Chapter 12: The Adversarial System as Manifold Exploration

Litigation as Adversarial Pathfinding

Chapter 6 established that legal disputes are pathfinding problems on the judicial complex. The plaintiff seeks the minimum-cost path from her current legal position to the desired remedy; the defendant seeks to block that path, lengthen it, or redirect it to a different goal.

This chapter examines the adversarial structure in depth. Litigation is not single-agent pathfinding — it is a two-player game in which each side actively modifies the search landscape faced by the other.

Definition (Adversarial Legal Search). An adversarial legal search is a two-player game on \mathcal{K}:

• Plaintiff (path-seeker): Seeks to find a minimum-cost path from c_0 (the complaint) to G_\pi (the desired remedy).
• Defendant (path-blocker): Seeks to either (a) demonstrate that no path to G_\pi exists (motion to dismiss), (b) increase the cost of every path to G_\pi above the burden-of-proof threshold (defense at trial), or (c) establish a shorter path to a different goal G_\delta (affirmative defense, counterclaim).

The two players alternate moves. The plaintiff proposes paths (legal arguments, evidence, witnesses). The defendant responds by blocking, deflecting, or countering. The game proceeds until one side runs out of moves (rests its case), the judge terminates the game (grants a motion), or both sides agree on a vertex (settlement).

The Minimax Value

The formal solution concept for two-player games is the minimax value: the outcome when both sides play optimally.

Definition (Minimax Value of a Legal Dispute). The minimax value of a legal dispute starting at c_0 is:

V(c_0) = \min_{\gamma_\delta} \max_{\gamma_\pi} \left[ \text{BF}_{\text{law}}(\gamma_\delta) - \text{BF}_{\text{law}}(\gamma_\pi) \right]

where \gamma_\pi is the plaintiff’s path and \gamma_\delta is the defendant’s path.

• If V(c_0) > 0: the plaintiff prevails — the plaintiff’s optimal path is shorter (less costly) than the defendant’s optimal blocking strategy.
• If V(c_0) \leq 0: the defendant prevails — the defendant can block, lengthen, or redirect every path the plaintiff might take.
• If V(c_0) \approx 0: the case is close — the outcome depends on execution, evidence quality, and the judge’s or jury’s assessment.

The minimax value is the theoretical “correct” outcome of the dispute, assuming perfect play by both sides. In practice, neither side plays perfectly, and the outcome depends on the quality of counsel, the availability of evidence, and the procedural context. But the minimax value provides a benchmark against which the actual outcome can be measured.

Why Adversarial?

The adversarial system is not the only possible method for exploring the judicial complex. The inquisitorial system, used in many civil law countries, assigns the exploration task to the judge rather than to the parties. The administrative hearing system uses a neutral fact-finder with relaxed procedural rules. Each system represents a different search algorithm operating on the same underlying complex.

The adversarial system has two properties that distinguish it as a search algorithm:

Completeness. Two adversarial parties, each motivated to find the strongest arguments for their side, are more likely to explore the full space of relevant paths than a single searcher. The plaintiff searches for paths to G_\pi; the defendant searches for paths away from G_\pi. Between them, they cover more of the complex than either would alone. This is the adversarial system’s exploration guarantee: the union of the two search trees is larger than either tree individually.

Robustness. Arguments that survive adversarial testing are more reliable than arguments that have not been tested. If the plaintiff’s path survives the defendant’s best efforts to block, lengthen, or redirect it, the path is robust — it does not depend on favorable assumptions, overlooked counterarguments, or unchallenged evidence. This is the adversarial system’s quality guarantee: surviving paths are higher quality than untested paths.

These properties come at a cost: the adversarial system is expensive. Two teams of lawyers, each conducting independent searches, consume far more resources than a single searcher. The adversarial system trades efficiency for completeness and robustness — a trade-off that is justified when the stakes are high (criminal cases, major civil rights actions) but may not be justified when the stakes are low (minor disputes, routine administrative matters).

