Comparing wealth distributions

Construct cumulative low-wealth populations and test a distribution-level claim over the complete finite support.

17 minutes · advanced · Original Poker4Quant lesson

What you will learn

Distinguish first-order stochastic dominance from a mean advantage, using explicit cumulative distributions and crossing witnesses.

Prerequisites: Expected utility and certainty equivalents · Complements and unions · Unequal outcome weights

A cumulative distribution counts low outcomes

For final wealth W, the cumulative distribution F(t)=P(W≤t) gives the probability at or below a wealth cutoff t. It includes equality. If wealth is preferable when it is larger, having less probability below a low cutoff can be favorable. This reverses the intuition that a larger value on every graph must be better. Read the axis and event before comparing curves.

In the activity, A has eight equally likely wealth cases 20,20,40,40,60,60,80,80; B has 10,10,30,30,50,50,70,70. At cutoff 35, A retains two cases and B four, giving F_A=.25 and F_B=.5. At cutoff 65, both retain six cases. Each retained count belongs to its own marginal law; matching case numbers do not identify a common underlying state.

Check every support jump, not one favorable point

We use a strict-improvement version of first-order stochastic dominance: A qualifies over B when F_A(t)≤F_B(t) for every t and the inequality is strict somewhere. In a finite distribution, the CDF changes only at its support values. Checking every support cutoff and the outer regions is therefore sufficient; the curves remain constant between adjacent support values.

For the first model, A’s CDF is never above B’s. This implies no smaller expected utility for A under any nondecreasing wealth-only utility function for these finite laws. Some utility functions can be indifferent, so strict improvement at one cutoff is not a promise that every possible preference strictly favors A. The claim does not cover nonmonotone objectives or constraints omitted from the wealth model.

Equal mean rankings can hide a crossing

Change A to four zero-wealth cases and four 100-wealth cases, while B always gives 40. A still has mean 50 and B mean 40, the same mean ranking as before. But at cutoff 0, F_A=.5 exceeds F_B=0; at cutoff 40, F_A=.5 is below F_B=1. Each direction has a counterexample, so neither distribution qualifies under the stated first-order criterion.

Linear utility favors A’s mean 50, whereas square-root utility gives A expected utility 5 and B utility√40, favoring B. This is a concrete preference disagreement, not proof that every utility ranks the crossing pair differently. More specialized comparisons require additional assumptions. A whole-distribution claim is stronger than a mean, a single cutoff comparison or a selected favorable realization.

Worked example

X gives wealth 20 or 60 equally likely; Y gives 10 or 50 equally likely. Compare the cumulative probabilities at 15,30 and 55.

At 15, F_X=0 and F_Y=1/2.
At 30, both CDFs equal 1/2.
At 55, F_X=1/2 and F_Y=1.
Checking the remaining support jumps and outer regions shows F_X≤F_Y everywhere, with strict inequality somewhere.

X qualifies under the stated first-order improvement criterion. This compares marginal wealth laws, not an asserted coupling of realized outcomes.

Hands-on: Construct cutoff cases and inspect every CDF jump

Build two low-wealth populations, inspect complete support regions and reconcile CDF differences before claiming distribution dominance.

Reading distribution workshop…

Common mistakes

Treating a larger low-wealth cumulative probability as automatically better.
Using only means or one selected cutoff to claim dominance.
Excluding equality from W≤t.
Interpreting matching case numbers as proof of a joint state or correlation.

Keep these ideas

A wealth CDF counts probability at or below a cutoff.
For higher-wealth preferences, the favorable CDF is no greater everywhere.
Finite support lets you check every jump and outer region.
Crossing curves can coexist with a clear mean advantage.