Accessible Unlicensed Requires Authentication Published by De Gruyter May 19, 2018

Non-Congruent Views about Signal Precision in Collective Decisions

Addison Pan, Simona Fabrizi and Steffen Lippert

Abstract

We relax the standard assumptions in collective decision-making models that voters can not only derive a perfect view about the accuracy of the information at their disposal before casting their votes, but can, in addition, also correctly assess other voters’ views about it. We assume that decision-makers hold potentially differing views, while remaining ignorant about such differences, if any. In this setting, we find that information aggregation works well with voting rules other than simple majority: as voters vote less often against their information than in conventional models, they can deliver higher-quality decisions, including in the canonical 12 jurors case. We obtain voting equilibria with many instances, in which other voting rules, including unanimity, clearly outperform simple majority.

JEL Classification: C63; C7; C92; D7; D8

Funding statement: This work was supported by Royal Society of New Zealand, Funder Id: 10.13039/501100001509, Grant Number: Marsden Grant UOA1617 and Massey University, Funder Id: 10.13039/501100001554, Grant Number: Massey University Research Fund (MURF) 2015.

Acknowledgements

We are grateful to the editor, Andrés Perea, and three anonymous referees for their valuable comments and suggestions. This paper is based on, and extends, results developed in Chapters 1–2 of Pan’s PhD thesis, with Fabrizi as the main supervisor. Pan thanks her co-supervisors Thomas Pfeiffer and Matthew Ryan for their continuous support, encouragement and inputs; Ben Greiner, Clemens Puppe, and Oscar Lau, as PhD thesis examiners; participants in the 2nd Massey Business School PhD Symposium (Manawatu), August 2015; Economics Seminars, Massey University (Albany), November 2015, Karlsruhe Institute of Technology, May 2016, and Free University of Bozen, April 2017; 2016 Australasian Economic Theory Workshop, Monash University; 2016 Summer Workshop of the Centre for Mathematical Social Science, University of Auckland; 2016 European Meeting on Game Theory, Odense; and 2016 Conference on Logic and the Foundations of Game and Decision Theory, Maastricht. Pan is also especially grateful to Simon Grant, for the valuable suggestions during the 28th PhD Conference in Economics and Business, University of Queensland, November 2015. Fabrizi thanks participants in the 17th APET Conference, Rio de Janeiro, July 2016; APET Workshop on Democracy, Public Policy, and Information, Deakin University, July 2016; and 6th Microeconomic Theory Workshop, Victoria University of Wellington, October 2016. Support by the 2015 Massey University Research Fund to Fabrizi and Pan and by Marsden Grant UOA1617 to Fabrizi and Lippert are gratefully acknowledged. The authors are grateful to Patrick Girard, Gabriele Gratton, and John Hillas from their comments on an early stage idea of this work, and wish to thank Valery Pavlov for his inputs and support for the ethical clearance to conduct human-subject laboratory experiments at the University of Auckland; and for facilitating access to DECIDE (Laboratory for Business Decision Making), based at the University of Auckland. The authors only are responsible for any remaining errors.

A Proofs

Proof of Proposition 1.

If the voting equilibrium is informative, each vote will have to reveal exactly what signal each juror receives.

Assume there to be a skeptical juror who receives an innocent signal i. The posterior probability that this juror holds about a defendant being guilty, conditional on being pivotal, and given unanimity voting, evaluated at σ(s,p_)={σ(i,p_)=0,σ(g,p_)=1} would reduce to

(A)β(i,p_)(n)=11+p_(1p_)n1(1p_)p_n1.

Hence, this juror will vote for acquittal if and only if β(i,p_)(n)q.

Vice versa, if this juror received a guilty signal g, the posterior probability that this juror would hold about a defendant being guilty, conditional on being pivotal, and given unanimity voting evaluated at σ(s,p_)={σ(i,p_)=0,σ(g,p_)=1} would be

(B)β(g,p_)(n)=11+(1p_)np_n.

