Centipede Game¶

A sequential finite-horizon game in which each player can take the larger share or pass to grow the pot, where backward induction prescribes taking on the first move — isolating common knowledge of rationality as the load-bearing assumption cooperation depends on.

Core Idea¶

The Centipede Game, introduced by Robert Rosenthal in 1981, is a sequential two-player game with perfect information and a known finite horizon in which cooperation unravels by backward induction despite generating larger total payoffs the longer it is sustained. The structure is as follows: at each node one player chooses either to take — ending the game and claiming the larger share of the current pot, leaving the smaller share for the opponent — or to pass — handing the move to the opponent and doubling (or otherwise growing) the total pot, giving the opponent the same take-or-pass choice at a larger stake. The game ends either when a player takes or when the last node is reached. Because the pot grows with each pass, continued cooperation is collectively beneficial; but subgame-perfect equilibrium analysis via backward induction prescribes take on the very first move. The logic is inexorable: at the final node the mover should take; knowing this, the second-to-last mover should take pre-emptively at the penultimate node rather than pass and receive less; that reasoning propagates backward through every node until the first mover, anticipating defection at every subsequent node, takes immediately. The predicted outcome is immediate defection and the minimum possible payoff for both players. The empirical finding, first documented systematically by McKelvey and Palfrey in 1992, is that experimental subjects routinely pass for multiple rounds, often reaching the end or near-end of the game and splitting much larger total stakes than the backward-induction equilibrium would predict. This gap between equilibrium prescription and observed behavior is the game's diagnostic value: it isolates the common knowledge of rationality assumption as the load-bearing element, since the backward-induction chain requires not only that each player is rational but that each player knows the other is rational, knows the other knows, and so on without bound. When that common-knowledge chain is broken — by bounded rationality, by uncertainty about the opponent's reasoning depth, or by other-regarding preferences that attach value to the opponent's welfare — cooperation can be sustained for several rounds before unraveling. The game is the canonical laboratory illustration of finite-horizon backward-induction failure, and it closely parallels the chain-store paradox (Selten 1978), in which a monopolist facing a sequence of potential entrants should, by backward induction, never fight entry — yet empirically and strategically often does.

Structural Signature¶

Sig role-phrases:

the two players — sequential movers with strict turn order and perfect information
the take-or-pass move — at each node: take (end the game, claim the larger share of the current pot) or pass (hand the move to the opponent, growing the pot, granting them the same choice at a larger stake)
the growing pot — the joint payoff that increases with each pass, making continued cooperation collectively beneficial
the known finite horizon — a common-knowledge last move, both players knowing exactly how many nodes remain
the common-knowledge-of-rationality assumption — each player rational, each knowing the other is, each knowing the other knows, without bound — the isolated load-bearing premise
the backward-induction chain — solving from the terminal node: the last mover takes, so the penultimate takes pre-emptively, propagating back to the first mover
the unravelling equilibrium — the unique subgame-perfect prescription of take on the very first move, the minimum joint payoff despite cooperation being collectively better at every step
the empirical departure — the robust finding that subjects pass for multiple rounds, splitting much larger stakes than equilibrium predicts
the three-branch diagnosis — the only ways the remaining assumption can fail: bounded reasoning depth, belief-uncertainty about the opponent's depth, or other-regarding preferences, each implying a distinct testable model

