Elsewhere I explored a bargaining model describing a legislative body with a one or near one-seat majority, resembling the current makeup of the House of Representatives. Some plausible assumptions are necessary but I believe the model credibly describes the nature of individual caucus holdouts’ leverage in such a scenario based on predictions mirroring recent bill negotiations.
But that model treated the negotiation as existing between “leadership” and a mutually exclusive faction of individual holdouts, each bargaining separately for their own desired outcome.
A recent paper by colleagues james_wallner and Soren Dayton explores how what we would more properly think of as factions have and can function in Congress. Putting aside their normative claims for now, it prompted me to consider extending the earlier model to more realistically account for factions in a legislative bargaining model.
First I briefly recap the consistent elements which carry over and then restate the implications.
Then I describe the extension and characterize it’s predictions for legislative outcomes.
Base Bargaining Model
F is the set of pivotal holdout members — members who can’t be substituted for alternative bargainers. Each individual holdout is indexed i. Membership in Fi is established beforehand. Leadership (L) acts as the sole proposer.
Variables:
x — the policy outcome, running from 0 (leadership’s baseline bill, no concessions) to 1 (a fully-conceded bill matching a holdout’s ideal).
ui(x) — holdout i‘s payoff from the bill passing at concession level x; utility increasing in x.
ci(t) — holdout i‘s own cost of delay, as a function of time t remaining before the deadline. A safe-seat member has a low, close-to-flat ci(t); a vulnerable-seat member has a steep one.
di — the disagreement point: what member i gets if bargaining collapses entirely and no deal is reached.
cL(x) — leadership’s cost of granting concession x, increasing in x.
cL(t) — leadership’s own cost of delay. Convexly increasing in t.
π(t) — the probability, at time t, bargaining collapses to the disagreement point rather than resolving at baseline.
xi* — holdout i‘s equilibrium reservation demand: the minimum concession they’ll accept.
Implications:
An individual holdout’s leverage is determined by:
cost of delay (idiosyncratic to the member)
divisibility (a characteristic of the bill itself)
ratio of cost of concession (leadership) to value of concession (holdout)
Leadership cost of delay non-linearly increasing (convexity) - this is an assumption built in to the model
Extension: Multiple Factions, One Shared Capacity
The base model treated the bill as a single scalar x and F as one holdout vector. Not all bills are single-ask, and once you look at recent cases side by side, two structurally different kinds of concession show up — and the base model’s existing machinery can tell them apart without adding anything new.
A SALT-cap fight and a Speaker election are different kinds of bargain. SALT is divisible: only members from high-tax states actually value it; a member from a no-income-tax state is indifferent.1 A Speaker vote is indivisible: it’s binary, and pays off identically to every member who backs the winning side regardless of how many others join them.
The bill becomes a vector, X = (x₁, ..., xD), allowing severable dimensions, each one of the two types above.
Divisible dimensions get a support set — the subset of members who actually have a stake.
Outside Fj, ui(xj) = 0, the member is indifferent. Each divisible dimension then runs the original scalar bargain, unmodified, restricted to its own Fj. The faction on this dimension is the set of members for whom the concession is worth something.
Indivisible dimensions are where a faction’s size becomes the live question, and here the existing budget constraint (2n + p < m) supplies the threshold, just read from the faction’s side rather than leadership’s:
Below kmin, adding members increases the faction’s blocking weight. At kmin, the faction is already sufficient — leadership can’t pass the bill without them — and additional members past that point buy the faction nothing.
wF(k) is the aggregate “blocking” leverage. Extraction per member peaks exactly at sufficiency and dilutes past it.
Because leadership’s tolerance — cL(t), and the same vote-budget constraint — is one finite resource, dimensions with completely disjoint memberships and zero utility overlap are still implicitly drawing on the same capacity. Satisfying one dimension’s demand consumes some of what’s available for every other active dimension:
A bill with a single indivisible dimension at the one-seat margin collapses kmin to 1 — a faction of one is sufficient — and the extension returns the original single-holdout result unchanged.
The extension applies to margins wide enough that multiple severable asks can be live at once, but narrow enough that a rival coalition still can’t credibly bid for defectors.
Comparative statics (testable directional claims)
The number of concurrently active factions on a bill tracks the number of severable, divisible dimensions it carries — not the size of the margin.
A concession granted on one dimension tightens what’s available on every other simultaneously active dimension, even where factions share no members and no overlapping preferences, because leadership’s capacity is one shared, finite resource. Early concessions will be bigger.
An indivisible vote (e.g., Speaker vote) collapses to the base model’s single-faction case, with per-member extraction maximized at exactly the sufficiency threshold and falling past it.
Conclusion
Most of this is intuitive once stated, but a couple of things fall out that aren’t:
The relevant predictor of how contested a bill will be isn’t the margin alone, it’s how many severable asks the bill happens to bundle. A wide majority carrying one clean, indivisible ask (a debt-ceiling raise with a single up-or-down rider) behaves like a razor-thin majority with one holdout — the margin matters less than the bill’s structure.
“The faction” isn’t a fixed roster across bills — it’s whoever has a stake in that bill’s specific dimension. The functional membership on a SALT fight and on a debt-ceiling rider can be entirely different people, even when press coverage uses the same caucus label for both.2
Bundling is a lever leadership can pull deliberately, not just a feature of how a bill happens to get written. Combining several divisible asks into one vehicle forces factions that would otherwise bargain independently to draw on the same finite capacity — which can dilute any single faction’s leverage, or just as easily create a more fragile package that any one faction can sink.
Predictions
OBBBA-style tax bills: where there are several divisible elements (SALT, sector carve-outs), concessions should land unevenly by order of resolution — factions that lock in early get closer to their ask; factions still bargaining once earlier concessions have drawn down leadership’s capacity settle for less, independent of their own underlying leverage.
A debt-ceiling bill carrying a single rider: treat it as the indivisible case — expect a faction size near kmin among holdouts, no internal division of the concession, and a resolution profile matching the original model's shutdown/deadline predictions (clustered near the binding constraint) rather than the OBBBA pattern above.
Appropriations omnibuses: holding margin constant, expect smaller per-faction concessions as the number of bundled asks rises — a crowding effect distinct from the deadline-clustering prediction from the base model.
That’s the band Wallner and Dayton are actually describing: DSG, RSC, Freedom Caucus and 2023 Speaker fight — none of them one-vote margins. The model’s prediction for that regime: a bloc holding at exactly kmin (Freedom Caucus, enough to deny McCarthy, no more) should out-extract, per member, a larger bloc with equivalent aggregate blocking power but more internal heterogeneity (RSC, historically).
The author is Nonresident Senior Fellow at the Foundation for American Innovation
It should not change directionally the implication of the model if we acknowledge other members may in fact not be indifferent. In this example, a fiscal conservative non-SALT state member may still be opposed to the deficit impact of granting a more generous deduction. It is in fact the case often conceding to one faction loses another. But the constraint holds either way, and since “policy” is a scalar, we do not lose much of the generality while avoiding an incredible amount of complexity (and tractability) by not allowing for inter-policy interactions. A future extension could consider how to explicitly do so.
Here we do not impose an ideological or other unifying characteristic determining participation in the faction, only a unified policy dimension. That said it is not precluded as structured, and overlapping policy interest itself can be representative of an ideological unity. Future extensions could allow for a repeat game such that durable factions, which persist across different policy fights, introduce an additional element which both determines faction participation and changes holdout costs due to revealed credibility. Though doing so would certainly introduce a significant increase in complexity.