the decision process M(δ_F, Ω) is path-independent after δ_F has been determined in the manner outlined. But the path-dependence problem has not been resolved; it has been shifted one level up. The path-independence of M(δ_F, Ω) has been achieved at the cost of making the aggregation function F used for defining δ_F itself “internally” path-dependent. As F is defined by a sequential decision process—i.e. δ_F(ϕ)=1 if and only if

, Ω_F)—the collective AR-function δ_F is not generally invariant under changes of the path Ω_F used for defining F. All the problems I have discussed—particularly strategic manipulability—will therefore recur at the level of the aggregation function F.

So, if the aim is not just to shift path-dependence one level up, the escape route via relaxing propositionwise aggregation seems not very promising. An open challenge is to find a generalized aggregation function that both relaxes the assumption of propositionwise aggregation and is not itself “internally” path-dependent in any problematic way.

CONCLUDING REMARKS

I have modeled sequential decisions over multiple interconnected propositions and illustrated my model by investigating path-dependence in such decisions. The model is intended to complement, not replace, existing social-choice-theoretic models. It seeks to capture a class of decisions that are not straightforwardly captured by classical models: reason-based decisions, where the focus is not only on outcomes, but also on underlying reasons, beliefs, and constraints. Let me briefly summarize my main substantive results.

I have shown that certain violations of perfect rationality by the relevant agent are necessary and sufficient for path-dependence. I have discussed the implications of this result for both individuals and groups. Whereas a perfectly rational individual can avoid path-dependence, a boundedly rational individual may be susceptible to it. An example is an agent who never accepts a proposition and its negation simultaneously but who fails to foresee all the logical implications of the propositions he or she accepts. As path-dependence makes the agent manipulable by the presentation of the propositions in some strategic order, this finding suggests a possible link between bounded rationality and Rikerian “heresthetics,” the art of political manipulation. Exploring this link may be an interesting avenue for future work.

Path-dependence is particularly serious at the collective level. Whereas individuals might try to avoid path-dependence through a self-imposed “discipline” of rationality, no such option is easily available to groups. Under certain conditions, any group that determines its propositional attitudes by aggregation over its members' views will necessarily run the risk of those rationality violations that lead to path-dependence. Path-dependence makes collective decisions vulnerable to manipulation both by agenda-setting and by expression of untruthful views. I have identified some formal escape routes from path-dependence at the collective level, and future research might ask how substantively promising these routes are.

My results are also relevant to the debates on collective consistency referred to above. As noted, social choice theorists have traditionally been skeptical toward the feasibility of a consistent collective set of views. Related to the classical results on collective inconsistencies—usually defined as voting cycles—there are also several results suggesting that collective outcomes are inherently unstable. This, in turn, can be strategically exploited. Schwartz's (1981) universal instability theorem, for instance, shows that, under mild assumptions, any collective outcome involving certain generalized exchange among agents is unstable: Some group of agents has the preference and power to overturn that outcome in favor of some other outcome. Further, some results show that, even when equilibria exist in politics, these may not be unique, which raises the problem of equilibrium selection. The Muller–Satterthwaite theorem (Muller and Satterthwaite 1977; Myerson 1996), for instance, suggests that, if we interpret a political system as a voting game and there are three or more possible outcomes, then, absent a dictator, the game does not generally have a unique equilibrium (if any).

Are my results yet another addition to the large set of social-choice-theoretic inconsistency and instability results? Not quite. Given a decision-path, the outcomes of a priority-to-the-past decision process in my model are neither inconsistent nor unstable. They are consistent, as the priority-to-the-past rule does not permit the acceptance of propositions that conflict with propositions accepted earlier. They are stable, as propositions, once accepted, are not overruled by the priority-to-the-past rule; moreover, each separate decision along the path is itself binary. The constraints implemented by the priority-to-the-past rule may thus reduce the continual drift of collective outcomes that classical models predict in unconstrained settings. All this comes at a price—path-dependence: A group can generate consistent collective outcomes in a priority-to-the-past decision process; it can do so in a relatively democratic manner, by using an aggregation function such as majority voting; but the decision outcomes may be path-dependent and thus affected by the problems I have identified, such as strategic manipulation.

