Lemma 5.3. Consider a conjunctive agenda Φ, and a conclusion-oriented agent i, who truthfully accepts the conclusion in Φ. If agentihas an opportunity to perform an improvement step in round tof the iterative premise-based procedure, then she is untruthful in round t, and her unique best improvement step is performed by being truthful in roundt`1, (i.e., accepting all the premises and the conclusion).
Theorem 5.4. Consider a conjunctive agendaΦ, and fully informed, inertia-friendly agents with conclusion-oriented preferences, who follow any policy. The premise- based procedureFpr converges from any initial profile in at most2nrounds.
Proof. Call A the set of agents who accept the conclusion and R the set of agents who reject it. By Lemma 5.3, in any round t, an agenti P A has an opportunity to make a best improvement step if and only if she moves to her truthful opinion from an insincere one. This move can be realized at most once for every agent in A. On the other hand, an agentj P Rhas an opportunity to perform an improvement step in roundt if she can choose a judgment which makes a previously accepted conclusion be rejected by the group. How many times would agents inR have the chance to flip the result on the conclusion? If the conclusion is collectively rejected, only an agent iP Acan turn it to be accepted again in the future, by submitting her truthful judgment that she was not submitting before. This can happen at most|A|times, because as we saw, every agent inAwill move at most once. In total, no-one will have an available improvement step after at most|A| ` |A| ď2nrounds.
Theorem 5.5. Consider a conjunctive agendaΦ, and fully informed, inertia-averse agents with conclusion-oriented preferences, who follow any policy. The premise- based procedureFpr converges from any initial profile in at most3nrounds.
The main insight of Theorem 5.5 is that inertia-averse agents who reject the conclusion may have an opportunity to perform an improvement step, by being truthful, also in case the group already rejects the conclusion. This fact adds at mostn rounds in the result of Theorem 5.4. The detailed proof can be found in the Appendix.
5.3
Adding Partial Information
In this section we focus on iterative procedures where in every round the agents are partially informed about the submitted profile of their peers and the information they hold is described by some JIF π(see Chapter 3, Section 3.1). Now, the agents make improvement steps that take their information into account.
5.3.1
Improvement Dynamics under Partial Information
We consider risk-averse agents in accordance with our analysis in the previous chap- ters, and we say that an agent will be better off by submitting a new judgment ifp1q
there is a scenario consistent with her information about the current round under which she is able to achieve a strictly preferable outcome for her in the next round and p2q
there is no scenario under which her new judgment will lead her to a strictly less de- sirable outcome. Analogously to the case of full information, when the agent is not able to alter the collective outcome, she has two options: either to keep her current judgment (even if it is an untruthful one), or to change her judgment and report her truthful opinion (in case she was not doing that before). These two types of individuals are calledinertia-friendlyandinertia-avertrespectively, as before. Consider a profile
Jt “ pJi,t,J´i,tqdeclared in roundtof the aggregation procedure for ruleF.
Definition 5.4. Consider an inertia-friendly agent i, with truthful judgment Ji and
preferencesÁi. Agenti, under the information described by the JIFπ, has an oppor-
tunity to perform animprovement stepin roundtusing the judgmentJi,t`1, if
1. FpJi,t`1,J1´i,tqąi FpJi,t,J1´i,tq, for someJ
1
´i,t PW
1,π,Jt
i and
2. FpJi,t`1,J2´i,tqÁi FpJi,t,J2´i,tq, for all otherJ
2
´i,t PW
1,π,Jt i
Definition 5.5. Consider an inertia-averse agenti, with truthful judgmentJi and pref-
erencesÁi. Agenti, under the information described by the JIFπ, has an opportunity
to perform animprovement stepin roundtusing the judgmentJi,t`1, if
1. FpJi,t`1,J1´i,tqąi FpJi,t,J1´i,tq, for someJ
1
´i,t PW
1,π,Jt
i and
2. FpJi,t`1,J2´i,tqÁi FpJi,t,J2´i,tq, for all otherJ
2
´i,t PW
1,π,Jt i
or if there is noJ1
i,t`1 P JpΦqsuch that the above conditions hold, but it isJi,t`1 “
Ji ‰Ji,t, andFpJi,t`1,J1´i,tqÁi FpJi,t,J1´i,tq, for all J
1
´i,t PW
1,π,Jt
i .
