Continuation-Performance Decomposition in Dynamic Games with Irreversible Failure

Once failure is irreversible, continuation payoffs cannot be meaningfully aggregated across strategies that differ in their survival properties. Standard scalar evaluation sidesteps this by arbitrarily completing payoffs beyond termination, but such completions are extrinsic to the game form. This paper introduces continuation-performance decomposition (CPD), proving that any evaluation satisfying natural regularity conditions, such as failure-completion invariance, survival locality, and local expected-utility coherence – must separate continuation from performance lexicographically. Continuation priority thus emerges as a consequence of well-posed evaluation, not as a behavioral assumption. We establish equivalence between CPD and the limit of games with diverging failure penalties, show that viability is a game-form invariant independent of payoffs, and apply the framework to bank runs: preemptive withdrawals reflect rational viability vetoes rather than coordination failure when continuation is distributively asymmetric. CPD resolves a representational problem, not a preference problem.

💡 Research Summary

The paper tackles a fundamental representation problem that arises in dynamic games where failure is irreversible—once a failure state is reached the game terminates forever and the continuation payoff domain disappears. Conventional intertemporal evaluation aggregates outcomes over time with a single scalar, typically discounted expected utility. This scalar aggregation implicitly assumes that outcomes leading to termination and outcomes that merely reduce future payoffs are commensurable at a fixed rate. The authors argue that this assumption breaks down when failure is absorbing because the payoff function is undefined beyond the failure time, and strategies that differ in their survival probabilities are defined on different payoff domains.

To resolve this, the authors introduce Continuation‑Performance Decomposition (CPD). CPD separates the evaluation of a strategy into two components: (i) a continuation profile C(σ) that records the probability of surviving each period, and (ii) a conditional performance eU_i(σ) that is the expected discounted utility conditional on survival (i.e., on the event T=∞). The continuation profile is compared lexicographically across strategies; only when two strategies have identical continuation profiles is the conditional performance used to break ties. Thus, continuation is given lexical priority over performance, but this priority emerges from the structure of the evaluation rule rather than from any behavioral assumption.

The core of the paper is a set of three intrinsic regularity conditions that any well‑posed evaluation rule must satisfy:

1. Failure‑completion invariance – the ranking of strategies must be unchanged by arbitrary ways of completing payoffs after failure. In other words, the evaluation must depend only on objects intrinsic to the game form (states, actions, transition kernel, and failure set) and not on extrinsic extensions of the payoff function.

2. Survival locality – conditional on survival, the evaluation may depend only on the induced continuation distribution and not on off‑path behavior or on payoffs assigned to histories that end in failure.

3. Local expected‑utility coherence – on any subset of strategies that share the same continuation profile, the induced ranking must coincide with the ranking generated by standard discounted expected utility evaluated conditional on survival.

The authors prove Theorem 1 (Canonical intrinsic evaluation): any evaluation rule satisfying the three conditions can be represented as a lexicographic pair (T(σ), U(σ)), where T depends solely on the law of the failure time (the “tail” component) and U depends only on the conditional distribution over surviving histories. Moreover, the minimal tail functional that satisfies the conditions is precisely the continuation profile C(σ). Hence, CPD is the unique minimal representation that respects the intrinsic nature of the game.

The paper then introduces the concept of tail‑performance separability (Definition 9‑10) and shows in Theorem 2 (Decoupling theorem) that any preference relation satisfying this separability must admit a lexicographic representation with a tail component and a performance component. CPD appears as the special case where the tail component is exactly the continuation profile.

A further contribution is the penalty‑limit equivalence. By imposing a large penalty M on the continuation loss L(σ) (the discounted probability of eventual failure) and letting M→∞, the authors recover CPD as the limit of standard discounted‑utility games. Lemma 1 shows that for sufficiently large M, any strategy with a strictly lower continuation loss dominates any strategy with a higher loss, regardless of conditional performance. Theorem 3 then states that any accumulation point of Nash equilibria of the penalized games is a Nash equilibrium of the CPD representation. This establishes a bridge between the CPD framework and the more familiar approach of assigning arbitrarily large penalties to failure.

The paper also discusses viability‑preserving strategies Π_viab, defined as strategies that keep the state out of the failure set with probability one. Proposition 1 shows that both the continuation profile and the set Π_viab depend only on the game form, not on payoffs or equilibrium selection. Consequently, if Π_viab is empty, any equilibrium is irrelevant to payoffs because the game inevitably fails; if Π_viab is non‑empty, the continuation profile becomes the decisive factor for equilibrium selection.

An illustrative application is given to bank runs. When continuation (the ability to stay solvent) is distributed asymmetrically across banks, pre‑emptive withdrawals are interpreted not as a coordination failure but as rational “viability vetoes”: each bank withdraws to preserve its own continuation, which is lexicographically prior to any payoff considerations. This reinterpretation challenges the standard narrative that bank runs are driven solely by informational cascades or coordination problems.

In summary, the paper makes three major theoretical advances:

1. It identifies a representation problem inherent to dynamic games with absorbing failure and shows that any intrinsic evaluation must decompose into a lexicographic continuation‑performance pair.
2. It proves that continuation priority is a logical consequence of natural regularity conditions, not an exogenous behavioral axiom.
3. It establishes equivalence between CPD and the limit of games with diverging failure penalties, thereby linking the new framework to existing literature on safety‑first, non‑Archimedean utilities, and constrained control.

The work opens several avenues for future research, including extensions to stochastic games with multiple failure absorbing sets, incorporation of learning dynamics where agents update beliefs about continuation probabilities, and policy design that manipulates the continuation structure (e.g., through insurance or liquidity provision) to mitigate systemic risk.