Editing Game theory (section)

==Different types of games==
{{see also|List of games in game theory}}

===Cooperative / non-cooperative===
{{main|Cooperative game theory|Non-cooperative game}}
A game is ''cooperative'' if the players are able to form binding commitments externally enforced (e.g. through [[contract law]]). A game is ''non-cooperative'' if players cannot form alliances or if all agreements need to be [[self-enforcing agreement|self-enforcing]] (e.g. through [[credible threat]]s).<ref>{{cite web |url=http://www.gametheory.net/dictionary/Non-CooperativeGame.html |title=Non-Cooperative Game |last=Shor |first=Mike |website=GameTheory.net |access-date=2016-09-15 |archive-date=1 April 2014 |archive-url=https://web.archive.org/web/20140401211601/http://www.gametheory.net/Dictionary/Non-CooperativeGame.html |url-status=live }}</ref>

Cooperative games are often analyzed through the framework of ''cooperative game theory'', which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs. It is different from ''non-cooperative game theory'' which focuses on predicting individual players' actions and payoffs by analyzing [[Nash equilibria]].<ref>{{cite web |url=http://www.utdallas.edu/~chandra/documents/6311/coopgames.pdf |archive-url=https://web.archive.org/web/20160418101712/http://www.utdallas.edu/~chandra/documents/6311/coopgames.pdf |archive-date=2016-04-18 |url-status=live |title=Cooperative Game Theory |last=Chandrasekaran |first=Ramaswamy |publisher=University of Texas at Dallas}}</ref><ref>{{cite web|last=Brandenburger|first=Adam|title=Cooperative Game Theory: Characteristic Functions, Allocations, Marginal Contribution|url=http://www.uib.cat/depart/deeweb/pdi/hdeelbm0/arxius_decisions_and_games/cooperative_game_theory-brandenburger.pdf|url-status=dead|archive-url=https://web.archive.org/web/20170829084624/http://www.uib.cat/depart/deeweb/pdi/hdeelbm0/arxius_decisions_and_games/cooperative_game_theory-brandenburger.pdf|archive-date=29 August 2017|access-date=14 April 2020}}</ref>

Cooperative game theory provides a high-level approach as it describes only the structure and payoffs of coalitions, whereas non-cooperative game theory also looks at how strategic interaction will affect the distribution of payoffs. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation.

===Symmetric / asymmetric===
{{Payoff matrix |Name=An asymmetric game |2L=E |2R=F |1U=E |UL=1, 2 |UR=0, 0 |1D=F |DL=0, 0 |DR=1, 2}}
{{main|Symmetric game}}
A symmetric game is a game where each player earns the same payoff when making the same choice. In other words, the identity of the player does not change the resulting game facing the other player.<ref>{{Cite web |last=Shor |first=Mike |date=2006 |title=Symmetric Game |url=https://www.gametheory.net/dictionary/Games/SymmetricGame.html |website=Game Theory.net}}</ref> Many of the commonly studied 2×2 games are symmetric. The standard representations of [[game of chicken|chicken]], the prisoner's dilemma, and the [[stag hunt]] are all symmetric games.

The most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the [[ultimatum game]] and similarly the [[dictator game]] have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured in this section's graphic is asymmetric despite having identical strategy sets for both players.

===Zero-sum / non-zero-sum===
{{Payoff matrix |Name=A zero-sum game |2L=A |2R=B |1U=A |UL=–1, 1 |UR=3, −3 |1D=B |DL=0, 0 |DR=–2, 2}}
{{main|Zero-sum game}}
Zero-sum games (more generally, constant-sum games) are games in which choices by players can neither increase nor decrease the available resources. In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, and always adds to zero (more informally, a player benefits only at the equal expense of others).<ref>{{cite book |title=Game Theory: Third Edition |last=Owen |first=Guillermo |author-link=Guillermo Owen |publisher=Emerald Group Publishing |year=1995 |isbn=978-0-12-531151-9 |location=Bingley |page=11}}</ref> [[Poker]] exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include [[matching pennies]] and most classical board games including [[Go (board game)|Go]] and [[chess]].

Many games studied by game theorists (including the famed prisoner's dilemma) are non-zero-sum games, because the [[Outcome (game theory)|outcome]] has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.

Furthermore, ''constant-sum games'' correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential [[gains from trade]]. It is possible to transform any constant-sum game into a (possibly asymmetric) zero-sum game by adding a dummy player (often called "the board") whose losses compensate the players' net winnings.

