Pirate game¶

A toy sequential-bargaining model showing how backward induction in a propose-vote-or-eliminate mechanism lets the most senior proposer capture nearly the whole prize, because each voter's price is their continuation payoff and downstream subgames impoverish enough players to buy a cheap minimum coalition.

Core Idea¶

The pirate game (Stewart, 1999) is a toy multi-stage bargaining model illustrating how backward induction in a sequential proposer-and-vote mechanism with elimination threat produces a highly inegalitarian equilibrium in which the most senior proposer captures nearly the entire prize. In the standard setup, N pirates ranked by seniority must divide M gold coins: the most senior proposes a distribution; all surviving pirates vote including the proposer; if at least half vote yes the distribution stands; otherwise the proposer is thrown overboard and the next-most-senior pirate proposes. Pirates hold lexicographic preferences — survival first, then maximising own gold, then (all else equal) preferring to throw others overboard. Backward induction from the two-pirate base case yields the counterintuitive result that the senior proposer can retain M minus a small number of minimal bribes: with five pirates and 100 coins under standard tie-breaking rules, the most senior pirate keeps 98 coins and pays 1 each to the third- and fifth-ranked pirates.

The load-bearing structural insight is that each voter's price in a sequential proposer mechanism is their continuation payoff — what they would receive if the current proposal fails and the game passes to the next proposer. Voters whose continuation payoff is zero under the next subgame will accept any non-negative bribe rather than cause the proposer's elimination; the proposer needs only a minimum winning coalition of such cheap voters, not a majority of all players. The apparent vulnerability of the most senior proposer — who is eliminated first on a failed vote — is illusory: the hierarchy of elimination-driven continuation values produces a structure in which the senior proposer faces the cheapest coalition to buy, because the downstream subgames progressively impoverish more and more players. The result is exquisitely sensitive to tie-breaking rules, the discreteness of the prize, and the exact preference ordering, which makes the game a clean demonstration that equilibrium in sequential mechanisms can depend critically on fine details of the rules rather than just the broad payoff structure.

Structural Signature¶

Sig role-phrases:

the ordered proposers — a finite set of agents ranked by seniority, who propose in turn
the divisible prize — a finite, integer-discrete pot of value to be allocated
the propose-vote-or-eliminate mechanism — the senior agent proposes a distribution; on a failed vote the proposer is eliminated and the next proposes
the lexicographic preferences — survival first, then own share, then a tie-breaking secondary preference
the backward-induction solution — the equilibrium is built from the trivial base case and propagated up the seniority ladder
the continuation payoff — each voter's price: what they would collect if the current proposal failed and the game passed to the next proposer
the cheap-voter coalition — voters whose downstream continuation payoff is zero accept any minimal bribe, so the proposer buys a minimum winning coalition, not a true majority
the inegalitarian capture — the most-exposed senior proposer faces the cheapest coalition and retains nearly the entire prize
the rule-sensitivity knobs — tie-breaking convention, integer discreteness of the prize, and exact preference ordering, any of which shifts the equilibrium sharply

What It Is Not¶

Not a case where the most-exposed proposer is the weakest. The senior pirate, thrown overboard first on a failed vote, looks most vulnerable, but that exposure is illusory: because the downstream subgames progressively impoverish more players, he faces the cheapest coalition to buy and captures nearly the entire prize. Apparent exposure and actual bargaining power run in opposite directions here.
Not an egalitarian or majority-bought outcome. The division is extreme (98 of 100 to the proposer in the canonical case), and the proposer buys only a minimum winning coalition of cheap voters, not a strict majority. Each voter is priced at their continuation payoff, so those the next subgame drives to zero are secured with the minimal bribe — a true majority is never needed.
Not a robust result. The 98-for-the-proposer headline is exquisitely sensitive to fine rule details — the tie-breaking convention, the integer discreteness of the coins, and the exact lexicographic preference ordering. Change any one and the equilibrium shifts sharply. The game is, deliberately, a demonstration that sequential-mechanism equilibria can hinge on the rules rather than the broad payoff structure.
Not a fresh structural mechanism. The pirates, gold, and plank-walking are a memorable vehicle, not a new commitment. Stripped of costume it is a corollary of backward_induction plus the sequential-bargaining result that a swing voter's price is exactly their next-subgame payoff; the cross-domain substance lives in those parents, not in anything proprietary to the game.
Not a description of how real bargaining unfolds. It is a subgame-perfect equilibrium under perfectly rational players with lexicographic survival-first preferences solving by backward induction. It is not a behavioural prediction about actual negotiators, who depart from it; the construction is a clean toy for the recursion, not an empirical claim about boardrooms or crews.

