# Nontransitive game

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A non-transitive game is a game for which the various strategies produce one or more "loops" of preferences. In a non-transitive game in which strategy A is preferred over strategy B, and strategy B is preferred over strategy C, strategy A is not necessarily preferred over strategy C.

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A prototypical example non-transitive game is the game rock, paper, scissors which is explicitly constructed as a non-transitive game. In probabilistic games like Penney's game, the violation of transitivity results in a more subtle way, and is often presented as a probability paradox.

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Dice are small, throwable objects with uniquely marked sides that can rest in multiple positions. They are used for generating random numbers and are commonly used in tabletop games. Such games include dice games, board games, role-playing games, and games of chance.

Rock paper scissors is a hand game usually played between two people, in which each player simultaneously forms one of three shapes with an outstretched hand. These shapes are "rock", "paper", and "scissors". "Scissors" is identical to the two-fingered V sign except that it is pointed horizontally instead of being held upright in the air. A simultaneous, zero-sum game, it has only two possible outcomes: a draw, or a win for one player and a loss for the other.

A Condorcet method is one of several election methods that elects the candidate that wins a majority of the vote in every pairing of head-to-head elections against each of the other candidates, that is, a candidate preferred by more voters than any others, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner.

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".

In mathematics, a homogeneous relation R over a set X is transitive if for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Transitivity is a key property of both partial orders and equivalence relations.

The game of chicken, also known as the hawk–dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the outcome is ideal for one player to yield, but the individuals try to avoid it out of pride for not wanting to look like a 'chicken'. So each player taunts the other to increase the risk of shame in yielding. However, when one player yields, the conflict is avoided, and the game is for the most part over.

In mathematics, intransitivity is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.

Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.

In mathematics, an asymmetric relation is a binary relation on a set X where

In game theory, a player's strategy is any of the options which he or she chooses in a setting where the outcome depends not only on their own actions but on the actions of others. A player's strategy will determine the action which the player will take at any stage of the game.

Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.

In game theory, strategic dominance occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.

Pairwise comparison generally is any process of comparing entities in pairs to judge which of each entity is preferred, or has a greater amount of some quantitative property, or whether or not the two entities are identical. The method of pairwise comparison is used in the scientific study of preferences, attitudes, voting systems, social choice, public choice, requirements engineering and multiagent AI systems. In psychology literature, it is often referred to as paired comparison.

A set of dice is nontransitive if it contains three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. In other words, a set of dice is nontransitive if the binary relation – X rolls a higher number than Y more than half the time – on its elements is not transitive.

In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras.

A game is a structured form of play, usually undertaken for enjoyment and sometimes used as an educational tool. Games are distinct from work, which is usually carried out for remuneration, and from art, which is more often an expression of aesthetic or ideological elements. However, the distinction is not clear-cut, and many games are also considered to be work or art.

Horsengoggle is a method of selecting a random person from a group. Unlike some other methods, such as rock paper scissors, one of the features of horsengoggle is that there is always a winner; it is impossible to tie.

Barca is a two-player strategy board game invented by Andrew Caldwell. It is played on a 10x10 checkered board with three types of animal playing-pieces that move like the queen, bishop and rook in chess. Two distinguishing features from a typical chess variant are the absence of capture, and the fundamental role of a rock-paper-scissors dominance relationship among the three types of pieces: elephant, lion and mouse.

## References

• Gardner, Martin (2001). The Colossal Book of Mathematics. New York: W.W. Norton. ISBN   0-393-02023-1 . Retrieved 15 March 2013.