The Irrationality Of Disagreement: Robin Hanson Associate Professor Of Economics George Mason University

The Irrationality of Disagreement Robin Hanson Associate Professor of Economics George Mason University

We Disagree, Knowingly • Stylized Facts: – Argue in science/politics, bets on stocks/sports – Especially regarding ability, when hard to check • Less on “There’s another tree”

– Dismiss dumber, but not defer to smarter – Disagree not embarrass, its absence can – Given any free pair, find many disagree topics • Even people who think rationals should not disagree

• Precise: we can publicly predict direction of other’s next opinion, relative to what we say

We Can’t Agree to Disagree Nobel Prize 2005

his most cited paper by x2

Aumann 1976 assumed: • Any information • Of possible worlds • Common knowledge • Of exact E1[x], E2[x] • • • •

Would say next For Bayesians With common priors If seek truth, not lie, josh, or misunderstand

Common Knowledge Set

Agent 1 Info Set

Agent 2 Info Set

John estimates car age E1[x] “It wasn’t shiny” “It sounded old”

“I’ve never been wrong before” “I can still picture it” “I had a good viewing angle”

“Fred said so” age 7 here age 2 here

“Mary is blind”

Mary estimates car age E2[x]

age 7 here age 2 here

Agree If Averages Same E1[x] = E2[x]

Aumann (1976) Annals Stat. 4(6):1236-9.

We Can’t Agree to Disagree Aumann in 1976: Since generalized to: • Any information • Of possible worlds Impossible worlds • Common knowledge Common Belief A f(•, •), or who max • Of exact E1[x], E2[x] 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 or misunderstand

We Can’t Agree to Disagree Aumann in 1976: Since generalized to: • Any information • Of possible worlds Impossible worlds • Common knowledge Common Belief A f(•, •), or who max • Of exact E1[x], E2[x] 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 or misunderstand

Disagreement Is Unpredictable I (ω ) knows X ω = v if I(ω) ⊆ {ω : X ω = v} meet IΛ J (ω ) is common knowledge (c.k.) Vω = E µ [ X ω | I (ω )], Yω = E µ [ X ω | J (ω )] Z ω = E µ [ Yω | I (ω )], Pω = 1[ Z ω − Vω > 0] Note :

E µ [ Z ω − Vω | IΛ J (ω )] = 0

If c.k. that J knows P, then c.k. that Z ω = Vω ! Hanson (2002) Econ. Lett. 77:365–369.

Experiment Shows Disagree E.g.: What % of U.S. say dogs better pets than cats? time

Example

• A gets clue on X • A1 = A’s guess of X

• B gets clue on X

30%

• B told A1 70% • B1 = B’s guess of X 40% • B2 = B’s guess of A2 • A told Sign(B2-B1) “low” • A2 = A’s guess of X 40% • Loss (A1-X)2+(A2-X)2 • Loss (B1-X)2+(B2-A2)2 A neglects clue from B

B reliably predicts neglect

Sample Percent Questions • What percent of people in the U.S. agree with this opinion? “God created humans in basically their present form in the last 10,000 years.” (Gallup,1999) • What percent of people in the U.S. agree with this opinion? “The U.S. government is hiding that it knows of the existence of aliens.” (CNN 1994) • By weight, what percent of cheddar cheese is protein? (U.S. Department of agriculture) • What percent of the population of India is literate? (Nation of India)

Experiment Features • All answers integers in [0,100], either real % or XA + XB, each from 6s dice: [0,10,20,30,40,50] • All by hand, subjects roll dice first, for credibility • Subjects told all after each round, to help learning • Zipper design, to minimize strategic interactions • Lottery payoff, to reduce risk aversion • Double dice, for easy squared-error penalty • Only tell B-sign, to reduce signaling ability

Complexity of Agreement Can exchange 100 bits, get agree to within 10% (fails 10%). Can exchange 106 bits, get agree to within 1% (fails 1%).

“We first show that, for two agents with a common prior to agree within ε about the expectation of a [0,1] variable with high probability over their prior, it suffices for them to exchange order 1/ε2 bits. This bound is completely independent of the number of bits n of relevant knowledge that the agents have. … we give a protocol ... that can be simulated by agents with limited computational resources.” Aaronson (2005) Proc. ACM STOC, 634-643.

We Can’t Agree to Disagree Aumann in 1976: Since generalized to: • Any information • Of possible worlds Impossible worlds • Common knowledge Common Belief A f(•, •), or who max • Of exact E1[x], E2[x] 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 or misunderstand

Generalized Beyond Bayesians • Possibility-set agents: if balanced (Geanakoplos ‘89), or “Know that they know” (Samet ‘90), … • Turing machines: if can prove all computable in finite time (Medgiddo ‘89, Shin & Williamson ‘95) • Ambiguity Averse (maxact minp in S Ep[Uact]) • Many specific heuristics … • Bayesian Wannabes

Consider Bayesian Wannabes ~ X i (ω ) = E [ X (ω ) | I i (ω )] + eiω [ X ] µi Pure Agree to Disagree?