RUNNING EXAMPLE — THE ADVERSARIAL GAME IN EJAC v. CITY OF SAN FRANCISCO

The adversarial game in EJAC’s case begins with the complaint — the plaintiff’s initial path proposal. The complaint identifies the start node (c_0: the current policing practices of the San Francisco Police Department), the goal region (G_\pi: a finding of pattern-or-practice liability and a consent decree requiring comprehensive reforms), and the proposed path (evidence of racial disparities in stops, searches, and force, combined with evidence of inadequate supervision and training).

The City’s first move is a motion to dismiss under Rule 12(b)(6). The City argues that the complaint fails to state a claim — that even taking all alleged facts as true, no path from c_0 to G_\pi exists within the legal constraints. Specifically, the City argues that statistical disparities alone, without evidence of intentional discrimination, do not establish a constitutional violation under Washington v. Davis.

This motion is a graph-connectivity argument: the City claims that G_\pi is unreachable from c_0 in \mathcal{K}. If the court grants the motion, the game ends — no path exists, and the plaintiff loses without trial.

Rivera denies the motion. Her reasoning: the complaint alleges not only statistical disparities but also specific instances of discriminatory conduct, evidence of supervisory knowledge, and a failure to investigate or discipline. These allegations, taken as true, are sufficient to establish a path from c_0 to G_\pi. The path may be long and costly, but it exists. The game continues.

The Geometry of Procedural Motions

Procedural motions are the legal system’s mechanism for pruning the search space — for eliminating paths that should not be explored, either because they are legally invalid or because the evidence does not support them.

Each type of motion corresponds to a specific search-pruning operation:

Motion to Dismiss (Rule 12(b)(6)). The defendant claims that even taking all facts as true, no path from c_0 to G_\pi exists within the legal constraints. This is a graph-connectivity assertion: the defendant argues that G_\pi is unreachable from c_0 in \mathcal{K}.

In A* terms: the defendant claims that the search space contains no path to the goal, so the search should terminate immediately. The motion is granted if the court agrees that the legal framework does not support the plaintiff’s theory — if the legal arguments, even with all factual allegations assumed true, do not connect the start to the goal.

Summary Judgment (Rule 56). The defendant (or plaintiff) claims that on the undisputed facts, the shortest path from c_0 leads to a specific outcome — the moving party’s favored outcome — and no reasonable jury could find otherwise. This is a claim that the edge weights are determined by the stipulated facts, and the optimal path can be computed without adversarial exploration.

In A* terms: the moving party claims that the heuristic is tight enough to determine the optimal path without further search. The motion is granted if the court agrees that no genuine dispute of material fact exists — if the edge weights are known and the shortest path is unambiguous.

Motion in Limine. Exclusion of evidence corresponds to removing edges from the judicial complex. If a piece of evidence is inadmissible (hearsay, fruit of the poisonous tree, unduly prejudicial), the edges that rely on that evidence are deleted from \mathcal{K}, and any paths that traverse those edges become impassable.

In A* terms: the motion removes successor nodes from the search frontier. Arguments that relied on the excluded evidence can no longer be explored.

Motion for Directed Verdict / Judgment as a Matter of Law (Rule 50). At the close of the plaintiff’s case, the defendant moves for judgment as a matter of law: the plaintiff has not established a path to the goal, and the remaining search cannot produce one. This is a claim that the plaintiff’s search tree, at its current depth, contains no viable paths — the search should be abandoned.

The Pruning Cascade

In a complex case, motions build on each other, creating a pruning cascade that progressively narrows the search space:

1. The motion to dismiss prunes legally invalid theories — entire branches of the search tree are removed.
2. Discovery reveals the factual landscape, which determines edge weights. Some paths that seemed viable at the pleading stage turn out to have high cost (weak evidence) or are blocked (missing elements).
3. Summary judgment prunes paths where the edge weights are determined and the outcome is clear.
4. Motions in limine remove specific edges (evidence rulings), further narrowing the search.
5. By the time the case reaches trial, the search space has been pruned to a manageable size: a small number of viable paths, each requiring adversarial testing.

This pruning cascade is why trials are manageable despite the enormous size of the underlying judicial complex. Without pruning, a complex civil rights case would require exploring millions of possible paths. With pruning, the search is narrowed to a handful of legal theories, each supported by a specific body of evidence, each following a specific chain of precedent.

