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{{infobox equilibrium|
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
name=Quantal response equilibrium|
subsetof = [[Bayesian game|Bayes Nash equilibrium]]|
supersetof = [[Nash equilibrium]], [[Logit equilibrium]]|
discoverer = [[Richard McKelvey]] and [[Thomas Palfrey]]|
usedfor = [[Non-cooperative game]]s|
example = [[Traveler's dilemma]]}}


'''Quantal response equilibrium''' ('''QRE''') is a [[solution concept]] in [[game theory]]. First introduced by [[Richard McKelvey]] and [[Thomas Palfrey]], it provides an equilibrium notion with [[bounded rationality]]. QRE is not an equilibrium refinement, and it can give significantly different results from [[Nash equilibrium]]. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.
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In a quantal response equilibrium, players are assumed to make errors in choosing which pure strategy to play. The probability of any particular strategy being chosen is positively related to the payoff from that strategy. In other words, very costly errors are unlikely.
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The equilibrium arises from the realization of beliefs. A player's payoffs are computed based on beliefs about other players' probability distribution over strategies. In equilibrium, a player's beliefs are correct.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


== Application to data ==
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


When analyzing data from the play of actual games (particularly from [[Experimental economics|laboratory experiments]]), Nash equilibrium can be unforgiving. Any non-equilibrium move can appear equally "wrong", but realistically should not be used to reject a theory. QRE allows every strategy to be played with non-zero probability, and so any data is possible (though not necessarily reasonable).
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


== Logit equilibrium ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


By far the most common specification for QRE is '''logit equilibrium''' ('''LQRE'''). In a logit equilibrium, player's strategies are chosen according to the probability distribution:
==Demos==


<math>P_{ij} = \frac{\exp(\lambda EU_{ij}(P_{-i}))}{\sum_k{\exp(\lambda EU_{ik}(P_{-i}))}}</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


<math>P_{ij}</math> is the probability of player i choosing strategy j.
<math>EU_{ij}(P_{-i}))</math> is the expected utility to player i of choosing strategy j given other players are playing according to the probability distribution <math>P_{-i}</math>.


Of particular interest in the logit model is the non-negative parameter λ (sometimes written as 1/μ). λ can be thought of as the rationality parameter. As λ→0, players become "completely irrational", and play each strategy with equal probability. As λ→∞, players become "perfectly rational", and play approaches a Nash equilibrium.
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
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** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


== For dynamic games ==
==Test pages ==


For dynamic ([[extensive form game|extensive form]]) games, McKelvey and Palfrey defined '''agent quantal response equilibrium''' ('''AQRE'''). AQRE is somewhat analogous to [[subgame perfection]]. In an AQRE, each player plays with some error as in QRE. At a given decision node, the player determines the expected payoff of each action by treating their future self as an independent player with a known probability distribution over actions.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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As in QRE, in a AQRE every strategy is used with nonzero probability. This provides an additional advantage of AQRE over perfectly rational solution concepts. Since every path is followed with some probability, there is no concern about defining beliefs "off the equilibrium path".
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
== Critiques ==
==Bug reporting==
=== Free parameter ===
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
LQRE has the free parameter λ. As λ→∞, LQRE→Nash equilibrium, so LQRE will always be at least as good a fit as Nash equilibrium. Changes in the parameter can result in large changes to equilibrium behavior.
 
However, the theory is incomplete without describing where λ comes from. Estimates of λ from experiments can vary significantly. Sometimes this variance seems to be a result of individual characteristics (for instance, λ sometimes increases with learning). Other times it appears that λ varies from game to game.
 
==References==
* {{Citation
  | last1 = McKelvey | first1 = Richard
  | author1-link = Richard McKelvey
  | last2 = Palfrey | first2 = Thomas
  | author2-link = Thomas Palfrey
  | title = Quantal Response Equilibria for Normal Form Games
  | journal = Games and Economic Behavior
  | volume = 10
  | pages = 6–38
  | year = 1995
  | doi = 10.1006/game.1995.1023 }}
* {{Citation
  | last1 = McKelvey | first1 = Richard
  | author1-link = Richard McKelvey
  | last2 = Palfrey | first2 = Thomas
  | author2-link = Thomas Palfrey
  | title = Quantal Response Equilibria for Extensive Form Games
  | journal = Experimental Economics
  | volume = 1
  | pages = 9–41
  | year = 1998
  | doi = 10.1007/BF01426213 }}
 
{{Game theory}}
{{gametheory-stub}}
 
[[Category:Game theory]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

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