For Bayesian Wannabes, Are
Disagreements Not About Info?
Robin Hanson
Economics, GMU
The Puzzle of Disagreement

Persistent disagreement ubiquitous



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Speculative trading, wars, juries, …
Argue in science, politics, family, …
Theory seems to say this irrational
Possible explanations
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
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We’re “just joshing”
Infeasible epistemic rationality
Fixable irrationality: all will change!
Other rationality – truth not main goal
My Answer: We Self-Deceive

We biased to think better driver, lover, …
“I less biased, better data & analysis”
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Evolutionary origin: helps us to deceive


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Mind “leaks” beliefs via face, voice, …
Leak less if conscious mind really believes
Beliefs like clothes

Function in harsh weather, fashion in mild
We Can’t Agree to Disagree
Aumann in 1976
Since generalized to
 Re possible worlds

Impossible worlds
 Common knowledge 
Common Belief
 Of exact E1[x], E2[x] 
A f(•, •), or who max

Last ±(E1[x] - E1[E2[x]])
 Would say next
 For Bayesians

At core, or Wannabe
 With common priors 
Symmetric prior origins
 If seek truth, not lie
Generalize to Bounded Rationality
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

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Bayesians (with common prior)
Possibility-set agents: balanced
(Geanakoplos ‘89), or “Know that
they know” (Samet ‘90), …
Turing machines: prove all
computable in finite time (Medgiddo
‘89, Shin & Williamson ‘95)
Many more specific models …
Consider Bayesian Wannabes
~
X i ( )  E [ X ( ) | I i ( )]  ei [ X ]
i
Disagree Sources
Pure Agree
to Disagree?
Prior 1()  2 () Yes
Info I1 ()  I 2 () No
Errors e1  e2
Yes
Ex: E1[p]
@
A.D. X( ) 
A.D. Y(  )  Y
Either combo
implies pure
version!
3.14, E2[p] @ 22/
Theorem in English

If two Bayesian wannabes




nearly agree to disagree about any X,
nearly agree that both think they nearly unbiased,
nearly agree that one agent’s estimate of other’s bias
is consistent with a certain simple algebraic relation
Then they nearly agree to disagree about Y,
one agent’s average error regarding X.
(Y is state-independent, so info is irrelevant).
Notation
State
   (finite)
Random variable
X ( )  [ X , X  X ]
Informatio n
I i ( )  I i (a partition)
Bayesian estimate
X i ( )  E i [ X ( ) | I i ( )]
~
~
X i ( )  Ei [ X ]  X i ( )  ei [ X ]
Wannabe estimate
Assume : ei  ei     I i ( )
More Notation
Expect unbiased
ei [ X | S ]  E  [ei [ X ] | S ]
~
Ei [ei [ X | S ]]  0
Calibrated error
ei [ X ]  mi [ X ]  ci [ X ]
Choose ci at
Di ( )  Di coarsens I i
Bias
Lemma 1 The ci which mins
~
E[( X i  X )2 | Di ( )] sets ei [ X | Di ( )]  0.
Still More Notation
~q
~
Estimation set
Bi (E )  { | Ei [ i ( E | I i ( ))]  q}
~q
N q-agree that E in CE
C   iN Bi (C  E )
~
~
i,j -disagree about X
{ | X i ( )  X j ( )   }  F [ X ]
~
~
i,j  , -disagree about X { | X i ( )      X j ( )}
i, j q-agree to -disagree
{i, j} q-agree that i, j -disagree
Let 1,2 Agree to Disagree Re X
~q
A  CF  F [ X ], Bi  Bi (A) (coarsens Di )
ei  ei [ X | Bi ], pi   ( A|Bi )
p0  min( p1, p2 ), ˆ( p)  p  2(1  p)X
Lemma 4 :   0 and e2  0 imply e1  ˆ( p0 )
~
~p  E
2
2 [ p0 ]
~
E2 [e1 ]  ˆ( ~
p2 )
~
E1 [e1 ]  0
(3)
(4)
Theorems
Re agents 1,2 q-agreeing to -disagree about X ,
IF
at some  equations 3 and 4 are satisfied,
1
THEN at  agents 2,1 q-agree to
ˆ( ~
p2 ),0-disagree about e1
2
IF agents 1,2 q-agree to -disagree (within C ) that
they -disagree about X and satisfy eqns 3 and 4,
THEN ( within C ) agents 2,1 q-agree to
ˆ( ~p2 ),0-disagree about e1
Theorem in English

If two Bayesian wannabes




nearly agree to disagree about any X,
nearly agree that both think they nearly unbiased,
nearly agree that one agent’s estimate of other’s bias
is consistent with a certain simple algebraic relation
Then they nearly agree to disagree about Y,
one agent’s average error regarding X.
(Y is state-independent, so info is irrelevant).
Consider Bayesian Wannabes
~
X i ( )  E [ X ( ) | I i ( )]  ei [ X ]
i
Disagree Sources
Pure Agree
to Disagree?
Prior 1()  2 () Yes
Info I1 ()  I 2 () No
Errors e1  e2
Yes
Ex: E1[p]
@
A.D. X( ) 
A.D. Y(  )  Y
Either combo
implies pure
version!
3.14, E2[p] @ 22/
Conclusion





Bayesian wannabes are a general model
of computationally-constrained agents.
Add minimal assumptions that maintain
some easy-to-compute belief relations.
For such Bayesian wannabes, A.D.
(agreeing to disagree) regarding X(w)
implies A.D. re Y(w)=Y.
Since info is irrelevant to estimating Y,
any A.D. implies a pure error-based A.D.
So if pure error A.D. irrational, all are.