What It Is Not¶

Not a prisoner's dilemma. There is no dominant-strategy defection here. The centipede is sequential, finite, perfect-information, and cooperation fails through the backward-induction chain — the last mover would take, so the penultimate takes pre-emptively, and the logic propagates to move one — not through single-shot dominance. The failure mode is unravelling from a known last move, not a temptation that pays regardless of the partner.
Not evidence that players who pass are irrational. Passing for several rounds is the robust empirical finding, and the game's value is to force a specific reading of it rather than a wave at "irrationality": players reason to bounded depth, or reason fully but doubt the opponent does, or value the opponent's payoff. The deviation indicts the unbounded common-knowledge-of-rationality chain, not the rationality of the people who cooperate.
Not a recommendation to "take immediately." Backward induction prescribes taking on move one as the unique subgame-perfect equilibrium, but that outcome is the minimum joint payoff while passing is collectively better at every step. The prescription is a diagnostic about what the rationality chain implies, not strategic advice to emulate; in practice, cooperating past the equilibrium is what splits the larger stakes.
Not driven by greed or temptation. Nothing in the structure makes defection dominant; the unravelling is produced purely by the known finite horizon plus mutual rationality being common knowledge. This is exactly why obscuring the horizon, adding uncertainty about the last move, or building reputation across rounds can sustain cooperation — the collapse is a property of the common-knowledge chain, not of a payoff temptation.
Not a cross-substrate "centipede dynamic." The mechanism requires reasoning agents propagating a common-knowledge chain; it has no counterpart in chemistry, ecology, or engineering, which contain no such agents. What recurs across the family — the chain-store paradox, finitely-repeated prisoner's dilemmas with a known endpoint — is the parent pattern of finite-horizon unravelling / backward-induction collapse, of which this game is the cleanest two-player illustration, not a structure that travels under its own name.

Scope of Application¶

The Centipede Game lives across game theory and strategic-interaction analysis wherever a situation is genuinely centipede-shaped — a sequential, perfect-information, common-knowledge finite horizon with a growing joint payoff and a known last move; its reach is one substrate (reasoning agents) at many social scales, and the broader unravelling family travels under the parent finite_horizon_unravelling / backward_induction, not under this named game.

Game theory proper — the canonical illustration of finite-horizon backward-induction failure, studied beside the chain-store paradox and finitely-repeated prisoner's dilemmas with a common-knowledge endpoint.
Behavioral economics — the standard testbed where equilibrium concepts diverge from experimental play (the McKelvey–Palfrey pass-for-rounds finding), a staple of teaching.
Bounded-rationality modeling — the proving ground for level-k reasoning, quantal-response equilibrium, and other refinements predicting the observed pass-then-take pattern.
Reputation and trust experiments — designs letting reputation accumulate across rounds, framing the centipede as a repeated trust game under backward-induction pressure.
Finite-horizon negotiation and treaty analysis — the unravelling of long-running cooperative arrangements once the end-date becomes common knowledge.

Clarity¶

The centipede game's clarifying force is that it isolates a single load-bearing assumption and makes its failure observable. Many cooperation breakdowns are overdetermined — information asymmetry, coordination problems, risk aversion, and temptation to defect all press at once, and one cannot say which is doing the work. The centipede strips those away: perfect information, a known horizon, no dominant-strategy temptation (passing is collectively better at every step), so the only thing driving the unique subgame-perfect prescription of "take on move one" is the backward-induction chain, and that chain in turn rests entirely on common knowledge of rationality — each player rational, each knowing the other is, each knowing the other knows, without end. Naming the game gives a theorist a clean instrument for that assumption: it converts an abstract worry about how much idealized common knowledge equilibrium reasoning smuggles in into a concrete, replicable measurement of how far real reasoning actually propagates.

That isolation sharpens what an observed deviation can mean. When subjects pass for several rounds — the robust McKelvey–Palfrey finding — the game forces the modeller to commit to a specific reading rather than waving at "irrationality": either players do not reason backward all the way (bounded depth), or they reason fine but do not believe their opponent reasons all the way (uncertainty about the other's depth), or they value the opponent's payoff (other-regarding preferences). Each branch carries distinct, testable content, and the centipede is the setting where they are pried apart — which is exactly why level-k models, quantal-response equilibrium, and reputation designs are tried here. The sharper question it licenses is not "are people rational?" but "how deep does the common-knowledge-of-rationality chain actually run before it breaks, and which break is operating?" — and by exhibiting the same unravelling skeleton as the chain-store paradox, it tells the field that the diagnosis travels to any finite-horizon cooperation with a known last move, not just to this toy.

Manages Complexity¶

Cooperation breakdowns, taken as they come, are overdetermined messes: information asymmetry, coordination failure, risk aversion, and the temptation to defect all press at once, and an analyst confronting a real unravelling rarely knows which force is responsible. The centipede game compresses that tangle by constructing a setting in which all but one of the forces is shut off by design — perfect information, a known finite horizon, and no dominant-strategy temptation, since passing is collectively better at every node — so that exactly one mechanism remains to drive the unique "take on move one" prescription: the backward-induction chain, which itself rests entirely on common knowledge of rationality. The sprawl of candidate causes collapses to a single load-bearing assumption, isolated and made observable, so the analyst no longer has to disentangle competing explanations of a given breakdown but can instead measure how far one chain propagates.

