An intransitive or non-transitive game is a
zero-sum
Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two competing entities, where the result is an advantage for one side and an equivalent loss for the other. In other words, player on ...
game in which pairwise competitions between the strategies contain a cycle. If strategy A beats strategy B, B beats C, and C beats A, then the binary relation "to beat" is intransitive, since transitivity would require that A beat C. The terms "transitive game" or "intransitive game" are not used in game theory.
A prototypical example of an intransitive game is the game
rock, paper, scissors
Rock, Paper, Scissors (also known by #Names, several other names and word orders) is an Intransitive game, intransitive hand game, usually played between two people, in which each player simultaneously forms one of three shapes with an outstret ...
. In
probabilistic
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
games like
Penney's game, the violation of transitivity results in a more subtle way, and is often presented as a probability
paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...
.
Examples
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Rock, paper, scissors
Rock, Paper, Scissors (also known by #Names, several other names and word orders) is an Intransitive game, intransitive hand game, usually played between two people, in which each player simultaneously forms one of three shapes with an outstret ...
*
Penney's game
*
Intransitive dice
A set of dice is intransitive (or nontransitive) if it contains X>2 dice, ''X1'', ''X2'', and ''X3''... with the property that ''X1'' rolls higher than ''X2'' more than half the time, and ''X2'' rolls higher than ''X3'' etc... more than half the ...
* ''
Fire Emblem
is a Video games in Japan, Japanese fantasy tactical role-playing game franchise developed by Intelligent Systems and published by Nintendo. First produced and published for the Nintendo Entertainment System in 1990, the series currently con ...
'', the video game franchise that popularized intransitive cycles in unit weapons: swords and magic beats axes and bows, axes and bows beat lances and knives, and lances and knives beat swords and magic
See also
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Stochastic transitivity Stochastic transitivity models are stochastic versions of the transitivity property of binary relations studied in mathematics. Several models of stochastic transitivity exist and have been used to describe the probabilities involved in experiments ...
References
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Game theory game classes
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