Example
The comparison can be more convenient represented if we describe investments in term of their returns.
State-by-state dominance: asset pays as much in all states of nature and strictly more in at least one state. For the above example, Asset 3 > Asset 1, 2. But for Asset 1 and Asset 2, there is no ranking possible on the basis of the dominance criterions. Hidden requirement here is that investors are all non-satiation.
Intuition for A FSD B: We are likely to have more wealth from investment A as comparing to investment B.
Decision rule: the cumulative distribution curve of B is above or equal A
Consider two assets A and B, security A FSD security B iff $F_A(x)\le F_B(x)$
$$ \begin{aligned} F_A(x)\le F_B(x) &\to p(w_A<x)\le p(w_B<x)\\ &\to p(w_A>x)\ge p(w_B>x) \end{aligned} $$
Which means: in every ordered state [a, b] of investment A and B, we receive higher wealth in investment A.
An example is shown as below
Therefore, investment A FSD investment B, which is equivalent to “someone’s choice of A over B”.
Intuition for C SSD D: C and D have the same return, but D has extra risk over C.
Decision rule: The area below the cumulative distribution curve of C is smaller than D
Consider 2 assets C and D, security C SSD security D iff