Cancelable Valuations: Fair Division & Finance

• Cancelable valuations are preference functions that satisfy the canceling property, ensuring that adding a common good preserves the relative order of bundle values.
• They are pivotal in discrete fair division, where they enable nearly full EFX allocations and support graph-theoretic methods for efficient mechanism design.
• In financial derivative pricing, cancelable valuations underpin optimal stopping models for American cancellable options and related contracts with explicit stopping boundaries.

Cancelable valuations specify preference functions that satisfy a "cancelling" property: for any two bundles $S,T$ and any good $g$ not in $S\cup T$, if adding $g$ increases the value of $S$ strictly more than $T$, then $S$ was already strictly preferred to $T$. This concept is central in both discrete fair division and stochastic optimal stopping/derivative valuation, as it underpins tractable mechanism design classes and corresponds to explicit stopping boundaries in financial mathematics.

1. Formal Definitions and Properties

A valuation $v:2^G \to \mathbb{R}_{\ge0}$ defined on a universe $G$ is called cancelable if for all $S,T\subseteq G$ and any $g\notin S\cup T$,

$$v(S \cup \{g\}) > v(T \cup \{g\}) \implies v(S) > v(T).$$

This property ensures that the relative ranking of two bundles cannot reverse by adding a single good to both. Cancelable valuations are monotone and normalized ($v(\emptyset)=0$).

Examples:

• Additive: $v(S)=\sum_{g\in S}v(g)$
• Unit-demand: $v(S)=\max_{g\in S} v(g)$
• Budget-additive: $v(S)=\min\{\sum_{g\in S}v(g),B\}$
• Multiplicative: $v(S)=\prod_{g\in S}v(g)$

The order-independence lemma: If $X=\{x_1,\dots,x_k\}$ and $Y=\{y_1,\dots,y_k\}$ with $v(x_j) \ge v(y_j)$ for each $j$, then $v(X) \ge v(Y)$, following repeated application of cancelability (Tao et al., 2023).

A valuation is nice cancelable if it is cancelable and there exists a non-degenerate, cancelable $v'$ strictly respecting $v$; this permits tie-breaking perturbations without losing cancelability (Berger et al., 2021).

2. Cancelable Valuations in Discrete Fair Division

Cancelable valuations strictly generalize additivity and are fundamental in fair allocation of indivisible goods. They encompass all classical additive, unit-demand, budget-additive, and multiplicative types.

Within the envy-free up to any good (EFX) allocation paradigm, Berger, Cohen, Feldman, and Fiat demonstrate that, for any $n$ agents holding nice cancelable valuations, one can construct EFX allocations leaving at most $n-2$ goods unallocated (Berger et al., 2021). For $n=4$, at most one good remains unallocated and no agent envies the discarded good. For three agents or for instances where agents' valuations are restricted to two possible nice cancelable types, full EFX allocations exist.

Proof techniques combine graph-theoretic and lexicographic potential arguments, notably using champion graphs encoding agent envy and generalized champion edges to systematically identify Pareto-improvable cycles in allocations.

Limitations and open questions:

• Eliminating all unallocated goods in the four-agent case is unresolved.
• Extending beyond cancelable to broader submodular classes (OXS, matroid ranks) and integrating truthfulness remains challenging (Berger et al., 2021, Tao et al., 2023).

3. Mechanism Design and Incentive Compatibility

The incentive properties under cancelable valuations are subtle. In (Tao et al., 2023), Amanatidis et al. prove:

• For arbitrarily small $\epsilon > 0$, every $\epsilon$-EF1 mechanism for cancelable valuations admits infinite incentive ratio—a single agent's benefit from misreporting can be unbounded.
• For subadditive cancelable valuations, Round-Robin yields EF1 with incentive ratio at most 2.
• Every $(\varphi-1)$-EF1 mechanism, with $\varphi = (1+\sqrt{5})/2 \approx 1.618$, has incentive ratio at least $\varphi$.

The table below summarizes these incentive properties:

Valuation Class	Mechanism	Incentive Ratio Bound
Additive	Round-Robin (EF1)	$2$
Cancelable (general)	$\epsilon$-EF1	$\infty$
Subadditive cancelable	Round-Robin (EF1)	$2$
Subadditive cancelable	$(\varphi-1)$-EF1	At least $\varphi$

This demonstrates that while the cancelable property generalizes tractable valuation classes, incentive compatibility is only attainable under further structural assumptions (subadditivity) (Tao et al., 2023).