This demonstrates that this same juror would have voted for conviction if and only if β(g,p_)(n)>q. Therefore, for a skeptical juror to always vote in accordance with the signal received, both conditions need to be satisfied at the same time, which happens if and only if β(i,p_)(n)q<β(g,p_)(n).

Consider equilibrium candidate σ(i,p_)=0 and 0σ(g,p_)<1. If this were an equilibrium, we would have

β(g,p_)(n)=11+(1p_)(γI(g,p_))n1p_(γG(g,p_))n1>11+(1pˉ)npˉnq,

which implies that a voter with a guilty signal would have a strict incentive to vote to convict, contradicting 0σ(g,p_)<1. So σ(i,p_)=0 and 0σ(g,p_)<1 is not an equilibrium.

Consider equilibrium candidate 0<σ(i,p_)1 and σ(g,p_)=1. If this were an equilibrium, we would have

β(i,p_)(n)=11+p_(γI(s,p_))n1(1p_)(γG(s,p_))n1<11+(1pˉ)npˉnq,

which implies that a voter with an innocent signal would have a strict incentive to acquit, contradicting 0<σ(i,p_)1. So 0<σ(i,p_)1 and σ(g,p_)=1 is not an equilibrium.

Similarly, we can derive conditions for the trusting juror type to vote informatively. These will dictate that β(i,pˉ)(n)q and β(g,pˉ)(n)>q, where

(C)β(i,pˉ)(n)=11+pˉ(1pˉ)n1(1pˉ)pˉn1q;

and

(D)β(g,pˉ)(n)=11+(1pˉ)npˉn>q,

which is equivalent to β(i,pˉ)(n)q<β(g,pˉ)(n).

Such interval for the posterior beliefs exists if and only if

max{β(i,p_)(n),β(i,pˉ)(n)}<min{β(g,p_)(n),β(g,pˉ)(n)}.

We know that (1) min{β(g,p_)(n),β(g,pˉ)(n)}=β(g,p_)(n) and (2) β(g,p_)(n)>β(i,p_)(n). Hence, there exists such a q for which informative voting is an equilibrium if

β(i,pˉ)(n)<β(g,p_)(n)11+1pˉpˉn2<11+1p_p_n1pˉpˉn2n>1p_p_.

For n finite there are pairs (p_,pˉ) for which this inequality holds.

Proof of Corollary 1

By Proposition 1, informative voting is an equilibrium if and only if q[β(i,p)(n),β(g,p)(n)[ for each p{p_,pˉ}. There is no such q if

max{β(i,p_)(n),β(i,pˉ)(n)}>min{β(g,p_)(n),β(g,pˉ)(n)}.

We know that (1) min{β(g,p_)(n),β(g,pˉ)(n)}=β(g,p_)(n) and (2) β(g,p_)(n)>β(i,p_)(n). Hence, there exists no q for which informative voting is an equilibrium if

β(i,pˉ)(n)>β(g,p_)(n)11+1pˉpˉn2>11+1p_p_n1pˉpˉn2n<1p_p_.

Because limn1pˉpˉn2n=1pˉpˉ, this inequality holds for any p_<pˉ.

Proof of Proposition 2.

If β(i,p_)(n)=q<β(g,p_)(n), we must have 0<σ(i,p_)<1, that is

11+p_(γI(s,p_))n1(1p_)(γG(s,p_))n1=q,

which implies

(γI(s,p_)γG(s,p_))n1=1qq1p_p_.

Thus, in the case when the pivotal voter receives g, it must be true that

11+(1p_)(γI(g,p_))n1p_(γG(g,p_))n1=11+1qq1p_p_1p_p_>q,

that is β(g,p_)(n)>q, which implies σ(g,p_)=1.

And when 0<σ(i,p_)<1 and σ(g,p_)=1,

β(i,p_)(n)=11+(p_1p_)(p_σ(i,p_)+(1p_)p_+(1p_)σ(i,p_))n1=q,

which implies that

σ(i,p_)=[(1q)(1p_)qp_]1(n1)p_(1p_)p_[(1q)(1p_)qp_]1(n1)(1p_).

