NationMaster - Encyclopedia: Pirate game

The pirate game is a simple mathematical game. It illustrates how, if assumptions conforming to a homo economicus model of human behaviour hold, outcomes may be surprising. It is a multi-player version of the ultimatum game. Image File history File linksMetadata Download high-resolution version (1840x2888, 1355 KB) Summary Howard Pyle illustration of pirates burying treasure, from Howard Pyles Book of Pirates. ... Image File history File linksMetadata Download high-resolution version (1840x2888, 1355 KB) Summary Howard Pyle illustration of pirates burying treasure, from Howard Pyles Book of Pirates. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... For other uses, see Game theory (disambiguation) and Game (disambiguation). ... Homo economicus, or Economic man, is the concept in some economic theories of man (that is, a human) as a rational and self-interested actor who desires wealth, avoids unnecessary labor, and has the ability to make judgments towards those ends. ... The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. ...

The Game

There are five rational pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them. Look up pirate and piracy in Wiktionary, the free dictionary. ...

The Pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The Pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates should then vote on whether to accept this distribution; the proposer is able to vote, and has the casting vote in the event of a tie. If the proposed allocation is approved by vote, it happens. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again. A casting vote is a vote given to the presiding officer of a council or legislative body in order to resolve a deadlock. ...

Pirates base their decisions on three factors. First of all, each pirate wants to survive. Secondly, each pirate wants to maximize the amount of gold coins they receive. Thirdly, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.^[1]

The Result

It might be expected intuitively that Pirate A will have to allocate little if any to himself for fear of being voted off so that there are fewer pirates to share between. However, this is as far from the theoretical result as is possible.

This is apparent if we work backwards: if all except D and E have been thrown overboard, D proposes 100 for himself and 0 for E. He has the casting vote, and so this is the allocation.

If there are three left (C, D and E) C knows that D will offer E 0 in the next round; therefore, C has to offer E 1 coin in this round to make E vote with him, and get his allocation through. Therefore, when only three are left the allocation is C:99, D:0, E:1.

If B, C, D and E remain, B knows this when he makes his decision. To avoid being thrown overboard, he can simply offer 1 to D. Because he has the casting vote, the support only by D is sufficient. Thus he proposes B:99, C:0, D:1, E:0. One might consider proposing B:99, C:0, D:0, E:1, as E knows he won't get more, if any, if he throws B overboard. But, as each pirate is eager to throw each other overboard, E would prefer to kill B, to get the same amount of gold from C.

Assuming A knows all these things, he can count on C and E's support for the following allocation, which is the final solution:

Also, A:98, B:0, C:0, D:1, E:1 or other variants are not good enough, as D would rather throw A overboard to get the same amount of gold from B.

Extension

The game can easily be extended to up to 200 pirates (or further if you increase the amount of gold). Ian Stewart extended it to an arbitrary number of pirates in the May 1999 edition of Scientific American, with further interesting results.^[1] Ian Stewart, FRS (b. ... Scientific American is a popular-science magazine, published (first weekly and later monthly) since August 28, 1845, making it the oldest continuously published magazine in the United States. ...

References

See also

Creative Problem Solving begins when knowledge and simply thinking about a problem fails. ... Lateral thinking is a term coined by Edward de Bono, a Maltese psychologist, physician and writer. ... For other uses, see Game theory (disambiguation) and Game (disambiguation). ... In game theory, normal form is a way of describing a game. ... It has been suggested that Game tree be merged into this article or section. ... A cooperative game is a game where groups of players (coalitions) may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players. ... In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed so far. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ... Price of market balance In economics, economic equilibrium is simply a state of the world where economic forces are balanced and in the abscence of external shocks the (equilibrium) values of economic variables will not change. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ... In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ... The trembling hand perfection is a notion that eliminates actions of players that are unsafe because they were chosen through a slip of the hand. ... Proper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson. ... In game theory, an Epsilon-equilibrium is a strategy profile that approximately satisfies the condition of Nash Equilibrium. ... In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. ... Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. ... Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme. ... In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ... Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. ... Pareto efficiency, or Pareto optimality, is an important notion in neoclassical economics with broad applications in game theory, engineering and the social sciences. ... In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ... In game theory, dominance occurs when one strategy is better or worse than another regardless of the strategies of a players opponents. ... A pure strategy is a term used to refer to strategies in Game theory. ... In game theory a mixed strategy is a strategy which chooses randomly between possible moves. ... Tit for Tat is a highly-effective strategy in game theory for the iterated prisoners dilemma. ... Grim Trigger is a trigger strategy in game theory for a repeated game, such as an iterated prisoners dilemma. ... Look up collusion in Wiktionary, the free dictionary. ... In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. ... Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ... In game theory, a sequential game is a game where one player chooses his action before the others chooses theirs. ... In game theory, a repeated game (or iterated game) is an extensive form game which consists in some number of repetitions of some base game (called a stage game). ... Signaling games are dynamic games with two players, the sender (S) and the receiver (R). ... Cheap Talk is a term used in Game Theory for pre-play communication which carries no cost. ... Zero-sum describes a situation in which a participants gain (or loss) is exactly balanced by the losses (or gains) of the other participant(s). ... Mechanism design is a sub-field of game theory. ... In game theory, a stochastic game is a competitive game with probabilistic transitions played by two players. ... A non-transitive game is a game for which the various strategies produce one or more loops of preferences. ... Game theory studies strategic interaction between individuals in situations called games. ... This article contains mathematical terminology from game theory, which should not be confused with the common usage. ... In game theory, the travelers dilemma (sometimes abbreviated TD) is a type of non-zero-sum game in which two players attempt to maximise their own payoff, without any concern for the other players payoff. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... For other uses, see Chicken (disambiguation). ... The Volunteers dilemma game models a situation in which each of N players faces the decision of either making a small sacrifice from which all will benefit or freeriding. ... On eBay, where an auction has a starting price of $1 ... The Battle of the Sexes is a two player game used in game theory. ... In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ... Matching Pennies is the name for a simple example game used in game theory. ... The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. ... Minority Game is a game proposed by Yi-Cheng Zhang and Damien Challet from the University of Fribourg. ... Rock, Paper, Scissors chart Rock, Paper, Scissors (sometimes with the elements in its name permuted and/or Rock replaced with Stone and/or Paper with Cloth, but also known as Roshambo, Rochambeau, Ick-Ack-Ock, Janken, Mora, Morra Cinese, Gawi-Bawi-Bo, JanKenPon or Farkle) is a popular hand game... The dictator game is a very simple game in experimental economics, similar to the ultimatum game. ... The Public goods game is a standard of experimental economics; in the basic game subjects secretly choose how many of their private tokens to put into the public pot. ... Blotto games (or Colonel Blotto games) constitute a class of two-person zero-sum games in which the players are tasked to simultaneously distribute limited resources over several objects, with the gain (or payoff) being equal to the sum of the gains on the individual objects. ... In game theory the War of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while accumulating costs at a constant rate c. ... -1... In game theory, the purification theorem was contributed by Nobel laurate John Harsanyi in 1973[1]. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them... In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players minimax conditions are satisfied. ... The revelation principle of economics can be stated as, To any equilibrium of a game of incomplete information, there corresponds an associated revelation mechanism that has an equilibrium where the players truthfully report their types. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox demonstrates the impossibility of designing a set of rules for social decision making that would meet all of a certain set of criteria. ...

Contents

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The Result

Extension

References

See also

The Game