Scope of Application¶

The pirate game lives within a single home discipline — game theory and its toy political-science modelling — restaged across its uses; its reach is bounded to sequential propose-vote-or-eliminate mechanisms with backward-induction-rational players, and the genuinely transferable substance (continuation values price the swing voter; small rule changes shift the outcome) belongs to its parent primes backward_induction, sequential bargaining, and mechanism_design, not to the pirate framing.

Game-theory pedagogy — the home turf: the canonical worked example for teaching subgame-perfect equilibrium, backward induction, and the counterintuitive distributive outcomes of sequential bargaining-with-survival.
Corporate-control modelling — stylised models of board votes, proxy contests, and coalition formation that read swing-voter power off continuation values and the minimum winning coalition (Riker's minimum-winning-coalition theory).
Recreational mathematics — the family of variants tuning the majority threshold, tie-breaking convention, and preference ordering, a staple of olympiad-adjacent puzzle literature.

Clarity¶

The pirate game makes legible the dissociation between apparent and actual bargaining power in sequential proposer mechanisms: the agent most exposed to rejection — the senior pirate, thrown overboard first on a failed vote — turns out to be the agent best positioned to extract the surplus, because the agents downstream have rational continuation values to bid against. The construction sharpens its lesson precisely by demolishing the intuitions students arrive with. "Surely the senior proposer is the most vulnerable" is wrong; "surely the division will come out roughly egalitarian" is wrong; "surely the proposer must buy a strict majority" is wrong. What the worked recursion makes unmissable is that the price of any voter is their continuation payoff — what they would collect if the current proposal failed and the game passed to the next proposer — and that voters whom the downstream subgames impoverish to zero can be bought for the minimal bribe, so the proposer needs only a cheap minimum winning coalition rather than a true majority.

The sharper question the concept hands the analyst is therefore not "who looks powerful here?" but "what is each agent's outside option if they refuse, and what is the minimum bribe needed to clear it?" That reframing is what carries to board votes, proxy contests, and coalition formation, where swing-voter power is set by continuation values rather than face-value exposure. The game equally makes legible a second, cautionary clarity for mechanism design: the extreme result is exquisitely sensitive to tie-breaking conventions, the integer discreteness of the prize, and the exact preference ordering, so it stands as a clean demonstration that equilibrium in sequential mechanisms can hinge on fine details of the rules rather than on the broad payoff structure — a warning about the rule-sensitivity of equilibrium predictions that the memorable framing drives home.

Manages Complexity¶

Asked directly, "how will N selfish rational agents divide M coins under a propose-vote-or-eliminate mechanism?" is a forbidding combinatorial question: the space of possible distributions, coalitions, and vote profiles explodes with the number of players. The pirate game compresses it to a one-line recursion. Solve the trivial two-pirate base case, then propagate a single quantity — each agent's continuation payoff, what they would collect if the current proposal failed and the game passed to the next proposer — up the seniority ladder, and the whole equilibrium falls out. The analyst stops reasoning about who looks powerful or exposed and instead tracks just that scalar per agent, reading off the outcome directly: any voter whose continuation payoff the downstream subgames drive to zero is bought for the minimal bribe, the proposer assembles the cheapest minimum winning coalition rather than a true majority, and the senior proposer captures nearly the entire prize. A high-dimensional bargaining problem thereby collapses to a chain of continuation values and a minimum-bribe rule, the same compression that carries to board votes, proxy contests, and coalition formation, where swing-voter power is read off continuation values rather than face-value exposure. The construction also exposes, in compressed form, exactly which knobs the outcome hangs on — the tie-breaking convention, the integer discreteness of the prize, and the precise preference ordering — so that the second-order question "how robust is this equilibrium?" reduces to inspecting that short list of rule parameters, a clean demonstration that equilibrium in sequential mechanisms can hinge on a few fine details of the rules rather than the broad payoff structure. What would otherwise be a per-instance combinatorial slog becomes the reading of one propagated quantity plus a small set of rule sensitivities.

