La Bourse / Volume VI / Nº VI.01 / Ten concept briefs

Four colleagues walking toward a long table set with dried grasses, the room bright with sun. — Plate VI.01 · Ten concept briefsThe Table Before the Meeting.A decision rule is furniture. It decides who can be in the room, how long they will be there, and who is not in the room at all — and it does this before anybody has said a word.

TEN CONCEPT BRIEFS · Chapter VI.01 — Who Decides

One page each. A reader who reads only these ten pages has the chapter.

BRIEF 1 — The Two Curves

The idea. Every decision rule spends in two currencies at once, and they move in opposite directions. External cost is the harm done to the people a decision goes against. Decision cost is the time, attention and bargaining it takes to reach agreement. Raise the number of people who must agree and the first falls while the second rises. Lower it and the reverse.

        total cost(k)  =  external(k)  +  decision(k)

  k          how many of n members must agree
  external   falls as k rises, reaching zero at unanimity
  decision   rises as k rises, steeply near unanimity

Worked example. On a board of nine where a member left outside the winning coalition bears £30,000 and a round of negotiation costs £3,240, a 4-of-9 rule costs £243,169 a year and an 8-of-9 rule costs £591,146. Neither is the answer. The bottom of the sum is at 6 of 9, £200,562 a year.

Why it matters. Almost every governance argument you have ever heard is one curve arguing with the other. "We need broader agreement" is the external-cost curve speaking. "We need to be able to move" is the decision-cost curve. They are both right, they are not opposed, and the argument has a numerical answer.

You already know this because you have sat in a meeting that took four hours and produced a decision nobody minded, and in another that took ten minutes and produced a decision two people are still angry about. You paid in both. The question was only which currency.

BRIEF 2 — External Cost

The idea. External cost is what a decision does to the people it was taken against. It is not regret and it is not politics; it is a measurable loss borne by identifiable people who did not agree.

        external(k)  =  decisions x loss x (n - k) / n

The share of the body outside the winning coalition is (n − k)/n. Multiply by what a member loses when a decision goes against them, and by how many such decisions there are in a year.

Worked example. Nine members, twelve contested decisions a year, £30,000 of loss to a member left outside. Under a 5-of-9 rule, four of nine are outside: external cost is £160,000 a year. Under 7-of-9, two of nine are outside: £80,000. Under unanimity it is £0, which is the entire reason unanimity keeps being proposed.

Why it matters. This is the term that people feel and never compute, so it gets argued in the language of fairness, where it cannot be settled. Compute it and the conversation changes register: we are no longer asking whether the minority matters, we are asking what it costs them, and whether that is more or less than what protecting them costs everybody.

You already know this because you have been the two-of-nine, and you can still name the decision and roughly what it cost you.

BRIEF 3 — Decision Cost

The idea. Decision cost is the price of assembling agreement, and it rises faster than the number of agreements you need. Each extra supporter must be won from a shrinking pool, so the last few are far more expensive than the first few.

        rounds(k)  =  1 / P(X >= k),     X ~ Binomial(n, p)
        decision(k) = decisions x cost per round x rounds(k)

p is the probability that a given member accepts a given proposal.

Worked example. Nine members, p = 0.60, one round costing 9 × 3 hours × £120 = £3,240. At 5-of-9 the tail probability is 0.7334, so 1.36 rounds. At 8-of-9 it is 0.0705 — 14.18 rounds. At unanimity it is 0.6⁹ = 0.010078 — 99.23 rounds, ninety-nine meetings for one decision, and £3,858,025 a year for twelve of them.

Why it matters. The curve is not steep; it is explosive, and it explodes in the region where most idealistic governance proposals live. Anyone proposing unanimity is proposing ninety-nine rounds and has not usually noticed.

You already know this because you have watched a group of four agree in a corridor and the same question take a group of forty an entire quarter.