That isolation imposes a clean, small branch structure on what an observed deviation can mean. When subjects pass for several rounds against the equilibrium — the robust empirical finding — the explanation space is not open-ended; it is exactly three branches, because the only assumption left to fail can fail in only a few ways: players do not reason backward all the way (bounded depth), or they reason fine but do not believe the opponent does (uncertainty about the other's depth), or they value the opponent's payoff (other-regarding preferences). The modeller tracks which branch operates and reads off the rest — which is precisely why level-k models, quantal-response equilibrium, and reputation designs are all tested in this one setting, each a commitment to a specific branch. The compression also travels: because the same unravelling skeleton appears in the chain-store paradox and in any finite-horizon cooperation with a known last move, recognizing a situation as centipede-shaped imports the whole apparatus — the single load-bearing assumption, the three-way diagnosis, the depth-of-reasoning parameter — without re-deriving it. A high-dimensional, overdetermined question about why cooperation fails reduces to one assumption, one measurable quantity (how deep the common-knowledge-of-rationality chain runs before it breaks), and a three-way branch from which the qualitative behavior follows.

Abstract Reasoning¶

The centipede game licenses reasoning moves a game theorist runs on any finite-horizon cooperation problem, all flowing from the backward-induction chain and the common-knowledge-of-rationality assumption it rests on.

The constructive move is backward-induction prediction from the last node. Given a sequential game with a known finite horizon, the analyst reasons from the end: identify what the final mover does (take), then propagate that determination backward — if the last mover takes, the penultimate mover should take pre-emptively rather than pass into a smaller share; if the penultimate takes, so should the one before; and the chain runs unbroken to the first mover, who takes immediately. The predicted outcome is immediate defection at the minimum joint payoff, derived not by looking forward from the start but by solving from the terminal node and folding the solution back. This is the move that generates the unique subgame-perfect prescription, and its characteristic signature is that more cooperation is collectively better at every step yet the equilibrium still prescribes none — the analyst predicts the welfare-destroying outcome precisely because each link in the chain is individually sound.

The second move is isolation-by-design diagnosis: using the game's stripped structure to pin a deviation on one assumption. Because perfect information, a known horizon, and the absence of any dominant-strategy temptation are built in, the analyst reasons that nothing else can be driving the unique prescription except the backward-induction chain, which rests entirely on common knowledge of rationality. So when behavior departs from "take on move one," the deviation cannot be blamed on information asymmetry or coordination failure (those are designed out) and must implicate the common-knowledge chain itself. This converts a vague worry about idealized equilibrium reasoning into a concrete measurement: the game becomes an instrument for asking how far the common-knowledge-of-rationality chain actually propagates before it breaks, with the round at which taking typically occurs reading off the effective depth of mutual reasoning.

The third move is three-branch attribution of an observed deviation. When subjects pass for several rounds against the equilibrium, the analyst does not wave at "irrationality" but commits to exactly one of three readings, because the single remaining assumption can fail in only a few ways: either players do not reason backward all the way (bounded reasoning depth), or they reason fully but do not believe the opponent reasons all the way (uncertainty about the other's depth), or they attach value to the opponent's payoff (other-regarding preferences). Each branch carries distinct, testable content and implies a different model — bounded depth points to level-k reasoning, belief-uncertainty to quantal-response equilibrium, other-regarding preferences to social-utility designs — so the analyst's move is to design the manipulation that pries the branches apart (vary the opponent's known sophistication, vary the payoff structure's distributional stakes) and read which branch the data support. The reasoning is interventionist: each candidate fix is simultaneously a hypothesis about which break is operating, confirmed or refuted by whether it shifts the observed unraveling point.