Abstract Reasoning¶

The pirate game licenses a distinctive style of bargaining inference, all keyed to one propagated scalar — each agent's continuation payoff — and to the minimum-bribe-minimum-coalition rule that falls out of it.

Predictive (solve the base case, propagate continuation values, read off the split). The signature move is backward induction over a single quantity. The analyst reasons FROM the trivial two-pirate base case TO each agent's continuation payoff at every rung of the seniority ladder — what they would collect if the current proposal failed and the game passed to the next proposer — and from those values directly to the equilibrium distribution. Reasoning runs FROM "this voter's continuation payoff under the next subgame is zero" TO "they accept any non-negative bribe," FROM "the proposer needs only a minimum winning coalition of such cheap voters" TO "the senior proposer captures nearly the entire prize." A combinatorial N-player divide-the-coins problem collapses to a chain of continuation values plus a minimum-bribe rule.

Diagnostic (price a swing voter at their outside option, not their face-value exposure). The framework's load-bearing inference reframes the question of power: the analyst reasons FROM "who looks powerful or exposed here?" TO "what is each agent's outside option if they refuse, and what is the minimum bribe needed to clear it?" The apparent vulnerability of the most-exposed agent (eliminated first on a failed vote) is diagnosed as illusory, because the downstream subgames progressively impoverish more players, so the senior proposer faces the cheapest coalition to buy. This is the inference that carries to board votes, proxy contests, and coalition formation: swing-voter power is read off continuation values, not face-value exposure, and the minimum winning coalition, not a true majority, is the relevant unit.

Boundary-drawing (the result hangs on a short list of rule parameters). The framework predicts its own fragility: the extreme outcome is exquisitely sensitive to a small set of knobs, so the second-order question "how robust is this equilibrium?" reduces to inspecting them. The analyst reasons FROM "change the tie-breaking convention (do ties pass?), the integer discreteness of the prize (coins cannot be split, so a minimal bribe has a floor), or the exact lexicographic preference ordering (does the proposer prefer to throw a pirate at no gold gain?)" TO "the equilibrium shifts dramatically." This draws a sharp boundary: equilibrium in sequential mechanisms can hinge on fine details of the rules rather than on the broad payoff structure — a warning the analyst carries to real mechanism design as a caution about rule-sensitivity.

Boundary-drawing (the substrate where the recursion runs). The inferences require a finite ordered set of proposers, a divisible prize, a propose-vote-or-eliminate mechanism with proposer rotation on failure, lexicographic survival-first preferences, and backward-induction-rational players. The analyst reasons FROM the absence of any of these — irrational or boundedly-rational voters, no elimination penalty, an infinite horizon with discounting instead — TO the inapplicability of the clean recursion. The substantive residue that does travel is the continuation-value-prices-the-swing-voter insight, which belongs to sequential bargaining and backward induction generally; the pirate-specific framing (gold, seniority, plank-walking) is a memorable vehicle, not a fresh mechanism, so stripped of it the game is the corollary "finite-horizon sequential bargaining with elimination threat yields a minimum-coalition distribution at continuation-value prices."

Knowledge Transfer¶

Within game theory and its applied offshoots the pirate game's substantive content transfers as mechanism, carried by one propagated quantity. The diagnostic (price each agent at their continuation payoff, not their face-value exposure), the prediction (the proposer buys the cheapest minimum winning coalition rather than a true majority, so the most-exposed agent can capture nearly the whole prize), and the robustness check (inspect the tie-breaking convention, the integer discreteness of the prize, and the exact preference ordering) carry intact wherever a sequential propose-vote-or-eliminate mechanism with backward-induction-rational players obtains. So the reasoning moves without translation from the game-theory classroom (its canonical role in teaching subgame-perfect equilibrium and backward induction), to corporate-control modelling (stylised board votes, proxy contests, and coalition formation where swing-voter power is set by continuation values and the relevant unit is the minimum winning coalition, formalised by Riker's minimum-winning-coalition theory), to recreational mathematics (the family of variants tuning majority threshold, tie-breaking, and preference ordering). The trappings vary; the continuation-value recursion and the minimum-bribe rule read the same — though it should be said plainly that these are the same game-theoretic substrate restaged, not three genuinely distinct domains.

