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| {{for|the business strategy|Dominance (economics)}}
| | The title of the writer is Numbers. One of the very best issues in the globe for me is to do aerobics and now I'm attempting to earn money with it. My day occupation is a librarian. Years in the past he moved to North Dakota and his family members enjoys it.<br><br>Check out my page; [http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw www.youronlinepublishers.com] |
| In [[game theory]], '''strategic dominance''' (commonly called simply '''dominance''') occurs when one [[strategy (game theory)|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.
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| 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. | |
| <!-- I assume [[perfect knowledge]] games. If someone knows how to extend this to imperfect knowledge games, I would find that interesting. -->
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| ==Terminology==
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| When a player tries to choose the "best" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better.
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| The result of the comparison is one of:
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| * B '''dominates''' A: choosing B always gives as good as or a better outcome than choosing A. There are 2 possibilities:
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| ** B '''strictly dominates''' A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do.
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| ** B '''weakly dominates''' A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give B the same payoff as A.
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| * B and A are '''intransitive''': B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing B is better in other cases, depending on exactly how the opponent chooses to play. For example, B is "throw rock" while A is "throw scissors" in [[Rock, Paper, Scissors]].
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| * B is '''dominated''' by A: choosing B never gives a better outcome than choosing A, no matter what the other player(s) do. There are 2 possibilities:
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| ** B is '''weakly dominated''' by A: There is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give A the same payoff as B. (Strategy A weakly dominates B).
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| ** B is '''strictly dominated''' by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B).
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| This notion can be generalized beyond the comparison of two strategies.
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| * Strategy B is '''strictly dominant''' if strategy B ''strictly dominates'' every other possible strategy.
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| * Strategy B is '''weakly dominant''' if strategy B ''dominates'' all other strategies, but some (or all) strategies are only ''weakly dominated'' by B.
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| * Strategy B is '''strictly dominated''' if some other strategy exists that strictly dominates B.
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| * Strategy B is '''weakly dominated''' if some other strategy exists that weakly dominates B.
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| ==Mathematical definition==
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| For any player <math>i</math>, a strategy <math>s^*\in S_i</math> '''weakly dominates''' another strategy <math>s^\prime\in S_i</math> if
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| :<math>\forall s_{-i}\in S_{-i}\left[u_i(s^*,s_{-i})\geq u_i(s^\prime,s_{-i})\right]</math> (With at least one <math>s_{-i}</math> that gives a strict inequality)
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| <math>s^*</math> '''strictly dominates''' <math>s^\prime</math> if
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| :<math>\forall s_{-i}\in S_{-i}\left[u_i(s^*,s_{-i})> u_i(s^\prime,s_{-i})\right]</math>
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| where <math>S_{-i}</math> represents the product of all strategy sets other than player <math>i</math>'s
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| == Dominance and Nash equilibria ==
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| {| border="1" align=right cellpadding="4" cellspacing="0" style="margin: 1em 0 1em 1em; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
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| |-
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| ! C
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| ! D
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| |-
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| ! C
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| | 1, 1
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| | 0, 0
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| |-
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| ! D
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| | 0, 0
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| | 0, 0
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| |}
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| If a strictly dominant strategy exists for one player in a game, that player will play that strategy in each of the game's [[Nash equilibrium|Nash equilibria]]. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium. However, that Nash equilibrium is not necessarily [[Pareto optimal]], meaning that there may be non-equilibrium outcomes of the game that would be better for both players. The classic game used to illustrate this is the [[Prisoner's Dilemma]].
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| Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such, it is irrational for any player to play them. On the other hand, weakly dominated strategies may be part of Nash equilibria. For instance, consider the [[payoff matrix]] pictured at the right.