4. Cancellable Valuations in Financial Derivative Pricing

In financial mathematics, cancelable contracts correspond to early-termination rights, analyzed via optimal stopping in spectrally negative Lévy models (Palmowski et al., 2022, Palmowski et al., 2017, Leung et al., 2010, Giada et al., 2012):

(a) American Cancellable Options

For perpetual American cancellable puts, the holder may exercise at any time, but the contract cancels at the last passage of the underlying above a fixed level $h$. The value function is

$$\overline V(s) = \sup_{\tau\in\mathcal{T}} \mathbb{E}_s[ e^{-r\tau} (K - S_\tau)^+ \mathbb{1}_{\{\tau < \theta\}} ],$$

where $\theta = \sup \{ t \ge 0: S_t \ge h \}$ is the last passage time. The problem reduces to a one-dimensional optimal stopping with a payoff

$G(s) = (K-s)^+ (h/s)^\alpha \wedge 1,$

and the solution involves spectrally negative scale functions $W^{(r)}$, $Z^{(r)}$ with closed-form representations and smooth-fit for the stopping boundary (Palmowski et al., 2022).

(b) Cancellable Drawdown/Drawup Contracts

For contracts contingent on asset drawdown and drawup, the protection buyer may cancel early for a penalty $g(D_\tau)$, and the contract ends at a drawdown $\tau_a$ or drawup $\sigma_b$. The optimal stopping boundary for cancellation is determined by value-matching and smooth-pasting for $d^*$ in the drawdown variable, using scale-function machinery parallel to American option theory (Palmowski et al., 2017).

(c) Default Swap Embedded Options

In credit risk models, cancellable default swaps (with step-up, step-down, or cancellation rights) entail solving stopping problems for the value of switching contract terms via thresholds in the underlying Lévy asset process. The spread adjustment and exercise boundaries are fully characterized by scale functions and associated exit identities (Leung et al., 2010).

(d) Counterparty Risk with Bermudan Option Clauses

Unilateral or bilateral break clauses in swaps are modelled as embedded Bermudan options. Under deterministic intensities and independence, the fair value under a single exercise date $\hat t$ is given by

$$\hat{V}_B^{AB}(t_0) = V_B^0(t_0) - BCVA_B(t_0, \hat{t}) + BDVA_B(t_0, \hat{t}) + \mathbb{E} [ \mathbb{1}_{\tau > \hat{t}} D(t_0, \hat{t}) [ BDVA_B(\hat{t},T) - BCVA_B(\hat{t},T) ]^+ | \mathcal{F}_{t_0} ],$$

quantifying the benefit of break clauses in mitigating CVA charges and P&L volatility (Giada et al., 2012).

5. Methodological Frameworks

The analysis of cancelable valuations in both discrete and stochastic settings relies critically on:

• Optimal stopping theory: Closed-form stopping boundaries via smooth-fit/value-matching.
• Scale functions for spectrally negative Lévy processes: Laplace transforms enable explicit integral solution representations.
• Graph-theoretic allocation frameworks: Champion graphs, Pareto-improvable cycles, and lexicographic potential functions govern allocation updates (Berger et al., 2021).
• Resolution via nonlinear equations and resolvent kernels: Boundary conditions for optimal stopping thresholds.

Implementation requires numerical Laplace inversion or analytic expressions for $W^{(q)}$ in specific models (Brownian motion, Cramér–Lundberg, jump-diffusion), root-finding for boundaries, and adjustment of premium/spread to match starting contract value constraints (Palmowski et al., 2017, Palmowski et al., 2022, Leung et al., 2010).

6. Significance and Open Problems

Cancelable valuations unify tractable classes for fair division and optimal contract design. In allocation problems, they permit nearly full EFX allocations for large agent sets, pushing beyond additive and unit-demand. In financial contracts, they enable analytical solutions for contracts with termination rights.

Open directions include:

• Achieving fully allocated EFX for four or more agents.
• Extending mechanisms to richer valuation classes (XOS, OXS).
• Integrating truthfulness/incentive compatibility with cancelable valuations in allocation mechanisms.
• Computationally efficient valuation for contracts with complex cancellation/termination rights under realistic market frictions.

The concept thus forms a foundation for both mechanism design in economic theory and the pricing of sophisticated contingent claims in quantitative finance.