Some Material Connected to Machina (1987)

Mark Machina, “Choice under Uncertainty: Problems Solved and Unsolved.” Journal of Economic Perspectives 1(1): 121-154, Summer 1987.

• A short review of choice under uncertainty: The general prospect (or lottery) is (x1, x2, …, xn; p1, p2, …, pn), where the xi’s are monetary outcomes and the pi’s are the associated probabilities.


• The expected value of the prospect (x1, x2, …, xn; p1, p2, …, pn) is p1x1+p2x2+…+pnxn = Σ pixi. 


• The expected utility of the prospect (x1, x2, …, xn; p1, p2, …, pn) is p1 U(x1) + p2 U(x2) +…+ pn U(xn) = Σ pi U(xi), where U(x) is the von Neumann-Morgenstern utility function defined over monetary outcomes xi. The standard model of choice under uncertainty is that a person will choose among prospects in such a manner as to maximize her expected utility.


• A person is risk averse if, when endowed with a riskless prospect, she always declines fair bets (bets that offer her the same expected value as her riskless prospect) – and this is equivalent to diminishing marginal utility of income.


• Machina (1987) and the Triangle Diagram: Fix the (three) dollar outcomes at x1, x2, x3; let x1<x2<x3. 


• With outcomes fixed but probabilities variable, every prospect (x1, x2, x3; p1, p2, p3) can be represented by a point in the unit simplex, which can be graphed as a triangle on p1-p3 axes (because p2 must equal 1-p1-p3, we only need a two-dimensional graph to indicate every prospect).


• Indifference curves for an expected utility maximizer will be linear in this space. Iso-expected value lines also will be linear. 


• We can use the diagram to speak about stochastic dominance; mean-preserving spreads; and risk preferences.


• Expected utility maximization requires that individual choices adhere to the "independence axiom": If the prospect P* is preferred to the prospect P, then the compound prospect aP* + (1-a)P’ is preferred to aP + (1-a)P’, for all prospects P’ and for all 0<a<1. 


• The independence axiom implies indifference curves that are linear in the probabilities, and hence, are straight, parallel lines within the triangle diagram. Nonetheless, many different choices seem to indicate that people have indifference curves that “fan out,” as opposed to being parallel lines.


• One common departure from the independence axiom (and hence from expected utility maximization) is the Allais Paradox (which can be neatly illustrated within the Triangle Diagram). Here's the setting:
  
  
                       Alternative 1                 Alternative 2
  
  
  Situation A:         ($1M;1)                           ($5M, $1M, $0; .1,.89,.01)
  
  
  Situation B:         ($1M,$0; .11,.89)          ($5M,$0; .1,.9)
  
  
  The "M" indicates that all dollar payoffs above involve millions of dollars. People typically choose alternative 1 in situation A, and alternative 2 in situation B. These two choices are inconsistent with expected utility maximization. [Why? To prefer alternative A1 to alternative A2, as an expected utility maximizer, you must have EU(A1) > EU(A2). This inequality can be rewritten as 1u($1) > .1u($5) + .89u($1) + .01u($0), and this inequality can be further simplified to .11u($1) > .1u($5) + .01u($0)*. If you also prefer  alternative B2 to alternative B1, then, as an expected utility maximizer, EU(B2) > EU(B1). This inequality can be rewritten as .1u($5) + .9 u($0) > .11u($1) + .89u($0), and this inequality can be further simplified to .1u($5) + .01u($0) > .11u($1), or equivalently, .11u($1) < .1u($5) + .01u($0). Compare this with inequality *; they contradict each other. Therefore, you cannot be an expected utility maximizer: there are no values for u($5), u($1), and u($0) such that your choices could be consistent with expected utility maximization.]