Basic Settings

The main difference between our model and that of Feddersen and Pesendorfer is that we

assume the quality of the signal to be ambiguous to jurors rather than unique and commonly known (measured). We assume that the correlation between the private signal and the true

state of the world even though positive is nevertheless ambiguous or partially ambiguous as defined in Chew et al. (2013). Introducing ambiguity in the accuracy with which signals

are obtained allows us to depart from Feddersen and Pesendorfer set-up and to consider far more common and realistic situations, where the objective probability of the reliability of

the evidence is unmeasured, in which jurors cannot assign a specific level of reliability to the evidence they are provided with, say, during a trial, and, furthermore, even if they were

able to do so with a certain degree of confidence they would not necessarily agree on what that level of reliability ought to be. Our next crucial assumption in this model, is to consider

jurors who are different by nature, with respect to their beliefs about the accuracy of the information they are confronted with. We will provide more details on this, when describing

jurors’ types.

Here, we assume that the accuracy of the signal each juror receives about the defendant being guilty or innocent belongs to the setP, wherePneeds not be a singleton – contrary to Feddersen and Pesendofer set-up. In this chapter, we concentrate on the case of an exogenous two-point non-common prior for the accuracy with which signals are believed to come about,

captured byP ={p,p}¯ . Alternatively, in chapter 3 we consider the entire set of values this accuracy can take within a closed interval, thereby capturing the extent of its potentially

continuous degree of ambiguity within that interval, withP = [p,p¯].

Hence, if the non-common prior about the quality of private information can take only one of two possibly distinct values,P={p,p}¯ , we can concentrate on the following infor- mative signal cases, for which 1/2<p<p¯<1. For example, the evidence hints toward the

1.4 Non-common Prior: The Two-point Prior Model 14

defendant being guilty with either probability 60% or 90%. This could be so, because the accuracy of a test/evidence is not universally accepted to be at a particular level, so that in

the absence of any extra information regarding the probability distribution of pand ¯p, jurors are free to believe either of those levels for the accuracy to be true. Only one would be, for

a specific case, but jurors have no way to verify that information when making their decisions.

We retain the assumption as in Feddersen and Pesendorfer (1998) that jurors have the same preferences with respect to reaching a right verdict (convicting the guilty, or acquitting

the innocent) or a wrong verdict (convicting the innocent, or acquitting the guilty).

We distinguish two types of jurors/voters - sceptic and trusting - according to which level of accuracy for their private information/signal they eventually adopt. We assume

voters form their subjective belief regarding the probability measure as follows:Prj(g|G) = Pr_j(i|I) = (1−δj)p+δjp¯, whereδj is the index of trust for voter j. Next, we concentrate attention to the extreme belief case, for which δj∈ {0,1}, which means that each of the

jurors can either be of an extreme sceptic or a fully trusting type. The radical trusting

juror believesPr(g|G) =Pr(i|I) = p¯, with δj =1; whereas the extreme sceptic believes

Pr(g|G) =Pr(i|I) = p, with δj=0. Assume there exists a proportion m of voters who

belong to the extreme sceptic-type and a proportion 1−mof voters who belong to the fully

trusting-type, withm∈[0,1].

To be specific, ourambiguous jury voting gamecan be described as follows:

(i) Nature first chooses the true state of the world, either "Guilty"-G, or "Innocent"-I, with

equal probability, 1/2.

(ii) Nature makesnindependent random draws of signalss∈ {g,i}fornvoters, j=1,· · ·,n, from random variable with precisionP ={p,p}¯ , where 1/2<p<p¯<1. No further probabilistic information about the signal precision is provided.

1.4 Non-common Prior: The Two-point Prior Model 15

(iii) Nature assigns types randomly and independently to all voters, such that each voter has

probabilityµ of being sceptic and 1−µ of being trusting. And the realised shares of

extreme sceptics and fully trusting jurors within a jury/committee are denoted bymand

1−m, respectively.

(iv) Each voter observes his private signal and votes according to his strategy(σj(i),σj(g)), which represents the probability with which that voter jvotes for conviction conditional on receiving a signal hinting toward innocence or guiltiness, respectively.

(v) After all voters simultaneously cast their votes, the collective decision is determined according to the given voting rule.

In the remainder of this chapter, we use this minimal set-up to explore the question of

how collective deliberation, such as voting by jurors in a trial, is affected by the reality that jurors have multiple/distinct priors. We seek to address how the results obtained under the

common prior assumption change when multiple priors are allowed for, and, more impor- tantly, whether such changes can ever offset the negative impact that more demanding voting

rules have in exacerbating the occurrence of mistakes in the judicial system. Do multiple priors help mitigate or even eliminate the paradoxical result that the more demanding the

hurdle for conviction – on the spectrum from simple majority to unanimity – the more likely it is that a jury will convict an innocent defendant?

1.4 Non-common Prior: The Two-point Prior Model 16