Decision brief: How to formalize holdings
Status: Decision needed before formal results can proceed. Date: Feb 12, 2026 Context: ChatGPT referee report identified that the current entailment definition is backwards and must be fixed. Three alternative formalizations proposed. Choice determines which propositions are provable, how the model relates to competing theories, and how legally realistic the framework is.
The problem with the current definition
The paper currently defines entailment as:
$$\mathcal{F}_t(d_t; z_t) \subseteq H_t$$
This requires the holding $H_t$ to contain all rules in $\mathcal{F}_t$ that produce the chosen outcome. That means the holding cannot exclude any outcome-consistent rule. Two consequences:
The breadth/citation channel collapses. If a holding must include all rules that justify the outcome, it cannot shape doctrine — it merely records the outcome. The judge's only real choice is the outcome $d_t$, not the holding. This makes the $\gamma$ (citation) and $\rho$ (risk) parameters inert.
It doesn't match how courts work. Real holdings are selected rationales — courts pick one justification from among multiple permissible ones. A holding that "race-conscious admissions must satisfy strict scrutiny" excludes some outcome-consistent rules (e.g., rules that would apply intermediate scrutiny) even though those rules also yield the same outcome.
The fix is conceptually simple but has multiple possible implementations, each with different modeling consequences.
The three options
Option C: "Single case-normal halfspace" (pure geometry)
What it is: Each case generates exactly one constraint — a halfspace in parameter space determined by the case facts $z_t$ and the chosen outcome $d_t$:
$$H_t = \{(w,c) : (-1)^{1-d_t}(w^\top z_t - c) \geq 0\}$$
If the judge rules for the plaintiff ($d_t = 1$), all admissible rules must classify $z_t$ on the plaintiff's side. If for the defendant ($d_t = 0$), the reverse. There is no discretion in the holding — only in the outcome.
Variant C2 adds one additional halfspace with a normal tied to $z_t$ but with a free threshold $b_t$, giving a single "breadth knob."
What the judge decides: Only the outcome $d_t$ (in C1). Outcome plus one scalar breadth parameter (in C2).
Consequences for the model:
- Maximally tractable. $\mathcal{F}_t$ is a polytope defined by halfspaces with known normals (the case fact vectors). All cutting-plane and version-space results apply directly.
- Propositions 1–3 (plausibility, monotone determinacy, breadth = penumbra shrinkage) are essentially immediate.
- The VC dimension corollary falls out for free: $k+1$ cases needed to pin down law in $\mathbb{R}^k$.
- Prop 5 (drift/path dependence) works cleanly because the geometry of intersecting halfspaces is well-understood.
- But: in C1, there is no strategic holding-writing. The judge's only instrument is the outcome. Holdings are mechanically generated. This means citations ($\gamma$) and risk ($\rho$) don't apply in C1, and there is no distinction between what the judge decides and what the judge says. C2 partially fixes this with one breadth knob.
Legal realism: Low for C1 (courts do more than announce outcomes). Medium for C2. Neither captures the idea that courts choose which rationale to emphasize.
Best for: A clean geometric spine that no referee can object to technically. Produces the strongest formal results with the least effort.
Variant A: "Chosen rationale" (strategic holding-writing)
What it is: The judge selects an intended rule $(\hat{w}_t, \hat{c}_t) \in \mathcal{F}_t$ — a specific legal theory that justifies the outcome. The holding is then a constraint set centered on or generated from this chosen rule. Two sub-variants:
- A1 (ball): $H_t = \{(w,c) : \|(w,c) - (\hat{w}_t, \hat{c}_t)\| \leq \varepsilon\}$. The judge picks a rationale and a radius $\varepsilon$ (breadth). Smaller $\varepsilon$ = tighter constraint = broader holding.
- A2 (halfspaces from rationale): $H_t = \bigcap_{j=1}^{m} \{(w,c) : a_{t,j}^\top(w,c) \geq b_{t,j}\}$ where the normals $a_{t,j}$ are functions of $(z_t, \hat{w}_t, \hat{c}_t)$. More "legal" than balls — the constraints are generated by the case facts and the chosen rationale.
No-dicta constraint: The holding must be consistent with the chosen rule ($(\hat{w}_t, \hat{c}_t) \in H_t$) and case-anchored (generated from $z_t$ and the rationale, not arbitrary). This prevents dicta without the backwards entailment problem.
What the judge decides: The intended rule $(\hat{w}_t, \hat{c}_t)$, the outcome (which follows mechanically from the rule at $z_t$), and the breadth $\varepsilon$ (in A1) or the specific constraint coefficients (in A2).
Consequences for the model:
- This is where the strategic behavior lives. The judge has meaningful choices: which rationale, how broad a holding.
- The citation incentive ($\gamma$) and risk cost ($\rho$) are meaningful: broader holdings (smaller $\varepsilon$) yield higher citation payoff but higher risk.
- Prop 4 (equilibrium holding choice) requires this variant. It gives the "microfoundation of breadth" result: $\gamma$ and $\rho$ map to equilibrium breadth via FOC $\gamma B'(\varepsilon) = \rho R'(\varepsilon)$.
- Drift (Prop 5) is richer here because the chosen rationale biases the direction of the cut, not just its sign.
- But: the holding language is less disciplined than Option C. A referee may push back on "balls in parameter space" as not corresponding to legal holdings. A2 is more realistic but harder to analyze.
Legal realism: Medium-high. Courts do select rationales and articulate them as holdings. The idea that a holding is "a constraint centered on a chosen legal theory" captures something real. But the specific forms (balls, generated halfspaces) are stylized.
