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 showing how backward induction in a sequential proposer-and-vote mechanism with elimination threat produces a highly inegalitarian equilibrium. N pirates ranked by seniority divide M coins: the senior proposes, all vote, and on a failed vote the proposer is thrown overboard. The load-bearing insight is that each voter's price is their continuation payoff, so the proposer buys a cheap minimum winning coalition — with five pirates and 100 coins, the senior keeps 98.

Scope of Application¶

The pirate game lives within a single home discipline — game theory and its toy political-science modelling — restaged across its uses, bounded to sequential propose-vote-or-eliminate mechanisms with backward-induction-rational players.

Game-theory pedagogy — the home: the canonical worked example for subgame-perfect equilibrium and backward induction.
Corporate-control modelling — stylised board votes, proxy contests, and coalition formation reading swing-voter power off continuation values (Riker's minimum-winning-coalition theory).
Recreational mathematics — the family of variants tuning majority threshold, tie-breaking, and preference ordering.

Clarity¶

The game makes legible the dissociation between apparent and actual bargaining power: the agent most exposed to rejection turns out best positioned to extract the surplus. It sharpens its lesson by demolishing intuitions — the senior proposer is not most vulnerable, the split is not egalitarian, the proposer need not buy a strict majority. The sharp question becomes not "who looks powerful?" but "what is each agent's outside option, and what is the minimum bribe to clear it?"

Manages Complexity¶

"How will N agents divide M coins under propose-vote-or-eliminate?" is a forbidding combinatorial question. The game compresses it to a one-line recursion: solve the two-pirate base case, then propagate a single scalar — continuation payoff — up the seniority ladder, and the equilibrium falls out. The analyst tracks one quantity per agent plus a short list of rule sensitivities (tie-breaking, integer discreteness of the prize, preference ordering), reducing the second-order robustness question to inspecting those knobs.

Abstract Reasoning¶

The game licenses predictive reasoning (solve the base case, propagate continuation values, read off the split); a diagnostic move pricing a swing voter at their outside option, not their face-value exposure; boundary-drawing over the short list of rule parameters the extreme result hangs on; and boundary-drawing over the substrate where the recursion runs — a finite ordered set, a divisible prize, elimination on failure, lexicographic survival-first preferences.

Knowledge Transfer¶

Within game theory the substantive content transfers as mechanism, carried by one propagated quantity: price each agent at their continuation payoff, buy the cheapest minimum winning coalition, and check the rule-sensitivity knobs — moving intact wherever a sequential propose-vote-or-eliminate mechanism obtains, though these are the same substrate restaged. Beyond it the pirate framing (gold, seniority, plank-walking) is costume, not fresh structure. Stripped, the game is a corollary of backward_induction plus sequential bargaining (continuation values price the swing voter) and minimum_winning_coalition / mechanism_design — where the cross-domain lesson properly belongs.

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)

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