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A group of people are choosing between two investments A and B. Both have these payoff distributions:

A:

$$\langle.2, .1, .2, .4, .1 \mid 1, 2, 3, 4, 5\rangle$$

B:

$$\langle.1, .3, .1, .3, .2 \mid 1, 2, 3, 4, 5\rangle$$

(IE for A, there is a .2 chance of getting 1, .1 chance of getting 2 etc. Sorry if my notation is bad).

All the people are risk averse but may have different utility functions. Can we say anything about any of the projects being unanimously preferred using notions of first and second order stochastic dominance?

I believe I understand the basics of what first and second order stochastic dominance means but I'm having trouble applying it to this question. Please help on how one might approach this type of problem.

Thanks

Michael Hardy
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Matt
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    I changed $<\cdots|\cdots>$ to $\langle\cdots|\cdots\rangle$, and then to $\langle\cdots\mid\cdots\rangle$ (the latter uses \mid and has built-in spacing to its left and right). That is standard usage. – Michael Hardy Feb 24 '13 at 00:21

1 Answers1

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First, in terms of microeconomic theory, there are 2 equivalent statements of stochastic dominance (of the 1st order):

  1. For every weakly increasing utility function $u$, $\int u(x) dF \ge \int u(x) dG$ , i.e., for a function $u(x)$ which obeys $u(t) \ge u(s)$ if $t \ge s$ (definition of weakly increasing utility function)
  2. $\forall x$, $F(x) \le G(x)$

Thus, for 1st order stochastic dominance, all that needs to be shown is that $F(x) \le G(x)$ $\forall x$ in the support. So try to calculate the cumulative density function (cdf) for each investment A and B:

$F_A(x) = <0.2, 0.3, 0.5, 0.9, 1.0 | 1, 2, 3, 4, 5>$

$F_B(x) = <0.1, 0.4, 0.5, 0.8, 1.0 | 1, 2, 3, 4, 5>$

So, based on this, $F_A(x) \ge F_B(x)$ when $x = 1, 3, 4, 5$, while $F_A(x) \le F_B(x)$ when $x = 2, 3, 5$.Thus, 1st order stochastic dominance does not hold $\forall x$ in the support.

Now, for 2nd order stochastic dominance, this has to do with a weakly increasing and concave utility function. Now, there is a condition which holds for the case when the mean (expected value) is the same for both cases. This doesn't hold, since $E_A(X) = 3.1$, while $E_B(X) = 3.2$. Thus, the concept of mean-preserving spread may not necessarily be useful here. However, from the perspective of risk-reward tradeoff, looking at the expected value and standard deviation can be a way to establish the better investment (which doesn't necessarily utilize stochastic dominance, but is a standard way of evaluating investments).

qxzsilver
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