The fourth move is skeleton-recognition transfer within the domain: identifying a real situation as centipede-shaped and importing the whole apparatus. The analyst reasons that any finite-horizon cooperation with a known last move and a growing joint payoff carries the same unraveling pressure — most directly the chain-store paradox, where a monopolist facing a known finite sequence of entrants should by backward induction never fight, yet strategically and empirically often does. So recognizing the shape lets the analyst predict the equilibrium (unravel to immediate defection), anticipate the empirical departure (cooperation sustained well past the equilibrium), and reach for the same three-way diagnosis and the same depth-of-reasoning parameter — without re-deriving any of it. The interventionist payoff travels too: to sustain cooperation in a finite-horizon setting, the analyst predicts one must break the common-knowledge chain on purpose — obscure the horizon, introduce uncertainty about the last move, or build reputation across rounds — because as long as the known last move and mutual rationality are common knowledge, the backward-induction collapse is the predicted fate.

Knowledge Transfer¶

Within game theory and strategic-interaction analysis the centipede game transfers as model — functioning here as transfer-as-mechanism — wherever a situation is genuinely centipede-shaped: a sequential, perfect-information, common-knowledge finite horizon with a growing joint payoff and a known last move. Where that precondition holds, the whole apparatus applies literally: the backward-induction prediction (unravel to immediate defection), the anticipated empirical departure (cooperation sustained well past the equilibrium), the three-way attribution of any deviation (bounded depth / belief-uncertainty about the opponent's depth / other-regarding preferences), the depth-of-reasoning parameter, and the interventionist corollary that sustaining cooperation requires breaking the common-knowledge chain on purpose (obscure the horizon, introduce uncertainty about the last move, build reputation). So it carries without translation to the chain-store paradox (a monopolist facing a known finite sequence of entrants), to finite-horizon negotiation and treaty exits once the end-date is known, to end-of-tournament cooperation, and to finitely-repeated prisoner's dilemmas with a common-knowledge endpoint — each a real instance of the same unravelling pressure, recognized rather than likened.

The honest qualification is that this apparent breadth is one substrate at many social scales, not transfer across structurally distinct domains. Every one of those cases — bilateral bargaining, business-cycle endings, employee retention near retirement, treaty termination — is still cooperation among reasoning agents under a finite horizon, so what looks like cross-domain reach is the model applying within a single substrate (strategically reasoning agents) at varying scales, which is why the transfer is mechanism-preserving rather than analogical there. Where the model genuinely stops is the boundary of that substrate: stripped of game-theoretic vocabulary (sequential moves, payoffs, equilibrium, backward induction), the centipede is the substrate-specific observation that "in a finite cooperative situation with a known last move, the rational-prescription chain unravels cooperation from the back" — and that does not port to chemistry, ecology, or engineering, because those have no reasoning agents propagating a common-knowledge chain. Any invocation of "a centipede dynamic" outside reasoning-agent cooperation would be borrowing the unravelling shape without the mechanism, and should be marked as analogy.

The genuinely portable content is one level up: the broader pattern finite-horizon unravelling / backward-induction collapse of cooperation, of which the centipede game is the cleanest two-player illustration and the chain-store paradox and finitely-repeated PD are siblings. That parent (the side-captured finite_horizon_unravelling, kin to backward_induction) is the level at which the lesson recurs across the family, and it is what should be carried — with the centipede cited as the canonical instrument that isolates common knowledge of rationality as the load-bearing assumption — rather than the name "centipede game," whose specific cargo (Rosenthal's construction, the McKelvey–Palfrey pass-for-rounds finding, the level-k/quantal-response/social-utility refinements tested on it) is a game-theory artifact best kept as the famous illustration. Mechanism within reasoning-agent cooperation at every scale; the genuine cross-family reach resident in the parent unravelling pattern, not in this named game. This is exactly the boundary Structural Core vs. Domain Accent draws.

Examples¶

Canonical¶

McKelvey and Palfrey's 1992 experiment gives the precise construction. Two players face a game with a large pile and a small pile — start them at $0.40 and $0.10 — and every time a player passes, both piles double; whoever takes grabs the current large pile and leaves the small pile for the other. Over four decision nodes the piles double up to $6.40 and $1.60 at the end. Backward induction solves from the last node: the final mover takes rather than passes; knowing that, the third mover takes; the reasoning folds back to Player 1, who is prescribed to take at the very first node for $0.40, leaving $0.10 — the minimum joint outcome, even though passing all the way would split $6.40 and $1.60. Empirically the opposite happened: almost no pair took at the first opportunity, and play typically continued several rounds before someone took, splitting far larger stakes than the equilibrium predicts.

