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Making Decisions in Large Worlds Ken Binmore [email protected] 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 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 principle is preposterous if carried to extremes. Leonard Savage, Foundations of Statistics 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 John Harsanyi’s problem What is the rational solution of the Battle of the Sexes in a symmetric environment? The symmetric Nash equilibrium only yields the players’ security levels. So why don’t they play their security strategies instead of their equilibrium strategies? Analogous problems were solved by extending the set of pure strategies to the set of mixed strategies. Can we similarly resolve Harsanyi’s problem by extending the set of mixed strategies to a larger set of muddled strategies? traditional mixing devices probability p 1 1 0 1 0 0 01 mixing box Richard von Mises probability p 1 1 0 1 0 0 01 mixing box no probability 1 1 0 1 0 0 01 muddling box? No probability 1 1 0 1 0 0 01 upper probability p*(x) lower probability p*(x) muddling box? Randomizing boxes When evaluating a muddling box x, I want only p*(x) and p*(x) to be relevant. What criterion makes this reasonable? Randomizing boxes When evaluating a muddling box x, I want only p*(x) and p*(x) to be relevant. What criterion makes this reasonable? k 1 * p (x) inf lim sup x n j m m k j1 k 1 p* (x) sup lim inf x n j m m k j1 where the inf and sup are taken over all finite sets n1, n 2 , ,n k Randomizing boxes When evaluating a muddling box x, I want only p*(x) and p*(x) to be relevant. What criterion makes this reasonable? k 1 * p (x) inf lim sup x n j m m k j1 k 1 p* (x) sup lim inf x n j m m k j1 A box is muddled if the inf and sup where the inf and sup are taken over all sets for all arefinite achieved {n1, n2,… nk} n1, n 2 , ,n k Upper and Lower Probability A (non-measurable) event E has a lower probability (inner measure) p* and an upper probability (outer measure) p*. Let G= gamble worst best E E Upper and Lower Probability A (non-measurable) event E has a lower probability (inner measure) p* and an upper probability (outer measure) p*. Let G= gamble worst best E E How to evaluate G? Upper and Lower Probability A non-measurable event E has a lower probability (inner measure) p* and an upper probability (outer measure) p*. Let G = worst best E E only probabilistic information used u(G) = U(p*, p*) John Milnor’s axioms for decisions under complete ignorance state action b a a c consequence compatible characterizing property ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication convexity special row adjunction principle of insufficient reason maximin criterion Hurwicz criterion minimax regret criterion compatible characterizing property ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication convexity special row adjunction principle of insufficient reason maximin criterion Hurwicz criterion minimax regret criterion compatible characterizing property principle of insufficient reason maximin criterion Hurwicz criterion minimax regret criterion ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication convexity special row adjunction U(p*,p*) = hp* + (1-h)p* Ellsberg paradox J= K= $1m $0m $0m $0m $1m $0m With the Hurwicz criterion: u(J) = 1/3 u(K) = 2(1-h)/3 u(L) = 2/3 u(M) = h/3 + (1-h) J K h 1/2 L M h 1/2 uncertainty (or ambiguity) aversion L= M= $0m $1m $1m $1m $0m $1m compatible characterizing property ordering symmetry strong domination continuity linearity row adjunction column linearity column duplication convexity special row adjunction principle of insufficient reason maximin criterion Hurwicz criterion minimax regret criterion 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: U(a*b* ,a b ) u(A B) U(a*,a )U(b* ,b ) * * * * ball box ball box 0 0 1 2 0 2 1 0 Battle of the Sexes 1-q 1-p p q 0 2 0 (1,2) * 1 0 2 1 Eve’s payoff Nash equilibrium outcomes 0 Battle of the Sexes .(0,0) * *(2,1) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff . (2,1) 0 Battle of the Sexes p=0 .(0,0) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff 0 Battle of the Sexes p=1/6 .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff . (2,1) 0 Battle of the Sexes p=1/3 .(0,0) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff . (2,1) 0 Battle of the Sexes p=1/2 .(0,0) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff 0 p=2/3 Battle of the Sexes .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 p=5/6 0 Battle of the Sexes . (1,2) 1 0 2 1 Eve’s payoff .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 (1,2) p=1 0 Battle of the Sexes . 1 0 2 1 Eve’s payoff .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 q=0 0 Battle of the Sexes . (1,2) 1 0 2 1 Eve’s payoff .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff 0 q=1/3 Battle of the Sexes .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff . (2,1) 0 Battle of the Sexes q=1/2 .(0,0) Adam’s payoff 1-q 1-p p q 0 2 0 0 Battle of the Sexes q=1/3 . (1,2) 1 0 2 1 Eve’s payoff q=2/3 .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff 0 q=5/6 Battle of the Sexes .(0,0) . (2,1) Adam’s payoff 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff . (2,1) 0 Battle of the Sexes .(0,0) q=1 Adam’s payoff q 1-p p 1-q 0 2 0 (1,2) * 1 0 2 1 Eve’s payoff Nash equilibrium outcomes 0 Battle of the Sexes (2/3,2/3) .(0,0) * * (3/4,3/4) (2,1) Adam’s payoff If h 1/2 is close enough to 1/2 and U(p,P)=phP1-h we can find a symmetric Nash equilibrium in muddled player more than the security strategies that pays each level of 2/3. In the case when h=1/2, each player uses a muddling box with p* 1/6 p* 5 /6 If h 1/2 is close enough to 1/2 and U(p,P)=phP1-h we can find a symmetric Nash equilibrium in muddled player more than the security strategies that pays each level of 2/3. In the case when h=1/2, each player uses a muddling box with p* 1/6 p* 5 /6 The corresponding payoffs to the players exceed the maximum symmetric payoff of 3/4 available if only mixed strategies are used. 1-q 1-p p q 0 2 0 . (1,2) 1 0 2 1 Eve’s payoff p* 5 /6 . (2,1) 0 Battle of the Sexes * p* 1/6 .(0,0) Adam’s payoff