===Simultaneous / sequential===
{{main | Simultaneous game | Sequential game}}
[[Simultaneous game]]s are games where both players move simultaneously, or instead the later players are unaware of the earlier players' actions (making them ''effectively'' simultaneous). [[Sequential game]]s (a type of dynamic games) are games where players do not make decisions simultaneously, and player's earlier actions affect the outcome and decisions of other players.<ref>{{cite book |doi=10.1016/b978-0-12-398512-5.00002-5 |chapter=Decisions in Engineering Design |title=Design Theory and Methods Using CAD/CAE |date=2015 |last1=Chang |first1=Kuang-Hua |pages=39–101 |isbn=978-0-12-398512-5 }}</ref> This need not be [[perfect information]] about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while they do not know which of the other available actions the first player actually performed.

The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, [[Normal form game|normal form]] is used to represent simultaneous games, while [[Extensive form game|extensive form]] is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see [[subgame perfection]].

In short, the differences between sequential and simultaneous games are as follows:
{| class="wikitable"
|-
! !! Sequential !! Simultaneous 
|-
| Normally denoted by || [[Decision tree]]s || [[Payoff matrix|Payoff matrices]]
|-
| {{longitem|style=line-height:1.3em;padding-right:0.65em|Prior knowledge<br />of opponent's move?}} || {{Yes}} || {{No}}
|-
| Time axis?    || {{Yes}} || {{No}}
|-
| Also known as || {{longitem|style=line-height:1.3em|[[Extensive-form game]]<br />Extensive game}} || {{longitem|style=line-height:1.3em|[[Strategy game]]<br />Strategic game}}
|}

===Perfect information and imperfect information===
{{main|Perfect information}}
[[File:PD with outside option.svg|thumb|upright=1.25|right|A game of imperfect information. The dotted line represents ignorance on the part of player 2, formally called an [[Information set (game theory)|information set]].]]

An important subset of sequential games consists of games of perfect information. A game with perfect information means that all players, at every move in the game, know the previous history of the game and the moves previously made by all other players. An imperfect information game is played when the players do not know all moves already made by the opponent such as a simultaneous move game.<ref name=":2" /> Examples of perfect-information games include [[tic-tac-toe]], [[Draughts|checkers]], [[chess]], and [[Go (board game)|Go]].<ref>{{cite web |url=https://www.math.ucla.edu/~tom/Game_Theory/mat.pdf#page=56 |archive-url=https://web.archive.org/web/20040730140255/http://www.math.ucla.edu/~tom/Game_Theory/mat.pdf |archive-date=2004-07-30 |url-status=live |title=Game Theory |first=Thomas S. |last=Ferguson|author-link= Thomas S. Ferguson |pages=56–57 |publisher=UCLA Department of Mathematics}}</ref><ref>{{cite book |first=Jan |last=Mycielski |author-link=Jan Mycielski |chapter=Games with Perfect Information |title=Handbook of Game Theory with Economic Applications |volume=1 |year=1992 |pages=41–70 |doi=10.1016/S1574-0005(05)80006-2 |isbn=978-0-4448-8098-7}}</ref><ref>{{cite web |url=https://www.youtube.com/watch?v=PN-I6u-AxMg&t=0m25s | archive-url=https://ghostarchive.org/varchive/youtube/20211028/PN-I6u-AxMg| archive-date=2021-10-28|title=Infinite Chess |work=PBS Infinite Series |date=2 March 2017 }}{{cbignore}} Perfect information defined at 0:25, with academic sources {{ArXiv|1302.4377}} and {{ArXiv|1510.08155}}.</ref>

Many card games are games of imperfect information, such as [[poker]] and [[contract bridge|bridge]].<ref>{{Cite book |title=Game Theory: Third Edition |last=Owen |first=Guillermo |publisher=Emerald Group Publishing |year=1995 |isbn=978-0-12-531151-9 |location=Bingley |page=4}}</ref> Perfect information is often confused with [[complete information]], which is a similar concept pertaining to the common knowledge of each player's sequence, strategies, and payoffs throughout gameplay.<ref>{{cite book |doi=10.1007/978-1-349-20181-5_22 |chapter=Perfect Information |title=Game Theory |date=1989 |last1=Mirman |first1=Leonard J. |pages=194–198 |isbn=978-0-333-49537-7 }}</ref> Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken, whereas perfect information is knowledge of all aspects of the game and players.<ref>{{Cite book|last=Mirman|first=Leonard|title=Perfect Information|publisher=Palgrave Macmillan|year=1989|isbn=978-1-349-20181-5|location=London|pages=194–195}}</ref> Games of [[Information asymmetry|incomplete information]] can be reduced, however, to games of imperfect information by introducing "[[Move by nature|moves by nature]]".{{sfnp|Shoham|Leyton-Brown|2008|p=60}}