Beyond that substrate the honest characterisation is that the pirate game proper does not transfer; what transfers is the general game-theoretic content it is a vehicle for, which already lives in its parent primes. The pirate-specific framing — gold, seniority, plank-walking, the 98-for-the-proposer headline — is a memorable vehicle, not a fresh structural commitment, and it does not travel. Stripped of it, the game dissolves into a corollary of more general results: finite-horizon sequential bargaining with an elimination threat, solved by backward induction, yields a minimum-coalition distribution at continuation-value prices. Each clause of that sentence belongs to a parent that genuinely recurs across domains as mechanism: backward_induction (the solution technique for any extensive-form game with perfect information), a sequential-bargaining pattern (continuation values setting bargaining power, the substantive theorem that a swing voter's price is exactly their next-subgame payoff), and minimum_winning_coalition / mechanism_design (the coalition unit and the rule-sensitivity warning). When the cross-domain lesson is needed — "in any sequential mechanism with proposer rotation, ask each agent's outside option and the minimum bribe to clear it," and "small rule changes can produce large outcome shifts, so design the rules with care" — it should be carried by those parents, because the lesson is the parent content, not anything specific to pirates. The home-bound cargo is merely the worked example's costume; there is no proprietary mechanism underneath it that the parents do not already supply. So invoking "the pirate game" outside game-theoretic bargaining is at best a pedagogical allusion and at worst analogy that mistakes a memorable illustration for an independent mechanism: the genuinely transferable substance is backward_induction plus the continuation-value-prices-the-swing-voter result of sequential bargaining, and that is where the cross-domain reach properly belongs (see Structural Core vs. Domain Accent).

Examples¶

Canonical¶

The classic version, popularised by Ian Stewart's 1999 Scientific American puzzle "A Puzzle for Pirates," has five pirates ranked A > B > C > D > E dividing 100 gold coins: the senior survivor proposes, all survivors vote, a proposal passes on at least half the votes (the proposer breaking ties), else the proposer is thrown overboard and the next proposes. Backward induction from the two-pirate base case (D keeps all 100, E gets 0) gives: with three pirates, C buys E for one coin (C 99, D 0, E 1); with four, B buys D (B 99, C 0, D 1, E 0); with five, A needs two extra votes and buys the two who would get nothing in the four-pirate subgame — C and E — for one coin each, keeping 98 (A 98, B 0, C 1, D 0, E 1).

Mapped back: A > B > C > D > E is the ordered proposers and the 100 coins the divisible prize under the propose-vote-or-eliminate mechanism; the zeros propagated up each subgame (E, then D, then C and E) are the continuation payoffs that price each voter; buying C and E for a single coin each is the cheap-voter coalition (a minimum winning coalition, not a true majority); and A retaining 98 is the inegalitarian capture by the most-exposed proposer.

Applied / In Practice¶

Baron and Ferejohn (1989, American Political Science Review) formalised legislative bargaining with the same structure the pirate game caricatures: a legislator is recognised to propose a division of a fixed budget, the chamber votes by majority, and on rejection recognition passes to another proposer under discounting. Their equilibrium shows the recognised proposer capturing a disproportionate share and assembling a minimum winning coalition, buying just enough colleagues at their continuation values rather than distributing broadly — the "proposal power" that anchors modern models of pork-barrel spending, government formation, and budget bargaining.

Mapped back: The recognised legislator is the active agent in the propose-vote-or-eliminate mechanism (here rejection rotates recognition rather than drowning the proposer); each legislator's price is their continuation payoff under future recognition; the minimum winning coalition is the cheap-voter coalition; and the proposer's disproportionate slice is the inegalitarian capture recovered in a realistic institution.

Structural Tensions¶

T1: Apparent exposure versus actual power (face-value vulnerability inverts under continuation-value pricing). The senior proposer is thrown overboard first on a failed vote and looks the most vulnerable agent at the table, yet captures nearly the entire prize — because power in a sequential mechanism is not set by who is most exposed but by each voter's continuation payoff, and the downstream subgames progressively impoverish enough players to leave the senior proposer facing the cheapest coalition to buy. The tension is that the intuitive reading (exposed first, therefore weakest) is exactly inverted, and the whole pedagogical value of the game is demolishing that intuition. But the inversion is also a trap for transfer: carry the game's headline as "the most exposed agent wins" and you have learned the wrong lesson; the portable content is "price each agent at their outside option," which happens to make the exposed agent strong here but need not elsewhere. Diagnostic: Is bargaining power being read off face-value exposure, or off each agent's continuation payoff if the current proposal fails?

