Obviously, if an iteration procedure terminates for unrestricted agents, then it also always reaches a convergence state for agents who belong in any other of the three cat- egories. This is easy to see thinking of the contrapositive case: if there is an improve- ment path that leads to a cycle for agents who use one of the more restricting policies, then the same path may happen to be followed by unrestricted agents too; thus, non- convergence in the restricting cases implies non-convergence for unrestricted agents. Moreover, strategy-proof aggregation rules can be vacuously said to converge after zero rounds, as their iterative procedure never commences. We know that all inde- pendent and monotonic rules are immune to manipulation (see Theorem 2.2 in the Background, by Dietrich and List, 2007c), and moreover that the quota rules are inde- pendent and monotonic (Theorem 2.1 in the Background, formulated by Dietrich and List, 2007b). Hence, the following corollary holds.
Corollary 5.1. Every profile J is an equilibrium profile with respect to the quota rules.
On the other hand, for some alternative rules it is easy to see that their iteration may never terminate.
Example 5.1. Consider the odd-parity rule F according to which a formula is ac- cepted by the group if and only if an odd number of agents accepts it. Let Φbe an agenda consisting of only one formula φ and its negation φ, and let the groupN contain only two agents, Alice (A) and Bob (B). Suppose that the opinions ofAand B onφdiffer, so thatAacceptsφandB rejects it. In addition, bothAandB strictly prefer the collective decision to agree with their individual judgment rather than to disagree with it. Then, whatever judgments the two agents submit, and independently of whether they are inertia-friendly or inertia-averse and which policy they follow, one of them will always have an opportunity to perform an improvement step. Specifically, if in roundt ruleF acceptsφ, then Bob will be better off by modifying his judgment in roundt`1, conditionally that Alice keeps her submitted opinion fixed. If in roundt rule F rejectsφ, the same holds for Alice. This means that the agents will never be
able to reach an equilibrium. 4
5.2
Iterative Premise-based Procedure under
Full Information
The various incentives of agents to misrepresent their truthful opinions when the premise-based procedure is applied are discussed extensively in Section 3.6. Now, considering a sequence of judgment aggregation rounds under the premise-based pro- cedure, the two central questions are whether and how fast agents will agree on a collective decision from which no-one has an incentive to deviate. We work with a
conjunctive agendaΦ(but note that all our results hold equivalently for a disjunctive agenda too), whose non-negated part is Φ` :
“ ta1, . . . , ak, cu, wherea1, . . . , ak are
propositional variables, and c Ø pa1 ^. . .^akq. Building on the insights of Sec-
tion 3.6, which demonstrates that substantial reasons for manipulability arise mainly when the agents have conclusion-oriented preferences, we will treat this case.31, 32
We begin with studying the iterative procedure when in the first round all the agents submit their truthful opinions. This assumption makes sense for agents that rely on the fact that if the collective outcome is not satisfactory, then in the future they will have the chance to change their judgments. Then, the iteration is guaranteed to terminate after at most one round.
Theorem 5.2. Consider a conjunctive agenda Φ, and fully informed agents with conclusion-oriented preferences. Independently of other assumptions on the agents, the premise-based procedure Fpr converges from the truthful profile in at most one round.
Proof. CallAthe set of agents who accept the conclusion andRthe set of agents who reject it. We assume that all the agents have conclusion-oriented preferences, hence in the first round they have an opportunity for an improvement step if and only if the collective decision oncdisagrees with their own judgment oncand they can change that. The agents inAshould accept all the premises in order to accept the conclusion and have no way to manipulate. Suppose now that an agentiinRhas an incentive to manipulate, which means that the collective result accepts the conclusion in round 1 and agentican make the group reject the conclusion in round 2. In the second round, agents inAhave still no way to manipulate, and all the agents inRare happy because they obtain their desirable outcome. Hence, everyone who is truthful has no reason to manipulate and the only untruthful agent cannot return to her truthful judgment, as that would make her worse off. We conclude that the procedure stops in one round. We continue with the investigation of agents who may not submit their truthful judg- ment at the beginning of the iterative procedure. This assumption makes sense for example when considering individuals who may get involved in more sophisticated planning and try to guide the procedure towards the most desirable result for them in the long-term. Theorems 5.4 and 5.5 indicate that the premise-based procedure is still guaranteed to reach an equilibrium, but it may take a linear number of rounds with respect to the number of the agentsn. Lemma 5.3 is quite intuitive, and plays an important role in the sequel. It is formally proven in the Appendix.
31Recall that an agentiwith truthful judgmentJihasconclusion-oriented preferencesif her prefer-
ence relationÁiis such thatJ ąi J1 if and only ifc PJiXJ andc R JiXJ1or c PJiXJ and cRJiXJ1; andJ„iJ1if and only ifcPJiXJandcPJiXJ1or cPJiXJand cPJiXJ1.
32Since conclusion-oriented agents are indifferent about the collective decision on all the premises,
it is meaningless to consider them outcome-focused. Thus, one could assume that they follow some of the other policies, but this will not affect our results by any means.