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Rational Choice Sociology Lecture 4 Rational Choice under Uncertainty

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Is the Problem of Rational Choice under Uncertainty Important? An actor chooses under uncertainty if she is not able to ascribe probabilities to the possible outcomes of her alternative actions How often the situations of choice under uncertainty happen? According to proponents of the theory subjective expected utility (seu), almost never, because actors almost always have subjective probabilities (it is even difficult to find an undisputable example of such situation). There are no situations where special rules for choice under uncertainty could be applied. The Bayesian rule (maximization of seu) is universal According to proponents of objectivistic version, Bayesian rule is applicable only if the actor has statistical data about the relative frequencies. If her subjective probabilities are not grounded in such data, she chooses under uncertainty. Situations of the choice of uncertainty include those where probabilities of the unique or rare events are involved. For such situations, special rule of choice is needed.

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Unsufficient reason rule Under uncertainty, consider all outcomes as equally probable. Multiply u by equal p and choose the action with greatest expected utility index; Or: choose the action with prospect that has greatest average utility C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 Action 1 12092 EU 1 =5,75 Action 2 5619 EU 2 =5,25

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Maximin/Minimax rule (or: radical pessimism rule) Under uncertainty, choose the action with best worst outcome (or: choose the best among worst) Alternatively: maximize the greatest minimal payoff (maximin) or Minimize the greatest maximal loss (minimax)

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Maximin/Minimax rule: example 1 C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 C5C5C5C5 Action 1 12 0 X 326 Action 2 82 1 X 34 Action 3 122233 4 X ! 6

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Maximin/Minimax rule: example 2 C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 C5C5C5C5 Action1-5-9-6-8 -10X Action2 -9X! 0-7 -8-8-8-8 Action3-5-8-9 -12X

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Maximax rule (or: radical optimism rule) C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 C5C5C5C5 Action122-9911 81 X ! Action23-71 7 X 4 Action3145-6 8 X

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What if there are two prospects with equally good worst or best outcomes? Apply leximin and leximax rules: choose the action which has better second worst outcome (leximin) or better second best outcome C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 C5C5C5C5 Action 1 89+3 0 XX -1 X Action 2 Best according leximax 1278+ 44 ++! -10 X 8 Action 3 Best according leximin 9 2 XX! 922+ -1 X Action 4 78+ 33 ++ -2 X 412

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Criticism of maximin/minimax and maximax rules C1C1C1C1 C2C2C2C2 Action10 1 000 000 Action2 0,01 X 1 C1C1C1C1 C1C1C1C1Action121 Action2 -1000 000 Or: to break the backbone 3 X

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Minimax regret rule: choose the action that brings minimal maximal regret if choice will be unsuccesfull: table left: utility indexes; table right: regret indexes C1C2 Action 1 0100 Action 2 X 11C1C2 Action 1 -1 X 0 Action 2 0-99

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Calculation of regret indexes (transformation of utility matrix into regret matrix) and choice according minimax regret rule NB: utility indexes should be interval scale Regret index R ij = u ij -u maxj Regret index R ij = u ij -u maxj In each column, find the outcome with greatest utility index; subtract this index from the utility indexes of each outcome in the same column. In each column, find the outcome with greatest utility index; subtract this index from the utility indexes of each outcome in the same column. Find in each prospect the greatest (worst) regret index (=maximal regret) Find in each prospect the greatest (worst) regret index (=maximal regret) Choose the action whose greatest regret index is least (=least maximal regret) Choose the action whose greatest regret index is least (=least maximal regret)

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Minimax regret rule: example C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 A1-25-8-7 A2-59011-12 A3420714 C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4A1-6 -85 X -19-21 A2-900 -26X! A30 -70 X -40

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Optimism-pessimism rule There are more candidates to play the role “middle between pessimism and optimism” rules, but they are increasingly complex and have demanding conditions of application (utility indexes measured at interval level and sometimes some complementary data). E.g. The optimism/pessimism rule: Choose the action with the greatest sum of the utilities of best and worst oucome weighted by optimism and pessimism indexes Action A i > A j, if a × u max (A i ) + 1-a ×u min (A i ) > a × u max (A j ) + 1-a ×u min (A j ) Where a is optimism index, 1-a is pessimism index If a=0, then optimism-pessimism rule collapses into maximin/minimax rule; if 1-a=0, then optimism-pessimism rule collapses into maximax rule

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Optimism-pessimism rule (example) C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 A1-25-8-7 A2 ! -59011-12 A3-420714 U op (A 1 )= 0,3×5 + 0,7×- 8=1,5+ (-5,6)= -4,1 U op (A 2 )= 0,3×90 + 0,7×-12 =27+ (-8,4)= 18,6 U op (A 3 )= 0,3×20 + 0,7×-4 =6+ (-2,8)= 3,2

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Which rules people follow choosing under uncertainty? It is empirical question! Which rule is best? (normative question) Undecided: this is question, who is more rational, optimist or pessimist, or which mix of them is rational… Most decision analysts say we should accept a and 1-a as data, like we accept preferences

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