On problem set 4, part D, most of you had no apparent difficulty in correctly establishing that the sum of the differences between the cumulative distribution (CDF) for risk 2 and the CDF for risk 1 is positive. However, many of you drew the wrong conclusion, claiming that since the sum of differences between and came out to a positive number, it followed that risk 2 second order stochastically dominates risk 1. Actually, this result implies the opposite; i.e., that risk 1 second order stochastically dominates 2. This blog posting aims to clarify everyone’s understanding of the logic behind the stochastic dominance model.

The one-page exam formula sheet includes section 4, which explains that risk *i *dominates risk *j, *in both the first and second cases when 1) the cumulative distribution function (CDF) of the *i*^{th} risk is either less than or equal to the CDF of the *j*^{th} risk for all states (first order dominance), or 2) the sum of the differences between the *j*^{th} risk CDF and the *i*^{th} risk CDF for all states is positive (second order dominance):

While the *math* behind first and second order stochastic dominance is summarized in my optional reading entitled “Technical Note on Stochastic Dominance and Expected Utility”, the *intuition* for first and second order stochastic dominance can be seen in the figures featured on pages 9 and 12 of my Decision-Making under Risk and Uncertainty, part 4 lecture note.

In the above figure from page 9 of my Decision-Making under Risk and Uncertainty, part 4 lecture note, the *G* risk has 50% of a $0 payoff, and 25% each of a $10 payoff and a $100 payoff. The *F* risk involves removing 25 percentage points off the $0 payoff and adding 25 percentage points extra to the $100 payoff, and both *F* and *G* have a 25% probability of $10 payoffs. Graphically, this ensures that *F* first order stochastically dominates *G*; i.e., *G*(*W _{s}*) is greater than or equal to

*F*(

*W*) for all

_{s}*s*, which also implies that

*E*

_{F}[

*U*(

*W*)] >

*E*

_{G}[

*U*(

*W*)]. Intuitively, the picture which gets rendered by this analysis shows that most of the probability mass of the stochastically dominant risk (in this case,

*F*) lies below the probability mass of the stochastically dominated risk (in this case,

*G*). Furthermore, since risk

*F*first order stochastically dominates risk

*G*, risk

*F*also second order stochastically dominates risk

*G*because

*G*(

*W*) –

_{s}*F*(

*W*) > 0 for $0 and $10 payoffs, and

_{s}*G*(

*W*) –

_{s}*F*(

*W*) = 0 for the $100 payoff.

_{s}Next consider the figure from page 12 of my Decision-Making under Risk and Uncertainty, part 4 lecture note:

Here, *G*(*W _{s}*) –

*F*(

*W*) > 0 for payoffs ranging from 1-5,

_{s}*G*(

*W*) –

_{s}*F*(

*W*) < 0 for payoffs ranging from 5-8, and

_{s}*G*(

*W*) –

_{s}*F*(

*W*) = 0 payoffs ranging from 8-12. Thus, there is no first order dominance. However, since the positive difference between

_{s}*G*(

*W*) –

_{s}*F*(

*W*) for payoffs ranging from 1-5 exceeds the negative difference between

_{s}*G*(

*W*) –

_{s}*F*(

*W*) for payoffs ranging from 5-8, the sum of

_{s}*G*(

*W*) –

_{s}*F*(

*W*) over the entire range of payoffs comes out positive. Thus, risk

_{s}*F*second order stochastically dominates risk

*G*, which also implies that

*E*

_{F}[

*U*(

*W*)] >

*E*

_{G}[

*U*(

*W*)].