The Renegotiation Triangle

A planner's framework for pricing the contingent fiscal cost of a PPP renegotiation offer — and the option value of a credible termination threat — with a reference implementation on the Austral platform, calibrated to the Chilean concession panel

Abstract

A renegotiation is the silent fiscal event of a public-private partnership (PPP). Doctrine treats it as exceptional; the evidence is that it is the base case. In the Chilean concession programme — the cleanest long-run panel in the world — 63% of 123 concessions were renegotiated (78 of 123), and where the renegotiated amount is disclosed the median draw is 11.7% of the tendered budget, the mean 25.6%, and the right tail reaches 431%. Guasch (2004), across more than a thousand Latin-American concessions, found renegotiation rates climbing from roughly a third overall to a majority of transport concessions, on average about two years after award — with the soft-budget account of why set out by Engel, Fischer and Galetovic. A demand that arrives this often cannot be handled ad hoc.

When it arrives, the Ministry of Finance has exactly three moves: ACCEPT the amendment as offered, COUNTER with a re-priced settlement, or TERMINATE the contract. This paper puts all three on a single cost-to-government axis and prices them from the project's own financial model and termination-value engine. The contribution is to make the middle move analytic. A renegotiation is a bargain, and the planner's leverage in that bargain is the credible threat to terminate. We define and compute its option value,

where $C_{\mathrm{ACCEPT}}$ is the discounted incremental fiscal cost of granting the demand, $K$ is the termination compensation the State would owe, $\beta$ is the concessionaire's bargaining power, and $p$ is the credibility of the threat. The threat is credible only when terminating is no dearer than capitulating ($G \equiv C_{\mathrm{ACCEPT}} - K \ge 0$); where $G<0$ the option is out of the money and $p$ collapses to zero — and the framework reduces exactly to the Engel–Fischer–Galetovic soft-budget result, in which the planner pays the full demand. The soft-budget trap is the $p=0$ corner of the triangle, not a separate theory.

One subtlety makes the triangle a genuine three-way choice rather than a binary. Actually terminating does not only cost the compensation $K$; it destroys the service — a continuity loss the planner would rather avoid. So a credible threat is most often worth using to counter, not executing: the State keeps the asset and bargains the price down. Whether COUNTER or TERMINATE wins turns on how essential the service is. We show this on three concessions from the Chilean panel, each run end to end through the shipped engine: a mild interurban road where the demand is below the termination cost, so the threat is not credible and the planner simply accepts; an essential hospital where the demand is large enough to make the threat credible but the facility cannot be allowed to go dark, so the planner counters; and a runaway urban road where the cumulative demand dwarfs the cost of walking away and the asset is re-tenderable, so the planner terminates. Accept, counter, terminate — ordered by the severity of the demand. Two prices in this bargain, though, are not the State's to set: the concessionaire can reject a counter and go to arbitration, whose binding award floors the settlement, and the termination compensation $K$ is adjudicated in that same forum rather than fixed by the clause. Both narrow the credible threat — they cap its option value and depress its credibility — so both are worked in as comparative statics that tighten the framework's conclusion rather than soften it (§2.7).

The honest limitation is stated up front: $p$, the credibility of the threat, is not observed in any renegotiation register — settlements and terminations are recorded, the counterfactual threat probability is not — so the option value is presented as a comparative static in $p$, never a point estimate, and the worked cases are calibrated model demonstrations, not reconstructions of the actual settlements. The policy lead is to book the renegotiation-response protocol ex ante: price the triangle before the demand arrives, cap the cumulative renegotiated value, and competitively re-price scope additions — because a credible termination threat is the cheapest discipline a ministry has, and it only exists if the contract built it in. It must also be the right actor's protocol: the demand usually lands first on the contracting authority — the sector ministry that owns the project and, when the claim stems from its own preparation weakness, is conflicted toward granting it — so the response decision belongs with the Ministry of Finance that bears its cost (§1.2).

Keywords: PPP renegotiation · contract amendment · termination payment · handback · option value · credible threat · arbitration · dispute resolution · adjudicated compensation · soft-budget constraint · fiscal commitments and contingent liabilities · contracting-authority conflict · intra-government principal–agent · concession · Ministry of Finance.

1. The decision problem

A renegotiation demand lands on the Ministry of Finance desk. It may be dressed as a convenio complementario, a contract amendment, a request for additional works, a tariff adjustment, or a claim — and it may be initiated by the concessionaire or by the State itself. However it is dressed, it forces a choice with a price tag attached to each branch:

Doctrine treats this as an exceptional, bilateral, largely legal event. The empirical record says otherwise. The Intelligent Gate (Paper 5) assembled the Chilean concession panel and showed two things that frame this paper: renegotiation is the base case — 63% of concessions — and it is not forecastable from the structural procurement flags (number of bidders, single-bidder award, below-norm pricing) that a gatekeeper can see at tender. The hazard of renegotiation on those flags was a robust null.

That null is not a dead end; it is a handoff. If you cannot predict which concessions will renegotiate, you must be ready for any of them to — and "ready" means having priced the response before the demand arrives. The Gate framed itself as the empirical front-end of this Renegotiation Triangle idea and called for a renegotiation-control-clause checklist as the gate's own companion deliverable; the response rule — provision for the draw at the base rate, then dispose of each branch in a disciplined way — is what this paper supplies.

1.1 What the triangle replaces

Today, a renegotiation demand arrives dressed as a convenio complementario and is routed to legal and contract counsel. It is handled as a bilateral negotiation over entitlement and precedent; the fiscal cost of accepting is booked, if at all, as ordinary spending once agreed, while the cost of the counter that was never run — and the value of the termination threat that was never priced — stay invisible. There is rarely a number on the table for what walking away would cost, so there is rarely a credible alternative to accommodation. The incentive runs backwards: the cheapest-looking move (accept, and book it as spend) is the one that forecloses the discipline, and the move that would have cost less over the contract (a threat-disciplined counter) is the one no one computed.

The triangle replaces the bilateral-negotiation default with a priced decision the planner can take, book, and defend.

1.2 Who decides — the contracting authority and the Ministry of Finance

	Today	The triangle
Framing	bilateral legal negotiation over entitlement	priced three-branch decision to government
What is costed	the accepted amendment, after the fact	all three branches, before the choice
The termination threat	rhetorical, unpriced	the cash-rational \(G \ge 0\) test + its option value \(OV(p)\)
The counter	ad hoc	the threat-discounted settlement \(S^{*}(p)\)
The record	a signed addendum	a tamper-evident audit chain of branch costs and parameters

The opening line — "a renegotiation demand lands on the Ministry of Finance desk" — is a simplification this section repays. The State is not a unitary planner. A PPP has at least two principals inside government with different objective functions, and the demand usually lands on the contracting authority first: the line ministry or sector agency that sponsored the project, signed the contract, and owns the service (in Chile, the Ministry of Public Works for roads, the Ministry of Health for hospitals). The Ministry of Finance is its fiscal counterpart — it carries the budget cost of every branch but holds none of the project's operational mandate.