T2: A crisp determinate result versus exquisite rule-sensitivity (what makes it teachable is what makes it fragile). The 98-for-the-proposer headline is memorable precisely because it is a single, sharp, fully determinate number — the kind of clean answer that makes the game a canonical teaching device. But that determinacy is razor-thin: change the tie-breaking convention, the integer discreteness of the coins (a minimal bribe has a floor only because coins cannot be split), or the exact lexicographic preference ordering, and the equilibrium shifts dramatically. The very feature that makes the result quotable — a crisp corner solution — is what makes it a poor guide to robust prediction, and the game is deliberately constructed to demonstrate that sequential-mechanism equilibria can hinge on fine rule details rather than the broad payoff structure. Its clarity and its fragility are the same property. Diagnostic: Does the conclusion survive perturbing the tie-break rule, the prize's divisibility, and the preference ordering — or is it a corner solution that only the exact rules produce?

T3: Idealized rationality versus real bargaining (the recursion runs on assumptions negotiators violate). The clean backward-induction solution presumes perfectly rational players with lexicographic survival-first preferences who compute every downstream subgame's continuation values as common knowledge. Those idealizations are exactly what lets the combinatorial N-player problem collapse to a one-scalar recursion — and exactly what real boardrooms, crews, and legislatures depart from, where outside options are contested, rationality is bounded, and survival is rarely the lexical first priority. The tension is that the transferable insight (continuation values price the swing voter) is genuinely real, while the literal prediction (98-2) is not a behavioral claim at all. Treat it as description and it fails; treat it as a diagnostic lens and it illuminates. Diagnostic: Is the game being used as a prediction of how agents will actually divide the prize, or as a lens for asking what each agent's outside option and minimum bribe are?

T4: Broad compression versus narrow substrate (the assumptions that make it solvable are its scope limit). The game's power is that it compresses a forbidding combinatorial bargaining problem to a chain of continuation values plus a minimum-bribe rule. But that solvability depends on a tight substrate: a finite ordered set of proposers, a divisible prize, a propose-vote-or-eliminate mechanism with rotation on failure, and backward-induction-rational players. Remove the elimination penalty, admit an infinite horizon with discounting, or drop perfect information, and the clean recursion no longer runs. The tension is that the same assumptions delivering the elegant closed-form answer are precisely the ones that bound where the method applies — the model is exactly as general as its preconditions, no more, and its neat solvability is a symptom of that narrowness rather than of broad reach. Diagnostic: Does the situation actually have a finite ordered elimination mechanism with backward-inducible subgames, or is the clean recursion being applied where its structural preconditions fail?

T5: Autonomy versus reduction (a memorable vehicle, or a costume over parents that already carry the content). The pirate game is vivid, named, and canonical — gold, seniority, plank-walking, the 98-2 punchline — and within game theory its substantive content transfers as mechanism across classroom, corporate-control, and legislative-bargaining settings. But those are the same game-theoretic substrate restaged, and the framing is explicitly costume: strip it and the game is a corollary of backward_induction, the sequential-bargaining result that a swing voter's price is their next-subgame payoff, and minimum_winning_coalition / mechanism_design. Unusually for a domain-specific entry, there is no proprietary mechanism left under the costume — the parents supply all of it. So invoking "the pirate game" outside game-theoretic bargaining is at best a pedagogical allusion and at worst an analogy that mistakes a memorable illustration for an independent mechanism. The tension is between a beloved standalone example and the fact that its every transferable clause already belongs to a parent. Diagnostic: Resolve toward the parents (backward induction, continuation-value bargaining, minimum winning coalition) whenever the cross-domain lesson is wanted; toward the named game only as a teaching vehicle or an in-situ puzzle.

