1. Rational analysis of the selection task
Oaksford and Chater (1994)
Presented by
Bryan C. Russell

2. A K 2 7 Wason selection task
Rule: if there is a vowel on one side, then there is an even number on the other side A K 2 7

3. Last time… Humans are stupid Cosmides et al.
Social contract theory (i.e. innate ability to detect cheaters)
Depends on perspective (enforcer or actor)

4. Outline Probabilistic account of Wason selection task
Information-theoretic background
Application to abstract selection task
Application to thematic selection task
Discussion

5. Another view of the task
Let rule be “if p then q”
Four types of cards
(p,q), (not-p,q), (p,not-q), (not-p,not-q)
Two hypotheses
MD: p,q are dependent (rule is true)
MI: p,q are independent (rule is false)

6. Assumptions We can assign probabilities to the cards
Should reflect natural statistics of “if p then q” statements in nature
P(p | MD) = P(p | MI) = a
P(q | not-p,MD) = P(q | not-p,MI) = b

7. Card probabilities

8. Card probabilities
Task: Select card that maximally reduces hypothesis uncertainty

9. Entropy/uncertainty

10. Entropy/uncertainty

11. Entropy/uncertainty

12. Another experiment…
Suppose you observe:
TTHTHHTHHHHTHHTHTHHT

13. Another experiment… Suppose you observe: TTHTHHTHHHHTHHTHTHHT
HTHHTTHTHHTTTTTTTTTH

14. Another experiment… Suppose you observe: TTHTHHTHHHHTHHTHTHHT
HTHHTTHTHHTTTTTTTTTH

15. Mutual information

16. Application to selection task
a = Pr(p)

17. Application to selection task
a = Pr(p)

18. Model behavior

19. Observations If Pr(q) is low, then choosing p card is informative
If Pr(p) and Pr(q) is low, then choosing q card is informative
If Pr(p) is high, then choosing not-q card is informative
not-p card is not informative (results in zero information)
P(MI) only scales information values

20. Model behavior R

21. Rarity assumption
For selection task, in humans Pr(p) and Pr(q) are low
Expectation over region R
choose p: 0.76
choose q: 0.20
choose not-q: 0.09
choose not-p: 0

22. How do humans compare?

25. Analysis
Both humans and model accounts for the following information relationship:
choose p > choose q > choose not-q > choose not-p

26. Thematic selection task
If a passenger form says “ENTERING” on one side then the other side must include cholera information
“ENTERING”
checked
“ENTERING”
not checked
cholera
information
no cholera
information

27. Rule types Obligations: if action (p), then must condition (q)
Permissions: if condition (p), then may action (q)

28. Subject perspective for permission rule
Enforcer
Pretend you are an immigrant officer…
Actor
Pretend you are a traveler…
Objective inquirer
Check the validity of the statement…

29. Human performance Obligation: action (p) => must condition (q)
Permission: condition (p) => may action (q)

30. Utility-based model Focus on rule-use, not rule-testing
Associate cost with turning over a card

31. Utility-based model

32. Performance of utility-based model

33. Anderson’s (1990) steps for rationality
Specify precisely the goals of the cognitive system
Develop a formal model of the environment to which the system is adapted
Make minimal assumptions about computational limitations

34. Anderson’s (1990) steps for rationality
Derive the optimal behavior function given the previous steps
Examine the empirical evidence to see whether the predictions of the behavioral function are confirmed
Rinse, lather, repeat, and refine the theory

35. Conclusions Bayesian approach more principled than other accounts
Applies to both abstract and thematic versions of the problem (and their variants)
Behavior is adapted to the environment and does not necessarily follow logic/mathematic theory

36. Discussion questions
Is the rarity assumption valid? How do we test it?
Is the selection task representative of accounting for rational thought? Is it exhaustive?
How is this model realized?
Does one learn explicitly the utility costs and Pr(p)/Pr(q)?