System 2 human prefrontal cortex allocates attention

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System 2 (“human”, prefrontal cortex ): allocates attention, exerts mental effort, requires concentration. Pupils are dilated, the heart rate goes up, glucose level decreases in the blood.(MENTAL EFFORT REQUIRED) Compute objective probabilities correctly (lotteries, stock market)(future rewards) Thus anchoring , availability bias , hindsight bias can be explained by failures System 1 to collect information and the unwillingness to use System 2. Compute expected value (say in the card games) Keep the broad frame and combine the effects of several decisions. Uncertainty described in 2 ways: Numerical probabilities States of the world (state of nature) 1.mutually exclusive 2.exhaustive 3.sufficiently detailed Probabilities 2 objective definitions: Symmetry across states - classical definition Symmetry across data points - frequentist definition The word objective means that reasonable people should agree to use a formula in a particular situation Outcomes (payoffs, rewards, prizes) can be Monetary : payoffs described in numbers Non-monetary : may have purely verbal descriptions 4 types of decision problems Type I: Numerical probabilities and monetary payoffs Type II: Numerical probabilities and general payoffs Type III: Verbal states and monetary payoffs Type IV: Verbal states and general payoffs. Standard Economic Model Expected value requires: Numerical probabilities are given for all events Payoffs are monetary
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Law of large numbers: says an agent who maximizes expected value in a long series of choices will be more wealthy than another agent who faces the same series of decisions but doesn’t maximize expected value Expected utility is Eu(f) = p 1 u(x 1 ) + p 2 u(x 2 ) +…+ p n u(x n ) where the function u(x) is called the Bernoulli utility index(defined in monetary=Bernoulli utility for money) Risk Premium Rp(f) = Ev(f) – C(f). If U(f) = Ev(f), then the certainty equivalent of f is equal to Ev(f) because U(Ev(f)) = Ev(Ev(f)) = Ev(f) = U(f). Allais Paradox: refers to preference reversals that violate Independence ( common ratio ) effect Expected utility and Risk aversion The following functions are increasing and concave(and all these utilities produce risk aversion) • quadratic : u(x) = -ax 2 + bx for x < b/2a • power utility : u(x) = x α for 0< α < 1 and x> 0 • constant relative risk aversion (CRRA) : u(x) = log x for x> 0 (only utility that preserves the risk premia) constant absolute risk aversion (CARA): u(x) = -(2 -αx ) for 0< α Note that the utility for money can be negative! It does not mean that money is bad! Mixtures : are well-defined even if the payoffs are not numbers at all Independence Axiom (von Neumann and Morgenstern 1944) If a gamble f is better than a gamble g then for any probability 1> α > 0 and any third gamble h α f + (1- α) h should be better than α g + (1- α) h Independence is normative : it prescribes a rule of behavior.
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