One might speculate whether my results suggest another possible answer to Tullock's (1981) question “Why so much stability?” Perhaps political outcomes are stabilized not only by constraints created by institutional arrangements—as in a structure-induced equilibrium (Shepsle 1979)—but also by constraints created by prior decisions or commitments—as in a priority-to-the-past decision process. Based on my research here, however, it would be premature to propose a new equilibrium concept, but there are interesting avenues for future research. For example, see Page's (2003) work on path-dependent equilibria.

How threatening is path-dependence? Some factors might make it less of a threat. A decision's subject-matter might itself single out the appropriate decision-path. Some propositions might be unambiguously “weightier than,” or “prior to,” others, and the order in which the propositions are to be considered might be uncontroversial. Alternatively, history or empirical contingencies might determine the decision-path, with few opportunities for intervention by agents with a strategic interest. But even then it is worth asking whether or not the given path makes a difference to the outcome. If there is no path-dependence, the legitimacy of an outcome will be under no threat, whether or not the decision-path is disputed; that path is simply irrelevant. But, if there is path-dependence, a justification of the chosen path is crucial. Further, in such cases, even if agents agree on a decision-path, this will not rule out manipulation by the expression of untruthful views. As shown, the mere existence of an alternative decision-path that would change the outcome—although that path is not adopted—may create incentives for strategic expression of untruthful views. So, curiously, path-dependence may matter even when the decision-path is fixed.

I have already mentioned several questions for further research. Others include how far the present results can be generalized to conflict resolution rules other than the priority-to-the-past rule; under what restrictions on the set of admissible, or naturally occurring, decision-paths there is no path-dependence; and whether any domain restrictions such as unidimensional alignment are empirically realistic.

Improving our understanding of path-dependence is an important challenge in the theory of democracy. Many democratic decision processes are sequential, and hence it is important to learn whether, and how, the decision-path matters and what the implications of path-dependence are.

APPENDIX

Proof of Lemma 2: (i) implies (ii).

Suppose that (i) holds. Let Ω be a decision-path such that ϕ∈M(δ, Ω). Choose t such that ϕ is accepted at time t in M(δ, Ω). Let

and ψ, is accepted at some time s < t in M(δ, Ω). As ϕ is accepted at time t, either

or

. If

, then ϕ has the properties required by (ii): ϕ is logically consistent, as ϕ is not a contradiction. If

, then

has the properties required by (ii):

is logically consistent, as

M (δ, Ω), which is logically consistent.

(ii) implies (i). Suppose that (ii) holds. Define Ω as follows. Let

. On

be any one-to-one mapping from

such that

. To continue the path

, k, let Ω be any one-to-one mapping from

, k onto

), where k

. Then Ω has the properties required by (i).

Proof of Theorem 1

Theorem 1 follows immediately from Lemmas 1 and 2

Lemma 3

Suppose that M (δ, Ω) is invariant under changes of Ω. Then, for every

(and any decision-path

if and only if

.

Proof of Lemma 3

Let ϕ∈X. Suppose that M(δ, Ω) is path-independent. We can use any decision-path Ω. For each ϕ∈X, define Ωϕ such that

, and

be any one-to-one mapping from

, where

. Suppose that

. Then, by the definition of M(

. Suppose, conversely, that

, but

. As

does not entail

. But ϕ or

do not occur elsewhere in

, so

, a contradiction. Hence

, as required.

Proof of Theorem 3

Let (i) and (ii) denote the left and right sides of the biconditional. Suppose that

is strongly consistent.

