en>Monkbot: /* Expression */Fix CS1 deprecated date parameter errors

2014-02-11T08:36:44Z

Expression: Fix CS1 deprecated date parameter errors

82.186.243.243: /* Expression */

2013-02-25T08:21:32Z

Expression

← Older revision		Revision as of 10:21, 25 February 2013
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	~~They're always ready~~ to ~~help~~, and ~~they're always making changes to~~ the ~~site to make sure you won't have troubles~~ in the ~~first place~~. ~~Medical word press themes give you~~ the ~~latest medical designs. These templates are professionally designed and are also Adsense ready. Transforming your designs~~ to ~~Word - Press blogs~~ is ~~not~~ that ~~easy because of the simplified way in creating your very own themes~~. It~~'s as simple as hiring a Wordpress plugin developer or learning how~~ to ~~create what is needed~~. <br>~~<br>purcase and download~~ - ~~WPZOOM Tribune wordpress Theme, find and use~~ the ~~WPZOOM Discount Code~~. ~~Best of all, you can still have all the functionality that you desire when you use the Word~~ - ~~Press platform~~. ~~Well Managed Administration The Word~~ - Press ~~can easily absorb the high numbers~~ of ~~traffic~~ by ~~controlling~~ the ~~server load to make sure that~~ the ~~site works properly~~. ~~By purchasing Word~~ - ~~Press weblogs you can acquire your very own domain title and have total command~~ of ~~your web site~~. ~~But in case you want some theme or plugin in sync with your business needs~~, ~~it is advisable that you must seek some professional help.~~ <br><br>~~If you loved this posting~~ and ~~you would like~~ to ~~obtain much more data pertaining to [http~~://~~topne~~.ws/~~WordpressDropboxBackup608707 wordpress backup plugin~~] ~~kindly pay~~ a ~~visit~~ to ~~our own page~~. ~~You can down load it here at this link~~: ~~and utilize your FTP software program~~ to ~~upload it~~ to ~~your Word - Press Plugin folder~~. ~~But if you are not willing to choose cost to~~ the ~~detriment of quality~~, ~~originality and higher returns~~, ~~then go for a self-hosted wordpress blog~~ and ~~increase~~ the ~~presence~~ of ~~your business in this new digital age.~~ I ~~hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine~~. ~~You or your web designer can customize it as per your specific needs~~. ~~Socrates:~~ (~~link to~~ ) ~~Originally developed for affiliate marketers~~, I'~~ve used this theme again~~ and ~~again~~ to ~~develop full-fledged web sites that include static pages~~, ~~squeeze pages~~, ~~and a blog~~. <br><br>~~Numerous bloggers are utilizing Word - Press~~ and ~~with good reason~~. ~~But the Joomla was created as the CMS over years~~ of ~~hard work. Next you'll go by way~~ of ~~to your simple Word - Press site~~. ~~If you just want to share some picture~~ and ~~want to use it as a dairy~~, that ~~you want to share~~ with ~~your friends and family members~~, ~~then blogger would be~~ an ~~excellent choice. If your site does well you can get paid professional designer to create a unique Word - Press theme.~~ <br><br>~~Millions~~ of ~~individuals~~ and ~~organizations are now successfully using this tool throughout~~ the ~~world~~. ~~s ability~~ to ~~use different themes and skins known as Word - Press Templates~~ or ~~Themes~~. ~~Useful Plugins~~ ~~Uber - Menu Top Megamenu Now it~~ is the ~~time of sticky Top navbar~~. ~~Working with a Word~~ - ~~Press blog and the appropriate cost~~-~~free Word~~ - ~~Press theme, you can get~~ a ~~professional internet site up and published~~ in ~~no time at all. As for performing online business~~, ~~websites and blogs are~~ the ~~only medium that are available to interact with customers and Word - Press perform this work with~~ the ~~help of cross-blog communication tools, comments and full user registration plug-ins~~.		[[Image:Game with no value.svg\|thumb\|240px\|Game square (that is, the payoff to player I) for a game with no value, due to Sion and Wolfe. The payoff is 0.5 along the two diagonal lines]]

			In [[game theory]], and in particular the study of [[zero-sum]] [[continuous game]]s, it is commonly assumed that a game has a [[minimax]] value. This is the [[expected value]] to one of the players when both play a perfect strategy (which is to choose from a particular [[probability density function\|PDF]]).

