A market made from two probabilities

The Pool That Cannot Sit Still

Give two people the same full insurance contract, then offer one carefully chosen cheaper contract. The safer person leaves, the riskier person stays, and the price of the original pool jumps from 15 to 24 on a 100-unit wealth scale. Every preference, expected claim, and insurer profit is recomputed live from the declared two-risk Rothschild-Stiglitz model.

Layer one: break the pool

Make the cheaper offer.

Both buyers begin in one actuarially fair full-insurance pool. The offer does not label either person. Each moves only if the expected utility shown in the ledger is larger.

Exclusive contracts, one policy per buyer ● recomputing in this browser
Original poolboth types

...premium for full coverage

LOW RISKchooses this contract
HIGH RISKchooses this contract
Cheaper rivalpartial cover

...premium, with ... indemnity

LOW RISKoffered the same menu
HIGH RISKoffered the same menu

The pool is waiting. Both types hold the original contract.

BuyerOriginal EURival EURival minus originalRival profitChoice
Low risk...............
High risk...............

EU(p,a,I) = (1-p)u(W-a) + p u(W-d-a+I). Insurer profit is a-pI. The premium jump is one repricing after composition changes, not an iterated death spiral.

Layer two: search beyond the toy

Can the only possible separation survive?

The 1976 result is stronger than one cream-skimming offer. The browser first solves the incentive-compatible separating pair, then searches the bounded contract plane for a pooling deviation that both types prefer and that earns a profit.

Premium and indemnity contract plane Fair-odds lines and the computed separating contracts, plus a profitable pooling deviation when one exists.
Horizontal: indemnity from none to the full loss. Vertical: premium. The search scans every indemnity in this displayed domain, solves the highest premium each type will accept, and refines the best cell.
Solving... The numerical search is running.
High-risk separating contract...
Low-risk separating contract...
Best pooling deviation found...

The low-risk contract stays on its own fair-odds line and moves toward less coverage until the high-risk type is exactly indifferent. A deviation counts only if both types strictly prefer it. When no profitable deviation is found, that is a result inside the declared search domain, not a theorem about contracts outside it.

The instrument checks its own teeth

The check

These are not three copies of one calculation. One route counts a finite population, one reproduces the paper's fair-odds and incentive-compatibility geometry, and one feeds the chooser a deliberately corrupted type. The offline verifier repeats the work with separate code and sweeps the controls.

Pass · independent route...Direct expected-cost arithmetic is compared with an explicit event count over a finite synthetic population.
Pass · published anchor...Rothschild and Stiglitz's fair-odds full-insurance result and the high-risk indifference boundary are evaluated as residuals.
Pass · poison rejected...The high-risk probability is overwritten with the low-risk probability. The claimed chooser split must disappear, so the checker must reject it.

Free choices and limits

  • One period, two privately known accident probabilities, identical wealth and loss, concave utility, risk-neutral competitive insurers, diversifiable losses, no costs, no moral hazard.
  • Contracts are exclusive and limited to nonnegative premiums and indemnities no larger than the loss. Both state consumptions must remain positive.
  • The deviation search uses a dense indemnity grid, then a local refinement and bisection for willingness-to-pay. The offline verifier uses a different nested exhaustive grid and analytic utility inversions.
  • Changing the utility selector changes the separating contract and survival boundary. It does not change fair premiums or the layer-one chooser split for the three utilities offered.
  • This models adverse selection, private information about risk before purchase. It does not model moral hazard, behavior changed by coverage.

Sources and uncertainty