The two want different branches. The contracting authority is invested in the asset: its mandate is delivery, its political capital is in the ribbon-cutting, and a stalled or terminated project is its failure, not the concessionaire's. It leans, structurally, toward ACCEPT — keep the project moving, settle the claim, avoid the dispute — and it almost never wants to TERMINATE, because terminating is an admission that its own project failed. The Ministry of Finance has the opposite incentive on the one axis it owns, cost, and is the actor with a reason to compute the counter, price the termination threat, and hold the credibility $p$ above zero.

The bias sharpens when the renegotiation traces to a weakness in project preparation — an optimistic demand forecast, an incomplete design, unresolved land or permitting, a risk the public side should have retired before tender. Then the contracting authority is conflicted: the cost it is now being asked to grant is, in part, the price of its own preparation failure, and accommodating the claim quietly is also the cheapest way to keep that failure off the record. The party that caused the cost is the party advocating to pay it — and it is not the party that bears the fiscal cost. This is a moral-hazard externality inside government, and it is exactly the configuration that produces the soft-budget corner of §2.3: the effective credibility $p \approx 0$ that the panel reveals by its zero terminations (§11.2) is not only "no commitment technology" in the abstract — it is a decision held by the actor with the least reason to make the threat believable.

This is why the framework is a Ministry-of-Finance instrument, and why "the planner" throughout this paper names the fiscal counterpart, not the project's sponsor. It is also the institutional content of Paper 3's re-partition of responsibility (From Gatekeeper to Orchestrator): the renegotiation-response decision belongs with the actor that bears its cost and can hold the threat, advised by — not delegated to — the authority that owns the service. The triangle does not assume this separation; it argues for it, by pricing what is lost when the advocate for the project is also the one who answers the claim. The continuity loss $D$ is real, and the contracting authority is right that it matters — but $D$ belongs inside the priced decision, as the term that makes COUNTER keep the asset (§2.4), not as a thumb on the scale toward unconditional ACCEPT.

2. The general principle — the triangle and the option value of a credible threat

2.1 The three branches on one axis

Write every branch as a cost to government in present value, positive meaning the State pays. The three are not commensurable as offered — ACCEPT is a stream of future payments, TERMINATE is a lump-sum compensation plus frictions — so the work is to put them on the same axis.

ACCEPT. Run the proposed amendment through the project's financial model and take the incremental fiscal cost: the discounted government cash under the amended contract minus the discounted government cash under the baseline. Granting a higher availability payment, a longer term, or additional government-funded works raises this number; that increment, $C_{\mathrm{ACCEPT}}$, is the price of saying yes.

TERMINATE. The termination clause fixes a formula for the compensation $K$ — typically the greater of net book value and outstanding debt plus net equity, the standard that protects lenders — though the number that formula yields is itself contested in arbitration, a refinement we take up in §2.7. To actually walk away the State also pays reprocurement and transaction costs and, crucially, suffers a service-continuity loss $D$: a closed hospital, an unbuilt road, a dark water plant. The all-in cost of executing is $C_{\mathrm{TERMINATE}} = K + D$.

2.2 The option value of a credible termination threat

A renegotiation is a bargain, and in a bargain the relevant question is not "what will the State pay if it does nothing" but "what is the State's outside option." The planner's outside option is termination — but only if the threat to use it is credible.

Credibility is an economic, not a rhetorical, condition. The concessionaire believes the State might terminate only if terminating is something the State would rationally choose: that paying the compensation is no dearer than capitulating to the demand. Define the gap

The threat is credible when $G \ge 0$ — the demand has grown larger than the cost of paying the concessionaire off and walking. (Note that credibility rests on the cash compensation $K$, the number the State would actually write a cheque for; the continuity loss $D$ is what the State feels, and it shapes whether the State wants to execute, not whether the concessionaire believes it could.)

When the threat is credible, it disciplines the settlement, and it is worth writing the bargain down rather than asserting the result. Model the resolution as a split of the cost the State will bear, anchored on a credibility-weighted disagreement point. If talks break down, with probability $p$ the State terminates and pays the compensation $K$, and with probability $1-p$ it capitulates and pays the full demand $C_{\mathrm{ACCEPT}}$; its expected break-down cost is therefore

The parties settle between this threat point and the concessionaire's ask $C_{\mathrm{ACCEPT}}$, with the concessionaire's bargaining weight $\beta$ on its own ask and $1-\beta$ on the State's threat point. Under risk-neutral, transferable-utility bargaining with no asymmetric information over $K$, the surplus split is linear in the threat point, so the settlement (the COUNTER cost) is

\[S^{*}(p) \;=\; (1-\beta)\,T(p) + \beta\,C_{\mathrm{ACCEPT}} \;=\; C_{\mathrm{ACCEPT}} - (1-\beta)\,p\,G.\]

We deliberately avoid the bare label "generalised Nash": a symmetric Nash split of a fixed surplus would be 50/50, whereas here the asymmetry enters through the weight $\beta$ on the threat point, and the credibility $p$ is exactly what a believable threat moves. (The same $S^{*}(p)$ is the limit of an alternating-offers game in which $\beta$ maps to relative patience.) The saving the credible threat buys — its option value — is the gap between capitulating, $S^{*}(0) = C_{\mathrm{ACCEPT}}$, and the disciplined settlement:

\[\boxed{\,OV(p) \;=\; S^{*}(0) - S^{*}(p) \;=\; (1-\beta)\,p\,G \;=\; (1-\beta)\,p\,\bigl(C_{\mathrm{ACCEPT}} - K\bigr)\,} \qquad (G \ge 0),\quad 0 \text{ otherwise.}\]

Here $\beta \in [0,1]$ is the concessionaire's bargaining power ($\beta = 1$ means the concessionaire holds all the cards and the threat buys nothing; $\beta = 0$ means the State imposes its floor) and $p \in [0,1]$ is the credibility of the threat — the concessionaire's assessed probability that the State actually terminates if talks break. The COUNTER cost is the discounted settlement,

with $\kappa_R$ the renegotiation's own transaction cost. The State keeps the asset; it simply pays a price reduced by the credible threat.

2.3 The soft-budget corner: EFG as the $p=0$ case

Set $p=0$ — the threat is not believed, because the State has no commitment technology, or because $G<0$ and termination is genuinely more expensive than capitulating. Then $OV=0$ and the COUNTER settlement collapses to $S^{*} = C_{\mathrm{ACCEPT}}$. Where termination is not even cash-rational ($G<0$, the soft-budget case), TERMINATE is ineligible and the State pays the full demand — exactly the soft-budget-constraint equilibrium of Engel, Fischer and Galetovic, in which the concessionaire renegotiates precisely because it expects accommodation. (Where the cash threat is credible, $G \ge 0$, but the State will not make it believable and the continuity loss is small, TERMINATE can still be the cheapest branch; $S^{*}=C_{\mathrm{ACCEPT}}$ pins only the COUNTER cost, not the whole decision. The EFG identity is the no-credible-threat corner.) The triangle does not contradict that result; it contains it, as the corner where credibility is zero. Everything interesting in this paper is what happens as $p$ rises off that corner: the planner buys discipline by making the threat believable.