BRIEF 4 — The Optimal Majority

The idea. There is a k that minimises the sum of the two curves, it is computable, and it is almost never 50 per cent plus one. Buchanan and Tullock named it in 1962. The useful part is that it is not a constant: it moves with the stake and with nothing else.

Worked example. Hold the board, the people and the cost of a meeting fixed. Move only what a member loses when a decision goes against them.

   loss to an excluded member      optimal rule
   £5,000                          4 of 9   (44.4 %)
   £10,000                         5 of 9   (55.6 %)
   £30,000                         6 of 9   (66.7 %)
   £100,000                        7 of 9   (77.8 %)
   £300,000                        8 of 9   (88.9 %)

Why it matters. It kills the idea of a decision rule for an organisation. The same nine people should decide the coffee supplier below a majority and the pension buy-out at near-unanimity, and a single rule applied to both is over-paying on one and under-protecting on the other. The rule belongs to the decision, not to the body.

You already know this because you require two signatures on a large payment and none on a small one, and you have never thought of that as a constitutional position. It is one.

BRIEF 5 — The Blocking Coalition

The idea. Every threshold rule has a shadow: the smallest group that can stop things. Under a k-of-n rule the minimal blocking coalition is n − k + 1. The threshold is the sentence everybody argues about; the blocking coalition is the sentence that describes what actually happens.

Worked example, and it is the clearest in the chapter. The United States Constitution requires three-fourths of the states to ratify an amendment: 38 of 50, so 13 states can block. The thirteen least populous states hold 14,618,613 people — 4.41 per cent of the country. The Senate's cloture rule needs 60 of 100, so 41 senators block, and forty-one senators come from at least 21 states holding 37,174,921 people, or 11.22 per cent.

Why it matters. A supermajority is a transfer of decision rights to a minority, and the honest way to set one is to name that minority out loud and say whether you meant to protect exactly them. Since 1789 roughly 11,848 measures to amend the Constitution have been proposed and 27 ratified — 0.228 per cent. That is the blocking coalition doing its job, at its stated price.

You already know this because you have seen one person on a committee stop something the other eight wanted, and you noticed that the rule said nothing about them and that they were nevertheless the whole story.

BRIEF 6 — Consent, and the Objection Grammar

The idea. Consent rules — sociocracy, and the systems descended from it — are not unanimity. They keep the veto and narrow what counts as using it. The question is not do you prefer this? but do you have an argued and paramount objection: a reason this will not work, or will harm the aim? A preference is not an objection.

        P(carry)  =  (1 - q) ^ n         q = share holding an admissible objection

Worked example. Nine members. True unanimity at p = 0.60 needs 99.23 rounds. Consent at q = 0.10 carries with probability 0.9⁹ = 0.3874 — 2.58 rounds, faster by 38.4 times. Now find the ordinary threshold rule with the same expected rounds: it is k = 6.23 of 9, or 69.2 per cent.

Why it matters. Consent is a two-thirds supermajority wearing unanimity's clothes, and the difference is billed to somebody. The external cost at an equivalent 6.23-of-9 rule is £110,918 a year, against £0 at true unanimity, and it falls on the members whose objection is a real preference they cannot phrase as a workability argument. That is a good trade. It is a trade.

You already know this because you have been told your objection "wasn't a real objection," and you knew, correctly, that a rule had just been applied to you.

BRIEF 7 — Sortition and the Counterfactual Sample

The idea. Sortition fills a deciding body by lot rather than election. Its claim is frequently misstated as statistical representativeness, and at realistic assembly sizes that claim does not survive contact with the arithmetic.

        margin of error  =  1.96 x sqrt(0.25 / n)

Worked example. Ireland's Citizens' Assembly seated 99 members: a 95 per cent margin of error of ± 9.8 points. America in One Room (September 2019) seated 526: ± 4.3 points. A 1,200-member mini-public would reach ± 2.8.

