St. Petersburg Paradox, Savage and StatePreference

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St. Petersburg Paradox, Savage and
State-Preference

• Lecture III

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St. Petersburg Paradox

• Most of what we know about risk started from
  games of chance.
• One of the first problems was with a bet known as
  the St. Petersburg Paradox.

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• To start the question, we pose gamble based on a
  coin toss. If the coin comes up heads, you earn
  1. If two heads in a row come up, you earn 2.
  The expected payoff is then the sum
• The expected return of this gamble approaches
  infinity as N goes to infinity.
• Would you pay infinity for this gamble?

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Friedman and Savage

• Friedman, M. and L.J. Savage (1948) The Utility
  Analysis of Choices Involving Risk. Journal of
  Political Economy 56(4) 279304.
• Friedman and Savage follow a description of the
  expected utility hypothesis similar to that
  presented in the preceding lecture. Taking B as
  the certain alternative and A as a risky
  alternative, we define the expected utility of A
  as

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• Next, we define the actuarial value of the risky
  alternative as
• Based on this formulation, we have three
  possibilities
• U(I) gt U(A) or the utility of the average income
  is greater than the utility of the gamble. In
  this case the individual avoids or is averse to
  risk.

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• U(I) U(A) the individual is indifferent between
  the gamble and a certain payoff or risk neutral.

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• U(I) lt U(A) the individual prefers the gamble to
  the certain payoff.

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• Friedman and Savage then consider five different
  behaviors
• Consumer units prefer larger to smaller incomes
• Low-income consumer units buy, or are willing to
  buy, insurance
• Low-income consumer units buy, or are willing to
  buy, lottery tickets

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• Many low-income consumer units buy, or are
  willing to buy, both insurance and lottery
  tickets and
• Lotteries typically have more than one prices

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State Preference Setup

• Let xj?RS be a vector of returns to investment
  or asset j in states s?1,S.
• X?MJ X S is the matrix of all such returns in
  the economy.

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• h is a portfolio of these possible investments or
  securities.
• The payoff of a porfolio is also a vector

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• u(c0,c1) is the utility of consumption defined on
  this state-space.
• c0 is consumption at time t0.
• c1 is the vector of possible payoffs at time t1.

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• State Dependent Expected Utility
• State Independent Expected Utility