Mapped back: The two experimental subjects are the two players; grab-the-large-pile-or-double-it is the take-or-pass move over a growing pot, with four nodes the known finite horizon. Solving from the last node forward is the backward-induction chain yielding the unravelling equilibrium (take at move one). Subjects passing for rounds is the empirical departure that indicts the common-knowledge-of-rationality assumption, not the players' rationality.

Applied / In Practice¶

Selten's chain-store paradox applies the same skeleton to industrial organization. A chain-store monopolist faces a known, finite sequence of potential entrants into its many towns; in each town it can fight entry (a price war costly to both) or acquiesce. Backward induction says: in the last town, fighting is pure loss, so acquiesce; knowing the incumbent will not fight the last entrant, the second-to-last enters unafraid; the chain unravels and the monopolist should never fight anywhere. Yet real incumbents do build predatory reputations and fight early entrants — behavior later rationalized (Kreps–Wilson) by injecting a small doubt about the incumbent's "type," precisely the common-knowledge break the centipede isolates.

Mapped back: The incumbent and successive entrants are reasoning agents facing a known finite horizon of towns with a growing stake in reputation. Backward induction delivers the unravelling equilibrium (never fight), and the observed fighting is the empirical departure. The reputation resolution is three-branch diagnosis in action — belief-uncertainty about the opponent's type breaks the common-knowledge-of-rationality assumption, and deliberately obscuring one's type is the interventionist move to sustain the aggressive stance the equilibrium forbids.

Structural Tensions¶

T1: Rational prescription versus self-defeating outcome (the equilibrium is the worst result). Backward induction delivers a unique subgame-perfect prescription — take on move one — in which each link of the chain is individually sound, yet the prescribed outcome is the minimum joint payoff while passing is collectively better at every step. So the concept simultaneously certifies "take immediately" as the uniquely rational play and shows that following it guarantees the worst result, which real players beat by cooperating well past it. This makes the normative status of the equilibrium genuinely ambiguous: is it rational to take, when taking is what secures the loss, and when the players who "irrationally" pass do better? The tension is not that the analysis is wrong but that its own conclusion turns rationality against welfare, leaving unclear whether the prescription is advice, a reductio of the assumptions that produce it, or merely a diagnostic. Diagnostic: Is "take on move one" being treated as what a player should actually do, or as the sign that the common-knowledge-of-rationality premises generating it are the thing to distrust?

T2: Isolation-by-design versus external validity (a clean instrument that is a contrived toy). The game's diagnostic power comes entirely from shutting off every force but the backward-induction chain — perfect information, a known horizon, no dominant-strategy temptation — so that a deviation can be pinned on the common-knowledge assumption and nothing else. But that surgical isolation is exactly what makes it a laboratory toy: real finite-horizon cooperation is overdetermined, tangled with information asymmetry, coordination, and payoff temptation, which the game deliberately removes. So the mechanism it isolates so cleanly rarely operates in isolation in the wild, and the precision of the instrument is bought against the representativeness of the situations it is meant to illuminate. The tension is that the cleaner the isolation of common knowledge of rationality, the more artificial the setting and the more caution the transfer to messy real cooperation requires. Diagnostic: Is the conclusion resting on the game's isolated common-knowledge chain, or being carried to a real situation where the forces the game designed out are also present and possibly dominant?

T3: A crisp three-branch diagnosis versus observationally-close branches. The game promises to convert a vague "irrationality" into a specific reading — bounded reasoning depth, belief-uncertainty about the opponent's depth, or other-regarding preferences — each with distinct testable content and its own model. But the three branches are notoriously hard to separate, because all three produce the same surface behavior: pass for several rounds, then take. Bounded depth ("I stop reasoning early") and belief-uncertainty ("I reason fully but doubt you do") are especially close, and even a purpose-built manipulation often leaves the pass-then-take data consistent with more than one. The tension is that the concept's payoff — forcing a specific attribution rather than a wave at irrationality — is undercut by the observational equivalence of the branches it forces a choice among, so the promised precision frequently dissolves back into underdetermination. Diagnostic: Does the manipulation actually discriminate which branch is operating, or is the observed unravelling point consistent with bounded depth, belief-uncertainty, and other-regarding preferences alike?

