formulasearchengine - User contributions [en]

Q value (nuclear science)

2014-01-30T20:43:36Z

90.185.64.69:

{{Infobox equilibrium|
name=(Normal form) trembling hand perfect equilibrium|
subsetof=[[Nash Equilibrium]]|
supersetof=[[Proper equilibrium]]|
discoverer=[[Reinhard Selten]]}}

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.

==Definition==

First we define a '''perturbed game'''. A perturbed game is a copy of a base game, with the restriction that only [[mixed strategy|totally mixed]] strategies are allowed to be played.
A totally mixed strategy is a mixed strategy where ''every'' [[pure strategy]] is played with non-zero probability.
This is the "trembling hands" of the players; they sometimes play a different strategy than the one they intended to play. Then we define a strategy set S (in a base game) as being trembling hand perfect if there is a [[sequence]] of perturbed games that converge to the base game in which there is a series of [[Nash equilibria]] that converge to S.

==Example==

The game represented in the following [[normal form game|normal form matrix]] has two pure strategy [[Nash equilibrium|Nash equilibria]], namely <math>\langle \text{Up}, \text{Left}\rangle</math> and <math>\langle \text{Down}, \text{Right}\rangle</math>. However, only <math>\langle \text{U},\text{L}\rangle</math> is trembling-hand perfect.

{{Payoff matrix |
UL = 1, 1 | UR = 2, 0 |
DL = 0, 2 | DR = 2, 2 |
Float = right}}

Assume player 1 is playing a [[mixed strategy]] <math>(1-\varepsilon, \varepsilon)</math>, for <math> 0<\varepsilon <1</math>. Player 2's expected payoff from playing L is:

:<math>1(1-\varepsilon) + 2\varepsilon = 1+\varepsilon</math>

Player 2's expected payoff from playing the strategy R is:

:<math>0(1-\varepsilon) + 2\varepsilon = 2\varepsilon</math>

For small values of <math>\varepsilon</math>, player 2 maximizes his expected payoff by placing a minimal weight on R and maximal weight on L. By symmetry, player 1 should place a minimal weight on D if player 2 is playing the mixed strategy <math>(1-\varepsilon, \varepsilon)</math>. Hence <math>\langle \text{U},\text{L}\rangle</math> is trembling-hand perfect.

However, similar analysis fails for the strategy profile <math>\langle \text{D}, \text{R}\rangle</math>.

Assume player 2 is playing a [[mixed strategy]] <math>(\varepsilon, 1-\varepsilon)</math>. Player 1's expected payoff from playing U is:

:<math>1\varepsilon + 2(1-\varepsilon) = 2-\varepsilon</math>

Player 1's expected payoff from playing D is:

:<math>0(\varepsilon) + 2(1-\varepsilon) = 2-2\varepsilon</math>

For all positive values of <math>\varepsilon</math>, player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence <math>\langle \text{D}, \text{R}\rangle</math> is not trembling-hand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating most often to L if there is a small chance of error in the behavior of player 1.

== Trembling hand perfect equilibria of two-player games ==

For two-player games, the set of trembling hand perfect equilibria coincides with the set of [[admissible decision rule|admissible]] equilibria, i.e., equilibria consisting of two undominated strategies. In the example above, we see that the imperfect equilibrium <nowiki><</nowiki>D,R<nowiki>></nowiki> is not admissible, as L (weakly) dominates R for Player 2.

== Trembling hand perfect equilibria of extensive form games ==

{{Infobox equilibrium|
name=Extensive-form trembling hand perfect equilibrium|
subsetof=[[Subgame perfect equilibrium]], [[Perfect Bayesian equilibrium]], [[Sequential equilibrium]]|
discoverer=[[Reinhard Selten]]|
usedfor=[[Extensive form game]]s
}}

There are two possible ways of extending the definition of trembling hand perfection to [[extensive form game]]s.

* One may interpret the extensive form as being merely a concise description of a normal form game and apply the concepts described above to this normal form game. In the resulting perturbed games, every [[strategy (game theory)|strategy]] of the extensive-form game must be played with non-zero probability. This leads to the notion of a '''normal-form trembling hand perfect equilibrium'''.

* Alternatively, one may recall that trembles are to be interpreted as modelling mistakes made by the players with some negligible probability when the game is played. Such a mistake would most likely consist of a player making another [[strategy (game theory)|move]] than the one intended at some point during play. It would hardly consist of the player choosing another [[strategy (game theory)|strategy]] than intended, i.e. a wrong plan for playing the entire game. To capture this, one may define the perturbed game by requiring that every [[strategy (game theory)|move]] at every [[information set]] is taken with non-zero probability. Limits of equilibria of such perturbed games as the tremble probabilities goes to zero are called '''extensive-form trembling hand perfect equilibria'''.

The notions of normal-form and extensive-form trembling hand perfect equilibria are incomparable, i.e., an equilibrium of an extensive-form game may be normal-form trembling hand perfect but not extensive-form trembling hand perfect and vice versa.
As an extreme example of this, Jean-François Mertens has given an [[Quasi-perfect equilibrium|example]] of a two-player extensive form game where no extensive-form trembling hand perfect equilibrium is admissible, i.e., the sets of extensive-form and normal-form trembling hand perfect equilibria for this game are disjoint.

