Game theory studies strategic interaction between individuals in situations called games. Classes of these games have been given names. This is a list of the most commonly studied games

Games can have several features, a few of the most common are listed here.

**Number of players**: Each person who makes a choice in a game or who receives a payoff from the outcome of those choices is a player.**Strategies per player**: In a game each player chooses from a set of possible actions, known as pure strategies. If the number is the same for all players, it is listed here.**Number of pure strategy Nash equilibria**: A Nash equilibrium is a set of strategies which represents mutual best responses to the other strategies. In other words, if every player is playing their part of a Nash equilibrium, no player has an incentive to unilaterally change his or her strategy. Considering only situations where players play a single strategy without randomizing (a pure strategy) a game can have any number of Nash equilibria.**Sequential game**: A game is sequential if one player performs her/his actions after another player; otherwise, the game is a simultaneous move game.**Perfect information**: A game has perfect information if it is a sequential game and every player knows the strategies chosen by the players who preceded them.**Constant sum**: A game is constant sum if the sum of the payoffs to every player are the same for every single set of strategies. In these games one player gains if and only if another player loses. A constant sum game can be converted into a zero sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.**Move by nature**: A game includes a random move by nature.

Game | Players | Strategies per player | No. of pure strategy Nash equilibria | Sequential | Perfect information | Zero sum | Move by nature |
---|---|---|---|---|---|---|---|

Battle of the sexes | 2 | 2 | 2 | No | No | No | No |

Blotto games | 2 | variable | variable | No | No | Yes | No |

Cake cutting | N, usually 2 | infinite | variable^{ [1] } | Yes | Yes | Yes | No |

Centipede game | 2 | variable | 1 | Yes | Yes | No | No |

Chicken (aka hawk-dove) | 2 | 2 | 2 | No | No | No | No |

Gift-exchange game | N, usually 2 | variable | 1 | Yes | Yes | No | No |

Commune game | 3 | Yes | |||||

Coordination game | N | variable | >2 | No | No | No | No |

Cournot game | 2 | infinite^{ [2] } | 1 | No | No | No | No |

Deadlock | 2 | 2 | 1 | No | No | No | No |

Dictator game | 2 | infinite^{ [2] } | 1 | N/A^{ [3] } | N/A^{ [3] } | Yes | No |

Diner's dilemma | N | 2 | 1 | No | No | No | No |

Dollar auction | 2 | 2 | 0 | Yes | Yes | No | No |

El Farol bar | N | 2 | variable | No | No | No | No |

Game without a value | 2 | infinite | 0 | No | No | Yes | No |

Guess 2/3 of the average | N | infinite | 1 | No | No | Maybe^{ [4] } | No |

Kuhn poker | 2 | 27 & 64 | 0 | Yes | No | Yes | Yes |

Matching pennies | 2 | 2 | 0 | No | No | Yes | No |

Muddy Children Puzzle | N | 2 | 1 | Yes | No | No | Yes |

Nash bargaining game | 2 | infinite^{ [2] } | infinite^{ [2] } | No | No | No | No |

Optional prisoner's dilemma | 2 | 3 | 1 | No | No | No | No |

Peace war game | N | variable | >2 | Yes | No | No | No |

Pirate game | N | infinite^{ [2] } | infinite^{ [2] } | Yes | Yes | No | No |

Platonia dilemma | N | 2 | No | Yes | No | No | |

Princess and monster game | 2 | infinite | 0 | No | No | Yes | No |

Prisoner's dilemma | 2 | 2 | 1 | No | No | No | No |

Public goods | N | infinite | 1 | No | No | No | No |

Rock, paper, scissors | 2 | 3 | 0 | No | No | Yes | No |

Screening game | 2 | variable | variable | Yes | No | No | Yes |

Signaling game | N | variable | variable | Yes | No | No | Yes |

Stag hunt | 2 | 2 | 2 | No | No | No | No |

Traveler's dilemma | 2 | N >> 1 | 1 | No | No | No | No |

Truel | 3 | 1-3 | infinite | Yes | Yes | No | No |

Trust game | 2 | infinite | 1 | Yes | Yes | No | No |

Ultimatum game | 2 | infinite^{ [2] } | infinite^{ [2] } | Yes | Yes | No | No |

Vickrey auction | N | infinite | 1 | No | No | No | Yes^{ [5] } |

Volunteer's dilemma | N | 2 | 2 | No | No | No | No |

War of attrition | 2 | 2 | 0 | No | No | No | No |

- ↑ For the cake cutting problem, there is a simple solution if the object to be divided is homogenous; one person cuts, the other chooses who gets which piece (continued for each player). With a non-homogenous object, such as a half chocolate/half vanilla cake or a patch of land with a single source of water, the solutions are far more complex.
- 1 2 3 4 5 6 7 8 There may be finite strategies depending on how goods are divisible
- 1 2 Since the dictator game only involves one player actually choosing a strategy (the other does nothing), it cannot really be classified as sequential or perfect information.
- ↑ Potentially zero-sum, provided that the prize is split among all players who make an optimal guess. Otherwise non-zero sum.
- ↑ The real value of the auctioned item is random, as well as the perceived value.

**Game theory** is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

In game theory and economic theory, a **zero-sum game** is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a larger piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally.