			This article gives an example of a [[zero sum game]] that has no [[minimax\|value]]. It is due to Sion and Wolfe.<ref>
			{{citation \| chapter = On a game without a value \| first1 = Maurice \| last1 = Sion \| first2 = Phillip \| last2 = Wolfe \| pages = 299-306 \| title = Contributions to the Theory of Games III \| editor1-first = M. \| editor1-last = Dresher \| editor2-first = A. W. \| editor2-last = Tucker \| editor3-first = P. \| editor3-last = Wolfe \| year = 1957 \| isbn = 9780691079363 \| publisher = Princeton University Press \| series = Annals of Mathematics Studies 39}}</ref>

			Zero sum games with a finite number of pure strategies are known to have a [[minimax]] value (originally proved by [[John von Neumann]]) but this is not necessarily the case if the game has an infinite set of strategies. There follows a simple example of a game with no minimax value.

			The existence of such zero-sum games is interesting because many of the results of [[game theory]] become inapplicable if there is no minimax value.

			==The game==

			Players I and II each choose a number, <math>x</math> and <math>y</math> respectively, with <math>0\leq x,y\leq 1</math>; the payoff to I is

			:<math>K(x,y)=
			\begin{cases}
			-1 & \text{if } x<y<x+1/2 \\
			0 & \text{if } x=y \text{ or } y=x+1/2\\
			1 & \text{otherwise}
			\end{cases}</math>

			(i.e. player II pays <math>K(x,y)</math> to player I;the game is [[zero-sum]]). Sometimes player I is referred to as the ''maximizing player'' and player II the ''minimizing player''.

			If <math>(x,y)</math> is interpreted as a point on the unit square, the figure shows the payoff to player I. Now suppose that player I adopts a mixed strategy: choosing a number from [[probability density function\|probability density function (pdf)]] <math>f</math>; player II chooses from <math>g</math>. Player I seeks to maximize the payoff, player II to minimize the payoff. Note that each player is aware of the other's objective.

			== Game value ==

			Sion and Wolfe show that

			:<math>
			\sup_f \inf_g \iint K\,df\,dg=\frac{1}{3}
			</math>

			but

			:<math>
			\inf_g \sup_f \iint K\,df\,dg=\frac{3}{7}.
			</math>

			These are the maximal and minimal expectations of the game's value of player I and II respectively.

			The <math>\sup</math> and <math>\inf</math> respectively take the supremum and infimum over pdf's on the unit interval (actually [[Borel measure\|Borel probability measures]]). These represent player I and player II's (mixed) strategies. Thus, player I can assure himself of a payoff of at least 3/7 if he knows player II's strategy; and player II can hold the payoff down to 1/3 if he knows player I's strategy.

			There is clearly no [[epsilon equilibrium]] for sufficiently small <math>\epsilon</math>, specifically, if <math>\epsilon < \frac{1}{2}\left(\frac{3}{7}-\frac{1}{3}\right)\simeq 0.0476</math>. Dasgupta and Maskin<ref>{{cite journal \| author=[[Partha Dasgupta\|P. Dasgupta]] and [[Eric Maskin\|E. Maskin]] \| title=The Existence of Equilibrium in Discontinuous Economic Games, I: Theory \| journal= [[Review of Economic Studies]]\| year=1986 \| volume=53 \| pages=1–26 \| doi=10.2307/2297588 \| issue=1 \| jstor=2297588}}</ref> assert that the game values are achieved if player I puts probability weight only on the set <math>\left\{0,1/2,1\right\}</math> and player II puts weight only on <math>\left\{1/4,1/2,1\right\}</math>.

			[[Glicksberg's theorem]] shows that any zero-sum game with [[upper semicontinuous\|upper]] or [[lower semicontinuous]] payoff function has a value (in this context, an upper (lower) semicontinuous function ''K'' is one in which the set <math>\{P\|K(P)<c\}</math> (resp <math>\{P\|K(P)>c\}</math>) is [[open set\|open]] for any [[real number\|real]] ''c'').

			Observe that the payoff function of Sion and Wolfe's example is clearly not semicontinuous. However, it may be made so by changing the value of ''K''(''x'', ''x'') and ''K''(''x'', ''x'' + 1/2) [i.e. the payoff along the two discontinuities] to either +1 or −1, making the payoff upper or lower semicontinuous respectively. If this is done, the game then has a value.

			==Generalizations==

			Subsequent work by Heuer <ref>{{cite journal \| author=G. A. Heuer \| title=Three-part partition games on rectangles \| journal= Theoretical Computer Science\| year=2001 \| volume=259 \| pages=639–661 \| doi=10.1016/S0304-3975(00)00404-7}}</ref> discusses a class of games in which the unit square is divided into three regions, the payoff function being constant in each of the regions.

			== References ==
			{{reflist}}

			[[Category:Game theory]]
			[[Category:Mathematical examples]]

221.135.143.46: /* Examples */

2012-07-26T11:34:55Z

Examples

New page

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en>Monkbot: /* Expression */Fix CS1 deprecated date parameter errors

82.186.243.243: /* Expression */

221.135.143.46: /* Examples */