Disagree Sources

Prior µ1 (ω ) ≠ µ2 (ω ) Info I1 (ω ) ≠ I 2 (ω ) e1ω ≠ e2ω Errors Ex: E

Yes No Yes

  3.14, E

A.D. X(ω ) ⇒ A.D. E[Y(ω ) | Ω]

Either combo implies pure version! 2 2/7

A.D. X(ω ) ⇒ A.D. Y(ω ) = Y

Notation State

ω ∈ Ω (finite)

Random variable

X (ω ) ∈ [ X , X + ∆X ]

Information

I i (ω ) ∈ I i (a partition)

X i (ω ) ≡ E µi [ X (ω ) | I i (ω )] ~ ~ Wannabe estimate X i (ω ) = Eiω [ X ] = X i (ω ) + eiω [ X ] Bayesian estimate

Assume : eiω = eiω ′ ∀ω ′ ∈ I i (ω )

More Notation ei [ X | S ] ≡ E µ [eiω [ X ] | S ] ~ Expect unbiased E iω [ei [ X | S ]] = 0 Bias

Calibrated error

eiω [ X ] = miω [ X ] + ci [ X ]

Choose ci at

Di (ω ) ∈ Di coarsens I i

Lemma 1 The ci which mins ~ E[( X i − X ) 2 | Di (ω)] sets ei [ X | Di (ω)] = 0.

Still More Notation Estimation set

~q ~ Bi (E ) ≡ {ω | E iω [ µ i ( E | I i (ω ))] ≥ q}

N q-agree that E in C E

~ C ⊂ i∈N Biq (C  E )

i,j ε-disagree about X

~ ~ {ω | X i (ω ) ≥ X j (ω ) + ε } ≡ Fε [ X ]

~ ~ i,j α , β -disagree about X {ω | X i (ω ) ≥ α > β ≥ X j (ω )} i, j q-agree to ε-disagree

{i, j} q-agree that i, j ε-disagree

Let 1,2 Agree to Disagree Re X ~q A = C F  Fε [ X ], Bi = Bi (A) (coarsens Di )

~ ~ ei = ei [ X | Bi ], pi = µ ( A|Bi ) p2 ≡ E 2ω [ p0 ] p0 ≡ min( p1 , p2 ), εˆ( p ) ≡ pε − 2(1 − p ) ∆X

Lemma 4 : ε ≥ 0 and e2 = 0 imply e1 ≥ εˆ( p0 )

~ ~ ˆ E 2ω [e1 ] ≥ ε ( p2 ) ~ E1ω [e1 ] = 0

(3) (4)

Theorems Re agents 1,2 q-agreeing to ε-disagree about X ,

1

2

IF

at some ω equations 3 and 4 are satisfied,

THEN at ω agents 2,1 q-agree to εˆ( ~ p2 ),0-disagree about e1 IF agents 1,2 q-agree to ε-disagree (within C ) that they ε-disagree about X and satisfy eqns 3 and 4, THEN ( within C ) agents 2,1 q-agree to εˆ ( ~p2 ),0-disagree about e1

Theorem in English • If two Bayesian wannabes – nearly agree to disagree about any X, – nearly agree each thinks himself 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). Hanson (2003) Theory & Decision 54(2):105-123.

Wannabe Summary • 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( ) implies A.D. re Y( )=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.

We Can’t Agree to Disagree Aumann in 1976: Since generalized to: • Any information • Of possible worlds Impossible worlds • Common knowledge Common Belief A f(•, •), or who max • Of exact E1[x], E2[x] 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 or misunderstand

Which Priors Are Rational? Prior = counterfactual belief if same min info • Extremes: all priors rational, vs. only one is – Can claim rational unique even if can’t construct (yet)

• Common to say these should have same prior: – Different mental modules in your mind now – You today and you yesterday (update via Bayes’ rule)

• Common to criticize self-favoring priors in others – E.g., coach favors his kid, manager favors himself – “I (Joe) beat Meg, but if I were Meg, Meg beats Joe”

• Prior origins not special => priors same

Origins of Priors • Seems irrational to accept some prior origins – Imagine random brain changes for weird priors

• In standard science, your prior origin 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

• Turns out you must think differing prior is special! • Can’t express these ideas in standard models

Standard Bayesian Model

Agent 1 Info Set A Prior Agent 2 Info Set Common Kn. Set

An Extended Model

Multiple Standard Models With Different Priors

Standard Bayesian Model State ω ∈ Ω (finite)

Agent i ∈ {1,2 ,...,N }

Prior pi (ω ) ,

p ≡ ( p1,p2 ,...,pN )

Info Π it (ω ), Π t ≡ ( Π1t ,Π t2 ,...,Π tN ), Π ≡ ( Π t )t∈T Belief

pitω ( E ) = pi ( E | Π it (ω )),

E⊂Ω

In Model (Ω, p, Π ), p is common knowledge

Extending the State Space Possible priors

pi ∈ P, p ∈ PN

~ N ~ Pre - state ω ≡ (ω , p ) ∈ Ω ≡ Ω × P As event ~ E ≡ {(ω, p ) : ω ∈ E}, p ≡ { (ω, p′) : p′ = p}