T4: Cooperation sustained by breaking common knowledge (the perverse dependence of cooperation on ignorance). The interventionist corollary is genuinely useful: to sustain finite-horizon cooperation, break the common-knowledge chain on purpose — obscure the horizon, inject doubt about the last move, build reputation. But its implication cuts against everything usually prized. It means desirable cooperation depends on epistemic degradation: uncertainty about the endgame, doubt about the opponent's rationality, opacity about one's own type. More transparency about the horizon, more mutual confidence in each other's rationality, more common knowledge — the goods we normally seek — are precisely what unravel the cooperation from the back. So achieving the collectively better outcome requires deliberately withholding or muddying knowledge that the participants would, in other contexts, want. The tension is that the concept ties cooperation's survival to the absence of the very common knowledge that rational-agent analysis idealizes. Diagnostic: Is the cooperation here surviving because the horizon and mutual rationality are genuinely uncertain, or is it one clarification away from the backward-induction collapse that fuller common knowledge would trigger?

T5: Autonomy versus reduction (a named game or the finite-horizon-unravelling pattern it illustrates). Within reasoning-agent cooperation the centipede transfers as model wherever a situation is genuinely centipede-shaped, and it carries across scales — chain-store paradox, finite-horizon negotiation, end-of-tournament play, finitely-repeated PD with a known endpoint — without translation, because each is the same substrate (strategically reasoning agents) at a different scale, not an analogy. But that breadth is one substrate, not cross-domain reach: strip the game-theoretic vocabulary and there is no "centipede dynamic" in chemistry or ecology, which have no agents propagating a common-knowledge chain. The genuinely portable content is one level up — the parent finite_horizon_unravelling / backward_induction pattern — of which the centipede is the cleanest two-player illustration, its specific cargo (Rosenthal's construction, the McKelvey–Palfrey finding, the level-k/QRE/social-utility refinements) being a game-theory artifact. The tension is between a famous, self-contained game and the recognition that its recurring lesson belongs to the unravelling parent. Diagnostic: Resolve toward finite_horizon_unravelling / backward_induction when carrying the collapse-of-cooperation lesson across the family; toward the centipede game when a concrete reasoning-agent interaction is centipede-shaped and common knowledge of rationality is the assumption to probe.

Structural–Framed Character¶

The Centipede Game sits on the structural side of the spectrum but stops short of the pole — best read as mixed-structural, on much the same footing as the Byzantine Generals Problem and the CAP theorem: a genuine formal result wearing vocabulary pinned to one substrate. Its structural credentials on the first criteria are strong. Evaluative_weight is essentially nil — the game is a formal model and backward induction a valid derivation, praising and blaming nothing; the "take on move one" prescription is a decision-theoretic consequence, not a moral verdict (and the entry's T1 leaves even its normative status deliberately open). Institutional_origin points structural: the game is a mathematical construction (Rosenthal 1981) and the unravelling equilibrium a proved consequence of the model's assumptions, not an artifact legislated by an agency. And within its proper range cross-substrate reuse is recognition, not import: the chain-store paradox, finite-horizon negotiation, end-of-tournament play, and finitely-repeated prisoner's dilemmas with a known endpoint are literal co-instances of the same model, the backward-induction chain and three-branch diagnosis carried intact — "recognized rather than likened," as the entry puts it.

Two things hold it off the structural pole and keep it domain-specific. First, a mild agent-bound tint on human_practice_bound: unlike isostasy's lithospheres, the centipede's mechanism requires reasoning agents propagating a common-knowledge-of-rationality chain, so it has no counterpart in chemistry, ecology, or engineering — its substrate is intentional agents, which sits nearer the human side than a message-passing network or a lithosphere, even though the backward-induction derivation itself is a mathematical fact. Second, and decisively, vocab_travels fails outside that substrate: the game-theoretic apparatus — sequential moves, payoffs, subgame-perfect equilibrium, backward induction, common knowledge — is pinned to reasoning-agent cooperation, so any "centipede dynamic" invoked in a domain without agents borrows the unravelling shape without the mechanism, which the entry marks as analogy.

