Optimizing Voting Order on Sequential Juries: A Median Voter Theorem and Beyond—A Supplement

1. Definitions of voting functions

1.1. Definition of distribution functions and one-step threshold



1.2. Thresholds and reliability of honest majority verdict



1.3. Threshold and reliability of unanimous verdict



1.4. Reliability function of sequential strategic voting



1.5. Reliability function of simultaneous honest voting



2. Other algebraic calculations

2.1. Order of the first two voters



2.2. Order of the two end-voters



2.3. Sequential versus simultaneous majority voting



2.4. Reliability versus heterogeneity and homogeneity



2.5. Strategic optimality versus heterogeneity and homogeneity



3. Algebraic proofs

3.1:

Q

(θ;c,b)-

Q

(θ;b,c)>=0

s.t.

0<=b<c<=1

.



3.2:

f

1

(a,b,c)>0

s.t.

0<=a,b,c<=1

and

ρ(b,a)<c<=ρ(a,b)

and

b/2<a<b

.



3.3:

f

3

(a,b,c)>0

s.t.

0<=a,b,c<=1

and

c>ρ(a,b)

and

b/2<a<b

.



3.4:

Δ

2

(a,b,c)>0

s.t.

0<=a,b<=1

and

a/2<c<=b/2

.



3.5:

f

4

(a,b,c)>0

s.t.

0<=a<b<=1

and

a<=ρ(b,c)

and

b>ρ(a,c)

and

b/2<c<=1

.



3.6:

f

5

(a,b,c)>0

s.t.

0<=a<b<=1

and

a>ρ(b,c)

and

b>ρ(a,c)

and

b/2<c<=1

.



3.7: Monotonicity of

Q(a,b,c)



3.8:

h

1

(a,b,c)>0

s.t.

λ(a,b,c)>=6/7

.



3.9:

-

h

1

(a,b,c)>0

and

h

2

(a,b,c)>0

s.t.

μ(a,b,c)>=4/3

.



3.10: Function

w(m,μ)

is strictly increasing over

{μ>=1:w(m,μ)>=0};

so are functions

Q

1

(m,μ)

and

Q

2

(m,μ)

over

{μ:μ>=1}

.



3.11:

Ω(b,c,0,

y

1

,

y

2

,

z

1

,

z

2

)>=0

s.t.

μ>=7/4

and

-1<=

y

1

,

y

2

,

z

1

,

z

2

<=1

.



3.12:

Q

sim

(1/2,a,a,a)>=

Q

str

(1/2,a,a,a,0,

y

1

,

y

2

,

z

1

,

z

2

)>=0

s.t.

-1<=

y

1

,

y

2

,

z

1

,

z

2

<=1

and

0<=a<=1

.

