Judgement Aggregation
Liam Kofi Bright
May 24, 2016
1 / 31
What is the relationship between individual and group beliefs?
2 / 31
A Miscarriage of Justice
Judge 1
Judge 2
Judge 3
The Glove Fits
Yes
No
Yes
If The Glove Fits Then Acquit
No
Yes
Yes
Acquit
No
No
Yes
3 / 31
A Miscarriage of Justice
Judge 1
Judge 2
Judge 3
Majority
The Glove Fits
Yes
No
Yes
Yes
If The Glove Fits Then Acquit
No
Yes
Yes
Yes
Acquit
No
No
Yes
No
4 / 31
What Just Happened?
I
Some Logically Related Propositions
5 / 31
A Miscarriage of Justice
Judge 1
Judge 2
Judge 3
Majority
The Glove Fits
Yes
No
Yes
Yes
If The Glove Fits Then Acquit
No
Yes
Yes
Yes
Acquit
No
No
Yes
No
6 / 31
What Just Happened?
I
Some Logically Related Propositions
I
Logically Consistent Agents
7 / 31
A Miscarriage of Justice
Judge 1
Judge 2
Judge 3
Majority
The Glove Fits
Yes
No
Yes
Yes
If The Glove Fits Then Acquit
No
Yes
Yes
Yes
Acquit
No
No
Yes
No
8 / 31
What Just Happened?
I
Some Logically Related Propositions
I
Logically Consistent Agents
I
Majoritarian Election
9 / 31
Anybody Notice Change of Slide Title?
Judge 1
Judge 2
Judge 3
Majority
The Glove Fits
Yes
No
Yes
Yes
If The Glove Fits Then Acquit
No
Yes
Yes
Yes
Acquit
No
No
Yes
No
10 / 31
What Just Happened?
I
Some Logically Related Propositions
I
Logically Consistent Agents
I
Majoritarian Election
I
Disaster
11 / 31
How Bad Is It? I
I
Arbitrariness in the Law [Kornhauser and Sager, 1993]?
12 / 31
How Bad Is It? I
I
Arbitrariness in the Law [Kornhauser and Sager, 1993]?
I
Undermining scientific consensus [Solomon, 2011]?
12 / 31
How Bad Is It? I
I
Arbitrariness in the Law [Kornhauser and Sager, 1993]?
I
Undermining scientific consensus [Solomon, 2011]?
I
Subverting democracy [List, 2006]?
12 / 31
How Bad Is It? II
When is (interesting) group belief formation possible?
13 / 31
Some Tools
I
Voters V: (at least 2) people who have opinions about
propositions.
I
Agenda A: propositions about which people have opinions.
14 / 31
The Agenda
I
At least two atomic propositions: Q, R.
15 / 31
The Agenda
I
At least two atomic propositions: Q, R.
I
Their conjunction: (Q&R)
I
The negation of their conjunction: ¬(Q&R)
15 / 31
Some Tools
I
Agenda A: propositions about which people have opinions,
I
Voters V : people who have opinions about propositions.
I
Profile P: opinions of people concerning propositions.
16 / 31
The Profile
I
Every voter has a personal profile
I
P is a list of all
of votes on A.
.
17 / 31
The Profile
I
Every voter has a personal profile
I
P is a list of all
I
Each is complete: for all propositions q in A, either q is in
or ¬q is in .
I
I
of votes on A.
.
Each is consistent: no propositions q in A is such that both
q and ¬q are in .
Each is deductively closed: for all propositions q, r in A, if q
entails r and q is in then r is in .
17 / 31
How Bad Is It? III
Given an A, V , and P as stated, is there a procedure for (safely)
forming group beliefs?
18 / 31
Group Belief
Want a function that takes P as input and outputs
I
I
I
G:
G
is complete: for all propositions q in A, either q is in
¬q is in G .
or
G
is consistent: no propositions q in A is such that both q
and ¬q are in G .
G
is deductively closed: for all propositions q, r in A, if q
entails r and q is in then r is in G .
19 / 31
How Bad Is It? IV
Given an A, V , is there a (safe) function from P to
G?
20 / 31
Safe Functions: Universal Domain
I
The function should take as input any P for any A.
21 / 31
Safe Functions: Anonymity
I
The function should be invariant under permutations of V .
22 / 31
Safe Functions: Systematicity
I
If all voters makes same judgements about Q as R, function
makes the same judgements about Q as R.
23 / 31
Safe Functions
I
The function should take as input any P for any A. [Universal
Domain]
I
The function should be invariant under permutations of V .
[Anonymity]
I
If all voters makes same judgements about Q as R, function
makes the same judgements about Q as R. [Systematicity]
24 / 31
How Bad Is It? V
Given an A, V , is there a function satisfying Universal Domain,
Anonymity, and Systematicity, from P to G ?
25 / 31
No.
26 / 31
Lemma
If Q is accepted with n votes then R is accepted with n votes.
27 / 31
Proof [List and Pettit, 2002]
Voter 1
Voter 2
Voter 3
Voter i > 3 & i is even
Voter i > 3 & i is odd
Q
Yes
Yes
No
Yes
No
R
Yes
No
Yes
Yes
No
(Q&R)
Yes
No
No
Yes
No
¬(Q&R)
No
Yes
Yes
No
Yes
28 / 31
Welp.
I
Relax Universal Domain? (Scientific conferences.)
29 / 31
Welp.
I
Relax Universal Domain? (Scientific conferences.)
I
Relax Anonymity? (Legal deliberation.)
29 / 31
Welp.
I
Relax Universal Domain? (Scientific conferences.)
I
Relax Anonymity? (Legal deliberation.)
I
Relax Systematicity? (Constitutional democracy.)
29 / 31
Forming group beliefs is hard, but maybe that’s ok.
30 / 31
Bibliography I
Lewis A. Kornhauser and Lawrence G. Sager. The one and the
many: Adjudication in collegial courts. California Law Review,
81(1–59), 1993.
Christian List. The discursive dilemma and public reason. Ethics,
116(2):362–402, 2006.
Christian List and Philip Pettit. Aggregating sets of judgments: An
impossibility result. Economics and Philosophy, 18:89–110, April
2002. ISSN 1474-0028. URL http:
//journals.cambridge.org/article_S0266267102001098.
Miriam Solomon. Group judgement and the medical consensus
conference. In Fred Gi↵ord, editor, Philosophy of Medicine,
pages 239–254. Elsevier, 2011.
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