If no voting rule is perfect (Arrow) and simple majority can cycle and oppress minorities, how should a society choose its decision rule? Buchanan and Tullock's The Calculus of Consent (1962) gives the canonical answer, and it is more subtle than 'majority rule is democratic, so use it'. The optimal rule trades off two costs, and the answer depends on the stakes.
The two costs of any decision rule
External costs vs decision-making costs
For a rule requiring K of N people to agree before the group acts: • External costs fall as K rises. These are the costs imposed on you by a collective decision you opposed. The more people who must consent, the less likely a decision harms you — at K = N (unanimity), no one can be forced to bear a net loss, so external costs are zero. • Decision-making costs rise as K rises. These are the time, effort, and strategic hold-out costs of reaching agreement. Getting one person to decide is instant; getting a bare majority takes bargaining; getting unanimity can be impossible as each holdout extracts concessions.
The optimal majority
The optimal decision rule minimises the sum of the two costs. Since external costs fall and decision costs rise with K, the sum is U-shaped and the minimum is at some interior K*. The central insight is that K* is generally not one-half-plus-one. Simple majority is not a sacred constant; it is one point on a curve, optimal only for a particular balance of the two costs.
Why the stakes decide the rule
The higher the potential harm from a wrong or oppressive collective decision — the larger the external-cost curve — the further right K* moves, toward supermajority or unanimity. So we rationally require: a simple majority for ordinary legislation (moderate stakes, frequent decisions where decision costs matter), but a two-thirds supermajority to amend a constitution, expel a member, or override a veto (high stakes, where the external cost of getting it wrong is large and we accept higher decision costs to guard against it). The same logic explains why juries require unanimity for criminal conviction — the external cost of a wrongful conviction is enormous.
Constitutional vs in-period choice
Buchanan and Tullock's deepest move is to separate two levels of decision. In-period (post-constitutional) choice is the ordinary game: given the rules, which policy passes? Constitutional choice is the prior question: which rules shall we adopt for making in-period decisions? The two are analysed differently because the information and incentives differ.
The veil of uncertainty
At the constitutional stage, you are choosing rules that will govern many future decisions whose effects on you, specifically, you cannot foresee — you don't know whether you'll be in the majority or minority, taxpayer or beneficiary, governing or opposition. This uncertainty about your future position pushes you toward impartial, general rules that you could accept whichever side you end up on. It is the contractarian counterpart to Rawls's veil of ignorance, and it is what gives constitutional rules their legitimacy: they command consent precisely because no one can yet tilt them in their own favour.
Why this matters for institutional design
The framework explains a great deal of real constitutional architecture: entrenched bills of rights (some things are removed from ordinary majority reach because their external costs are too high), supermajority requirements for constitutional amendment, bicameralism and separation of powers (multiple consents raise K for major changes), and the practice of negotiating reforms 'behind the veil' — agreeing on rules before the specific winners and losers are known, which is far easier than agreeing once identities are revealed (the Fernandez-Rodrik lesson from the last course, seen from the design side).
Exercise
A new regional bloc is writing its founding treaty. Members must decide the voting rule for three kinds of decision: (a) routine administrative matters, (b) the annual budget, and (c) admitting new members or amending the treaty. (1) Using the two-cost framework, recommend a decision rule for each and justify it in terms of external and decision-making costs. (2) Explain why negotiating these rules now, in the treaty, is easier than deciding each future case ad hoc — invoke the veil of uncertainty. (3) A large member demands a veto (unanimity) over all decisions. Analyse the trade-off it is imposing on the bloc. (4) Connect this to Arrow and the previous module: what problem do supermajority rules mitigate, and at what price?