Methods
Setup
- Cases arrive as fact vectors $z_t \in \mathbb{R}^k$, drawn i.i.d. from distribution $G$
- Judges drawn i.i.d. from finite set $J$, each with ideal affine rule $(w_j, c_j)$
- Legal state is a feasible set $\mathcal{F}_t \subset \mathbb{R}^{k+1}$ of admissible affine decision rules
Key assumptions
- Decision rules are affine: $f_{w,c}(z) = w^\top z - c$, outcome $d = \mathbf{1}\{w^\top z \geq c\}$
- Holdings are finite sets of linear inequalities on $(w, c)$
- Feasible set updates by intersection: $\mathcal{F}_{t+1} = \mathcal{F}_t \cap H_t$
- Holdings must be entailed by the case: $\mathcal{F}_t(d_t; z_t) \subseteq H_t$
Key parameters
- $K > 0$: sanction cost for implausible outcomes or infeasible holdings
- $C > 0$: cost per overruled holding
- $\gamma \geq 0$: citation payoff from broader holdings
- $\rho \geq 0$: risk cost from broader holdings
- $\alpha > 0$: strength of outcome preferences
Equilibrium concept
- Per-period payoff: $U_t = u_{j_t}(d_t, z_t) + (\gamma - \rho) B(H_t; \mathcal{F}_t) - C|\Delta\mathcal{R}_t| - K \cdot \min\{1, \mathbf{S}_t\}$
- Breadth measure: $B(H_t; \mathcal{F}_t) = \log(\mathrm{Vol}(\mathcal{F}_t) / \mathrm{Vol}(\mathcal{F}_t \cap H_t))$
- Equilibrium not yet fully characterized (open task)
Identification strategy (for future empirical work)
- Ideology should predict outcomes more when doctrine is unsettled (large feasible set)
- Random judge assignment provides quasi-experimental variation
- NLP-based measures of holding breadth as empirical proxies