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| {{Infobox equilibrium|
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| name=Proper equilibrium| | |
| subsetof=[[Trembling hand perfect equilibrium]]|
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| discoverer=[[Roger B. Myerson]]}}
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| '''Proper equilibrium''' is a refinement of [[Nash Equilibrium]] due to [[Roger B. Myerson]].
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| Proper equilibrium further refines [[Reinhard Selten]]'s notion of a
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| [[trembling hand perfect equilibrium]] by assuming that more costly trembles are made with
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| significantly smaller probability than less
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| costly ones.
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| ==Definition==
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| Given a [[Normal form (abstract rewriting)|normal form]] game and a parameter <math>\epsilon > 0</math>, a [[mixed strategy|totally mixed]] strategy profile <math>\sigma</math> is defined to be '''<math>\epsilon</math>-proper''' if, whenever a player has two pure strategies s and s' such that the expected payoff of playing s is smaller than the expected payoff of
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| playing s' (that is <math> u(s,\sigma_{-i})<u(s',\sigma_{-i})</math>), then the probability assigned to s
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| is at most <math>\epsilon</math> times the probability assigned to s'.
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| A strategy profile of the game is then said to be a proper equilibrium
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| if it is a limit point, as <math>\epsilon</math> approaches 0, of a sequence of <math>\epsilon</math>-proper strategy profiles.
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| == Example ==
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| The game to the right is a variant of [[Matching Pennies]].
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| {| align=right border="1" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 1em; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
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| |+ align=bottom |''Matching Pennies with a twist''
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| |-
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| ! ''Guess heads up''
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| ! ''Guess tails up''
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| ! ''Grab penny''
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| |-
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| ! ''Hide heads up''
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| |align=center|-1, 1
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| |align=center|0, 0
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| |align=center|-1, 1
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| |-
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| ! ''Hide tails up''
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| |align=center|0, 0
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| |align=center|-1, 1
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| |align=center|-1, 1
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| |}
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| Player 1 (row player) hides a
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| penny and if Player 2 (column player) guesses correctly whether it is heads up or tails up, he gets the penny. In
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| this variant, Player 2 has a third option: Grabbing the penny without guessing.
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| The [[Nash equilibrium|Nash equilibria]] of the game are the strategy profiles where Player 2 grabs the penny
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| with probability 1. Any mixed strategy of Player 1 is in (Nash) equilibrium with this pure strategy
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| of Player 2. Any such pair is even [[trembling hand perfect equilibrium|trembling hand perfect]].
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| Intuitively, since Player 1 expects Player 2 to grab the penny, he is not concerned about
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| leaving Player 2 uncertain about whether it is heads up or tails up. However, it can be seen
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| that the unique proper equilibrium of this game is the one where Player 1 hides the penny heads up with probability 1/2 and tails up with probability 1/2 (and Player 2 grabs the penny).
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| This unique proper equilibrium can be motivated
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| intuitively as follows: Player 1 fully expects Player 2 to grab the penny.
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| However, Player 1 still prepares for the unlikely event that Player 2 does not grab the
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| penny and instead for some reason decides to make a guess. Player 1 prepares for this event by
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| making sure that Player 2 has no information about whether the penny is heads up or tails up,
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| exactly as in the original [[Matching Pennies]] game.
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| == Proper equilibria of extensive games ==
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| One may apply the properness notion to [[extensive form game]]s in two different ways, completely analogous to the
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| two different ways [[trembling hand perfect equilibrium|trembling hand perfection]]
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| is applied to extensive games. This leads to the notions of '''normal form proper equilibrium'''
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| and '''extensive form proper equilibrium''' of an extensive form game. It was shown by van
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| Damme that a normal form proper equilibrium of an extensive form game is behaviorally equivalent to
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| a [[quasi-perfect equilibrium]] of that game.
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| == References ==
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| {{unreferenced|date=September 2013}}
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| == Further reading ==
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| * Roger B. Myerson. Refinements of the Nash equilibrium concept. ''International Journal of Game Theory'', 15:133-154, 1978.
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| * [[Eric van Damme]]. "A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games." ''International Journal of Game Theory'' 13:1--13, 1984.
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|
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| {{Game theory}}
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| [[Category:Game theory]]
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| [[Category:Non-cooperative games]]
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Claude is her name and she totally digs that name. Kansas is our beginning place and my parents live close by. One of the things I love most is greeting card gathering but I don't have the time lately. Interviewing is what I do for a living but I strategy on changing it.
Here is my blog post :: extended auto warranty