Because σ(i,p_) has to be strictly between 0 and 1, we require q>1p_.

If β(i,p_)(n)<q<β(g,p_)(n), we must have σ(i,p_)=0 and σ(g,p_)=1 following the equilibrium condition.

If β(g,p_)(n)=q, we must have 0<σ(g,p_)<1, that is

11+(1p_)(γI(s,p_))n1p_(γG(s,p_))n1=q,

which implies

(γI(s,p_)γG(s,p_))n1=1qqp_1p_.

Then, in the case when the pivotal voter’s private signal is i, it must be true that

11+p_(γI(i,p_))n1(1p_)(γG(i,p_))n1=11+1qqp_1p_p_1p_<q,

that is β(i,p_)(n)<q, which implies σ(i,p_)=0.

However, this is not an equilibrium as we have shown in the proof of Proposition 1, i.e., given σ(i,p_)=0, we must have σ(g,p_)=1, which contradicts the assumption 0<σ(g,p_)<1. Therefore, σ(i,p_)=0 and 0<σ(g,p_)<1 is not a responsive equilibrium.

If q<β(i,p_)(n), according to the equilibrium condition, we must have σ(i,p_)=1 and σ(g,p_)=1. However, this is not a responsive voting strategy.

If β(g,p_)(n)<q, we must have σ(i,p_)=0 and σ(g,p_)=0, which is not a responsive voting strategy either.

The same logic applies to type (i,pˉ) voters and type (g,pˉ) voters. Therefore, any responsive voting equilibrium for those who believe in pˉ must be of the form 0σ(i,pˉ)<1 and σ(g,pˉ)=1. And σ(i,pˉ) is between 0 and 1 as long as q>1pˉ, where

σ(i,pˉ)=[(1q)(1pˉ)qpˉ]1(n1)pˉ(1pˉ)pˉ[(1q)(1pˉ)qpˉ]1(n1)(1pˉ).

Proof of Proposition 5.

Assume a supermajority rule, n2+1<kˆ. Then β(i,p_)(αn)<β(i,pˉ)(αn) and β(g,p_)(αn)<β(g,pˉ)(αn). We have β(i,pˉ)(αn)<β(g,p_)(αn) if and only if

β(i,pˉ)(αn)=11+pˉ1pˉ1pˉpˉkˆ1pˉ1pˉnkˆ<11+1p_p_1p_p_kˆ1p_1p_nkˆ=β(g,p_)(αn)1pˉpˉ2kˆn2>1p_p_2kˆn.

Suppose β(i,pˉ)(αn)q<β(g,p_)(αn) and 1pˉpˉ2kˆn2>1p_p_2kˆn are violated. In this case, we can demonstrate that voters who believe in pˉ prefer to randomize their vote to convict.

First assume that, if the juror knew all n signals, kˆ1 of which were g, he would strictly prefer to vote to convict, that is,

q<(1pˉ)(pˉ)kˆ1(1pˉ)nkˆ(1pˉ)(pˉ)kˆ1(1pˉ)nkˆ+pˉ(1pˉ)kˆ1pˉnkˆ.

Then, if β(i,pˉ)(αn)=q<β(g,pˉ)(αn), we must have 0<σ(i,pˉ)<1, that is

11+pˉ1pˉ(γI(s,pˉ)γG(s,pˉ))kˆ1(1γI(s,pˉ)1γG(s,pˉ))nkˆ=q,

and we can derive

(γI(s,pˉ)γG(s,pˉ))kˆ1(1γI(s,pˉ)1γG(s,pˉ))nkˆ=(1q)(1pˉ)qpˉ.

Thus, in the case when the pivotal voter receives signal g, it must be true that

11+1pˉpˉ(γI(s,pˉ)γG(s,pˉ))kˆ1(1γI(s,pˉ)1γG(s,pˉ))nkˆ=11+1qq1pˉpˉ1pˉpˉ>q,

that is β(g,pˉ)(αn)>q, which implies σ(g,pˉ)=1.

