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Making Decisions in Large Worlds

Explore Bayesian decision theory, Nash equilibrium, and strategies in large worlds to make rational decisions. Learn innovative approaches with practical implications.

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Making Decisions in Large Worlds

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  1. Making Decisions in Large Worlds Ken Binmorek.binmore@ucl.ac.uk

  2. What is a small world? Bayesian decision theory applies only in a small world, where you can always: Look before you leap Leonard Savage, Foundations of Statistics

  3. What is a small world? Bayesian decision theory applies only in a small world, where you can always: Look before you leap But… The look-before-you-leap principleis preposterous if carried to extremes. Leonard Savage, Foundations of Statistics

  4. What is a large world? In a small world, you can always Look before you leap. In a large world, you must sometimes Cross that bridge when you come to it. Leonard Savage, Foundations of Statistics

  5. John Harsanyi’s problem What is the rational solution of the Battle of the Sexes in a symmetric environment? The symmetric Nash equilibrium only yieldsthe players’ security levels. So why don’t they play their security strategies instead of their equilibrium strategies?

  6. Analogous problems were solved by extending the set of pure strategies to the set of mixed strategies. Can we similarly resolve Harsanyi’s problem byextending the set of mixed strategies to a largerset of muddled strategies?

  7. traditionalmixingdevices

  8. probability p 1 1 0 0 0 1 0 1 mixing box

  9. Richard von Mises probability p 1 1 0 0 0 1 0 1 mixing box

  10. no probability 1 1 0 0 0 1 0 1 muddling box?

  11. No probability 1 1 0 0 0 1 0 1 upper probability p*(x) lower probability p*(x) muddling box?

  12. Randomizing boxes When evaluating a muddling box x, I want only p*(x) and p*(x) to be relevant. What criterion makes this reasonable?

  13. Randomizing boxes When evaluating a muddling box x, I want only p*(x) and p*(x) to be relevant. What criterion makes this reasonable? where the inf and sup are taken over all finite sets

  14. Randomizing boxes When evaluating a muddling box x, I want only p*(x) and p*(x) to be relevant. What criterion makes this reasonable? A box is muddled if the inf and sup are achieved for all {n1, n2,… nk} where the inf and sup are taken over all finite sets

  15. Upper and Lower Probability A (non-measurable) event E has a lower probability(inner measure) p* and an upper probability (outermeasure) p*. Let worst best G = gamble

  16. Upper and Lower Probability A (non-measurable) event E has a lower probability(inner measure) p* and an upper probability (outermeasure) p*. Let worst best G = gamble How to evaluate G?

  17. Upper and Lower Probability A non-measurable event E has a lower probability (inner measure) p* and an upper probability (outermeasure) p*. Let worst best G = only probabilisticinformation used u(G) = U(p*, p*)

  18. John Milnor’s axioms fordecisions under complete ignorance state action b a a c consequence

  19. compatible principle ofinsufficientreason maximincriterion Hurwiczcriterion minimaxregret criterion characterizingproperty ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication convexity special row adjunction

  20. compatible principle ofinsufficientreason maximincriterion Hurwiczcriterion minimaxregret criterion characterizingproperty ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication convexity special row adjunction

  21. compatible principle ofinsufficientreason maximincriterion Hurwiczcriterion minimaxregret criterion characterizingproperty ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication U(p*,p*) = hp* + (1-h)p* convexity special row adjunction

  22. Ellsberg paradox $1m $0m $0m J = $0m $1m $0m K = With the Hurwicz criterion: $0m $1m $1m u(J) = 1/3 u(K) = 2(1-h)/3 L = u(L) = 2/3 u(M) = h/3 + (1-h) $1m $0m $1m M = uncertainty (or ambiguity) aversion

  23. compatible principle ofinsufficientreason maximincriterion Hurwiczcriterion minimaxregret criterion characterizingproperty ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication convexity special row adjunction

  24. Upper and Lower Probability With some mild extra assumptions, the product form U(p*,p*) = {p*}h{p*}1-h follows from retaining the multiplicative property of the probabilities of independent events:

  25. ball box 0 1 box 0 2 0 2 ball 1 0 Battle of the Sexes

  26. Nashequilibrium outcomes Eve’spayoff 1-q q (1,2) * 0 1 1-p 0 2 0 2 p 0 1 * (2,1) Battle of the Sexes * . (0,0) Adam’spayoff

  27. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) Battle of the Sexes p=0 . (0,0) Adam’spayoff

  28. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) p=1/6 Battle of the Sexes . (0,0) Adam’spayoff

  29. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) p=1/3 Battle of the Sexes . (0,0) Adam’spayoff

  30. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) p=1/2 Battle of the Sexes . (0,0) Adam’spayoff

  31. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) p=2/3 Battle of the Sexes . (0,0) Adam’spayoff

  32. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p p=5/6 . 0 1 (2,1) Battle of the Sexes . (0,0) Adam’spayoff

  33. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p=1 p . 0 1 (2,1) Battle of the Sexes . (0,0) Adam’spayoff

  34. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p q=0 . 0 1 (2,1) Battle of the Sexes . (0,0) Adam’spayoff

  35. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) q=1/3 Battle of the Sexes . (0,0) Adam’spayoff

  36. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) q=1/2 Battle of the Sexes . (0,0) Adam’spayoff

  37. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) q=2/3 Battle of the Sexes . q=1/3 (0,0) Adam’spayoff

  38. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) q=5/6 Battle of the Sexes . (0,0) Adam’spayoff

  39. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) Battle of the Sexes q=1 . (0,0) Adam’spayoff

  40. Nashequilibriumoutcomes Eve’spayoff (1,2) q 1-q * 0 1 1-p 0 2 0 2 p 0 1 * Battle of the Sexes * (2,1) . (2/3,2/3) (0,0) Adam’spayoff (3/4,3/4)

  41. If is close enough to and U(p,P)=phP1-h we can find a symmetric Nash equilibrium in muddled strategies that pays each player more than the securitylevel of 2/3. In the case whenh=1/2, each player uses amuddling box with

  42. If is close enough to and U(p,P)=phP1-h we can find a symmetric Nash equilibrium in muddled strategies that pays each player more than the securitylevel of 2/3. In the case whenh=1/2, each player uses amuddling box with The corresponding payoffs to the players exceed the maximum symmetric payoff of 3/4 available if only mixed strategies are used.

  43. Eve’spayoff 1-q q . (1,2) 0 1 1-p 0 2 0 2 p . 0 1 (2,1) * Battle of the Sexes . (0,0) Adam’spayoff

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