2.4 Why the triangle is a triangle, not a line

If the planner minimised cash alone, the triangle would collapse: whenever the threat is credible ($K \le C_{\mathrm{ACCEPT}}$), terminating at cost $K$ would beat any settlement above $K$, and the choice would be the binary "accept, or terminate." The triangle is genuinely three-sided because executing the threat destroys the service. The all-in walk-away cost $C_{\mathrm{TERMINATE}} = K + D$ carries the continuity loss; the COUNTER keeps the asset and pays $C_{\mathrm{COUNTER}} = C_{\mathrm{ACCEPT}} - OV + \kappa_R$. COUNTER beats TERMINATE precisely when

i.e. when the service is essential enough that losing it outweighs the residual gap. This is the paper's sharpest practical message: a credible termination threat is usually worth more as a threat than as an act. The State that has built a believable threat will, for most essential assets, use it to counter and keep the service — and only walk away when the demand is so large, or the asset so easily replaced, that executing dominates.

2.5 Prior art (synthesis, not discovery)

The pieces are known; the planner-side composition is not. Engel, Fischer and Galetovic (2014, and earlier work) give the soft-budget account of why renegotiation is endemic. Guasch (2004) documents that a large share of Latin-American renegotiations are government-led and quantifies the incidence by sector and regulatory regime. Guasch, Laffont and Straub (2008) model the bargaining under imperfect enforcement. A game-theoretic and real-options literature (Ho on bargaining-driven renegotiation; option-to-abandon termination pricing) values renegotiation flexibility — but from the concessionaire's side. What is missing, and what this paper supplies, is the Ministry-of-Finance view: the three branches priced on one cost-to-government axis, the option value of a credible termination threat computed directly off the platform's own termination-value engine, and the whole thing reproducible against a real concession panel.

Source	Whose side	What it models	Termination as outside option?	Continuity loss \(D\)?	Anchor
Engel–Fischer–Galetovic	State	the soft-budget equilibrium (why renegotiation is endemic)	implicit	no	theory + LAC
Guasch (2004)	both	incidence and determinants of renegotiation	no	no	>1,000 LAC concessions
Guasch–Laffont–Straub (2008)	both	renegotiation bargaining under imperfect enforcement	no	no	LAC panel
Real options / abandonment (Ho; option-to-abandon)	concessionaire	the value of the concessionaire's flexibility	— (its own option)	no	varies
This paper	State	the priced accept/counter/terminate choice + \(OV(p)\)	yes — the State's credible threat, on cash \(K\)	yes — the COUNTER/TERMINATE separator	Chilean panel + a shipped engine

Two distinctions a referee will want drawn. Against Guasch–Laffont–Straub (2008) — the closest model — the delta is the service-continuity loss $D$, which makes the keep-or-walk choice turn on the asset's essentiality (GLS price the bargain but do not separate COUNTER from TERMINATE), and the pricing of every branch off a live termination-value engine rather than a stylised payoff. Against the real-options line, $OV(p)$ here is the State's option, valued on the cash-compensation threat point — the bargaining surplus a credible threat captures, structurally different from a Black–Scholes-type option to abandon held by the concessionaire. The contribution is a decision framework and a reference implementation, not a new estimator.

2.6 What the cost axis holds fixed — and two assumptions made explicit

Putting ACCEPT (a stream) and TERMINATE (a lump sum) on one axis requires saying what is held common, and a careful reader is right to press on it.

The accounting convention. All three branches are priced as the incremental cost to government of resolving this demand, measured against the same baseline contract over the same remaining term. What is common to all three is therefore differenced away: the baseline payments the State was already committed to, and — on the ACCEPT and COUNTER branches — the continuation value of the service, which both keep. TERMINATE is the only branch that forfeits that continuation value, and the service-continuity loss $D$ is precisely its price. So the comparison is not "a stream versus a lump sum"; it is "the incremental cost of granting the demand and keeping the asset" against "the cost of compensation plus the service given up." $C_{\mathrm{ACCEPT}}$ is the increment over the baseline — not the whole amended stream — which is why a small amendment does not spuriously lose to a large termination payment. All three branch costs are present values at the decision date, so the stream and the lump sum are commensurable: $C_{\mathrm{ACCEPT}}$ is discounted at the government rate, and $K$ is the compensation owed at the assumed termination, discounted back to the same date.

Two assumptions, made explicit, because a sharp reader can otherwise construct an inconsistency. The credibility seam: credibility is judged on the cash compensation $K$ ($G = C_{\mathrm{ACCEPT}} - K$), while executing costs the all-in $K + D$. This is deliberate. Credibility is what the concessionaire believes the State could rationally do, and the State could rationally pay $K$ to be rid of a demand larger than $K$ — the cash comparison is the verifiable one. Whether the State then prefers to execute depends on $K + D$. The model thus describes a credible threat the State would rather not carry out — the standard logic of a deterrent that disciplines the outcome without being used, and a commitment assumption stated as one. The bargaining assumptions: $S^{*}(p)$ assumes risk-neutrality, transferable utility, and symmetric information over $K$; relaxing any of these (e.g. a concessionaire better-informed about its own continuation value) would bend the linear-in-$G$ form. We take the linear case as the transparent benchmark, relax the two prices the State does not set — the concessionaire's arbitral outside option and the adjudicated value of $K$ — in §2.7, and flag the further extensions in §11.

Scope: government-pays structures. The framework prices the fiscal cost of a renegotiation, so $C_{\mathrm{ACCEPT}}$ bites only where the State pays — an availability PPP, a subsidised concession, a minimum-revenue guarantee. For a pure user-pays toll road the State pays no availability and the $C_{\mathrm{ACCEPT}}$ of a tariff-only amendment is near zero (the engine emits an explicit warning); there the renegotiation transfers value between users and the concessionaire, not onto the budget, and the triangle's fiscal axis is silent on it by design. The natural extension — pricing a demand-guarantee or revenue-floor amendment on a user-pays road, where the State does carry contingent exposure — is in the roadmap. We flag this boundary so the framework is not read as claiming toll-road gouging is fiscally free: it claims only that the fiscal cost is what it prices, and on a pure user-pays road that cost runs through guarantees, not availability.