Therefore, since for 0<σ(i,pˉ)<1 and σ(g,pˉ)=1 we have γI(s,pˉ)=1pˉ+pˉσ(i,pˉ) and γG(s,pˉ)=pˉ+(1pˉ)σ(i,pˉ), we obtain

11+pˉ1pˉ(1pˉ+pˉσ(i,pˉ)pˉ+(1pˉ)σ(i,pˉ))kˆ1(pˉpˉσ(i,pˉ)1pˉ(1pˉ)σ(i,pˉ))nkˆ=q,

which implies that

σ(i,pˉ)=1pˉ(1+((1qq)(pˉ1pˉ)kˆn1)1kˆ1)(1pˉ)((1qq)(pˉ1pˉ)kˆn1)1kˆ1pˉ.

Thus,

σ(i,pˉ)=pˉ(1+f(pˉ,kˆ))1pˉf(pˉ,kˆ)(1pˉ),

where

f(pˉ,kˆ)=((1q)q((1pˉ)pˉ)nkˆ+1)1/(kˆ1).

Because σ(i,pˉ) has to be strictly between 0 and 1, we require q<11+(pˉ1pˉ)nkˆ+1.

The proofs for the minority and simple majority rules are analogous.

Proof of Corollary 2.

Informative voting is an equilibrium if and only if q[β(i,p)(αn),β(g,p)(αn)[ for each type p{p_,pˉ}. (1) Assume a minority rule, kˆ<n2. Then, keeping in mind 2kˆn2<2kˆn<0, for n, such an interval does not exist, because

limn1p_p_2kˆn2limn1pˉpˉ2kˆnp_pˉ,

holds. (2) Assume the simple majority rule, n2<kˆ<n2+1. Then, keeping in mind 2kˆn2<0<2kˆn, for n, such an interval does not exist, because

limn1p_p_2kˆn2limn1p_p_2kˆnp_p_,

holds. (3) Assume a supermajority rule, n2+1<kˆ. Then, keeping in mind 0<2kˆn2<2kˆn, for n, such an interval does not exist, because

limn1pˉpˉ2kˆn2limn1p_p_2kˆnp_pˉ,

holds.

Proof of Proposition 7.

With a proportion m of skeptical voters and a proportion 1m of trusting voters, the probability that a defendant is indeed guilty, given a conviction verdict is pronounced is

Pr(G|C)=11+((γˆI(s,p_))mα(γˆI(s,pˉ))(1m)α(1γˆI(s,p_))m(1α)(1γˆI(s,pˉ))(1m)(1α)(γˆG(s,p_))mα(γˆG(s,pˉ))(1m)α(1γˆG(s,p_))m(1α)(1γˆG(s,pˉ))(1m)(1α))n.

For n this probability that a defendant is guilty if convicted approaches one; and the probability that a defendant is innocent if convicted approaches zero.

This is so, since when n, we have

limnf(p_,kˆ)=limn((1q)q((1p_)p_)nkˆ+1)1/(kˆ1)=(1p_p_)1αα,

and

limnf(pˉ,kˆ)=limn((1q)q((1pˉ)pˉ)nkˆ+1)1/(kˆ1)=(1pˉpˉ)1αα.

and, therefore, we have

σ(i,p_)=p_(1+f(p_,kˆ))1p_f(p_,kˆ)(1p_)=p_(1+(1p_p_)(1α)/α)1p_(1p_p_)(1α)/α)(1p_),

and

σ(i,pˉ)=pˉ(1+f(pˉ,kˆ))1pˉf(pˉ,kˆ)(1pˉ)=pˉ(1+(1pˉpˉ)(1α)/α)1pˉ(1pˉpˉ)(1α)/α)(1pˉ).

