Saddle Point

Worked case · Chapter 3, Bargaining, Commitment, and Repeated Cooperation · p. 117 · sources checked 9 November 2026

The oil price cap coalition

how much discipline can such a coalition sustain, and which improvement in its monitoring is worth most?

The network

acts enforcement, unobserved by the otheracts enforcement, unobserved by the othersignal a clean or a bad periodsignal a clean or a bad periodacts forgiving trigger: after a bad signal, withdraws for good with probability πacts the same forgiving triggeracts can revise the ruleMember 1how strictly its service providers are held to the capMember 2two symmetric players in the model; many and asymmetric in factPublic signalattestations, shipping data, price reports; informativeness (p, q)The Councilcan revise the published rulesPublished adjustment rulecan be revised; not the signal the trigger uses
player an object of the game outside the game Model B4. A reading of the mapping table, drawn by the site. Hover a node or an arrow; the arrows in words:
  1. Member 1 → Public signal acts enforcement, unobserved by the other
  2. Member 2 → Public signal acts enforcement, unobserved by the other
  3. Public signal → Member 1 signal a clean or a bad period
  4. Public signal → Member 2 signal a clean or a bad period
  5. Member 1 → Member 2 acts forgiving trigger: after a bad signal, withdraws for good with probability π
  6. Member 2 → Member 1 acts the same forgiving trigger
  7. The Council → Published adjustment rule acts can revise the rule

The mapping

The six fields of the model card, as the book fills them in for this situation.

FieldIn the case
1 Decision-makersthe coalition's member jurisdictions, each choosing how strictly its service providers are held to the cap; two symmetric players in the model, many and asymmetric in fact
2 Timerepeated: voyages and enforcement decisions recur; patience is the weight a member puts on the coalition's future
3 Informationmembers do not observe each other's enforcement directly; they observe signals (attestations, shipping data, price reports) whose informativeness is the pair : the probability that a period looks clean when all comply, and when one member's providers service above-cap cargoes
4 Commitmentnone beyond the published rules; the automatic adjustment is a published rule, revisable by the Council (item 3(iv) below)
5 Payoffsa member's gain from its providers' above-cap business ( in the model) against the value of a functioning coalition ()
6 Strategiesforgiving-trigger public strategies: cooperate, and respond to a bad signal by withdrawing cooperation for good with a calibrated probability

What the mathematics licenses.

Six statements, each about the model under the card above; the mapping to the coalition is a reconstruction R, and no statement is an estimate of the real coalition's discipline.