=== Bayesian game ===
{{main|Bayesian game}}
One of the assumptions of the Nash equilibrium is that every player has correct beliefs about the actions of the other players. However, there are many situations in game theory where participants do not fully understand the characteristics of their opponents. Negotiators may be unaware of their opponent's valuation of the object of negotiation, companies may be unaware of their opponent's cost functions, combatants may be unaware of their opponent's strengths, and jurors may be unaware of their colleague's interpretation of the evidence at trial. In some cases, participants may know the character of their opponent well, but may not know how well their opponent knows his or her own character.<ref>{{Cite book|last=Osborne|first=Martin J.|title=An Introduction to Game Theory|publisher=Oxford University Press|year=2000|pages=271–272}}</ref>

[[Bayesian game]] means a strategic game with incomplete information. For a strategic game, decision makers are players, and every player has a group of actions. A core part of the imperfect information specification is the set of states. Every state completely describes a collection of characteristics relevant to the player such as their preferences and details about them. There must be a state for every set of features that some player believes may exist.<ref>{{Cite book|last=Osborne|first=Martin J|title=An Introduction to Game Theory|publisher=Oxford University Press|year=2020|pages=271–277}}</ref>
[[File:An_example_of_diagram.jpg|thumb|Example of a Bayesian game]]
For example, where Player 1 is unsure whether Player 2 would rather date her or get away from her, while Player 2 understands Player 1's preferences as before. To be specific, supposing that Player 1 believes that Player 2 wants to date her under a probability of 1/2 and get away from her under a probability of 1/2 (this evaluation comes from Player 1's experience probably: she faces players who want to date her half of the time in such a case and players who want to avoid her half of the time). Due to the probability involved, the analysis  of this situation requires to understand the player's preference for the draw, even though people are only interested in pure strategic equilibrium.

=== Combinatorial games ===
<!-- visible anchor to link to, for example at "Solving chess" -->

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and [[Go (game)|Go]]. Games that involve [[Perfect information|imperfect information]] may also have a strong combinatorial character, for instance [[backgammon]]. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve some particular problems and answer some general questions.<ref name="Bewersdorff2005" />

Games of perfect information have been studied in [[combinatorial game theory]], which has developed novel representations, e.g. [[surreal numbers]], as well as [[Combinatorics|combinatorial]] and [[Abstract algebra|algebraic]] (and [[Strategy stealing argument|sometimes non-constructive]]) proof methods to [[Solved game|solve games]] of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory.<ref>{{citation |last1=Albert |first1=Michael H. |author1-link=Michael H. Albert |last2=Nowakowski |first2=Richard J. |last3=Wolfe |first3=David |isbn=978-1-56881-277-9 |publisher=A K Peters Ltd |title=Lessons in Play: In Introduction to Combinatorial Game Theory |year=2007 |pages=3–4}}</ref><ref>{{cite book |last=Beck |first=József |author-link=József Beck |isbn=978-0-521-46100-9 |publisher=Cambridge University Press |title=Combinatorial Games: Tic-Tac-Toe Theory |title-link=Combinatorial Games: Tic-Tac-Toe Theory |year=2008 |pages=[https://archive.org/details/combinatorialgam00jbec/page/n15 1]–3}}</ref> A typical game that has been solved this way is [[Hex (board game)|Hex]]. A related field of study, drawing from [[computational complexity theory]], is [[game complexity]], which is concerned with estimating the computational difficulty of finding optimal strategies.<ref>{{citation |first1=Robert A. |last1=Hearn|author1-link=Bob Hearn |first2=Erik D. |last2=Demaine |title=Games, Puzzles, and Computation |title-link= Games, Puzzles, and Computation |year=2009 |publisher=A K Peters, Ltd. |isbn=978-1-56881-322-6}}</ref>

Research in [[artificial intelligence]] has addressed both perfect and imperfect information games that have very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics, like [[alpha–beta pruning]] or use of [[artificial neural network]]s trained by [[reinforcement learning]], which make games more tractable in computing practice.<ref name="Bewersdorff2005">{{cite book |author=Jörg Bewersdorff |title=Luck, logic, and white lies: the mathematics of games |year=2005 |publisher=A K Peters, Ltd. |isbn=978-1-56881-210-6 |pages=ix–xii |chapter=31 |author-link=Jörg Bewersdorff }}</ref><ref name="Jones2008">{{cite book |first=M. Tim |last=Jones |title=Artificial Intelligence: A Systems Approach |year=2008 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-7337-3 |pages=106–118}}</ref>

===Discrete and continuous games===
Much of game theory is concerned with finite, discrete games that have a finite number of players, moves, events, outcomes, etc. Many concepts can be extended, however. [[Continuous game]]s allow players to choose a strategy from a continuous strategy set. For instance, [[Cournot competition]] is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.