RUNNING EXAMPLE — THE PRUNING CASCADE IN EJAC v. CITY

Rivera’s management of the EJAC case follows the pruning cascade:

Phase 1: Motion to dismiss. The City moves to dismiss several claims. Rivera dismisses the Title VI claim against individual officers (Title VI applies only to entities receiving federal funding, not to individuals). She dismisses the negligent supervision claim (California has no standalone negligent supervision cause of action against municipalities). She sustains the Section 1983 pattern-or-practice claim and the Monell municipal liability claim. Pruning result: two of five legal theories are removed; three survive.

Phase 2: Discovery. Twelve months of discovery produce 200,000 pages of documents, including internal affairs files, stop-and-search data, use-of-force reports, training materials, and departmental policies. The plaintiffs’ expert statistician analyzes five years of stop data and finds that Black residents are 4.3 times more likely to be stopped than white residents after controlling for neighborhood crime rates, time of day, and other variables. The City’s expert disputes the methodology but does not dispute the underlying disparity. Edge weights are now calibrated: the statistical evidence makes the equal-protection path from c_0 to G_\pi shorter and less costly.

Phase 3: Summary judgment. The City moves for summary judgment on the Monell claim, arguing that the plaintiffs have not identified a specific municipal “policy or custom” that caused the constitutional violations. Rivera denies the motion: the plaintiffs have presented evidence of a departmental culture of tolerance for discriminatory policing, including internal emails acknowledging the disparity and declining to address it. The evidence creates a genuine dispute of material fact about whether the City had an unofficial policy or custom of discriminatory policing. The path to Monell liability is not blocked — the edge weights are contested, which means the adversarial process (trial) must resolve them.

Phase 4: Motions in limine. Rivera grants the City’s motion to exclude certain social media posts by individual officers (prejudicial, limited probative value). She denies the City’s motion to exclude the plaintiffs’ statistical expert (the methodology is reliable under Daubert). She grants the plaintiffs’ motion to exclude evidence of plaintiffs’ criminal histories (inadmissible character evidence under Rule 404(b)). Pruning result: several edges are removed from the search space, but the core paths remain intact.

Phase 5: Trial. The search space has been pruned to two viable theories (Section 1983 pattern-or-practice and Monell liability), supported by a specific body of evidence (statistical analysis, internal emails, specific incident reports), following a specific chain of precedent (Monell → Canton → Brown → Connick). The adversarial game at trial will explore these paths in depth.

The Distance Interpretation

Chapter 6 introduced the burden of proof as the shortest-path distance from the plaintiff’s initial vertex to the goal region. This chapter develops the interpretation more fully, connecting the different standards of proof to specific properties of the graph distance.

Proposition (Burden of Proof as Graph Distance). The burden of proof in a legal proceeding is the shortest-path distance from the plaintiff’s initial vertex c_0 to the nearest vertex in the goal region G:

\text{Burden}(c_0, G) = \min_\gamma \text{BF}_{\text{law}}(\gamma) \quad \text{s.t. } \gamma \text{ starts at } c_0, \; \gamma \text{ ends in } G

Different standards of proof correspond to different thresholds on the relative distance:

Preponderance of the evidence (civil cases): The plaintiff’s optimal path must be shorter than the defendant’s optimal path. The plaintiff wins if:

\text{BF}_{\text{law}}(\gamma^*_\pi) < \text{BF}_{\text{law}}(\gamma^*_\delta)

This is the “more likely than not” standard: the plaintiff’s story (path) is more plausible (shorter) than the defendant’s.

Clear and convincing evidence (civil cases with heightened stakes): The plaintiff’s optimal path must be substantially shorter than the defendant’s:

\text{BF}_{\text{law}}(\gamma^*_\pi) < \alpha \cdot \text{BF}_{\text{law}}(\gamma^*_\delta)

for some \alpha < 1 (e.g., \alpha = 0.7). The plaintiff must not merely prevail on the balance — she must prevail by a margin.

Beyond a reasonable doubt (criminal cases): The prosecution’s path must be so short — the evidence so strong — that no reasonable alternative path exists:

\text{BF}_{\text{law}}(\gamma^*_\pi) < \epsilon

for small \epsilon. The path to guilt must be so direct and well-supported that all alternative paths (paths to innocence) have costs that exceed any reasonable threshold.