The portable structural skeleton is finite-horizon unravelling — cooperation collapsing from a known last move through a chain of individually-sound rational steps, so a growing joint payoff goes unrealised. That skeleton is genuinely portable across the reasoning-agent family and recurs as real co-instances (the chain-store paradox, finitely-repeated PD), but it is precisely what the centipede instantiates from its parent (finite_horizon_unravelling, kin to backward_induction), not what makes the named game itself travel: the cross-family reach belongs to the parent unravelling pattern, while the game's specific cargo — Rosenthal's construction, the McKelvey–Palfrey pass-for-rounds finding, the level-k/quantal-response/social-utility refinements tested on it — stays home as the canonical illustration. Its character: a real, evaluatively neutral, provably grounded game-theoretic model recognised intact across reasoning-agent cooperation at every social scale, but stated in an equilibrium vocabulary so pinned to intentional-agent substrates that only the qualitative finite-horizon-unravelling skeleton it borrows from its parent reaches beyond the single named game.

Structural Core vs. Domain Accent¶

This section decides why the Centipede Game is a domain-specific abstraction and not a prime, and it carries the case for its domain-specificity — there is no separate section for that.

What is skeletal (could lift toward a cross-domain prime). Strip the specific game and a thin relational structure survives: cooperation with a known last move collapses from the back — because a chain of individually-sound steps, each anticipating defection at the next, propagates the decision to defect all the way to the first move, so a growing joint payoff goes unrealised. The pieces that travel are abstract — a finite horizon known to all parties, a joint payoff that rewards continued cooperation, and a terminal-node determination that folds backward through every step until the collapse reaches the present. That skeleton is genuinely substrate-portable across reasoning agents, which is exactly why it recurs as real co-instances in the chain-store paradox and finitely-repeated prisoner's dilemmas with a common-knowledge endpoint, and why it sits in the catalog as finite_horizon_unravelling, kin to backward_induction — the parent primes the entry instantiates. But it is the core it shares, not what makes the Centipede Game distinctive.

What is domain-bound. Everything that makes it the Centipede Game in particular is game-theory furniture and none of it survives extraction intact: Rosenthal's specific two-player take-or-pass construction with its doubling pot; the subgame-perfect-equilibrium apparatus and the formal backward-induction derivation of "take on move one"; the isolation of common knowledge of rationality as the single load-bearing premise; the McKelvey–Palfrey pass-for-rounds empirical finding; the three-branch diagnosis (bounded reasoning depth, belief-uncertainty about the opponent's depth, other-regarding preferences) and the models tested on it (level-k reasoning, quantal-response equilibrium, social-utility designs); and the interventionist corollary of breaking common knowledge to sustain cooperation. These are the worked construction, the derivation, and the empirical program that game theory actually studies. The decisive test: the mechanism requires reasoning agents propagating a common-knowledge chain, so strip the game-theoretic vocabulary and there is no "centipede dynamic" in chemistry, ecology, or engineering — those substrates contain no agents to run the chain, and any invocation there borrows the unravelling shape without the mechanism. Remove the intentional-agent substrate and the concept does not reach a new domain, it loses its very mechanism and decays into the parent unravelling pattern.

