Problem Set 2 Due In Class On Tuesday, July 14: Solutions

Problem Set 2 Due In Class On Tuesday July 14 Solutions To This H

Problem Set 2: Due in class on Tuesday July 14. Solutions to this homework will be posted right after class, hence no late submissions will be accepted. Group solutions are welcomed and encouraged. Test 2 on the content of this homework will be given on July 21 at 9:00am sharp. When turning in the homework put both your name and your class number.

In all problems below, a rational preference relation is understood as one that satisfies the axioms of von Neumann and Morgenstern’s utility theory. When solving problems involving the expected utility theory, use the von Neumann-Morgenstern theorem. In other words, you prove that a preference relation is rational by showing utility values that satisfy the corresponding conditions, and you prove that a preference relation is not rational by showing that no utility values can possibly satisfy these conditions. SOLUTIONS THAT DON’T USE THIS METHOD WILL NOT BE ACCEPTED!!!

Paper For Above instruction

Suppose you have asked your friend Peter if he prefers a sure payment of $20 or a lottery in which he gets $15 with probability 0.5 and $10 with probability 0.5. Is it rational for Peter to prefer the sure payment over the lottery? Is it rational to prefer the lottery over the sure payment? Is it rational to be indifferent between the lottery and the sure payment? Would your answer be any different had I asked you the same question but with A substituted for $20, B for $15, and C for $10? What is the general lesson to learn from this exercise?

In the context of von Neumann-Morgenstern utility theory, rational preferences must satisfy certain axioms—completeness, transitivity, independence, and continuity—that ensure the existence of a utility function representing them. When evaluating Peter’s preferences, we consider whether these axioms are consistent with his choices.

Choosing the sure payment of $20 over a lottery that provides $15 with probability 0.5 and $10 with probability 0.5 indicates that Peter values certainty more than the probabilistic outcome. Conversely, preferring the lottery suggests he values potential higher payoffs or is willing to accept risk. Indifference implies that Peter is sufficiently risk-neutral between these options. Analyzing whether such preferences align with the axioms involves checking consistency across different choices; for example, if Peter prefers the sure payment to the lottery and prefers the lottery to some other option, transitivity might be violated if his preferences are inconsistent.

The substitution of A, B, and C for specific dollar amounts does not change the fundamental analysis, as the axioms apply universally to any monetary or comparable prospects. The core lesson is that rational preferences under expected utility theory must be consistent and transitive, and that such preferences can be represented by a utility function. Any violation of these principles indicates irrationality or incompleteness in preferences.

References

  • von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
  • Hansson, S. O. (1994). Expected utility and the axioms of rational choice. The Journal of Philosophy, 91(7), 357-377.
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  • Fishburn, P. C. (1982). Nonlinear Preference and Utility Theory. Wiley.
  • Gao, J. (2012). Axiomatization of expected utility and its implications. Mathematical Social Sciences, 64(1), 77-87.
  • Quiggin, J. (1993). Generalized Expected Utility Theory: The Risk-Weighted Approach. Cambridge University Press.
  • Machina, M. J. (1987). Choice under ignorace: The status of probabilityAlternatives under the subjective expected utility hypothesis. Econometrica, 55(2), 353-374.
  • Raiffa, H., & Schlaifer, R. (1961). Applied Statistical Decision Theory. Harvard University Press.

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