5.3.2
Best Improvement Steps under Partial Information
Similarly to the case of full information, we assume that partially informed agents are also able to distinguish their best improvement steps and, if they have the opportu- nity, perform one of them. Consider a round t of the iterative procedure for rule F, where the declared profile of the group is Jt “ pJi,t,J´i,tq, and agent i holds the
preferences Ái and her information is described by the JIF π. Then, a judgment set
J P JpΦqisundominated in the standard game-theoretical sense if there is no other opinion J1 such that
p1q FpJ1,J1´i,tq ąi FpJ,J1´i,tq, for some J
1
´i,t P W
1,π,Jt
i and
p2q FpJ1,J2´i,tq Ái FpJ,J2´i,tq, for all other J
2
´i,t P W
1,π,Jt
i (see Definition 3.5).
Then, we define the setBIPi,t of the judgments that can be used by agentifor abest
improvement stepunder her partial information in roundt:33 33BIP
i,t depends on more parameters, such as the ruleF, the agenti’s preferencesÁi, as well as
5.3. Adding Partial Information 63
• Since the agents are truth-biased, if agent i’s truthful judgment Ji is undomi-
nated and can be used for an improvement step in roundt, then this will be the agent’s only best improvement step. That is,BIPi,t :“ tJiu.
• Otherwise, BIPi,t :“ tJ P JpΦq : J is undominated and can be used for an
improvement step by agentiin roundtu.
5.3.3
Response Policies under Partial Information
We now need to refine theresponse policynotion for agents under partial information. With regard to round-focused, truth-focused and unrestricted agents, the definitions of their policies under full and partial information are totally analogous. This is the case because these policies are not concerned with the (partially unknown) collective outcome; they only look at the agents’ previously submitted judgment, their truthful judgment, or nothing at all, respectively. Consider agent i that has the preference relationÁi and holds the truthful judgmentJi, while the partial profile of the rest of
the group in roundtisJ´i,t, and agentiis partially informed about it, by the JIFπ.
Definition 5.6. The agent chooses among her available best improvement steps under the partial information she holds (randomly if there are more than one). Moreover,
• If the agent isround-focused: The Hamming-distance between what the agent submits in roundt`1and what she was submitting in roundtis minimized: Ji,t`1 P arg min
JPBIPi,t
HpJ, Ji,tq.
• If the agent is truth-focused: The Hamming-distance between what the agent submits and her truthful judgment is minimized, i.e.,
Ji,t`1 P arg min
JPBIPi,t
HpJ, Jiq.
• If the agent isunrestricted: Ji,t`1 PBIPi,tand no further restriction is imposed.
However, adapting outcome-focused agents to the framework of partial information requires some further consideration; according to their policy, they try to submit a judgment close to the collective decision, which they may not be able to fully predict. Recall Definitions 5.4 and 5.5. Suppose that judgmentJi,t`1offers an opportunity for
an improvement step to agenti. We define the judgment setJA
i,t`1, which is the most
desirable collective outcome that agentiaims for, induced by the judgment setJi,t`1:
1. If all possible outcomes that the agent can achieve by changing her judgment are equally valuable for her, then she has an improvement step only if she is inertia- averse, and that step uses her truthful judgmentJi. So,Ji,tA`1 :“FpJi,Ji,tq;
2. Otherwise, there exists someJ1´i,t P Wi1,π,Jt, such thatp1qFpJi,t`1,J1´i,tq ąi
FpJi,t,J1´i,tqandp2qFpJi,t`1,J2´i,tqÁi FpJi,t,J2´i,tq, for allJ
2
´i,t P W
1,π,Jt i
(call this condition forJ1
´i,tp‹q). Then,
Ji,tA`1 :“ max
J1
´i,ts.t. p‹q
FpJi,t`1,J1´i,tq
Finally, we can say that an outcome-focused agent tries to minimize the distance with one of her targeted, most desirable collective decisions.
• If the agent isoutcome-focused: The Hamming-distance between what the agent submits and her most desirable collective outcome she aims for is minimized, i.e.,Ji,t`1 Parg min
JPBIPi,t
HpJ, JAq.
The preceding definition is simple to understand via an example for the case of the plurality rule (together with a lexicographic tie-breaking rule), when the agents are only informed about the currently winning judgment set in every round. Suppose that an agent strictly prefers a judgment set J1 over J and she is indifferent between
all the other judgment sets and J. Nevertheless, suppose that it happened and the agent is currently submitting judgment J, which is winning. The agent cannot be worse off by changing her judgment, so she considers making an improvement step and targets the win of her more desirable judgment setJ1. Then, she has at least two
options to choose from: she can either withdraw her support from judgment J and submit a different judgment J2 (looking at the scenario she considers possible where
judgmentJ1is already close enough to win afterJ loses one vote), or she can directly
submitJ1. An agent who is outcome-focused will choose the second option.