2.7 The two contested prices — the concessionaire's outside option, and the adjudicated value of $K$

The bargain of §2.2 gives the State an outside option — terminate — and quietly treats two numbers as if the State set them: the concessionaire's acceptance of the counter, and the compensation $K$. Neither is the State's to fix. Both are settled, in the last instance, by a third forum: the contract's dispute-resolution mechanism. In Chile that is the Panel Técnico de Concesiones, whose recommendation is technical and non-binding, escalating to the Comisión Arbitral, whose award binds both parties (Ley 20.410). The very panel técnico the policy lead recommends building (§11.5) is, seen from the other side of the table, the concessionaire's outside option.

The concessionaire's outside option floors the settlement. A concessionaire that dislikes the counter need not accept it; it can reject and go to arbitration, and the award binds the State too. Let $A$ be the concessionaire's expected award in that forum — the discounted value it rationally believes it would be handed. Then $A$ is a floor on the settlement: the State cannot counter below what the concessionaire expects to win anyway, so $S^{*}(p) \ge A$, and the option value is capped,

\[OV(p) \;=\; \min\!\bigl\{\,(1-\beta)\,p\,G,\;\; C_{\mathrm{ACCEPT}} - A\,\bigr\} \qquad (G \ge 0).\]

This also microfounds $\beta$: the concessionaire's bargaining weight is not a free primitive but a reduced form of the two sides' arbitral positions. Where the concessionaire's claim is strong ($A$ near $C_{\mathrm{ACCEPT}}$), even a fully credible termination threat buys almost nothing — the surplus a believable threat could have captured has already been claimed by the outside option. This floor is the concessionaire-side counterpart of the State-side credibility gap: §11.4's prediction that settlements do not exceed $C_{\mathrm{ACCEPT}}$ gains a lower bound — that they do not fall below $A$.

The compensation $K$ is adjudicated, not fixed. §2.1 wrote the clause as fixing $K$; it fixes a formula — the greater of book value and debt-plus-equity — but the number that formula yields is contested in the same forum: what the equity is worth, which capex was prudently incurred, whether lost-profit or fuerza mayor claims attach. The State therefore faces not a known $K$ but an expected adjudicated compensation $\hat{K} = \mathbb{E}[K \mid \text{dispute}] \ge K_{\text{formula}}$, right-skewed and payable only after a multi-year proceeding. Two consequences, both narrowing the threat:

A fourth actor. Behind the State, the concessionaire, and the conflicted contracting authority (§1.2) stands the arbitral tribunal, whose expected award bounds both prices — the settlement from below ($A$) and the compensation from above and uncertain ($\hat{K}$). Neither is set unilaterally by the State. This does not overturn the triangle; it tightens it in the direction the paper already argues. Both refinements shrink $OV$ and depress $p$, adding a second channel — contract-external, alongside the intra-government one of §1.2 — through which the credibility that would discipline a renegotiation is held down, and the incumbent favoured. We carry them as comparative statics: the engine already exposes $K$ as a parameter with a compensation-method haircut (§11.3), so $\hat{K}$ and its band drop straight in and $A$ enters as a floor on $OV$; endogenising the tribunal's award is left to the roadmap (§11.6).

3. Size the offer — the ACCEPT branch

The ACCEPT cost is the incremental fiscal net present value (NPV) of the amendment, and the platform already runs both sides: the Renegotiation analyzer re-runs the financial-model engine on the baseline and on the amended inputs and diffs the headline metrics. The government cash series — the availability payment in an availability PPP, the operational payment or subsidy in a user-pays-with-support structure — is discounted at the government rate and differenced:

\[C_{\mathrm{ACCEPT}} \;=\; \sum_{t} \frac{f_t(\theta_{\mathrm{amended}}) - f_t(\theta_{\mathrm{baseline}})}{(1+r_g)^t}.\]

A correctness note, surfaced rather than buried. The financial-model engine reports a field named fiscal_cost_npv that is in fact an undiscounted sum of the government cash series. Inheriting it would misstate every $C_{\mathrm{ACCEPT}}$, $G$ and $OV$ in this paper. The Renegotiation Triangle module therefore re-extracts the per-year government series and discounts it itself at the government rate (default 6%); all ACCEPT figures here are true NPVs. The mislabelled engine field is logged as a separate fix.

4. The cost distribution — renegotiation is endemic and right-skewed

Sizing one offer is necessary but not sufficient; the planner provisions for a distribution of draws, because the next renegotiation is a near-certainty and its size is uncertain. The Chilean panel gives the shape. Of 123 concessions, 78 (63%) were renegotiated. Where the renegotiated amount is disclosed (74 of 123 — the other 49 carry no disclosed figure, see §11 on the missingness; two of the 74 are net-negative settlements, retained as reported), the draw as a share of the tendered budget (presupuesto oficial) is heavily right-skewed (percentiles by linear interpolation):

Statistic	Renegotiated value ÷ tendered budget
Median	11.7%
Mean	25.6%
90th percentile	41.7%
95th percentile	63.6%
Maximum	430.9%

The mean is more than twice the median; a handful of runaway concessions dominate the aggregate. The same 74 draws sort by a severity-survival curve: 51% exceed 10% of the budget, 46% exceed 15%, 34% exceed 25%, but only 6.8% exceed half the budget and 4.1% exceed it entirely. Most renegotiations are modest; a thin tail is enormous. And the programme renegotiates repeatedly: among the 78 renegotiated concessions the median is 5 amendments (mean 6.7, maximum 25), 48 of them are amended five times or more, and the amendment count rises with realised severity (correlation $+0.16$). The serial-accept path — a concession amended again and again, each draw individually unremarkable, the cumulative total reaching the right tail — is the signature the policy lead (§11) is built to interrupt, and, as §11.2 argues, the observable fingerprint of a low-credibility (soft-budget) regime.

This is the distribution the planner provisions against at Stage E (§7), and the right tail is exactly where a credible termination threat earns its keep.

5. Price the three branches

The platform composes the three branch costs from two engines it already ships — the Renegotiation analyzer (ACCEPT) and the Termination-value engine (the compensation $K$) — plus the option-value arithmetic above. In the style of the Third-Party Guarantee paper's society-cost table:

The termination compensation $K$ is drawn from the four methods the termination engine computes — net book value, outstanding debt plus net equity, their maximum (the market-value proxy and the legally typical standard), or the DCF of remaining cash flows — at the assumed termination year, with the arbitration/award uncertainty around it Monte-Carlo'd into an expected value and a 95th-percentile tail. The continuity loss $D$ and the bargaining parameters $\beta$, $p$ are modelling inputs the planner sets and the audit chain records; they are not estimated from the contract, and the paper does not pretend otherwise.