Structural–Framed Character¶

The pirate game sits on the framed-leaning side of the structural–framed spectrum, and for an unusual reason: it is not a phenomenon in the world but a pedagogical vehicle — a memorable costume over a game-theoretic theorem — and, as the entry says plainly, there is no proprietary mechanism underneath the costume for it to own. Its framedness is not normative charge but constructed-illustration-ness. On human_practice_bound it is framed: "the pirate game" is constituted by the practice of game-theory teaching and puzzle-making; the pirates, the gold, and the plank-walking are a device someone authored to make a recursion vivid, and they dissolve the instant that pedagogical practice is set aside (what remains is the bare theorem). Institutional_origin is framed: it is a named artifact of a specific author and venue (Stewart's 1999 Scientific American puzzle) and of the recreational-math / game-theory-pedagogy tradition. On vocab_travels it fails outright: seniority, gold, walking the plank, and the 98-2 headline are pure costume that does not survive extraction. Import_vs_recognize is bimodal: within game theory the substantive content is recognized across the classroom, corporate-control modelling, and legislative bargaining (one substrate restaged), while beyond it "the pirate game" is at best a pedagogical allusion and at worst an analogy mistaking an illustration for a mechanism. The one criterion pulling structural is evaluative_weight, which is nil — the model convicts nothing, deriving an equilibrium rather than rendering a verdict — which, together with the fully mathematical nature of its content, keeps it from being framed in the normative sense even as it sits on the framed side as a construct.

Uniquely among these entries, the pirate game has no distinctive portable skeleton of its own: stripped of costume it dissolves entirely into its parents — finite-horizon sequential bargaining with an elimination threat, solved by backward induction, yields a minimum-coalition distribution at continuation-value prices — each clause of which belongs to backward_induction (the solution technique), the sequential-bargaining pattern (continuation values price the swing voter), and minimum_winning_coalition/mechanism_design (the coalition unit and the rule-sensitivity warning). That composite is what the pirate game instantiates — indeed merely illustrates — from those parents, with nothing left over: the cross-domain reach ("ask each agent's outside option and the minimum bribe to clear it"; "small rule changes shift the outcome") belongs wholly to the parents, and the game contributes only the worked example's costume. Its character: an evaluatively neutral, mathematically exact game-theoretic result wearing a memorable pedagogical costume, structural entirely in the backward-induction-plus-continuation-value-bargaining content it borrows from its parents — a vehicle for that content rather than a mechanism of its own, which is exactly why it stays domain-specific and its every transferable clause resolves upward to a prime.

Structural Core vs. Domain Accent¶

This section decides why the pirate game is a domain-specific abstraction and not a prime, and it carries the case for its domain-specificity in one place. It is the batch's limiting case: an entry whose portable content is entirely its parents, with the domain accent being pure costume rather than a proprietary mechanism.

What is skeletal (could lift toward cross-domain primes). Strip the costume and a thin relational structure survives, composed wholly of parents: finite-horizon sequential bargaining with an elimination threat, solved by backward induction, yields a minimum-coalition distribution at continuation-value prices. The portable pieces are abstract — the backward-induction solution technique (backward_induction), the sequential-bargaining result that a swing voter's price is exactly their next-subgame continuation payoff, and the minimum-winning-coalition unit plus the rule-sensitivity warning (minimum_winning_coalition/mechanism_design). That composite is genuinely substrate-portable — it recurs in any sequential propose-vote-or-eliminate mechanism with backward-induction-rational players — which is exactly why the entry instantiates those parents. Unusually, this composite is all there is: there is no residue the parents do not already supply.

What is domain-bound. What makes the concept the pirate game in particular is not a mechanism at all but a memorable vehicle: the ordered pirates ranked by seniority, the gold coins as the divisible prize, the walk-the-plank elimination, the lexicographic survival-first preferences, and the 98-for-the-proposer headline (with its exact tie-break, integer-coin, and preference-ordering knobs). This costume is what makes the recursion vivid and teachable, and it is precisely what does not travel. The decisive test: strip the pirates, the gold, and the plank-walking — keeping only "sequential bargaining with elimination, solved by backward induction, buying a minimum coalition at continuation values" — and nothing distinctive is lost, because there was no proprietary mechanism under the costume; what remains is the bare theorem the parents carry. This is the opposite of the usual domain-specific case, where stripping the accent leaves a portable skeleton the parents generalize: here stripping the accent leaves the parents themselves, with the entry contributing only the illustration.

