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Uncommon Priors Require
Origin Disputes
Robin Hanson
George Mason University
Rutgers Probability Seminar
October 15, 2018
Standard Econ Bayesian Model
1. Grand State Space - resolves all uncertainty
2. One Prior - complete, same for all times/cases
3. Update – diff. beliefs only via diff. info: EP[X|I]
4. Expected Utility – picks max EP[U|I], U consta.
5. Common Prior – all agents use same prior
(Gilboa, Postlewaite, & Schmeidler 2011, Synthese)
An argument: differing prior act-equivalent to differing utility
We Can’t Agree to Disagree
Aumann in 1976
• Any finite information
• Re possible worlds
• Common knowledge
• Of exact E1[x], E2[x]
• Would say next
• For Bayesians
• With common priors
• If seek truth, not lie
Since generalized to
® Infinite spaces
® Impossible worlds
® Common Belief
® A f(•, •), or who max
® Last ±(E1[x] - E1[E2[x]])
® At core, or Wannabe
® Symmetric prior origins
Two Faces of Bayesian Priors
• Prior = counterfactual probability would
assign to states given least possible info
– Prior gives any belief/expectation: EP[X|I]
• Can have beliefs on prior origins/causes
– Can this help constrain rational priors?
• Problem: all priors always common kn.
• Solution: embed standard model in
extended model
Origins of Priors
• Seems irrational to accept some priors
– Imagine random brain changes for weird priors
• In standard theories, your prior is not special
– Species-common DNA
• Selected to predict ancestral environment
– Individual DNA variations (e.g. personality)
• Random by Mendel’s rules of inheritance
• Sibling differences independent of everything else!
– Culture: random + adapted to local society
• But you must think differing prior special!
• Can’t express these ideas in standard models
Example of Standard Model
X=1
p=1/4
q=1/3
X=2
p=1/4
q=1/3
X=3
p=1/4
q=1/6
X=4
p=1/4
q=1/6
A not A
B
not B
P knows
if A or not
Q knows
if B or not
EP[X | A] = 2
EP[X | not A] = 3
EQ[X | B] = 1.5
EQ[X | not B] = 3.5
Example of Extended Model
X=1
p=1/4
q=1/3
X=2
p=1/4
q=1/3
X=3
p=1/4
q=1/6
X=4
p=1/4
q=1/6
A not A
B
not B
X=1
p=1/4
q=1/4
X=2
p=1/4
q=1/4
X=3
p=1/4
q=1/4
X=4
p=1/4
q=1/4
A not A
r=1/6 r=1/6
r=1/12 r=1/12
r=1/8
r=1/8
r=1/8
r=1/8
Standard Bayesian Model
Agent 1 Info Set
Agent 2 Info Set
Common Kn. Set
A Prior
An Extended Model
Multiple Standard
Models With
Different Priors
Standard Bayesian Model
knowledgecommonis),,,(ModelIn
)),(|()(Belief
)(),(),(Info
)()(Prior
}21{Agent(finite)State
21
21
pp
EEpEp
,...,,
,...,p,ppp,p
,...,N,iω
t
iiit
Tt
tt
N
tttt
i
Ni










Extending the State Space
)}(,:),({)),((
~
(1))()|
~
(~
}:),({},:),({
~
~
),(~state-Pre
,priorsPossible



t
i
t
i
ii
N
N
i
pppp
EppEp
ppppEpE
p
pp





P
PP
As event
An Extended Model
.,ynecessarilnotbut,,
~
torelative
knowledgecommonis),,,,,(ModelIn
(2))|
~
(~)|
~
(
allow),(),~(prior-Pre
),~(
~
)~(
)(),,...,,(),(info-Pre
21
21
StTt
pq
pEppEq
qq,...,q,qqqωq
STt
tt
ii
jiNi
t
i
t
i
St
tt
N
tttt
i





 
P


Combining (1) & (2)
𝑝𝑖( 𝐸|𝑝) ≡ 𝑝𝑖(𝐸) (1)
𝑞𝑖( 𝐸|𝑝) = 𝑝𝑖( 𝐸|𝑝) (2)
𝑞𝑖( 𝐸|𝑝1, 𝑝2, ⋯ , 𝑝𝑖 ⋯ , 𝑝 𝑁) = 𝑝𝑖(𝐸)
My Differing Prior Was Made Special
).()|
~
(viaondependsE
~
any,In
.given,anyoftindependenisE
~
any,In
.ifgivenofis,In
)|()
~
,,...,,...,,|
~
((2)&(1) 21
EppEqpq
ppq
P(A|C)P(A|BC)CBAP
BEpBppppEq
iiiii
iiji
iNii




2Theorem
1Theorem
tindependen
My prior and any ordinary event E are informative about
each other. Given my prior, no other prior is informative
about any E, nor is E informative about any other prior.
Corollaries
).()(then),|
~
()|
~
(If EPEPPpEqPpEq iiii
1Corollary
).()(then
),,|
~
(),|
~
(If
EPEP
PpPpEqPpPpEq jiijii

2Corollary
My prior only changes if events are more or less likely.
If an event is just as likely in situations where my prior is
switched with someone else, then those two priors
assign the same chance to that event.
Only common priors satisfy these and symmetric prior origins.
A Tale of Two Astronomers
• Disagree if universe open/closed
• To justify via priors, must believe:
“Nature could not have been just as
likely to have switched priors, both if
open and if closed”
“If I had different prior, would be in
situation of different chances”
“Given my prior, fact that he has a
different prior contains no info”
All false if they are brothers!

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Uncommon Priors Require Origin Disputes