===Differential games===
{{main|Differential game}}
Differential games such as the continuous [[Pursuit-evasion|pursuit and evasion game]] are continuous games where the evolution of the players' state variables is governed by [[differential equation]]s. The problem of finding an optimal strategy in a differential game is closely related to the [[optimal control]] theory. In particular, there are two types of strategies: the open-loop strategies are found using the [[Pontryagin's Minimum Principle|Pontryagin maximum principle]] while the closed-loop strategies are found using [[Hamilton–Jacobi–Bellman equation|Bellman's Dynamic Programming]] method.

A particular case of differential games are the games with a random [[time horizon]].<ref name="PM66">{{cite journal |language=ru |last1=Petrosjan |first1=L. A. |last2=Murzov |first2=N. V. |date=1966 |title=Game-theoretic problems of mechanics |journal=Litovsk. Mat. Sb. |volume=6 |pages=423–433}}</ref> In such games, the terminal time is a random variable with a given [[probability distribution]] function. Therefore, the players maximize the [[mathematical expectation]] of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

===Evolutionary game theory===
{{main|Evolutionary game theory}}
Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.<ref name=Newton2018renaissance>{{cite journal | last1=Newton | first1=Jonathan | year=2018 | title=Evolutionary Game Theory: A Renaissance | journal=Games | volume=9 | issue=2 | pages=31 | doi=10.3390/g9020031| doi-access=free | hdl=10419/179191 | hdl-access=free }}</ref>  In general, the evolution of strategies over time according to such rules is modeled as a [[Markov chain]] with a state variable such as the current strategy profile or how the game has been played in the recent past. Such rules may feature imitation, optimization, or survival of the fittest.

In biology, such models can represent [[evolution]], in which offspring adopt their parents' strategies and parents who play more successful strategies (i.e. corresponding to higher payoffs) have a greater number of offspring. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.{{sfnp|Webb|2007}}

===Stochastic outcomes (and relation to other fields)===
Individual decision problems with stochastic outcomes are sometimes considered "one-player games". They may be modeled using similar tools within the related disciplines of [[decision theory]], [[operations research]], and areas of [[artificial intelligence]], particularly [[AI planning]] (with uncertainty) and [[multi-agent system]]. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using [[Markov decision process]]es (MDP).<ref>{{cite book |last1=Lozovanu |first1=D |last2=Pickl |first2=S |title=A Game-Theoretical Approach to Markov Decision Processes, Stochastic Positional Games and Multicriteria Control Models |date=2015 |publisher=Springer, Cham |isbn=978-3-319-11832-1 }}</ref>

Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("[[Move by nature|moves by nature]]").{{sfnp|Osborne|Rubinstein|1994}} This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the [[Minimax|minimax solution]] is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.<ref name="McMahan">{{cite thesis |first=Hugh Brendan |last=McMahan |date=2006 |title=Robust Planning in Domains with Stochastic Outcomes, Adversaries, and Partial Observability |type=PhD dissertation |publisher=Carnegie Mellon University |url=https://www.cs.cmu.edu/~mcmahan/research/mcmahan_thesis.pdf |archive-url=https://web.archive.org/web/20110401124804/http://www.cs.cmu.edu/~mcmahan/research/mcmahan_thesis.pdf |archive-date=2011-04-01 |url-status=live |pages=3–4}}</ref> (See [[Black swan theory]] for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.)

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "[[gold standard]]" is considered to be partially observable [[stochastic game]] (POSG), but few realistic problems are computationally feasible in POSG representation.<ref name="McMahan" />

===Metagames===
These are games the play of which is the development of the rules for another game, the target or subject game. [[Metagame]]s seek to maximize the utility value of the rule set developed. The theory of metagames is related to [[mechanism design]] theory.

The term [[metagame analysis]] is also used to refer to a practical approach developed by Nigel Howard,{{sfnp|Howard|1971}} whereby a situation is framed as a strategic game in which stakeholders try to realize their objectives by means of the options available to them. Subsequent developments have led to the formulation of [[confrontation analysis]].

===Mean field game theory===
{{main|Mean field game theory}}
Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by [[Boyan Jovanovic]] and [[Robert W. Rosenthal]], in the engineering literature by [[Peter E. Caines]], and by mathematicians [[Pierre-Louis Lions]] and Jean-Michel Lasry.