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 pirate game's transfer is bimodal, and the beyond-mode is unusually thin. Within game theory the substantive content transfers as mechanism — the game-theory classroom, corporate-control modelling (board votes, proxy contests, coalition formation à la Riker), and recreational-mathematics variants are the same game-theoretic substrate restaged, so the continuation-value recursion, the minimum-bribe rule, and the rule-sensitivity check re-apply without translation. Beyond game-theoretic bargaining "the pirate game" is at best a pedagogical allusion and at worst an analogy that mistakes a memorable illustration for an independent mechanism, because the game contributes only costume. And when the cross-domain lesson is needed — ask each agent's outside option and the minimum bribe to clear it; expect small rule changes to shift the outcome — it is already carried, in fully general form, by backward_induction, the continuation-value-prices-the-swing-voter result of sequential bargaining, and minimum_winning_coalition/mechanism_design. The cross-domain reach belongs entirely to those parents; "the pirate game," as named, carries only the worked example's costume, which does not and should not travel.

Relationships to Other Abstractions¶

Current abstraction Pirate game Domain-specific

Parents (1) — more general patterns this builds on

Pirate game is part of Subgame Perfect Equilibrium Domain-specific

The pirate game contains a subgame-perfect solution in which each proposed coalition is priced from the next continuation game's equilibrium payoff.

Hierarchy paths (3) — routes to 2 parentless roots

Pirate game → Subgame Perfect Equilibrium → Nash Equilibrium → Equilibrium → Fixed Point

Show alternative paths (2)

Not to Be Confused With¶

Ultimatum game. A single-round bargaining game — one proposer offers a split, one responder accepts (both get the shares) or rejects (both get nothing) — with no seniority ladder, no proposer rotation, and no elimination. The ultimatum game is famous as a behavioural result (responders reject unfair offers, defying the rational prediction); the pirate game is a multi-stage subgame-perfect construction solved by backward induction. Tell: is there one take-it-or-leave-it offer whose interest is that people deviate from equilibrium (ultimatum), or a rotating propose-vote-or-eliminate chain solved by continuation values (pirate game)?
Dictator game. A degenerate allocation game in which the proposer simply dictates the split with no vote and no way for anyone to reject. The pirate game's entire mechanism is the voting-and-elimination threat that prices each voter at their continuation payoff; remove the vote and the strategic recursion vanishes. Tell: can the other players vote and unseat the proposer (pirate game) or is the allocation imposed unilaterally with no response (dictator)?
Rubinstein alternating-offers bargaining. The canonical infinite-horizon bargaining model where two players trade offers across rounds and discounting (impatience) — not elimination — drives the split. The pirate game is finite-horizon with a base case and an elimination penalty rather than a discount factor. Tell: is the pressure toward agreement time-discounting over an unbounded horizon (Rubinstein), or a finite ladder with proposer-elimination on rejection (pirate game)?
Baron–Ferejohn legislative bargaining. The realistic co-instance, not a rival: a legislator is recognized to propose a budget division, the chamber votes by majority, and on rejection recognition passes under discounting. It shares the pirate game's structure — disproportionate proposer power, a minimum winning coalition bought at continuation values — but rotates recognition rather than drowning proposers and is a working political-economy model rather than a puzzle. Tell: same continuation-value logic, but is the setting a formal legislative model with recognition-and-discounting (Baron–Ferejohn) or the toy elimination puzzle (pirate game)?
Shapley value / cooperative fair division. An axiomatic solution assigning each player a share by their average marginal contribution across coalitions — a normative fairness concept from cooperative game theory. The pirate game is non-cooperative and yields an extreme, inegalitarian capture, the opposite of an equity criterion. Tell: is the allocation derived from fairness axioms over marginal contributions (Shapley), or from strategic backward induction that lets the proposer capture nearly everything (pirate game)?
The backward_induction + minimum_winning_coalition + mechanism_design parents. The substrate-neutral content the pirate game merely illustrates — the solution technique, the continuation-value-prices-the-swing-voter result, the coalition unit, and the rule-sensitivity warning. Uniquely, stripping the costume leaves nothing proprietary behind; the parents carry all of it. Tell: the parents carry the cross-domain lesson ("ask each agent's outside option and the minimum bribe to clear it"); "the pirate game" is only the memorable teaching vehicle, valid as a puzzle or classroom device, not an independent mechanism.

Neighborhood in Abstraction Space¶

Pirate game sits in a crowded region of the domain-specific corpus (11^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Choice Paradoxes & Collective Decision-Making (14 abstractions)

Nearest neighbors

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