Download - Rational choice: An introduction Political game theory reading group 3/6-2008 Carl Henrik Knutsen
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Rational choice: An introduction
Political game theory reading group3/6-2008
Carl Henrik Knutsen
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Basics
• A “deceptively simple sentence” that summarizes the theory of rational choice: “When faced with several courses of action, people usually do what they believe is likely to have the best overall outcome” (Elster, 1989:22)
• 1)Thin and 2)instrumental rationality: Ad 1)No initial requirements on what type of goals that should be pursued. 2) Actions are chosen because of intended consequences. Actions are not valued because of themselves (contrast with Kant)
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Requirements on preferences
• Actors must be able to rank different outcomes. >, < or =. (Complete preferences)
• If x>y and y>z x>z (Transitivity)• Reflexive preferences: x≥x• “Weak ordering” is binary relation that is complete,
transitive and reflexive• Theorem (Debreu, 1959): Preferences are complete,
reflexive, transitive and continuous There will exist a continuous utility function that represents preferences
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The utility function and inter-person comparisons
• U(x1, x2, x3,…xn)• Utility functions as ordinal. We can only rank different alternatives,
and we can therefore only make claims like Utility of outcome a> Utility of outcome b and we can not make claims like the utility of a is twice as high as that of b in a strict metaphysical sense.
• Ordinality of utility functions makes inter-person comparisons problematic. We escape the “Utility-monster” problem and other problems that have been used against utilitarianism.
• Solutions to the inter-person comparison problem that is generated:– Pareto-optimality rather than social utility maximization– The “representative individual”– Back to the social welfare function and welfare weights
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The interesting properties of the utility function
• U’(x), First-order derivatives: Increasing or decreasing (U’(x)>0 or U’(x)<0). Marignal utility
• U’’(x), Second-order derivatives: Convexity or concavity of utility function (U’’(x)>0 or U’’(x)<0)
• Partial derivatives, notation.– ∂U(x,y)/∂x– ∂2U(x,y)/∂x2
– ∂2U(x,y)/∂x∂y (Young’s theorem)
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Optimization• Over discrete choices, rather simple: Pick the one which gives
highest utility/pay-off/etc (or minimal cost..depends on problem)• Unconstrained optimization (continuous): U’(x)=0 (and proper
second order condition)• Optimization under constraints. Economists disagree with
Leibniz: We are most often not in the best of all possible worlds. Different constraints (political, budgets, technological..etc). Problem is now: Maximize utility given that the constraints must hold.– Insert constraints into utility function before maximizing– Lagrange-functions– More complex problems..
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Indifference curves and preferences
• Indifference curves (2 goods): Combinations of goods that give the same level of utility
• Convex preferences (not to be confused with convex utility function) “Averages better than or equal to extremes”
λ element in (0,1) and indifferent x and y Convexity λx + (1-λ)y ≥ x (or y)
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Policy spaces and preferences
• General utility function for these purposes when one dimensional policy space:– U(x) = h(-|x-z|)
• Bliss point or ideal point (z). Utility loss when deviating from this point
• N-dimensions. Measure distances by Eucledian norm:||x-z|| =√(∑(xj-zj)2)
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Choice under uncertainty
• A priori knowledge of probability distribution related to outcomes, but does not know specific outcome. Way to model beliefs.
• Given beliefs, maximize expected utility• Von Neumann-Morgenstern utility functions:– EU(p) = p1u1 + p2u2+…+pnun
– Some needed assumptions: Doesn’t care about order in which lottery is described, cares only about net probabilities, independence of irrelevant alternatives, cardinal utility
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Risk
• Note that we do not in general maximize for example expected revenue or expected number of votes. We can take into account that actors are not risk neutral.
• Risk aversion, two outcomes: – u(px1+(1-p)x2)>pu(x1) + (1-p)u(x2)– Risk aversion related to concavitiy of utility function. Arrow-Pratt
measure of relative risk aversion: ρ= -(u’’(x)*x)/u’(x)• Risk aversion and satiable preferences. If expected outcome is
ideal point Avoid lotteries with larger spread around ideal point
• Risk premium: What one would be willing to give up in order to avoid randomness (same expected outcome, different risks)
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Learning and time preferences
• Rational actors incorporate new information after observing events. Update their beliefs.
• Bayes’ Rule P(A|B) = (P(B|A)*P(A))/P(B)– Game thoery: Actions and types. Signaling games.
• Optimization over time, discount factor: δ, between 0 and 1.
• How to compute a pay-off stream, infinite sequence starting in t=0: ∑δtu = u/(1- δ)
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Some criticisms
• Real people and beliefs/understanding of probabilities, some systematic biases
• Allais’ paradox (independence of irrelevant alternatives does not hold) and Ellsberg’s paradox. Based on experiments.
• Behavioral economics: Kahneman and Tversky: Prospect theory (loss aversion, reference points)
• Time preferences and hyperbolic discounting