~ ~ pi ( E | p ) ≡ pi ( E )

(1)

~t t ′ ′ ′ ′ Π i ((ω, p )) ≡ {(ω , p ) : p = p, ω ∈ Π i (ω )}

An Extended Model Pre - info Γit (ω ), Γ t = ( Γ1t , Γ2t ,..., ΓNt ), Γ = ( Γt )t∈S ~t ~ ~ Γ (ω ) = Π i (ω ), ∀t ∈ T ∩ S t i

~ ), q = ( q ,q ,...,q ), allow q ≠ q Pre - prior qi (ω 1 2 N i j ~ ~ ~ qi ( E | p ) = pi ( E | p )

(2)

In Model (Ω, P, Π, q, Γ), p is common knowledge ~t t relative to Π , t ∈ T , but not necessarily Γ , t ∈ S .

My Differing Prior Was Made Special 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.

(1) & (2) ⇒

~ ~ qi ( E | p1 , p2 ,..., pi ,..., pN , B ) = pi ( E | B )

In P, A is independent of B given C if P(A|BC) = P(A|C). ~ Theorem 1 In qi , any E is independent of any p j ≠i , given pi . ~ ~ Theorem 2 In qi , any E depends on pi via qi ( E | pi ) = pi ( E ).

Corollaries ~ ~ Corollary 1 If qi ( E | pi = P ) = qi ( E | pi = P′), then P ( E ) = P′( E ). My prior only changes if events are more or less likely.

~ ~ ′ Corollary 2 If qi ( E | pi = P, p j = P ) = qi ( E | pi = P′, p j = P ), then P( E ) = P′( E ). 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 particular prior says nothing useful” All false for brothers’ genetic priors!

We Can’t Agree to Disagree Aumann in 1976: Since generalized to: • Any information • Of possible worlds Impossible worlds • Common knowledge Common Belief A f(•, •), or who max • Of exact E1[x], E2[x] 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 or misunderstand

Why Do We Disagree? • • • • • • • •

Theory or data wrong? Few know theory? Infeasible to apply? We lie? Exploring issues? Misunderstandings? We not seek truth? Each has prior: “I reason better” ?

• They seem robust • Big change coming? • Need just a few adds • We usually think not, and effect is linear • But we complain of this in others

Our Answer: We Self-Deceive • We biased to think better driver, lover, … “I less biased, better data & analysis”

• Evolutionary origin: helps us to deceive – Mind “leaks” beliefs via face, tone of voice, … – Leak less if conscious mind really believes

• Beliefs like clothes – Function in harsh weather, fashion in mild

• When made to see self-deception, still disagree – So at some level we accept that we not seek truth

How Few Meta-Rationals (MR)? Meta-Rational = Seek truth, not lie, not selffavoring-prior, know disagree theory basics • Rational beliefs linear in chance other is MR • MR who meet, talk long, should see are MR? – Joint opinion path becomes random walk

• We see no virtually such pairs, so few MR! – N each talk 2T others, makes ~N*T*(%MR)2 pairs – 2 billion ea. talk to 100, if 1/10,000 MR, get 1000 pairs

• None even among accept disagree irrational

When Justified In Disagree? When others disagree, so must you • Key: relative MR/self-deception before IQ/info • Psychology literature self-deception clues: – Less in skin response, harder re own overt behaviors, older kids hide better, self-deceivers have more self-esteem, less psychopathology/depression

• Clues?: IQ/idiocy, self-interest, emotional arousal, formality, unwilling to analyze/consider – Self-deceptive selection of clues use

• Need: data on who tends to be right if disagree! – Tetlock shows “hedgehogs” wrong on foreign events – One media analysis favors: longer articles, in news vs editorial style, by men, non-book on web or air, in topical publication with more readers and awards

We Can’t Agree to Disagree Aumann in 1976: Since generalized to: • Any information • Of possible worlds Impossible worlds • Common knowledge Common Belief A f(•, •), or who max • Of exact E1[x], E2[x] 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 or misunderstand

Implications • Self-Deception is Ubiquitious! • Facts may not resolve political/social disputes – Even if we share basic values

• Let models of academics have non-truth-seekers • New info institution goal: reduce self-deception – Speculative markets do well; use more?

• Self-doubt for supposed truth-seekers – “First cast out the beam out of thine own eye; and then shalt thou see clearly to cast out the mote out of thy brother's eye.” Matthew 7:5

Common Concerns • • • • • • • •

I’m smarter, understand my reasons better My prior is more informed Different models/assumptions/styles Lies, ambiguities, misunderstandings Logical omniscience, act non-linearities Disagree explores issue, motivates effort We disagree on disagreement Bayesian “reductio ad absurdum”

Counter Example • P(y) = exp(-by) • Asodpf

π ( y ) ∝ exp(− βy ) s = y +ε E[ε 2 ] = U ( s′, b) = bs′ + ln( P ( y* | s′))

ε ≈ N (0, σ 2 ),η ≈ N (0,θ 2 ) P ( s′ | s, b) ∝ exp( E[U ( s′, b) | s, b] / r ) P ( y | s ) = N ( s − βσ 2 , σ 2 )