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| Strategy ''C'' weakly dominates strategy ''D.'' Consider playing ''C'': If one's opponent plays ''C,'' one gets 1; if one's opponent plays ''D,'' one gets 0. Compare this to ''D,'' where one gets 0 regardless. Since in one case, one does better by playing ''C'' instead of ''D'' and never does worse, ''C'' weakly dominates ''D.'' Despite this, ''(D, D)'' is a Nash equilibrium. Suppose both players choose ''D''. Neither player will do any better by unilaterally deviating—if a player switches to playing ''C,'' they will still get 0. This satisfies the requirements of a Nash equilibrium. Suppose both players choose C. Neither player will do better by unilaterally deviating—if a player switches to playing D, they will get 0. This also satisfies the requirements of a Nash equilibrium.
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| == Iterated elimination of dominated strategies (IEDS) ==
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| The iterated elimination (or deletion) of dominated strategies is one common technique for solving games that involves [[iteration|iteratively]] removing dominated strategies. In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies. This results in a new, smaller game. Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on. The process stops when no dominated strategy is found for any player. This process is valid since it is assumed that rationality among players is [[common knowledge (logic)|common knowledge]], that is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum (see Aumann, 1976).
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| There are two versions of this process.
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| One version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium.
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| Another version involves eliminating both strictly and weakly dominated strategies. If, at the end of the process, there is a single strategy for each player, this strategy set is also a [[Nash equilibrium]]. However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the ''only'' Nash equilibrium. (In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.)
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| In any case, if by iterated elimination of dominated strategies there is only one strategy left for each player, the game is called a '''dominance-solvable''' game.
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| == See also ==
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| * [[Arbitrage]]
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| * [[Winning strategy]]
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| * [[Risk dominance]]
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| ==References==
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| *{{cite book |last=Fudenberg |first=Drew |authorlink2=Jean Tirole |first2=Jean |last2=Tirole |year=1993 |title=Game Theory |location= |publisher=MIT Press |isbn= }}
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| *{{cite book |last=Gibbons |first=Robert |year=1992 |title=Game Theory for Applied Economists |location= |publisher=Princeton University Press |isbn=0-691-00395-5 }}
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| *{{cite book |last=Ginits |first=Herbert |year=2000 |title=Game Theory Evolving |location= |publisher=Princeton University Press |isbn=0-691-00943-0 }}
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| * {{cite book | last2=Shoham | first2=Yoav | last1=Leyton-Brown | first1=Kevin | title=Essentials of Game Theory: A Concise, Multidisciplinary Introduction | publisher=Morgan & Claypool Publishers | isbn=978-1-59829-593-1 | url=http://www.gtessentials.org | year=2008 | location=San Rafael, CA}}. An 88-page mathematical introduction; see Section 3.3. [http://www.morganclaypool.com/doi/abs/10.2200/S00108ED1V01Y200802AIM003 Free online] at many universities.
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| *{{cite book |authorlink=Anatol Rapoport |last=Rapoport |first=A. |year=1966 |title=Two-Person Game Theory: The Essential Ideas |location= |publisher=University of Michigan Press |isbn= }}
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| * [http://www.virtualperfection.com/gametheory/Section2.1.html Jim Ratliff's Game Theory Course: Strategic Dominance]
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| * {{cite book | last1=Shoham | first1=Yoav | last2=Leyton-Brown | first2=Kevin | title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations | publisher=[[Cambridge University Press]] | isbn=978-0-521-89943-7 | url=http://www.masfoundations.org | year=2009 | location=New York}}. A comprehensive reference from a computational perspective; see Sections 3.4.3, 4.5. [http://www.masfoundations.org/download.html Downloadable free online].
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| {{PlanetMath attribution|id=3196|title=Dominant strategy}}
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| {{Game theory}}
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| [[Category:Game theory]]
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The title of the writer is Numbers. One of the very best issues in the globe for me is to do aerobics and now I'm attempting to earn money with it. My day occupation is a librarian. Years in the past he moved to North Dakota and his family members enjoys it.
Check out my page; www.youronlinepublishers.com