An extensive-form trembling hand perfect equilibrium is also a [[sequential equilibrium]]. A normal-form trembling hand perfect equilibrium of an extensive form game may be sequential but is not necessarily so. In fact, a normal-form trembling hand perfect equilibrium does not even have to be [[subgame perfect equilibrium|subgame perfect]].

== References ==

* [[Reinhard Selten|Selten, R.]] (1975) A reexamination of the perfectness concept for equilibrium points in extensive games. ''International Journal of Game Theory'' 4:25-55.

{{Game theory}}

[[Category:Game theory]]
[[Category:Non-cooperative games]]

Alexandru Proca

2012-07-07T21:21:52Z

90.185.67.22: /* Scientific achievements */

Many individuals favor them over conventional metal knives as a result of they are surely lighter and don't need to be honed almost as [http://Wikievo.Uv.es/mediawiki/index.php/Bear_Grylls_Paracord_Knife_Specs incessantly]. Many individuals like them as a result of they're typically out there with a black blade, which may make their pocket knives appear modern and trendy. No matter your cause, ceramic blade knives actually are on the "cutting [http://www.knifecenter.com bowie knives for sale] edge" of the latest pocket knives gadgetry. Leave the infinite struggling and grumbling over tightly sealed bins to others. Slit it open in a matter of moments together with your trusty pocket knife. Shipping information and data are extremely different for nearly any objects & by contract reminiscent of free delivery remedy, and so upon. Relating to the sting of the blade, there are serrated and non-serrated, also called plain edge. I sometimes recommend plain edge to nearly everybody, nonetheless serrated knives do have a number of special functions. If you are slicing large amounts of fibrous material, similar to rope, serrated knife would possibly make the most effective [http://thebestpocketknifereviews.com/bear-grylls-paracord-knife-review/ Bear Grylls Paracord Knife Reviews] pocket knife for you. For every part else, follow the non-serrated. Concerning partially serrated blades, I find that these have a tendency not to have sufficient plain edge or serrated edge to be notably useful in doing duties designed for either. Whether folding knife , or mounted blade, no matter your want could also be, an excellent customized knife would be the reply. The Rockwell scale is a measurement system used by industrial producers to determine the relative hardness of objects based mostly on the depth of indentation from a heavy object. In different words, they drop a heavy ball or diamond cone onto the metal and measure the dimensions of indentation. For knives they use the “C” scale in items labeled “HRC.” Excessive carbon steel kitchen knives generally [http://en.wiktionary.org/wiki/pocketknife folding knives] run from HRC fifty six-58. HRC 62 is a really hard knife. It might seem unusual for the TSA to chill out its guidelines concerning small knives, especially when it's onerous to imagine a concerted effort by lobbyists pushing to allow small blades on airplanes. Not solely do Kershaw knives feature the quality precision of Sandvik stainless-steel blades however their superior manufacturing course of is unmatched within the trade. Featuring different kinds like moderately curved stomach blades to extend reducing power and flat ground tanto tips to plain edge or serrated edge pocket knives you will see that a method that matches your wants. Now that you've an understanding of what goes into figuring out the best pocket knife , here's a list of the top rated knives available in the market right now. It is best to to discover a related merchandise to have the ability to compare because generally it could help you in shopping for options. Travelling round Australia and the world, teaching and public talking about permaculture, I have to be self-reliant in different ways. Organising laptops and projectors for displays often requires a knife (e.g. to chop gaffer tape, or to repair cables to floors). Not like the professional tech assist person who has their tools of trade [http://www.thebestpocketknifereviews.com/bear-grylls-paracord-knife-review/ bear Grylls paracord knife price] with them, the permaculture presenter typically works in venues the place there isn't a tech assist; a DIY scenario that displays the practicalities as well as the rules of permaculture. On more than one occasion I have used a knife blade to straighten bent pins to facilitate a connection between electrical equipment. For instance, common lighters were removed from the banned list in 2007, however torch lighters that create super-hot, needle-like flames are still prohibited. A e book of [http://wiki.ippk.ru/index.php/Where_To_Buy_Bear_Grylls_Paracord_Knife matches] is allowed into the cabin, however not the kind that a consumer can strike on any floor. Rare outdated Utica Kutmaster Advertising Knife "Northrup King Seeds" three blade stockman - Large clip blade has good long nail pull. Nickel silver bolsters. Brass liners. Good old American made ad knife Just the best little knife , Joseph Rodgers "Mini Sheepsfoot" - Deal with is rough-textured black bone with inlaid nickel silver oval protect (great to engrave on). Blade has tremendous-sturdy; it snaps open and shut. Okay, time for another credit card style knife and this time the Cardsharp2 designed by British designer Iain Sinclair. This child weighs lower than one ounce and carries a razor sharp blade which is lower than 0.1 inches thick. It opens up using an clever folding operation and folds back easily for storage in your wallet, pocket or purse. Now, let’s get something straight – you’re not going to be slicing down trees or carving a spear with this one. In fact, you gained’t even be dicing onions or tomatoes. So what is going to you be doing with something so elegant and cool wanting? CRKT Van Hoy Snap Lock.