6. The decision rule and the headline

\[a^{*} \;=\; \arg\min\,\bigl\{\,C_{\mathrm{ACCEPT}},\; C_{\mathrm{COUNTER}},\; C_{\mathrm{TERMINATE}}\,\bigr\},\]

Branch	What the State does	Cost to government (PV)
ACCEPT	Grant the amendment as offered	\(C_{\mathrm{ACCEPT}}\) = discounted incremental fiscal cost of the amended contract
COUNTER	Settle on re-priced terms, keep the asset	\(C_{\mathrm{COUNTER}} = C_{\mathrm{ACCEPT}} - (1-\beta)\,p\,G + \kappa_R\)
TERMINATE	Invoke handback, pay compensation, re-tender	\(C_{\mathrm{TERMINATE}} = K + D\) = compensation (book / debt+equity / market / DCF) + reprocurement + service-continuity loss

with TERMINATE eligible only when the threat is credible ($G \ge 0$). Reading the rule off the parameters gives a clean three-region map, scaled by the gap $G$ and governed by two quantities — the severity of the demand (which sets the sign and size of $G$) and the essentiality of the service (which sets $D$):

The single act-on number the framework returns is the recommended branch and its cost, together with $OV(p)$ — the saving the credible threat is worth, which the planner can read against the cost of building that threat into the contract (a cumulative-value cap, a panel técnico, a step-in right).

$Figure 1 — The Renegotiation Triangle's decision regions in the (severity, essentiality) plane. The horizontal axis is the demand relative to the termination compensation, $C_{\mathrm{ACCEPT}}/K$ — the threat is credible to the right of the hinge at 1. The vertical axis is the service-continuity loss $D/K$. The three worked concessions (§9) are plotted: A (the mild road) in ACCEPT, C (the essential hospital) in COUNTER, B (the runaway road) in TERMINATE. Drawn for $\beta=0.5$, $p=0.6$; the boundaries shift with $\beta$ and $p$ (§10.1).$

Figure 1 — Decision regions in the (severity, essentiality) plane, with the three worked concessions. Generated from the shipped engine's closed-form decision rule by make_decision_map.py.

7. Wire to the affordability envelope (Stage E)

The Triangle is not a standalone calculator; it is the action at one node of the orchestrated chain this series has built. From Gatekeeper to Orchestrator (Paper 3) placed fiscal-commitments-and-contingent-liabilities (FCCL) aggregation at Stage E, and The Intelligent Gate told the planner to provision there for the renegotiation draw at the endemic base rate. The chosen branch's cost is that draw, realised. When the demand arrives, the Triangle prices the branches; the recommended cost is booked against the Stage-E envelope the planner already set aside; and the audit chain records which branch was chosen, at what option value, against which credibility assumption. The macro-prudential layer (the value-at-risk, VaR, of the contingent stock, the subject of a companion paper on portfolio stress) sits above this: the envelope is sized for the distribution of draws, the Triangle disposes of each one.

8. Architecture and reproduce in ten minutes

The whole framework is one pure-Python module, fm/renegotiation_triangle.py, that composes two engines already in the platform — the Renegotiation analyzer (run_baseline_vs_amendment) for ACCEPT and the Termination-value engine (compute_termination_values) for the compensation — and adds the option-value arithmetic. It is the direct analogue of the fx_placement instrument behind Paper 6: a request in, a list of priced branches out, a recommended field, and a tamper-evident audit-chain entry. The endpoint,

POST /workspaces/{wid}/fm/renegotiation/triangle

is editor-gated and audit-chained; it runs the financial-model engine up to three times (baseline, amendment, and — if supplied — an explicit counter-offer), prices the three branches, and returns the recommendation, the option value, and the Guasch-style renegotiation-risk band for documentation. A worked example reproduces in one call (Appendix A); the per-branch formulas are in Appendix B; the module ships with thirteen unit tests pinning the closed-form flip thresholds ($p^\*$, $D^\*$), the option-value gate, the EFG $p=0$ corner, the seeded arbitration tail, and the full argmin across the three regions.

9. Worked examples — the three regions on three concessions

We run three concessions from the Chilean panel through the shipped engine — the importable core behind the /fm/renegotiation/triangle endpoint. Each baseline is a stylised availability PPP scaled to the concession's tendered budget; each amendment is a higher availability payment calibrated to the concession's realised renegotiation severity. These are calibrated model demonstrations, not reconstructions of the actual settlements — the per-event accept/counter/terminate decision and its matched compensation are in no public register (§11). Money is in thousands of UF (Unidad de Fomento, Chile's inflation-indexed accounting unit).

Concession (sector, severity)	\(C_{\mathrm{ACCEPT}}\)	\(C_{\mathrm{COUNTER}}\)	\(C_{\mathrm{TERMINATE}}\)	\(G\)	\(OV(p)\)	Recommended
A — 061_PMPA1 Vial interurbana, 0.265	774	827	3,314	−1,775	0	ACCEPT
C — 080_HSSG1 Hospitalaria, 0.421	5,789	5,173	6,138	+2,282	685	COUNTER
B — 027_SNSS1 Vial urbana, 1.801	23,254	20,284	14,093	+10,999	3,300	TERMINATE

A — the mild road (ACCEPT). The interurban road's renegotiation (severity 0.265 — above the panel median of 0.117, but still small relative to its mid-life termination compensation) asks for less than it would cost the State to pay the concessionaire off and walk away. $G = -1{,}775 < 0$: the threat is not credible, $OV = 0$, and a counter only adds transaction cost. The planner accepts. This is the soft-budget corner, and it is where most ordinary renegotiations live — which is exactly why they are accommodated.

C — the essential hospital (COUNTER). The hospital's demand is large enough to make the termination threat credible ($G = +2{,}282 > 0$). But a hospital cannot be allowed to go dark: the service-continuity loss makes the all-in cost of actually terminating (6,138) higher than the threat-disciplined counter (5,173). So the planner uses the credible threat to bargain — capturing an option value of 685 — and keeps the asset. This is the case the framework was built for: a credible threat worth using, not executing.

B — the runaway road (TERMINATE). The urban road's cumulative demand (severity 1.801 — it has been renegotiated to nearly twice its tendered budget) dwarfs the cost of walking away, and an urban road is re-tenderable, so the continuity loss is modest. The all-in termination cost (14,093) beats both the demand (23,254) and the best counter (20,284). The planner terminates — and the case stands as the argument for an ex ante cumulative-value cap (§11): had a 20–25% cap been written into the contract, the serial-accept path that produced a 180% draw would have been foreclosed.

10. Reading the three together

The three concessions are ordered by severity, and the recommendation moves with it: accept the small demand, counter the credible-but-essential one, terminate the runaway. The option value is zero in the first case and rises with the gap in the other two. The continuity loss is not a free dial that yields any answer one wants: in each case the parameter must clear a threshold — the residual gap $G\bigl(1-(1-\beta)p\bigr)+\kappa_R$ — to change the decision. The essential hospital (case C) counters at a continuity loss of 75% of its compensation and would flip to terminate only below about 47%; the runaway road (case B) terminates at 15% and would flip to counter only above about 66%. The lever moves the recommendation only when the service's essentiality crosses an economically meaningful bar, not at the analyst's whim. Two policy points fall straight out. First, the planner's leverage is manufactured, not given: $OV(p)$ is zero unless the contract made the termination threat credible and the State can make $p$ believable — which is what a cumulative-value cap, a step-in right, and a panel técnico are for. Second, the runaway case is the expensive one, and it is the one the Gate told you to fear precisely because you could not see it coming: the discipline has to be structural (the cap), booked before the demand arrives, because by the time the 180% draw is on the desk the cheapest remaining move is the one that loses the asset.