Therefore, in the limit case of n, we must have

γˆG(s,p_)=pˆ+(1pˆ)p_(1+(1p_p_)(1α)/α)1p_(1p_p_)(1α)/α)(1p_),γˆI(s,p_)=pˆp_(1+(1p_p_)(1α)/α)1p_(1p_p_)(1α)/α)(1p_)+(1pˆ),γˆG(s,pˉ)=pˆ+(1pˆ)pˉ(1+(1pˉpˉ)(1α)/α)1pˉ(1pˉpˉ)(1α)/α)(1pˉ),γˆI(s,pˉ)=pˆpˉ(1+(1pˉpˉ)(1α)/α)1pˉ(1pˉpˉ)(1α)/α)(1pˉ)+(1pˆ).

Hence, it is easy to verify that γˆG(s,p_)>γˆG(s,pˉ)>γˆI(s,p_)>γˆI(s,pˉ). Therefore, a sufficient condition for

limnPr(G|C)=limn11+(γˆI(s,pˉ)γˆG(s,pˉ))(1m)αn(γˆI(s,p_)γˆG(s,p_))mαn(1γˆI(s,pˉ)1γˆG(s,pˉ))(1m)(1α)n(1γˆI(s,p_)1γˆG(s,p_))m(1α)n=1

is (1m)γG(s,pˉ)+mγG(s,p_)>α and (1m)γI(s,pˉ)+mγI(s,p_)<α.

If γˆG(s,p_)>γˆG(s,pˉ)>α>γˆI(s,p_)>γˆI(s,pˉ), we have:

limn(γˆI(s,pˉ)γˆG(s,pˉ))(1m)αn(γˆI(s,p_)γˆG(s,p_))mαn(1γˆI(s,pˉ)1γˆG(s,pˉ))(1m)(1α)n(1γˆI(s,p_)1γˆG(s,p_))m(1α)n=0,

implying limnPr(G|C)=1.

B Simulations

Table 5:

Type I errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.52, q=0.9 and different voting rules.

kˆ789101112
σ(i,p_)000000
σ(g,p_)000000
σ(i,pˉ)00.0229760.1435520.2777690.4226450.5759
σ(g,pˉ)111111
n1Type I error
12000000
11000000
10000000
9000000
8000000
7000000
6000000
51.28E-0500000
48.45E-055.17E-060000
30.0003143.75E-053.04E-05000
20.0008640.0001510.0002180.0001800
10.0020.0004480.000860.00120.00110
00.00390.00110.00250.00450.00670.0069

  1. † The corresponding voting equilibrium for the case of n=12 and kˆ=7 is not responsive, that is σ(i,p_)=σ(g,p_)=σ(i,pˉ)=0 and σ(g,pˉ)=1, since 2kˆn2=0 and β(i,p_)=β(i,pˉ)<β(g,p_)q<β(g,pˉ). †The corresponding voting equilibrium for the case of n=12 and kˆ is either equal to 8, 9, 10, 11, or 12 is not responsive, that is σ(i,p_)=σ(g,p_)=0, 0<σ(i,pˉ)<1, and σ(g,pˉ)=1, since 2kˆn2>0, q<1/(1+(p_/(1p_))nkˆ1) and β(i,p_)<β(g,p_)q<β(i,pˉ)<β(g,pˉ), with p_<1/(1+((1pˉ)/pˉ)(2kˆn2)/(2kˆn)).

Table 6:

Type I errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.55, q=0.9 and different voting rules.

kˆ789101112
σ(i,p_)000000
σ(g,p_)000001
σ(i,pˉ)00.0229760.1435520.2777690.4226450.5759
σ(g,pˉ)111111
n1Type I error
12000004.1E-09
11000001.35E-08
10000004.47E-08
9000001.48E-07
8000004.88E-07
7000001.61E-06
6000005.33E-06
51.28E-0500001.76E-05
48.45E-055.17E-060005.81E-05
30.0003143.75E-053.04E-05000.000192
20.0008640.0001510.0002180.0001800.000634
10.0020.0004480.000860.00120.00110.0021
00.00390.00110.00250.00450.00670.0069

  1. † The corresponding voting equilibrium for the case of n=12 and kˆ=12 is semi-mixed, that is σ(i,p_)=0, σ(g,p_)=1, 0<σ(i,pˉ)<1, and σ(g,pˉ)=1, since 2kˆn2>0, q<1/(1+(p_/(1p_))nkˆ1) and β(i,p_)q<β(g,p_)β(i,pˉ)<β(g,pˉ), with p_1/(1+((1pˉ)/pˉ)(2kˆn2)/(2kˆn)).

