For Bayesian Wannabes, Are Disagreements Not About Info?

1. For Bayesian Wannabes, Are Disagreements Not About Info? Robin Hanson Economics, GMU

2. The Puzzle of Disagreement • Persistent disagreement ubiquitous • Speculative trading, wars, juries, … • Argue in science, politics, family, … • Theory seems to say this irrational • Possible explanations • We’re “just joshing” • Infeasible epistemic rationality • Fixable irrationality: all will change! • Other rationality – truth not main goal

3. My 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, voice, … • Leak less if conscious mind really believes • Beliefs like clothes • Function in harsh weather, fashion in mild

4. Aumann in 1976 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 Impossible worlds Common Belief A f(•, •), or who max Last ±(E1[x] - E1[E2[x]]) At core, or Wannabe Symmetric prior origins We Can’t Agree to Disagree

5. Generalize to Bounded Rationality • 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 …

6. Consider Bayesian Wannabes Prior Info Errors Pure Agree to Disagree? Disagree Sources Yes No Yes Either combo implies pure version! Ex: E1[p] @ 3.14, E2[p]@ 22/7

7. 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).

8. Notation

9. More Notation

10. Still More Notation

11. Let 1,2 Agree to Disagree Re X

12. Theorems 1 2

13. 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).

14. Consider Bayesian Wannabes Prior Info Errors Pure Agree to Disagree? Disagree Sources Yes No Yes Either combo implies pure version! Ex: E1[p] @ 3.14, E2[p]@ 22/7

15. 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.