The Logic of Decision Symposium in honor of Richard Jeffrey Philosophy of Science Association October 24, 1998 Ethan Bolker Department of Mathematics and Computer Science UMass-Boston Themes • Glimpses of history • Revisiting the existence theorem – two philosophical challenges: noncommutative logic doing without impartiality • Everyday probability is hard • Decision theory with a human face Revisiting the existence theorem • Find all binary relations pref that average: A, B incompatible, A pref B implies A pref (A union B) pref B • Canonical example: the ordering on sets determined by computing the expected value of a function with respect to a probability measure. The domain of pref • In quantum mechanics, lattice of subspaces of Hilbert space replaces set of subsets of phase space - the expected value of an observable still averages. • 196x goal: prove theorems in this noncommutative logic. • 1998 challenge: is there a noncommutative logic of decision? The Existence Question • Canonical example revisited value(A) = integral f(s) dprob(s) des(A) = value(A)/prob(A) • des induces pref • value(A) = des(A) prob(A) (value is discounted desirability) • When does pref come from value and prob? The jeffrey function • For each x in (0,1) (the range of des) let jeffrey(x,A) = x prob(A) - value(A) ° jeffrey(x,A) <= 0 just when x <= des(A) ° jeffrey is a (signed) measure: when A and B are incompatible, jeffrey (x,A union B) = jeffrey (x,A) + jeffrey (x,B) ° jeffrey(x,A) is an increasing function of x • Challenge: What does jeffrey (x,A) mean? (when we know we may rename it) The Existence Theorem • Given a jeffrey such that jeffrey(x, · ) is a signed measure jeffrey( ·, A) is an increasing function define des by des(A) = the unique x such that jeffrey(x,A) = 0. Then des averages, and every averaging des is of this form. The role of impartiality • Impartiality is equivalent to the existence of value and prob such that jeffrey(x,A) = x prob(A) - value(A) • That’s all it does – like the parallel postulate – thin disguise for fair coins and arbitrary gambles – how much (what kind of) decision theory can you do with weaker (other) assumptions about jeffrey? Into the third dimension • M = vector space of all signed measures • M+ = cone of positive measures • V = span of jeffrey • V+ = M+ intersect V • Impartiality if and only if dim V = 2 dim V+ = 1 implies classic uniqueness dim V+ = 2 implies weird (J/B) uniqueness • What does dim V = 3 mean? Everyday probability is hard • What is probability? • An elusive question in philosophy and psychology (only easy in mathematics) • Dick says “The skill consists … in coming to have appropriate degrees of belief between 0 and 1 when conditions are less than ideal …” • A short experiment under ideal conditions Count runs of 4 … HTHHHHTTTTTH … expect (# runs) * (2/16) = 61/8 = 7.6 0 7 1 8 2 9 3 10 4 11 5 12 6 13 Observations • Human instinct’s weak on probability in general and Poisson processes in particular • Short memory approximates memoryless • You can win stone/paper/scissors … • You can cross a busy street • When will wide ties return? • We need the Logic of Decision Radical probabilism • “It is a frequent complaint against academic moral philosophers …” (Braithwaite) • Dick argues that probability - a.k.a. uncertainty - is all there is • Life is a high risk enterprise • “… whenever there is a real issue between two of us, or whenever one of us is of two minds, both sides are ruled reasonable ...