10.1 Sensitivity, flip thresholds, and the one honest caveat

A point recommendation is worth little without knowing how far it sits from the boundary, and two outputs make that exact. First, the flip thresholds, which are closed form (no re-run): the credibility $p^{*} = \dfrac{\kappa_R}{(1-\beta)\,G}$ above which COUNTER overtakes ACCEPT, and the continuity loss $D^{*} = G\bigl(1-(1-\beta)p\bigr) + \kappa_R$ below which TERMINATE overtakes COUNTER. For the worked cases the hospital (C) flips from COUNTER to TERMINATE only if its continuity loss falls below 47% of compensation (it is calibrated at 75%), and the runaway road (B) flips from TERMINATE to COUNTER only above 66% (it is at 15%); in both, COUNTER overtakes ACCEPT for any credibility above ~5–6%. The demos are not knife-edge in $p$ or $D$.

Second — and this is the honest caveat — a robustness grid over the two choices that move every number, the government discount rate ($r_g \in \{4,6,8,10\}\%$) and the termination-value method (${$book, debt+equity, market, DCF$}$), shows the mild road (A) is robustly ACCEPT across the entire grid, but the recommendations for the two credible-threat cases (B, C) are method-sensitive: switching the compensation method changes $K$, and $K$ is the credibility hinge, so for B and C the recommended branch can move between ACCEPT, COUNTER and TERMINATE across the grid. This is not discretion to be hidden — the termination-compensation method is a contractual fact the planner reads off the contract, not a free dial — and the finding is itself useful: for a credible-threat case, which compensation standard the contract specifies is decision-relevant, and the robust objects are the decision logic and the flip thresholds, not a context-free verdict. The full grid is in out/sensitivity.json.

10.2 The whole panel under one rule — the regions are not cherry-picked

The sharpest version of the selection objection is: "you chose one concession per region, so of course you recovered three regions." To answer it, we apply one fixed, pre-declared rule — $\beta=0.5$, $p=0.6$, a continuity loss of 75% for essential sectors and 15% for re-tenderable ones, and the termination compensation set at a constant share of the budget — to every disclosed renegotiation in the panel (72 with a non-negative disclosed amount), with no per-case tuning. The result is the frequency distribution the framework predicts: under a compensation of 40–60% of budget, 86–94% of renegotiations fall in the ACCEPT region — the demand is below the compensation, the threat is not credible, the State accommodates — and only 6–14% reach credible-threat territory, splitting between COUNTER and TERMINATE by essentiality. By construction the credible share is simply the severity distribution thresholded at the compensation ratio $K/\text{budget}$ — there is no hidden machinery; the content of the exhibit is that the realised severities put most renegotiations below any plausible compensation, which is the soft-budget result, not an artefact of the rule. The three regions are not a manufactured equal split; they are a dominant soft-budget mass and a thin credible-threat tail, exactly as the Engel–Fischer–Galetovic account implies. The worked examples are one representative draw from each region — and case B is not a cherry: it is the panel's single most-renegotiated concession (25 amendments) and its largest by disclosed value (26% of all disclosed renegotiation), the actual extreme. The driver is whole_panel.py.

11. Empirical anchor, limitations, and the policy lead

11.1 What the panel validates — and what it cannot

The paper is calibrated to the same Chilean concession panel as The Intelligent Gate — 123 concessions, 63% renegotiated — for continuity and reproducibility, and every panel figure it cites is recomputed and golden-checked by panel_stats.py. The incidence is not Chile-specific: Guasch (2004), on over a thousand Latin-American concessions, found renegotiation rates of roughly 10% in electricity, 55% in transport and 75% in water, on average about two years after award; Bitran, Nieto-Parra and Robledo (2013) document widespread renegotiation across the Chilean, Colombian and Peruvian road programmes (with Colombia's road-PPP fiscal cost rising several-fold); and Peru's Ositran record shows a comparable share of concessions renegotiated. A 63% rate sits squarely in this range.

What the panel validates: renegotiation incidence (63%), the severity distribution and its right skew, the serial-renegotiation signature (median 5 amendments, up to 25), and the identity of the extreme. What it cannot validate is any per-event accept/counter/terminate decision or matched compensation — those are in no public register — and, tellingly, the panel contains zero recorded terminations: all 123 concessions are in Construcción, Construcción y Operación, Operación or Finalizado, with no caducidad or relicitación event. The TERMINATE branch is therefore grounded in the contractual compensation methodology and in documented relicitación episodes elsewhere (failed Chilean and Argentine concessions relicensed in the 2000s), not in a panel termination. That a programme renegotiating 63% of its concessions terminates none is itself a finding — almost by definition the behaviour of the soft-budget corner, which sharpens §11.2.

The time-to-first renegotiation in the panel is a median of 6.3 years (mean 8.9; only 23% within three years), markedly longer than Guasch's ~2.2 years. This is a survivorship and vintage artefact, not a contradiction: the panel is a surviving cross-section weighted toward long-lived concessions, and the fast, early renegotiators that were cancelled or relicensed are not in it. The bias runs toward the paper's conclusion — the true termination frequency is understated and the credible-threat tail is, if anything, larger than the survivors show.

11.2 Disciplining the unobserved $p$

The hinge parameter, the credibility $p$, is genuinely unobserved — and rather than wave at it, the panel can bound and sign the regime. Three observations do the work. First, $OV(p)$ is monotone and bounded: $OV \in [0,\,(1-\beta)G]$, with the ceiling at $p=1$, so every worked recommendation is robust to any $p$ above its flip threshold $p^{*}$ (§10.1) — and those thresholds are small (~5–6%). The decision does not need a point estimate of $p$; it needs to know which side of $p^{*}$ the regime sits, and the margin is wide. Second, the panel's behaviour reveals the regime directly, and the cleanest signal is the zero terminations: a State that has never once walked away, across 123 concessions and 78 renegotiations, holds no credible threat — its effective $p$ is $\approx 0$ by revealed preference. The serial-renegotiation count (median 5 amendments, 48 amended five times or more) corroborates this — repeated accommodation is what a soft-budget regime looks like — though, read alone, a long amendment chain is also consistent with long-lived staged infrastructure, so we lean on the termination fact and treat the serial count as supporting evidence. Either way the soft-budget corner is not assumed; it is read off the behaviour. Third, this turns the framework's central object into a regime classifier: where the contract carries no credible termination right (no step-in, no pre-agreed compensation, no panel técnico, a serial-accept history), the State should price its renegotiations at $p \approx 0$ — the EFG corner, $OV=0$ — and the whole value of the framework is then the prescription to move $p$ off zero by building the threat in. We cannot point-identify $p$; we can bound $OV$, sign the regime from observed serial behaviour, and report the $p^{*}$ at which the recommendation turns.