Burden-Shifting as Weight Re-Assignment

Burden-shifting frameworks — like the McDonnell Douglas framework from Chapter 6 — correspond to weight re-assignments during the search. When the burden shifts from plaintiff to defendant, the edge weights change: dimensions that the plaintiff previously had to traverse are now the defendant’s responsibility.

Formally, the burden shift at step k of the framework modifies the cost function:

\text{BF}_{\text{law}}^{(k+1)}(\gamma) = \text{BF}_{\text{law}}^{(k)}(\gamma) - \sum_{e \in \text{shifted edges}} w_\pi(e) + \sum_{e \in \text{shifted edges}} w_\delta(e)

The total path cost is conserved — only the allocation between parties changes. This is consistent with the liability conservation principle from Chapter 9: burden-shifting redistributes the cost of establishing the path, but the total cost of the path itself is unchanged.

The Geometry of Settlement

Settlement is a vertex in \mathcal{K} (or a new point interpolated between vertices) that both parties prefer to the expected outcome of continued litigation.

Definition (Settlement Region). A settlement is a vertex c_s \in \mathcal{K} that both parties prefer to the expected outcome of continued litigation:

c_s \in \{ c \in \mathcal{K} \mid U_\pi(c) > \mathbb{E}[U_\pi(\gamma^*_\pi)] \text{ and } U_\delta(c) > \mathbb{E}[U_\delta(\gamma^*_\delta)] \}

where U_\pi, U_\delta are the utility functions of plaintiff and defendant, and \gamma^*_\pi, \gamma^*_\delta are their respective optimal litigation paths.

Settlement occurs when the graph distance from the current vertex to a mutually acceptable vertex is shorter than either party’s litigation path. In geometric terms, settlement is a shortcut — a path through the complex that avoids the costly adversarial search and arrives at a mutually acceptable outcome directly.

Proposition (Settlement Existence). A non-empty settlement region exists whenever the sum of both parties’ expected litigation costs exceeds the graph distance between their goal regions:

\mathbb{E}[\text{BF}_{\text{law}}(\gamma^*_\pi)] + \mathbb{E}[\text{BF}_{\text{law}}(\gamma^*_\delta)] > d_{\mathcal{K}}(G_\pi, G_\delta)

This proposition explains the fundamental empirical observation of civil litigation: the vast majority of cases settle. Settlement exists whenever litigation is expensive relative to the distance between the parties’ positions — which is almost always.

When Settlement Fails

Settlement fails when the settlement region is empty — when no vertex exists that both parties prefer to litigation. This occurs in two geometric configurations:

Informational asymmetry. The parties disagree about the edge weights — they have different assessments of the evidence strength, the legal merits, or the likely outcome. The plaintiff believes her path is short (the evidence is strong); the defendant believes the plaintiff’s path is long (the evidence is weak). Each believes they will do better at trial than at any settlement point.

In geometric terms, the parties are navigating different copies of the judicial complex — complexes with the same topology but different edge weights. The settlement region in each party’s perceived complex may be non-empty, but the intersection of the settlement regions (the vertices that both parties, using their own edge weights, prefer to litigation) may be empty.

Indivisible outcomes. Some legal outcomes are binary and cannot be interpolated. A criminal defendant is either convicted or acquitted; a patent is either valid or invalid; a custody arrangement gives primary custody to one parent or the other. In these cases, the “settlement region” consists of a small number of discrete vertices, and the parties may not be able to find one that both prefer.

The Economics of Settlement

The geometric framework provides a clean derivation of the classic result from law and economics: cases settle when the sum of litigation costs exceeds the parties’ disagreement about the likely outcome.

Let p_\pi be the plaintiff’s estimate of winning at trial, p_\delta be the defendant’s estimate, D be the damages at stake, C_\pi be the plaintiff’s litigation cost, and C_\delta be the defendant’s litigation cost. The settlement region is non-empty when:

p_\pi \cdot D - C_\pi < \text{settlement amount} < (1 - p_\delta) \cdot D + C_\delta

This is equivalent to:

C_\pi + C_\delta > (p_\pi - (1 - p_\delta)) \cdot D = (p_\pi + p_\delta - 1) \cdot D

Settlement fails only when the parties’ combined optimism (p_\pi + p_\delta - 1 > 0, meaning both believe they are likely to win) exceeds the combined litigation costs. Since litigation costs are usually substantial, settlement is usually possible.

