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The Theory of Rational Choice: Preferences, Payoffs, and Decision-Making

This chapter introduces the theory of rational choice, which examines how decision-makers choose the best action based on their preferences and available options. It explores the concept of preferences, the role of payoffs in decision-making, and the consistency of choices.

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The Theory of Rational Choice: Preferences, Payoffs, and Decision-Making

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  1. Chapter 1 The Theory of Rational Choice

  2. Practice problems • Consider the problem: min 3x2 - 5x + 3. What is the FOC, how do we find the solution, how do we know this is a minimum? • We have a system of 2 simultaneous equations linear:5x + 3y = 192x + 6y = 22 What are x and y? • g(x) = 4x2 - 5x + 1 . What value(s) of x set g(x) = 0? • A random variable X takes the value of 1 with probability ¼ , 2 with probability ¼ , and 3 with probability ½ . What is E(X)? • A random variable Y is uniformly distributed between 0 and 2. What is the probability that a particular draw from this distribution is greater than or equal to 1.5? • What is the sum of 2 + 1 + 0.5 + 0.25 + …. ? • Event A occurs with probability 0.4. If event A occurs, event B may also occur. If the unconditional probability that B occurs is 0.2, what is the conditional probability that B occurs, given that A has occurred?

  3. Answers • FOC 6x – 5 = 0. x = 5/6 is soln. SOC: 6 > 0, so we have a minimum. • x=2, y=3 • x = [5+- (25-16)^0.5]/8 → x=1, x=1/4 • E(X) = 0.25 + 0.5 + 1.5 = 2.25 • Prob(Y1.5) = 0.25 • First term = 2, common ratio = 0.5, so sum to infinity = 2/(1 - 0.5) = 4 • Conditional probability = 0.5

  4. The Theory of Rational Choice A rational decision-maker chooses the best action according to her preferences, among the actions available to her. Set of available actions Preferences Complete Consistent (transitive) Rational ≠ Selfish

  5. The Theory of Rational Choice Payoff function: associates a number with each action in such a way that actions with higher number are preferred. For a and b in some setA u(a) > u(b) if and only if the decision-maker prefers a to b

  6. The Theory of Rational Choice Example: A = {Coke, Pepsi, Sprite} = {C, P, S} Decision-maker prefers C to P and P to S • u(C)=3, u(P)=2, u(S)=1 Or • u(C)=10, u(P)=0, u(S)=-2

  7. The Theory of Rational Choice Preferences  Ordinal information v is another payoff function that represents the same preferences as u if v(c) > v(p) u(c) > u(p) Any monotonically increasing function of u represents the same preferences

  8. The Theory of Rational Choice Example: u(C)=3, u(P)=2, u(S)=1 u(C)=3 > u(P)=2 > u(S)=1 f(x)=2*x v(x) = f(u(x)) v(C) = f(u(C)) = f(3)=6, v(P)=4, v(S)=2 v(C)=6 > v(P)=4 > v(S)=2

  9. The Theory of Rational Choice The action chosen by a decision-maker is at least as good, according to her preferences, as every other available action. Decisions should not be affected by irrelevant alternatives. Example: If A={P,C} and she always chooses C If A’={P,C,S} and she chooses P • Inconsistent with the Theory of Rational Choice To be consistent she must choose C or S - See the Weak Axiom of Revealed Preference (WARP)

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