11.3 Limitations, stated plainly

11.4 Testable predictions

The framework is not only descriptive; it generates predictions checkable against renegotiation registers, and states what would refute it. (1) Monotone-severity ordering — accommodation should dominate at low severity and termination/relicitación concentrate at high severity; a programme that terminated mild renegotiations and accepted runaway ones would refute the cost ordering. (2) Regime signature — weak-commitment regimes (no step-in, no panel técnico) should show the serial-accept tail and near-zero terminations (the Chilean panel does), and stronger-commitment regimes shorter amendment chains. (3) Essentiality and termination — at a given severity, terminations should be rarer in essential sectors (hospitals, water) than in re-tenderable ones (roads); terminations concentrated in essential services would refute the continuity-loss mechanism. (4) Settlements bounded by the threat — realised settlements should not exceed $C_{\mathrm{ACCEPT}}$ where a credible threat existed, and, symmetrically, should not fall below the concessionaire's expected arbitral award $A$ (§2.7); settlements above the full demand, or below what the concessionaire could have won in arbitration, would refute the bargaining account. None of these requires the unobserved $p$; each is a sign restriction on observables.

11.5 The policy lead

Book the renegotiation-response protocol before the contract is signed, not after the demand arrives. Three controls, all of which the planner can price against the option value the framework computes:

11.6 Roadmap

The limitations imply a concrete agenda. (1) Build a renegotiation-outcome register — the decision taken, the compensation method invoked, the settlement — so $p$ becomes estimable rather than only bounded and the §11.4 predictions can be run. (2) Wire each chosen branch's cost into the portfolio-VaR companion (§7), so the Stage-E envelope is sized from the realised draw distribution. (3) Extend the calibration to a second register (Peru's Ositran, Colombia's ANI) and regulatory regime, to test the regime-signature prediction. (4) Endogenise $p$ from a commitment technology, endogenise the arbitral award behind the settlement floor $A$ and the adjudicated compensation $\hat{K}$ (§2.7), and add the asymmetric-information and risk-aversion cases that bend the linear-in-$G$ settlement. (5) Extend the fiscal axis to user-pays structures by pricing demand-guarantee and revenue-floor amendments, closing the §2.6 scope boundary.

12. The practitioner's desk

The framework is built to be used on a live demand and defended afterwards. This is the desk kit.

Setting $p$ — a credibility ladder. $p$ is not observed, but it is signposted by the contract and the State's record. Read it off observable features rather than guessing a number:

Contract / regime feature	\(p\) band
No step-in right, no pre-agreed compensation, serial-accept history	\(p \approx 0\) (soft-budget corner — accept)
Pre-agreed compensation method + an arbitration record	low–moderate
Step-in right and/or a panel técnico	moderate
All of the above and a documented prior termination/relicitación	high

The point of the ladder is not precision; it is that a ministry with none of these features should price its renegotiations at $p \approx 0$ and recognise that its only lever is to build the threat in — the policy lead.

Setting $D$ — an essentiality tier. The continuity loss follows from the asset's substitutability and re-tenderability, in the units the engine uses (% of $K$):

Asset / service	\(D\) band (% of \(K\))
Single essential facility, no substitute (a region's only hospital, a water system)	high (≳ 60%)
Networked but essential (an urban transit line, a trunk road)	moderate (30–60%)
Re-tenderable with a ready market (an interurban toll road, a parking concession)	low (≲ 20%)

When $p$ and $\beta$ are genuinely unknown — bound the decision. Report the recommendation at both corners. The State-pessimistic corner ($p=0$, $\beta=1$) gives $OV=0$ and collapses COUNTER into ACCEPT — the soft-budget worst case and a safe upper bound on what the State pays. The State-optimistic corner ($p=1$, $\beta=0$) gives the maximum option value ($OV=G$) and the most aggressive COUNTER. If the same branch is recommended at both corners, the decision is robust to the unknowns; if it differs, the contract's commitment features (the credibility ladder) break the tie — and that is exactly the conversation to have.

How to defend the decision. The engine emits the artefacts each audience needs. To the minister: the recommended branch, its cost, and the option value — one number, one decision. To the supreme audit institution: the tamper-evident audit-chain entry, the per-branch cost detail, and the parameter block, so the choice is reproducible and the judgement inputs are explicit and overridable. To the concessionaire's counsel: the branch costs and the $G \ge 0$ test, which make the termination threat a documented, cash-rational position rather than a bluff. The minimum data to assemble: the baseline government cash series (from the FM scenario), the contract's termination-compensation method and the asset's depreciation profile (from the contract), and the three judgement inputs (declared, with their ladder/tier rationale). Everything that is a contractual fact is auditable; everything that is a modelling choice is flagged as one.

Appendices

Appendix A — Reproducible API call

# Seed the FM demo, then price the triangle for an amendment on the baseline scenario.
curl -X POST "$BASE/workspaces/$WID/fm/renegotiation/triangle" \
  -H "Authorization: Bearer $TOKEN" -H "Content-Type: application/json" \
  -d '{
        "baseline_inputs":  { ... baseline ModelInputs ... },
        "amendment_inputs": { ... amended ModelInputs ... },
        "gov_discount_rate_pct": 6.0,
        "termination_method": "market_value",
        "termination_year": 2039,
        "beta": 0.5, "p_credibility": 0.6,
        "service_continuity_loss_pct": 75.0,
        "tail": { "arbitration_vol_pct": 25.0, "reprocurement_pct": 12.0,
                  "n_sims": 20000, "seed": 42, "var_pct": 95.0 }
      }'
# -> { accept, counter, terminate, decision: { branch_costs, accept_terminate_gap,
#      credible_threat, option_value, recommended_branch, rationale }, risk_score, warnings }

Every number in this paper is reproducible from the committed scripts: the worked examples by worked_examples.py (→ out/worked_examples.json); the panel statistics by panel_stats.py (golden-checked, → out/panel_stats.json); the flip thresholds and the discount-rate / termination-method robustness grid by sensitivity.py (→ out/sensitivity.json); the whole-panel pre-registered rule by whole_panel.py (→ out/whole_panel.json); and Figure 1 by make_decision_map.py. All run against the shipped engine or the committed Chilean panel.