RUNNING EXAMPLE — SETTLEMENT NEGOTIATIONS IN EJAC v. CITY

Six months before trial, Rivera holds a settlement conference. The parties’ positions:

EJAC seeks: a consent decree requiring (1) revision of stop-and-search policies, (2) independent monitoring of racial disparity data, (3) mandatory bias training, (4) a civilian oversight board with subpoena power, and (5) $4.5 million in damages for the named plaintiffs. EJAC estimates its probability of prevailing at trial: 70%.

The City offers: (1) a voluntary review of stop-and-search policies (no consent decree), (2) internal monitoring (no independent monitor), (3) training (already planned), and (4) $800,000 in damages. The City estimates EJAC’s probability of prevailing: 35%.

The settlement gap is large. In geometric terms, the parties are navigating different complexes: EJAC’s complex has short paths to the pattern-or-practice liability vertex (the statistical evidence is strong, the internal emails are damning), while the City’s complex has longer paths (the statistical methodology is disputed, the emails are ambiguous).

Rivera, as the settlement judge, has a unique vantage point: she has seen the evidence in discovery, ruled on the motions, and formed her own assessment of the edge weights. She believes EJAC’s probability of prevailing is approximately 55% — between the parties’ estimates. She communicates this assessment (as she is permitted to do in a settlement conference) and suggests a framework: a consent decree with the key structural reforms plus $2.0 million in damages.

The parties negotiate. After two days, they reach agreement: a consent decree with policy reforms and independent monitoring, but without the civilian oversight board, plus $2.2 million in damages. The settlement vertex is located between the parties’ goal regions — closer to EJAC’s position on structural reforms, closer to the City’s position on damages.

In the geometry of the complex, the settlement is a shortcut: a path from the current vertex to a mutually acceptable outcome that bypasses the costly adversarial search of a three-week trial. The shortcut works because the combined litigation costs (estimated at $3 million for both sides) exceed the distance between the parties’ positions (approximately $2 million in damages plus the value of the disputed reforms). The settlement region is non-empty, and the parties have found a point in it.

The Three Standards of Appellate Review

An appeal is not a new search — it is a re-evaluation of the trial court’s path under modified conditions. The standard of review determines what kind of re-evaluation the appellate court performs.

De novo review. The appellate court re-runs the search from scratch with the same start and goal but potentially different heuristics (different legal interpretations). De novo review applies to questions of law — the appellate court owes no deference to the trial court’s legal conclusions. In geometric terms: the appellate court uses the same judicial complex but may apply a different connection (different precedential weights) to reach a different optimal path.

Abuse of discretion. The appellate court checks only whether the trial court’s path is within a neighborhood of the optimal path — whether the chosen path’s cost is within a tolerance \epsilon of the shortest path. Abuse-of-discretion review applies to evidentiary rulings, case management decisions, and other matters within the trial court’s discretion. In geometric terms: the appellate court verifies that \text{BF}_{\text{law}}(\gamma_{\text{trial}}) < \text{BF}_{\text{law}}(\gamma^*) + \epsilon, where \gamma_{\text{trial}} is the trial court’s path and \gamma^* is the optimal path.

Clearly erroneous. The appellate court checks whether the trial court’s path crosses a boundary it should not have crossed — a factual error that places the path in the wrong region of the complex. Clearly-erroneous review applies to findings of fact. In geometric terms: the appellate court verifies that \gamma_{\text{trial}} does not cross any regime boundaries that the evidence does not support.

Appeals as Error Correction

The appellate system functions as an error-correction mechanism for the adversarial search. The trial-level search may produce a suboptimal path due to:

• Errors in heuristic application (incorrect legal analysis)
• Errors in edge-weight estimation (incorrect factual findings)
• Errors in pruning (incorrect evidentiary rulings)
• Adversarial failure (inadequate representation on one side)

The appellate court identifies and corrects these errors, subject to the standard of review:

• Legal errors (heuristic application) are corrected de novo — the appellate court substitutes its own analysis.
• Factual errors (edge-weight estimation) are corrected only if clearly erroneous — the appellate court defers to the trial court’s direct observation of witnesses and evidence.
• Procedural errors (pruning) are corrected only if they constitute an abuse of discretion — the appellate court defers to the trial court’s case management.

