Rational choice: An introduction

Political game theory reading group3/6-2008

Carl Henrik Knutsen

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)

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

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

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)

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..

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)

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)

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

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)

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)

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