Appendix B — Branch-cost and option-value reference

Appendix C — Glossary

Symbol	Meaning	Source
\(C_{\mathrm{ACCEPT}}\)	Discounted incremental fiscal cost of the amendment	FM engine, baseline vs amended government cash series, discounted at \(r_g\)
\(K\)	Termination compensation (cash)	Termination engine: book / debt+equity / market / DCF at the termination year, Monte-Carlo'd for arbitration uncertainty
\(D\)	Service-continuity loss	Modelling input: `service_continuity_loss_pct` × \(K\)
\(C_{\mathrm{TERMINATE}}\)	All-in walk-away cost	\(K + D\) (+ reprocurement)
\(G\)	Accept–terminate gap	\(C_{\mathrm{ACCEPT}} - K\); threat credible iff \(G \ge 0\)
\(\beta\)	Concessionaire bargaining power	Modelling input \([0,1]\)
\(p\)	Threat credibility	Modelling input \([0,1]\) — unobserved; bounded/signed, never point-estimated
\(T(p)\)	Credibility-weighted disagreement point	\(p\,K + (1-p)\,C_{\mathrm{ACCEPT}}\)
\(S^{*}(p)\)	Threat-discounted settlement (the COUNTER cost)	\((1-\beta)T(p) + \beta C_{\mathrm{ACCEPT}} = C_{\mathrm{ACCEPT}} - (1-\beta)pG\)
\(OV(p)\)	Option value of the credible threat	\((1-\beta)\,p\,G\) when credible, else 0; bounded in \([0,(1-\beta)G]\)
\(C_{\mathrm{COUNTER}}\)	Settlement plus renegotiation cost	\(S^{*}(p) + \kappa_R\)
\(p^{*}\)	Credibility at which COUNTER beats ACCEPT	\(\dfrac{\kappa_R}{(1-\beta)\,G}\)
\(D^{*}\)	Continuity loss at which TERMINATE beats COUNTER	\(G\bigl(1-(1-\beta)p\bigr) + \kappa_R\)
\(a^{*}\)	Recommended branch	\(\arg\min\) over the eligible branches

Renegotiation — a post-award change to a concession's terms (amendment, convenio complementario, additional works, tariff adjustment, claim). Amendment — the specific change proposed. Termination payment / compensation ($K$) — what the State owes the concessionaire on termination, by the contracted method. Handback — return of the asset to the State at termination or contract end; the residual-value percentage measures unamortised asset life. Option value of a credible threat ($OV$) — the settlement saving a believable termination threat buys. Soft-budget constraint — the equilibrium in which a concessionaire renegotiates because it expects the State to accommodate (the $p=0$ corner). Stage E / FCCL — the fiscal-commitments-and-contingent-liabilities aggregation node where the chosen branch's cost is booked. $\beta$ — concessionaire bargaining power; $p$ — threat credibility; $G$ — the accept–terminate gap; $D$ — service-continuity loss; $\kappa_R$ — renegotiation transaction cost.

Appendix D — The triangle in pseudocode

run_triangle(baseline, amendment, params):
    C_ACCEPT  = discounted_fiscal(amendment) - discounted_fiscal(baseline)      # at r_g
    K, var    = montecarlo(compensation(baseline, method, year) + reprocurement) # cash threat point
    D         = continuity_loss_pct * K
    C_TERM    = K + D                                                            # all-in walk-away
    G         = C_ACCEPT - K
    credible  = (G >= 0)
    OV        = (1 - beta) * p * G   if credible else 0
    C_COUNTER = C_ACCEPT - OV + kappa_R
    eligible  = {ACCEPT: C_ACCEPT, COUNTER: C_COUNTER}
    if credible: eligible[TERMINATE] = C_TERM
    return argmin(eligible), OV, G, credible

Calibration table (Chilean panel, 2026-01-09 snapshot): 123 concessions, 63% renegotiated; disclosed severity (n=74; 2 net-negative) median 11.7%, mean 25.6%, p90 41.7%, p95 63.6%, max 430.9% (percentiles by linear interpolation). Worked-example concessions: 061_PMPA1 (Vial interurbana, 3 bidders, 0.265), 080_HSSG1 (Hospitalaria, 4 bidders, 0.421), 027_SNSS1 (Vial urbana, 3 bidders, 1.801).

Appendix E — Revision note

26 October 2026 — the three termination compensations, from an engine improvement. The worked examples of §9 and the two continuity-loss flip thresholds of §10.1 were re-run against the shipped engine after a correction to how the financial model capitalises interest during construction. Every recommendation is unchanged — ACCEPT for the mild road, COUNTER for the essential hospital, TERMINATE for the runaway road — as are the three ACCEPT costs, the credibility thresholds ($p^{*}\approx 5\text{–}6\%$), the robustness grid of §10.1, the whole-panel exhibit of §10.2, and every panel statistic in §4 and §11.

	\(C_{\mathrm{ACCEPT}}\)	\(C_{\mathrm{COUNTER}}\)	\(C_{\mathrm{TERMINATE}}\)	\(G\)	\(OV(p)\)	\(D^{*}\)
A — 061_PMPA1	774 (unchanged)	829 → 827	3,386 → 3,314	−1,831 → −1,775	0 (unchanged)	n/a
C — 080_HSSG1	5,789 (unchanged)	5,194 → 5,173	6,272 → 6,138	+2,205 → +2,282	662 → 685	45% → 47%
B — 027_SNSS1	23,254 (unchanged)	20,357 → 20,284	14,401 → 14,093	+10,731 → +10,999	3,219 → 3,300	63% → 66%

The cause is mid-year construction interest: interest during construction now accrues on the mid-year drawn balance and is capitalised pro rata, rather than on a full-year balance. Less interest is rolled into the debt at the end of works, so the asset base the termination formula compensates is smaller and the cash threat point $K$ falls in all three cases. Everything else in the table is that one movement propagating through identities the paper already states: $C_{\mathrm{TERMINATE}} = K + D$ falls with $K$ — and $D$, calibrated as a share of $K$, falls with it; $G = C_{\mathrm{ACCEPT}} - K$ rises, because $C_{\mathrm{ACCEPT}}$ is a difference of two availability-payment streams and is untouched; $OV = (1-\beta)pG$ rises with $G$; the threat-discounted settlement $C_{\mathrm{COUNTER}} = C_{\mathrm{ACCEPT}} - OV + \kappa_R$ therefore falls; and $D^{*} = G\bigl(1-(1-\beta)p\bigr) + \kappa_R$, read as a share of a smaller compensation, rises by two to three points. The direction is the economically sensible one: a smaller compensation makes the termination threat more credible, not less, which is why the gap widens on both credible-threat cases and the option value with it.

The revision is reproducible as the paper has always claimed — worked_examples.py and sensitivity.py against the shipped engine, writing out/worked_examples.json and out/sensitivity.json, both regenerated here. Figure 1 is redrawn from them: the hospital (C) and the runaway road (B) move slightly to the right — $C_{\mathrm{ACCEPT}}/K$ from 1.62 to 1.65 and from 1.86 to 1.90 — and every case stays in its region.

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The Renegotiation Triangle — Accept, Counter, or Terminate