The asymmetry in the standards reflects the comparative advantages of trial and appellate courts:

• Trial courts are better at fact-finding (they see witnesses, hear testimony, observe demeanor).
• Appellate courts are better at legal analysis (they have more time for research, see patterns across cases, and have the benefit of briefing focused solely on the legal issues).

The standard of review allocates the error-correction task to the court best positioned to perform it.

The Multi-Level Appellate System

The federal judicial system has three levels of appellate review: district court (trial) → circuit court (first appeal) → Supreme Court (final appeal). This multi-level structure creates a hierarchical error-correction system:

• The circuit court corrects errors in the trial court’s search, with deference to factual findings.
• The Supreme Court corrects errors in the circuit court’s search, typically focusing on legal questions of broad importance.
• Each level of review refines the path, removing errors that lower courts missed.

In the geometry of the complex, the hierarchical system performs iterative path optimization: each level of review improves the path’s quality by correcting errors that the lower court’s search introduced. The final path (after all appeals are exhausted) is the system’s best approximation of the optimal path through the complex.

RUNNING EXAMPLE — POTENTIAL APPEAL IN EJAC v. CITY

The settlement in EJAC v. City moots the appeal question — settlements are not appealable. But Rivera considers what an appeal would have looked like if the case had gone to trial.

Suppose the jury returned a verdict for the plaintiffs on the Section 1983 pattern-or-practice claim and the Monell claim, awarding $3.5 million in damages and supporting injunctive relief. The City would appeal on several grounds:

1. Legal error (de novo review): The City would argue that Rivera applied the wrong legal standard for Monell liability — that she should have required “deliberate indifference” rather than mere “knowledge and acquiescence.” The Ninth Circuit would review this de novo, applying its own analysis of the Monell framework.

2. Evidentiary error (abuse of discretion): The City would argue that Rivera erred in admitting the plaintiffs’ statistical expert (arguing Daubert violation). The Ninth Circuit would review for abuse of discretion, affirming unless Rivera’s decision was outside the range of permissible choices.

3. Factual error (clearly erroneous): The City would argue that the jury’s finding of a pattern or practice of discrimination was clearly erroneous — that the evidence, taken as a whole, did not support a finding of systematic discrimination as opposed to isolated incidents. The Ninth Circuit would review for clear error, affirming unless the finding was “implausible in light of the record viewed in its entirety.”

Each ground of appeal corresponds to a different claim about the trial court’s path through the complex:

- Ground 1: The path used the wrong heuristic (wrong legal standard). - Ground 2: The path traversed an edge that should have been pruned (inadmissible evidence). - Ground 3: The path crossed a regime boundary that the evidence does not support (factual insufficiency).

The appellate court’s task is to evaluate each claim under the appropriate standard of review and, if errors are found, to correct the path — either by remanding for a new trial (restart the search), reversing the judgment (declare a different path optimal), or modifying the remedy (adjust the endpoint).

Why Two Searchers Are Better Than One

The deepest justification for the adversarial system is informational: two adversarial searchers reveal more information about the judicial complex than a single neutral searcher.

Each party in a legal dispute has private information — facts, interpretations, legal theories — that the other party and the court do not have. The adversarial process forces the revelation of this information through discovery, cross-examination, and the presentation of competing arguments.

In the geometry of the complex, each party has a partial view of the edge weights. The plaintiff knows facts that make her path look short (evidence of liability). The defendant knows facts that make the plaintiff’s path look long (evidence of a defense). Neither has a complete map of the complex.

The adversarial process combines these partial views into a more complete picture. Discovery forces the revelation of facts. Cross-examination tests the reliability of evidence. Competing arguments reveal the range of plausible legal interpretations. The result is a better-calibrated map of the complex — better edge weights, better-defined regime boundaries, better heuristics — than either party could produce alone.