Why it matters. The defence of sortition is not this is what the public thinks — a thousand-person poll does that better and cheaper. It is this is what the public concludes when it has the evidence, the time and each other, which is a counterfactual no poll of any size can produce. The Irish assembly recommended repeal of the Eighth Amendment; the referendum of May 2018 carried at 66.40 per cent on 64.13 per cent turnout. The assembly decided nothing. It made a decision available. The OECD counted 289 such processes across member countries between 1986 and 2019 — 34 years, about 8.5 a year.

You already know this because you trust a jury of twelve strangers who heard the evidence more than a survey of twelve hundred who did not.

BRIEF 8 — Delegation and Agency Loss

The idea. Delegation is decision cost converted into agency loss at an exchange rate. It is almost always worth it, and the reason it is almost always worth it is enormous — which is precisely why the exceptions have to be written down in advance.

   break-even loss  =  decision cost saved / (divergence rate x decisions)

Worked example. An assembly of 1,200 deciding directly at a 60 per cent threshold expects 1.95 rounds at £432,000 a round: £10,114,446 a year for twelve decisions. The same twelve delegated to a board of nine at 6-of-9 cost £80,562 — a saving of £10,033,884. At a 25 per cent divergence rate, the loss per divergent decision at which delegation stops paying is £3,344,628.

Why it matters. It produces the schedule directly: delegate everything whose loss-if-wrong is below £3.34m and reserve everything above. And it names the cost that no rule change reaches — the House of Representatives carried 60,449 people per seat in 1789 and 761,952 in 2020, so the principal's stake in each agent has thinned by 12.6 times. Each European Parliament seat carries 623,898. Divergence is not a constant; it grows with that ratio.

You already know this because you do not attend your bank's decisions, and you do read the terms on anything above a certain size.

BRIEF 9 — May's Theorem, and the Price of the Status Quo

The idea. Kenneth May proved in 1952 that for a choice between two alternatives, simple majority rule is the unique rule that is anonymous (everyone's vote counts the same), neutral (the two alternatives are treated alike) and positively responsive. Every supermajority rule breaks neutrality — deliberately. It treats the status quo and the proposal differently.

Worked example. Buchanan and Tullock measure every cost from the status quo, so the status quo costs nothing by construction and the external-cost curve reaches zero at unanimity. Price the unmade decision instead. Give the standing arrangement an annual harm S borne by whoever it is already costing, and add the delay each rule creates. At S = £0 the optimum is 6 of 9 (66.7 per cent). At S = £300,000 a year it falls to 5 of 9 (55.6 per cent), and stays there as S rises.

Why it matters. A veto is not a right to be unharmed. It is a right to keep whatever harm is already in place, and the people that harm falls on are not in the model unless somebody puts them there. The external-cost curve does not reach zero at unanimity; it reaches the cost of not deciding.

You already know this because you have watched a committee protect a minority by doing nothing, and noticed that a different minority was paying for the nothing.

BRIEF 10 — The Reserved-Matters Schedule

The idea. The instrument that carries all of the above is already standard legal furniture: a schedule in a shareholders' agreement listing which decisions need whose consent. What is not standard is pricing it.

Worked example. Three classes. Class A: ordinary operating decisions, board simple majority. Class B: decisions above the computed threshold, board plus investor-director consent. Class C: constitutional matters at a special resolution — 75 per cent under section 283 of the Companies Act 2006. Then test every proposed veto: at a 20 per cent chance of a block, a 90-day delay and £4,000 a day at risk, the veto costs £72,000 against £150,000 of external cost prevented — 2.08 to 1, so it is granted. Items that fail the test are deleted.

Why it matters. It has a balance-sheet consequence most teams never see. Under IFRS 10, a minority holding substantive participating rights over an investee's relevant activities can prevent consolidation, while merely protective rights cannot. The drafting of the schedule decides whether a subsidiary appears line by line or as one equity-accounted investment — which moves reported gearing and covenant headroom.

You already know this because you have signed something with a consent list in it, and nobody costed a single line of that list before you signed.