Table 7:

Type I errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.6, q=0.9 and different voting rules.

kˆ789101112
σ(i,p_)0000.0325130.1452010.258831
σ(g,p_)001111
σ(i,pˉ)00.0229760.1435520.2777690.4226450.575916
σ(g,pˉ)111111
n1Type I error
12006.22E-051.45E-052.69E-052.07E-05
11008.7E-052.44E-054.35E-053.36E-05
10000.0001210.0000417.04E-055.45E-05
9000.0001686.84E-050.0001138.85E-05
8000.0002320.0001130.0001820.000144
7000.0003190.0001860.0002910.000233
6000.0004360.0003040.0004640.000379
50.00002800.0005910.0004910.0007360.000614
48.45E-055.17E-060.0007980.0007860.00120.000997
30.0003143.75E-050.00110.00120.00180.0016
20.0008640.0001510.00140.00190.00280.0026
10.0020.0004480.00190.0030.00440.0043
00.00390.00110.00250.00450.00670.0069

  1. † The corresponding voting equilibrium for the case of n=12 and kˆ equal to either 10, 11, or 12 is a randomizing one, that is 0<σ(i,p_)<1, σ(g,p_)=1, 0<σ(i,pˉ)<1, and σ(g,pˉ)=1, since 2kˆn2>0, q<1/(1+(pˉ/(1pˉ))nkˆ1) and q<β(i,p_)<β(g,p_)β(i,pˉ)<β(g,pˉ), with p_1/(1+((1pˉ)/pˉ)(2kˆn2)/(2kˆn)).

Table 8:

Type I errors of the 12-Person Jury case, given pˆ=0.8, pˉ=0.9, p_=0.7, q=0.9 and different voting rules.

kˆ789101112
σ(i,p_)000.0875150.216980.350960.488343
σ(g,p_)011111
σ(i,pˉ)00.0416750.1462840.2770940.4309390.575916
σ(g,pˉ)111111
n1Type I error
1200.0005810.000730.00150.00210.0018
1100.0006390.0008160.00170.00240.002
1000.0007020.000910.00180.00260.0023
900.000770.0010.0020.00290.0025
800.0008450.00110.00220.00330.0028
700.0009260.00130.00240.00360.0032
600.0010.00140.00260.0040.0035
51.28E-050.00110.00160.00290.00450.004
48.45E-050.00120.00170.00320.0050.0044
30.0003140.00130.00190.00340.00550.0049
20.0008640.00150.00210.00380.00610.0055
10.0020.00160.00240.00410.00680.0062
00.00390.00170.00260.00450.00750.0069

Table 9:

Type I errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.75, q=0.9 and different voting rule.

kˆ789101112
σ(i,p_)000.1234030.2562230.3961540.541545
σ(g,p_)011111
σ(i,pˉ)00.9961090.1435520.2777690.4226450.575916
σ(g,pˉ)111111
n1Type I error
1200.0005810.00160.00310.00440.0042
1100.0006130.00170.00320.00460.0043
1000.0006460.00180.00330.00480.0045
900.0006820.00180.00340.00490.0047
800.0007180.00190.00350.00510.0049
700.0007570.00190.00370.00530.0051
600.0007970.0020.00380.00540.0054
51.28E-050.000840.00210.00390.00560.0056
48.45E-050.0008840.00220.0040.00580.0058
30.0003140.0009310.00220.00410.0060.0061
20.0008640.000980.00230.00430.00620.0064
10.0020.0010.00240.00440.00640.0066
00.00390.00110.00250.00450.00670.0069