Best for: The strategic story — citations, breadth tradeoffs, ideological holding-writing. This is what makes the model different from Callander & Clark (where the judge only chooses an outcome).
Variant B: "Minimal doctrinal rule" (institutional constraint)
What it is: The holding is the least constraining element of a restricted holding language $\mathcal{H}$ that is consistent with the chosen outcome:
$$H_t \in \arg\min_{H \in \mathcal{H}} B(H; \mathcal{F}_t) \quad \text{s.t.} \quad \mathcal{F}_t(d_t; z_t) \subseteq H$$
This is the formal analog of "decide no more than necessary" or "narrowest grounds."
What the judge decides: Only the outcome $d_t$. The holding follows mechanically from the minimization. There is no breadth discretion — the holding is as narrow as the language permits.
Consequences for the model:
- The holding is not a strategic instrument for the judge. Breadth is determined by the holding language and the geometry of the case, not by judicial choice. This means $\gamma$ and $\rho$ have no role (no breadth tradeoff).
- This is attractive if the model's main message is "law binds because of the geometry of constraint accumulation" rather than "judges strategically shape doctrine through holding choice."
- Propositions 1–3 work well. Drift works but is slower (holdings are minimal, so each cut removes less).
- Prop 4 (equilibrium holding choice) does not apply — there is no holding choice.
- But: the direction of drift depends entirely on outcome selection and the case distribution, not on judicial strategy. The model becomes more "mechanical" — which some may see as a feature (law constrains despite judges, not because of their strategic choices).
- The tractability depends heavily on $\mathcal{H}$. If $\mathcal{H}$ is "any finite set of halfspaces," the minimization can be degenerate. Needs a disciplined language.
Legal realism: Highest. "Narrowest grounds" is a recognized judicial norm (cf. Sunstein's "judicial minimalism"). Many common-law traditions hold that courts should decide no more than the case requires.
Best for: A "law constrains despite judges" story. Clean pairing with the geometry lemmas. But less distinctive than Variant A for economics audiences who expect strategic agents.
The key tradeoff
The fundamental tension is between tractability + clean geometry and strategic richness + legal realism:
| Option C | Variant A | Variant B | |
|---|---|---|---|
| Judge chooses holding? | No (C1) / limited (C2) | Yes — rationale + breadth | No — follows from minimality |
| Citation/risk incentives ($\gamma, \rho$)? | Inert (C1) / limited (C2) | Fully operative | Inert |
| Tractability | Highest | Medium | Medium (depends on $\mathcal{H}$) |
| Legal realism | Low–medium | Medium–high | Highest |
| Main story | Constraint geometry | Strategic doctrine-shaping | Institutional constraint |
| Distinguishes from Callander & Clark? | Weakly | Strongly | Moderately |
| Props 1–3 (geometry) | Immediate | Immediate | Immediate |
| Prop 4 (equilibrium breadth) | No (C1) / partial (C2) | Yes | No |
| Prop 5 (drift) | Clean | Richer | Slower |
| Prop 6 (overruling) | Clean | Clean | Clean |
| VC corollary | Immediate | Needs assumptions | Needs assumptions |
The recommended packaging
ChatGPT (and I concur) recommends a two-layer approach:
Main model: Option C (single case-normal halfspace, possibly C2). This delivers the geometric spine — Propositions 1–3, the drift result, the overruling threshold, and the VC corollary. No referee can object to the math. The model is tight, clean, and produces the core insight: law binds without being determinate, and slippery slopes emerge without lawlessness.
Extension: Variant A (chosen rationale with a disciplined holding language). This adds the strategic breadth story — Proposition 4 and richer drift dynamics. It's what distinguishes the paper from Callander & Clark (where judges only choose outcomes). Present it as "what happens when judges can also shape doctrine."
This gives you a paper that:
- Has a bulletproof formal core (Option C)
- Has an interesting economics contribution (Variant A adds strategic behavior)
- Can be read at two levels (geometry fans get the spine; IO/polisci fans get the strategic extension)
Variant B could appear as a remark or discussion section: "if we impose judicial minimalism as a norm, drift slows but doesn't vanish."
Questions for Holger
Is the strategic holding-writing channel essential to the paper's contribution? If yes, the model needs Variant A (at least as extension). If the main point is about geometry of constraint accumulation, Option C alone might suffice.
How important is the "holdings ≠ outcomes" distinction? This was highlighted as the key value-added over Callander & Clark. Under Option C1, holdings are mechanically determined by outcomes — the distinction is lost. C2 and Variant A preserve it. Is this worth the additional complexity?
Should the model endogenize broad vs. narrow holdings? The citation incentive ($\gamma$) and the breadth tradeoff are prominent in the current paper. They only work under Variant A. If we drop them, the model is simpler but loses a distinctive feature. Is Holger attached to this channel?
Variant B as default norm? The "narrowest grounds" / minimal holding idea resonates with Holger's coordination-game framing (judges minimize conflict). Would Holger prefer a model where holdings are institutionally minimal, with strategic breadth as a deviation from the norm?
Is the two-layer (C then A) packaging strategy acceptable? Or does Holger want a single unified model?
Decision timeline
This decision blocks all formal work. The geometry lemmas (Props 1–3) are the same under all options, so those could be drafted now. But the equilibrium characterization, drift dynamics, and paper structure all depend on this choice.
Suggest: decide within one meeting with Holger, then proceed to formal results.