(i) implies (ii). Suppose that there exist

such that

and

, but δ is deductively closed with respect to ϕ. As δ is strongly consistent, M(δ, Ω) is weakly but not strongly path-dependent. As ϕ and

satisfy (i) in Lemma 2, there exists a logically consistent

such that

and

. Then

, by deductive closure of δ. Choose t such that

. Under path

, at time t, ϕ is not accepted, as

,

, this requires

. By the definition of

,

; so M(δ, Ω) is strongly path-dependent, a contradiction. Hence δ is not deductively closed with respect to ϕ.

(ii) implies (i). Suppose that δ violates deductive closure with respect to ϕ∈X, but M(δ Ω) is path-independent with respect to ϕ. There exist a logically consistent

and

such that

,

ϕ but

. By Lemma 2, there exists Ω such that

. As

is path-independent,

for every Ω. Consider

as in the proof of Lemma 3. As

and not

and

for

. But ϕ or

do not occur elsewhere in

, so

(

, contradicting

for every Ω. Hence M(δ, Ω) is weakly path-dependent with respect to ϕ.

Proof of Theorems 4 and 5

Let (i) and (ii) denote the left and right sides of the biconditional in Theorem 5. Theorem 4 follows from the proof that (i) implies (ii).

(i) implies (ii). Let δ be an aggregation function such that, for every

) is decisive and path-independent. By Lemma 3, for every ϕ∈X,

) if and only if

. As

) is decisive and logically consistent, δ_〈δi〉 is complete and strongly consistent, hence weakly consistent and deductively closed. So δ induces, for every

, a complete, weakly consistent, and deductively closed δ_〈δi〉. By Proposition 4, δ violates anonymity. More strongly, by Proposition 5, δ is a dictatorship of some

.

(ii) implies (i). Let δ be a dictatorship of some

. Take any

. Then

. Because

is complete, weakly consistent and deductively closed (hence strongly consistent), so is δ_〈δi〉. By Theorems 2 and 3, M(

) is invariant under changes of the decision-path. The decisiveness of M(

) follows from the completeness of δ_〈δi〉.

Proof of Theorem 6

Suppose that δ is weakly mono- tonic, and

) is path-independent for every

. Take any

, any

, and any

(with

). I prove that

) is at least as close to

). By path-independence of

) in D, Lemma 3 implies that, for every ϕ∈X,

) if and only if

and

) if and only if

, where

. I must show that, for every

) implies

), i.e.,

. Take any ϕ∈X. Let

, and

). Then

, and

(

. The weak monotonicity of δ implies that

, and thus

, as required.

Proof of Corollary of Theorem 6

Suppose that δ is weakly monotonic, and M($\delta,\Omega $) is path-{\it in}dependent for all

. Take any

, any

, and any

(with

). I show

, where

) and

). By Theorem 6,

is at least as close to

as

. By Assumption 1,

.

Proof of Theorem 7

Assume Theorem 7's conditions. Define

and

as in the section Relaxing the Universal Domain, above. Let δ be supermajority voting with parameter (

; δ satisfies anonymity. Take any

. Let

be the set of propositions accepted under the aggregation function

. I show that, for some

. As each

is weakly consistent, there is no

such that ϕ and

both obtain (

supermajorities, so δ_〈δi〉 is weakly consistent. Partition Φ into

equivalence classes of logically equivalent propositions. As δ_〈δi〉 is weakly consistent,

contains at most one member of each proposition-negation pair, so

. Let

be representatives for these equivalence classes. Now

. Then

Also,

. Continuing,

. Because

, hence

. So for some

. But every

is equivalent to one of

. As

is complete, weakly consistent, and deductively closed, it follows that, for every

; hence

is strongly consistent, so is δ_〈δi〉.

Proof of Theorem 8

Let δ be majority voting. By Proposition 6, δ satisfies anonymity and, for any

is almost complete, weakly consistent, and deductively closed, thus strongly consistent. Take any

. By Theorem 3,

) is path-independent. By Lemma 3,

. The almost-completeness of δ_〈δi〉 implies the almost-decisiveness of

); (i) and (ii) follow from (i) and (ii) in Proposition 6.