  1. †The corresponding voting equilibrium for the case of n=12 and kˆ=8 is semi-mixed, that is σ(i,p_)=0, σ(g,p_)=1, 0<σ(i,pˉ)<1, and σ(g,pˉ)=1, since 2kˆn2>0, q<1/(1+(p_/(1p_))nkˆ1) and β(i,p_)q<β(i,pˉ)<β(g,p_)<β(g,pˉ), with p_>1/(1+((1pˉ)/pˉ)(2kˆn2)/(2kˆn)). †The corresponding voting equilibrium for the case of n=12 and kˆ equal to either 9 or 10 is a randomizing one, that is 0<σ(i,p_)<1, σ(g,p_)=1, 0<σ(i,pˉ)<1, and σ(g,pˉ)=1, since 2kˆn2>0, q<1/(1+(pˉ/(1pˉ))nkˆ1) and q<β(i,p_)<β(i,pˉ)<β(g,p_)<β(g,pˉ), with p_>1/(1+((1pˉ)/pˉ)(2kˆn2)/(2kˆn)).

Table 10:

Type II errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.52, q=0.9, and different voting rules.

kˆ789101112
n1Type II error
12111111
11111111
10111111
9111111
8111111
7111111
6111111
50.790311111
40.49670.82441111
30.26180.54980.8157111
20.12090.30840.53150.789911
10.05040.15110.28810.48640.74071
00.01940.06660.13520.24520.41130.6548

Table 11:

Type II errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.55, q=0.9 and different voting rules.

kˆ789101112
n1Type II error
12111110.9313
11111110.9214
10111110.9101
9111110.8971
8111110.8823
7111110.8654
6111110.846
50.790311110.8238
40.49670.82441110.7984
30.26180.54980.8157110.7694
20.12090.30840.53150.789910.7362
10.05040.15110.28810.48640.74070.6982
00.01940.06660.13520.24530.41130.6548

Table 12:

Type II errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.6, q=0.9 and different voting rules.

kˆ789101112
n1Type II error
12110.20540.41850.63370.8542
11110.19910.4040.61770.8433
10110.19280.38930.60120.8316
9110.18660.37470.58420.8191
8110.18050.360.56670.8056
7110.17750.34530.54870.7912
6110.16860.33070.53030.7756
50.790310.16280.31610.51140.7589
40.49670.82440.15710.30170.49210.741
30.26180.54980.15140.28730.47240.7217
20.12090.30840.14590.27310.45240.701
10.05040.15110.14050.25910.4320.6787
00.01940.06660.13520.24530.41130.6548

Table 13:

Type II errors of the 12-Person Jury case, given pˆ=0.8, pˉ=0.9, p_=0.7, q=0.9 and different voting rules.

kˆ789101112
n1Type II error
1210.07260.16110.28710.4740.7262
1110.07160.15870.28360.46830.7209
1010.07070.15640.28010.46260.7154
910.06980.15410.27660.45680.7099
810.06890.15180.27310.45110.7042
710.0680.14950.26970.44530.6985
610.06710.14720.26620.43940.6926
50.79030.06630.1450.26280.43360.6866
40.49670.06540.14280.25930.42770.6805
30.26180.06450.14050.25590.42180.6742
20.12090.06370.13840.25250.41580.6679
10.05040.06280.13620.24910.40990.6614
00.01940.0620.1340.24570.40390.6548

Table 14:

Type II errors of the 12-Person Jury case, given pˆ=pˉ=0.8, p_=0.75, q=0.9 and different voting rules.

kˆ789101112
n1Type II error
1210.07260.14430.25990.43480.6846
1110.0720.14350.25870.43290.6823
1010.07150.14280.25740.43090.6799
910.0710.1420.25620.4290.6774
810.07050.14120.2550.4270.675
710.0710.14050.25380.42510.6725
610.070.13970.25250.42310.67
50.79030.06950.1390.25130.42110.6676
40.49670.06860.13820.25010.41920.665
30.26180.06810.13750.24890.41720.6625
20.12090.06760.13670.24770.41520.6599
10.05040.06710.1360.24640.41330.6574
00.01940.06660.13520.24520.41130.6548

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Published Online: 2018-05-19

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