Stochastic dominance and utility

Idea: Rank without a utility

Stochastic dominance compares whole distributions and ranks them for a whole family of utilities at once.

• First order: everyone who prefers more agrees
• Second order: every risk-averse holder agrees
• Both fail: the ranking depends on the utility

Formula: First-order dominance

$F_X(t)\le F_Y(t),\forall t$

At every threshold, X is no more likely than Y to end at or below it.

Example: Desks with the same payoffs

$A$ pays $0$, $10$, $30$ with chances $0.1$, $0.5$, $0.4$; $B$ with $0.3$, $0.5$, $0.2$. Does $A$ dominate? Gap in means?

1. Cumulative at $0$ and $10$: $A$ has $0.1$, $0.6$; $B$ has $0.3$, $0.8$
2. $A$ sits below $B$ everywhere, so $A$ dominates
3. Area between curves: $0.2\times 10+0.2\times 20=6$

Common mistake: Dominance is not winning every time

Reading $X⪰Y$ as "X beats Y in every scenario". The definition looks only at the distributions.

Instead On one coin $X$ pays $1$ or $3$ and $Y$ pays $2$ or $0$: $Y$ wins on heads, yet $X$ dominates.

Idea: A higher mean proves nothing

$C$ pays $12$ for sure; $D$ pays $0$ or $40$ with equal chances, mean $0.5\times 40=20$.

• The curves cross, so neither dominates
• With $u(w)=w$ you choose $D$
• With $u(w)=\sqrt{w}$ you choose $C$: $\sqrt{12}\approx 3.46$ beats $0.5\times \sqrt{40}\approx 3.16$

Formula: Second-order dominance

$S_X(t)=E[\max (t-X,0)]$

If X never has a larger expected shortfall at any target t, every risk-averse holder prefers X.

Example: Same mean, more noise

$G$ pays $14$ or $16$, $H$ pays $5$ or $25$, each with equal chances. Expected shortfall below $20$?

1. $G$: $0.5\times 6+0.5\times 4=5$
2. $G$ is never worse at any target: it dominates
3. $H$: $0.5\times 15=7.5$

Formula: Jensen for a concave u

$E[u(X)]\le u(E[X])$

A risk-averse holder values a gamble below its mean; the gap is the risk premium.

Quick check: Price of the risk

With $u(w)=\sqrt{w}$, a bet pays $100$ with chance $\frac{1}{4}$, else $16$. What is the risk premium?

Idea: Say it in order

Compare cumulative chances first: if one curve never rises above the other, no utility is needed.

• Curves cross: compare means, then shortfalls
• Both fail: the ranking depends on the utility
• Then show the answer for a named one