This is the game-theoretic justification for the adversarial system: it is an information revelation mechanism that produces better decisions by forcing the disclosure of private information that would otherwise remain hidden.

The Cost of Information Revelation

The adversarial system’s information revelation comes at a cost: it requires two full search operations (one per side), extensive discovery (revealing private information), and a trial (testing the information adversarially). These costs are justified when the value of better information exceeds the cost of obtaining it — which is typically true for high-stakes cases.

For low-stakes cases, the adversarial system may be unnecessarily expensive. The geometric framework suggests a criterion for when the adversarial system’s costs are justified:

\text{Value of adversarial information} = E[\text{Quality}(\gamma^*_{\text{adversarial}})] - E[\text{Quality}(\gamma^*_{\text{inquisitorial}})]

If this value exceeds the additional cost of adversarial proceedings, the adversarial system is justified. If not, a simpler procedure (inquisitorial, administrative, or alternative dispute resolution) may produce comparable results at lower cost.

Scenario

EJAC v. City of San Francisco goes to trial. Rivera must manage the adversarial process across a three-week bench trial (no jury — the parties have consented to a bench trial on the equitable claims).

The Adversarial Search Tree

The trial proceeds as a sequence of moves in the adversarial game:

Week 1: Plaintiff’s case-in-chief.

Move 1: Plaintiffs present their statistical expert, who testifies that Black residents are stopped 4.3 times more frequently than white residents, controlling for neighborhood crime rates. This establishes a short path from c_0 (current policing) to the equal-protection liability vertex — the statistical evidence significantly reduces the cost of the equal-protection path.

Defendant’s response (cross-examination): The City’s counsel challenges the expert’s methodology: the crime-rate controls are based on reported crime rather than actual crime, which may understate crime in minority neighborhoods. The cross-examination attempts to increase the cost of the plaintiff’s path by raising doubts about the edge weights (the reliability of the statistical evidence).

Move 2: Plaintiffs present internal departmental emails in which supervisors acknowledge the disparity and decline to investigate. This evidence shortens the Monell liability path — it establishes “deliberate indifference” by showing that the City knew of the problem and chose not to act.

Defendant’s response: The City argues the emails are taken out of context and reflect an ongoing policy discussion, not indifference. The City presents a narrative in which the department was actively studying the disparity and developing a response.

Week 2: Defendant’s case.

Move 3: The City presents its own statistical expert, who uses a different methodology (different controls, different baseline) and finds a smaller disparity (2.1 times rather than 4.3 times). The City’s expert also testifies that the remaining disparity is explained by deployment patterns (more officers in high-crime neighborhoods, which are disproportionately minority) rather than by discriminatory intent.

Plaintiff’s response (cross-examination): Plaintiffs challenge the City’s expert on the deployment-pattern explanation: if more officers are deployed to minority neighborhoods, and those officers make more stops, the deployment itself may be discriminatory. The deployment pattern is not an independent explanation but a mechanism through which the discrimination operates.

Week 3: Rebuttal and closing arguments.

Move 4: Plaintiffs present a rebuttal witness — a former officer who testifies about a departmental culture that tolerated racial profiling. The testimony is anecdotal rather than statistical, but it corroborates the statistical evidence with first-person narrative.

Closing arguments: Each side presents its optimal path through the complex, highlighting the edges that support its narrative and minimizing the edges that support the opposing narrative.

Rivera’s Decision

Rivera’s task is to evaluate both paths and determine which is more geometrically consistent — which path follows the legal geodesic more closely.

She finds for the plaintiffs on the Section 1983 pattern-or-practice claim (the statistical evidence, corroborated by the internal emails and the rebuttal testimony, establishes a pattern of racially discriminatory policing) and on the Monell claim (the City’s deliberate indifference is established by the supervisory emails and the failure to investigate).

She enters a consent decree requiring policy reforms, independent monitoring, and $2.8 million in damages.

In the geometry of the complex, Rivera’s decision selects the plaintiff’s path as the optimal path through the complex. The defendant’s path — which would have led to a different vertex (no liability, no reform) — was longer, less well-supported by the edge weights, and inconsistent with the constitutional constraints of the equal-protection subcomplex.