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Adjudicator — a veto, and a reserve that may refuse to exist#

Capstone · Actuarial

Source: Adjudicator.hs · Adjudicator.hp · Reserve.hb · AdjudicatorTests.cs · ReserveTests.cs

Disagreement is the design#

On the capstone's house, valuation says buy, condition says negotiate, credit says walk. Each is right about its own question. The adjudicator's job is not to average them — it is to decide what their disagreement means.

A veto, not a vote#

any agent walks      → walk
else any negotiates  → negotiate
else someone voted   → buy

Because the costs are not symmetric. A wrong walk costs a missed house — bounded. A wrong buy can cost a default, a repair bill the buyer cannot meet, or both — unbounded. So one agent seeing ruin outranks two seeing a bargain, and a majority vote would be the wrong shape entirely.

one_walk_is_enough is a theorem in Adjudicator.hp: a single walk vote marks the purchase walked, whatever anyone else said.

"Everyone said buy" has no Horn form#

A Horn rule has no universal in its body, so "every agent said buy" cannot be written directly. It is written as what it amounts to when every agent votes exactly once:

def verdict_buy(purchase: Case) :- voted(purchase) ∧ ¬ walked(purchase) ∧ ¬ haggled(purchase)

Somebody voted, nobody walked, nobody haggled. The two negations are stratified — walked and haggled depend on nothing that depends on them — the same shape as only_child in the Kinship theory.

And silence is not agreement: without voted(purchase), an empty panel would buy everything.

Every verdict test asserts that exactly one verdict fires. Dropping ¬ walked from the negotiate rule lets a purchase be walked and negotiated at once — and that was checked to fail a test.

The reserve, had it gone ahead#

The buyer walks. But suppose they bought anyway: what would it take to carry it?

Treat the predicted shortfalls — repair exposure plus default loss — as claims against a fund, and the buyer's yearly set-aside as its premium. Cramér and Lundberg worked out when such a fund survives:

safety loading   θ = c / (λ · m) − 1
adjustment       R = max(1/m − λ/c, 0)
Lundberg bound   ψ(u) ≤ exp(−R · u)
reserve          u = ln(1/ε) / R

A simulation can only say "not yet"#

Run the fund forward a thousand years and it survives. That proves nothing about year one thousand and one. The Lundberg bound answers the real question — the chance of ever running dry — and the reserve is sized against that. A test checks it both ways: the reserve for a 5% chance of ruin, fed back into the bound, gives exactly 5% again.

A thin loading makes the reserve explode#

On the capstone's house — claims averaging $51 323, arriving about once every two years (a rate of 0.5), against a set-aside of $30 000 a year:

  loading 16.9%
    reserve $1,063,225 keeps the chance of EVER running dry at or below 5%

A million-dollar reserve for a $172 150 house. That is not a bug. A loading of 16.9% makes the adjustment coefficient tiny, and the reserve is inversely proportional to it. The honest reading: this purchase is barely fundable, and ruin theory says so in a number too large to ignore.

And below the loading, no reserve at all#

At a $20 000 set-aside, expected claims beat the premium:

  loading -22.1%
    NO RESERVE IS REPORTED. The set-aside does not beat the expected claims, so ruin is certain
    however much is held back — and any figure printed here would be a comfortable lie.

When the loading is at or below zero, ruin is certain and R = 0. The reserve formula then divides by zero and returns positive infinity — a number, which would print as "a very large reserve" when the truth is "no reserve is enough". A test pins that infinity, so the refusal has something concrete to refuse. The safety loading is a precondition, not a diagnostic.