Why this does not clear the prime bar. A prime is a relational structure whose vocabulary travels and whose cross-domain transfer is recognition of the same mechanism, not analogy. The Centipede Game's transfer is bimodal, but with an important twist: its apparent breadth is one substrate at many social scales, not cross-domain reach. Within reasoning-agent cooperation it travels intact — the chain-store paradox, finite-horizon negotiation and treaty exits, end-of-tournament play, finitely-repeated PD with a known endpoint — because each is the same substrate (strategically reasoning agents) at a different scale, so the backward-induction prediction, the empirical-departure expectation, the three-branch diagnosis, and the depth-of-reasoning parameter all carry without translation; that is recognition, not analogy. Beyond the reasoning-agent substrate it does not travel at all: agentless systems have nothing for the common-knowledge chain to run on, and "a centipede dynamic" invoked there is analogy of the shape only. And when the bare structural lesson is needed across the family — finite-horizon cooperation with a known last move unravels from the back through individually-sound steps — it is already supplied in more general form by the parents the entry instantiates: finite_horizon_unravelling and backward_induction. The cross-family reach belongs to those parents; "Centipede Game," as named, carries Rosenthal's construction, the McKelvey–Palfrey finding, and its refinement literature as baggage that does not and should not travel — it stays home as the canonical instrument that isolates common knowledge of rationality.

Relationships to Other Abstractions¶

Current abstraction Centipede Game Domain-specific

Parents (2) — more general patterns this builds on

Centipede Game is part of Subgame Perfect Equilibrium Domain-specific

The canonical centipede construction contains its subgame-perfect solution, whose every-node credibility requirement yields take on the first move.
Centipede Game is part of Common Knowledge Prime

The centipede model contains common knowledge of rationality and the final move as the epistemic chain that propagates backward to immediate taking.

Not to Be Confused With¶

The finitely-repeated prisoner's dilemma (with a known endpoint). A sibling co-instance under the same parent — cooperation unravels from a known last round by backward induction — but the stage game differs decisively: in the PD, defection is a dominant strategy independent of what the partner does, so the unravelling rides on top of a per-round temptation. The centipede has no dominant-strategy defection; its collapse is produced purely by the known finite horizon plus common knowledge of rationality. Tell: is there a within-round payoff that makes defecting best regardless of the opponent's move (repeated PD), or does defection become attractive only because the horizon is known and the chain propagates back (centipede)?
The chain-store paradox (Selten). The other canonical illustration of the same finite-horizon unravelling — a monopolist facing a known finite sequence of entrants who, by backward induction, should never fight, yet often does. It is a sibling instance of the parent pattern, not a rival to the centipede: both isolate the common-knowledge-of-rationality chain and both are resolved empirically by injecting doubt about the opponent's type. Tell: is the interaction a symmetric two-player growing-pot take-or-pass sequence (centipede) or an incumbent-versus-successive-entrants reputation problem (chain store)? Same mechanism, different staging.
The trust game / investment game. A sequential game in which one player passes resources that get multiplied and the other decides how much to return. It shares the sequential, grow-the-pot flavor and the centipede is sometimes framed as a repeated trust game — but the classic trust game is a single exchange with a proposer-and-responder split and no known-finite-horizon backward-induction chain running to a first-move defection. Tell: does the game hinge on a chain of take-or-pass nodes unravelling from a common-knowledge last move (centipede), or on a one-shot invest-then-reciprocate decision about how much of a multiplied endowment to return (trust game)?
The dollar auction / war of attrition. Escalation games where players keep committing resources and the trap is sunk cost and pre-emption — each further bid is rational given what is already spent, driving both toward loss. The centipede's collapse is the opposite temporal logic: it unravels by backward induction from a known terminal node, not forward through accumulating commitments, and it involves no bidding-past-value dynamic. Tell: are players trapped by escalating sunk commitments with no fixed end (dollar auction, attrition), or by folding a terminal-node determination back to the present move under a known horizon (centipede)?
Backward induction / finite-horizon unravelling (the parent it instances). The substrate-neutral solution method and the general pattern — cooperation with a known last move collapsing from the back through a chain of individually-sound steps — that the centipede instantiates and isolates most cleanly. Backward induction is the technique; finite-horizon unravelling is the recurring outcome-shape; the centipede is the two-player laboratory instrument that pins the collapse on common knowledge of rationality. Tell: strip Rosenthal's take-or-pass construction and the McKelvey–Palfrey finding and what remains — a known-horizon cooperation unravelling from the terminal node — is the parent pattern, which is what carries the lesson across the family; the named game stays home. (Treated more fully in the sections above.)

Neighborhood in Abstraction Space¶

Centipede Game sits in a crowded region of the domain-specific corpus (1^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Strategic Interaction & Game Theory (23 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-07-12