In game theory, the **Nash equilibrium**, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

**Matching pennies** is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even keeps both pennies, so wins one from Odd. If the pennies do not match Odd keeps both pennies, so receives one from Even.

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, **battle of the sexes** (**BoS**) is a two-player coordination game. Some authors refer to the game as **Bach or Stravinsky** and designate the players simply as Player 1 and Player 2, rather than assigning sex.

**Backward induction** is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by first considering the last time a decision might be made and choosing what to do in any situation at that time. Using this information, one can then determine what to do at the second-to-last time of decision. This process continues backwards until one has determined the best action for every possible situation at every point in time. It was first used by Zermelo in 1913, to prove that chess has pure optimal strategies.

In game theory, **trembling hand perfect equilibrium** is a refinement of Nash equilibrium due to Reinhard Selten. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or **tremble,** may choose unintended strategies, albeit with negligible probability.

In game theory, **folk theorems** are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.

In game theory, a **repeated game** is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. *Single stage game* or *single shot game* are names for non-repeated games.

In game theory, a **correlated equilibrium** is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.

**Sequential equilibrium** is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. A sequential equilibrium specifies not only a strategy for each of the players but also a **belief** for each of the players. A belief gives, for each information set of the game belonging to the player, a probability distribution on the nodes in the information set. A profile of strategies and beliefs is called an **assessment** for the game. Informally speaking, an assessment is a perfect Bayesian equilibrium if its strategies are sensible given its beliefs **and** its beliefs are confirmed on the outcome path given by its strategies. The definition of sequential equilibrium further requires that there be arbitrarily small perturbations of beliefs and associated strategies with the same property.

In game theory, a **subgame perfect equilibrium** is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that if the players played any smaller game that consisted of only one part of the larger game, their behavior would represent a Nash equilibrium of that smaller game. Every finite extensive game with perfect recall has a subgame perfect equilibrium.

**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. A Nash equilibrium is considered **payoff dominant** if it is Pareto superior to all other Nash equilibria in the game. When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered **risk dominant** if it has the largest basin of attraction. This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.

In game theory, an **epsilon-equilibrium**, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.

In algorithmic game theory, a **succinct game** or a **succinctly representable game** is a game which may be represented in a size much smaller than its normal form representation. Without placing constraints on player utilities, describing a game of players, each facing strategies, requires listing utility values. Even trivial algorithms are capable of finding a Nash equilibrium in a time polynomial in the length of such a large input. A succinct game is of *polynomial type* if in a game represented by a string of length *n* the number of players, as well as the number of strategies of each player, is bounded by a polynomial in *n*.

A **Markov perfect equilibrium** is an equilibrium concept in game theory. It has been used in analyses of industrial organization, macroeconomics, and political economy. It is a refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be identified. The term appeared in publications starting about 1988 in the work of economists Jean Tirole and Eric Maskin.

**Jean-François Mertens** was a Belgian game theorist and mathematical economist.

**Mertens stability** is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens. This solution concept is now called Mertens stability, or just stability.

- Arthur, W. Brian “Inductive Reasoning and Bounded Rationality”,
*American Economic Review (Papers and Proceedings)*, 84,406-411, 1994. - Bolton, Katok, Zwick 1998, "Dictator game giving: Rules of fairness versus acts of kindness"
*International Journal of Game Theory*, Volume 27, Number 2 - Gibbons, Robert (1992) A Primer in Game Theory, Harvester Wheatsheaf
- Glance, Huberman. (1994) "The dynamics of social dilemmas."
*Scientific American.* - H. W. Kuhn, Simplified Two-Person Poker; in H. W. Kuhn and A. W. Tucker (editors), Contributions to the Theory of Games, volume 1, pages 97–103, Princeton University Press, 1950.
- Martin J. Osborne & Ariel Rubinstein: A Course in Game Theory (1994).
- McKelvey, R. and T. Palfrey (1992) "An experimental study of the centipede game,"
*Econometrica*60(4), 803-836. - Nash, John (1950) "The Bargaining Problem" Econometrica 18: 155-162.
- Ochs, J. and A.E. Roth (1989) "An Experimental Study of Sequential Bargaining" American Economic Review 79: 355-384.
- Rapoport, A. (1966) The game of chicken, American Behavioral Scientist 10: 10-14.
- Rasmussen, Eric: Games and Information, 2004
- Shor, Mikhael. "Battle of the sexes". GameTheory.net. Retrieved September 30, 2006.
- Shor, Mikhael. "Deadlock". GameTheory.net. Retrieved September 30, 2006.
- Shor, Mikhael. "Matching Pennies". GameTheory.net. Retrieved September 30, 2006.
- Shor, Mikhael. "Prisoner's Dilemma". GameTheory.net. Retrieved September 30, 2006.
- Shubik, Martin "The Dollar Auction Game: A Paradox in Noncooperative Behavior and Escalation," The
*Journal of Conflict Resolution*, 15, 1, 1971, 109-111. - Sinervo, B., and Lively, C. (1996). "The Rock-Paper-Scissors Game and the evolution of alternative male strategies". Nature Vol.380, pp. 240–243
- Skyrms, Brian. (2003) The stag hunt and Evolution of Social Structure Cambridge: Cambridge University Press.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.