  • Discipline is bounded by information, not by severity. Among forgiving-trigger profiles the best sustainable value is , the same for every above the threshold, and it is reached by the smallest incentive-compatible punishment probability (Proposition 3.9 (p. 133)). Within that class, a response harsher than the smallest one buys no discipline and, whenever compliant play can look bad (), destroys value through false alarms D [Proposition 3.9 (p. 133)]. At the chapter's base signal (, ) no strongly symmetric equilibrium of any kind does better (Proposition 3.10 (p. 133)). At other signals the trigger value is a floor for that wider class, not a ceiling: with , , and , a strongly symmetric equilibrium in which each member complies with probability earns , above the trigger value D [exact computation in the class of Proposition 3.10 (p. 133)].
  • There is a visibility threshold. If evasion moves the signal too little, that is if , which at the chapter's payoffs means , no forgiving trigger sustains full cooperation at any patience (Table 3.1 (p. 121)a). Partial cooperation can survive the threshold: at , with and , a strongly symmetric equilibrium with compliance probability earns , above mutual defection's D [exact computation in the class of Proposition 3.10 (p. 133)]. A purely documentary signal that an evading provider can produce at low cost pushes towards , and therefore towards that threshold D [Proposition 3.9 (p. 133); the mapping of attestations to is R].
  • False alarms cost more than missed evasion. At the margin, and : their ratio is , independent of the payoffs and greater than whenever the signal is informative and imperfect (). At the chapter's values (, ) the ratio is exactly : an improvement of in the probability that compliant play looks clean is worth , the same improvement in the probability that evasion is visible only (Table 3.1 (p. 121)). The reason is structural: under trigger-type enforcement, false alarms fire on the equilibrium path, while the visibility of evasion matters only at the incentive margin. In this model, then, an enforcement system that wrongly flags compliant trade erodes the coalition more, unit for unit, than one that misses some evasion D [prop:barg-b4,tab:barg-case]. Whether a unit of either is equally costly to achieve is outside the model.
  • A published automatic rule is inspectable, not committed. The adjustment mechanism made the cap's path predictable; its suspension in July 2026 shows it was a revisable default. Chapter 4's distinction applies: publication makes a rule inspectable, but commitment requires that revising the rule be costly enough not to pay (by law, by penalties or by reputation), and only then does the rule carry the commitment value of Proposition 4.7 (p. 171) R.
  • Terms re-derived from current conditions need provisions for moving conditions. The mechanism re-derived the cap from the current market price; when the market moved so that the re-derived term would loosen the constraint, the coalition declined the re-derived term. Model B5 makes the general point in its own setting: an agreement whose terms are recomputed from current conditions at every date must anticipate how those conditions move, or be renegotiated when they do (Proposition 3.12 (p. 135)) R.
  • Divergent rates break the symmetric model, not the logic. From September 2025 two caps coexisted. A transaction caught by both regimes, with the same definitions and exemptions, has to meet the stricter cap: a matter of logic, reflected in the insurance market's notices D [Lloyd's Market Association, 14 August 2025; West P&I, September 2025],*Lloyd's Market Association, “Oil Price Cap – Updated Market Guidance for Marine and Political Risk Insurers”, LMA25-033-AR, 14 August 2025, https://lmalloyds.com/oil-price-cap-updated-market-guidance-for-marine-and-political-risk-insurers/; West of England P&I Club, “Russian sanctions: lower price cap for Russian crude”, news item, September 2025, referring to its Notice to Members No. 9 2025/26, https://www.westpandi.com/News-and-Resources/News/September-2025/Russian-sanctions-lower-price-cap-for-Russian-crud/; the paragraph stating that, as an entity subject to the EU and UK regimes, the club can in practice cover only voyages priced at or below US$47.60. though not a statement about any particular contract. But Model B4's symmetric two-player structure no longer describes the coalition; coordination within a given coalition of asymmetric members is the subject of Chapter 8, and endogenous coalition formation lies outside the book's models R.

Where the mapping breaks.

The coalition has many members with unequal stakes and different exposures to the service markets; the model has two symmetric players. In the model's own terms, members that weigh false alarms, enforcement costs and market friction differently do not share one or one , and would prefer different caps; the divergence of rates from September 2025 and the freeze of the automatic mechanism in July 2026 are consistent with that asymmetry, and the record documents the decisions, not any member's reasons R. The target is a strategic player, not a source of noise: a “shadow” fleet operating outside coalition services does not raise ; it removes trade from the monitored set altogether, which changes the game rather than its parameters. Enforcement acts on firms, through regulators and courts, while the model's players are the members themselves; the link from a member's choice to its firms' behaviour is a principal–agent problem of its own (Chapter 6's inspection games). Payoffs move with world prices, as July 2026 showed; the model's payoffs are stationary. And the agreement is renegotiated in public, by political bodies, under pressures the model does not contain.

In the book

The full case runs through six steps: The situation, as documented; The model card; What the mathematics licenses; Where the mapping breaks; What one would have to measure; Sources. The numbers in step 3 are certified in the chapter's engine room, and the sources in step 6 are listed with the day they were read. The site shows the model mapping (step 2), what the mathematics licenses (step 3) and where the mapping breaks (step 4); the full case is on p. 117 of the book.