Ambipolar diffusion: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Materialscientist
fmt
en>WPGA2345
 
Line 1: Line 1:
In [[game theory]], '''normal form''' is a description of a ''game''. Unlike [[extensive-form game|extensive form]], normal-form representations are not graphical ''per se'', but rather represent the game by way of a [[matrix (mathematics)|matrix]]. While this approach can be of greater use in identifying [[strictly dominated strategies]] and [[Nash equilibrium|Nash equilibria]], some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable [[Strategy (game theory)|strategies]], and their corresponding payoffs, for each player.
== the moment somewhat suddenly nodded. ==


In static games of [[complete information|complete]], [[perfect information]], a normal-form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, whereas a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a [[cardinal utility|cardinal]] or [[ordinal utility]]—often cardinal in the normal-form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output.
Of the body is the Bodhi saliva? But the door to the heritage Hsiao, how could draw Jean Monnet day trip to take heart or something? Mind flashed wipe suspect this 'confusion', Xiao Yan's eyes suddenly catch [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ 掛け時計] a glimpse of the Soviet Union one thousand large elders, the moment somewhat suddenly nodded.<br><br>Xiao Li into the light beam lasts thirty-four minutes or so, that is out of line, watching its appearance, though not bitterly of 'color', but did not smile much, surely they have come up [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 時計] with something black emperor were not give their assessment is too high now.<br><br>mind flashed the idea of ​​this road, Xiao Yan is slowly stood up.<br><br>the time being as an object of [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html 腕時計 メンズ casio] attention, Xiao Yan so move naturally Lima will be a road around the eyes contains a variety of emotions pulled over.<br><br>For these eyes, Xiao Yan did not bother physique of a move, floating [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ 掛け時計] across the air, then it is to fall into a light beam into.<br><br>stature [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys] swept into the beam of light, glare
相关的主题文章：
<ul>
 
  <li>[http://www.onlineln.com/home.php?mod=space&uid=25832 http://www.onlineln.com/home.php?mod=space&uid=25832]</li>
 
  <li>[http://exclusivelibraryrelease.com/groupware/blogs/post/1740 http://exclusivelibraryrelease.com/groupware/blogs/post/1740]</li>
 
  <li>[http://mint.dee.cc/cgi-bin/40st-bbs1/40st.cgi http://mint.dee.cc/cgi-bin/40st-bbs1/40st.cgi]</li>
 
</ul>


== An example ==
== little girl in white this time ==


{| class="wikitable" style="float: right; text-align: center;"
, This display Ziyun wing, rest assured that back into the hospital.<br><br>With the days 'color' bleak for some reason, Xiao Yan again smooth out those who want to challenge him busybodies, coming into the pan straight into the door, swaggering back to the pavilion, but when it pushed the door Pops, in front of white shadow suddenly Shanlue, so too he hurried Police took [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 時計] a step back, take a closer [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波] look, could not help a bit of shock, the original character asks shadow in front of it, who actually is the so-called 'strong list' first, Warcraft purple research incarnation.<br><br>little girl in [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ腕時計 メンズ] white this time, his face looked at the door are looking forward to Xiao Yan, black eyes filled with mouth-watering flavor.<br><br>'Well, purple research, how do you come ah?' will bring to the heart [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 腕時計 ソーラー] to put it down, Xiao Yan cough loudly, laughing, 'knead' the 'rub' Purple research little [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ腕時計 メンズ] head, then that is the flash sideways into the room, into the room, it was found that the children smoked, Hu Jia, Wu Hao, who are in them, and this
|+ A normal-form game
相关的主题文章：
|-
<ul>
! Player 1 \ Player 2
 
! scope="col" style="color: #900; width: 90px;" | Player 2 chooses left
  <li>[http://42.121.114.132/plus/view.php?aid=234036 http://42.121.114.132/plus/view.php?aid=234036]</li>
! scope="col" style="color: #900; width: 90px;" | Player 2 chooses right
 
|-
  <li>[http://www.nobuzushi.com/freecgi/treebbs.cgi http://www.nobuzushi.com/freecgi/treebbs.cgi]</li>
! scope="col" style="color: #009; width: 100px;"| Player 1 chooses top
 
| <span style="color: #009">4</span>, <span style="color: #900">3</span>
  <li>[http://www.pseudodrom.com/cgi-bin/gaestebuch.cgi http://www.pseudodrom.com/cgi-bin/gaestebuch.cgi]</li>
| <span style="color: #009">−1</span>, <span style="color: #900">−1</span>
 
|-
</ul>
! scope="col" style="color: #009;"| Player 1 chooses bottom
| <span style="color: #009">0</span>, <span style="color: #900">0</span>
| <span style="color: #009">3</span>, <span style="color: #900">4</span>
|}


The matrix to the right is a normal-form representation of a game in which players move simultaneously (or at least do not observe the other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For example, if player 1 plays top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).
== some of the strength of incompetence ==


=== Other representations ===
Are all could not [http://nrcil.net/sitemap.xml http://nrcil.net/sitemap.xml] help but burst of raspberry, is indeed the order of the bucket were other strong, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ gps 時計] just the hand, the presence of strong imperial struggle is no one who can cast out.<br><br>huge green 'color' spin shrouded wind Yunshan whole body, roared high-speed rotation, gales, many people will be on the plaza beneath the rickety stature are blown, some of the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys] strength of incompetence, and even more are back again and again, finally and sat on the floor, his face looked up at the horror that almost [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html casio 腕時計 スタンダード] across the vast world of the wind spin, this is [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 腕時計 チタン] like the terrible force of nature like, Yunshan single-handedly that it would be the call out, when really terrible.<br><br>wutherin winds wreak havoc in the cloud Arashiyama sky and over, along with the mountains in the woodlands also Kuangmeng surging waves from a tree at the center of a ** green 'color' appeared, and finally into a rounded, swept the four parties .<br><br>suddenly looked so big action Yunshan, Xiao Yan is Yi Zheng, immediately surfaced on the pale face
相关的主题文章：
<ul>
 
  <li>[http://www.axzgaxzj.com/bbs/forum.php?mod=viewthread&tid=40508 http://www.axzgaxzj.com/bbs/forum.php?mod=viewthread&tid=40508]</li>
 
  <li>[http://166.111.7.59/luo/forum.php?mod=viewthread&tid=2527586&extra= http://166.111.7.59/luo/forum.php?mod=viewthread&tid=2527586&extra=]</li>
 
  <li>[http://xisyys.com/forum.php?mod=viewthread&tid=18036&fromuid=4313 http://xisyys.com/forum.php?mod=viewthread&tid=18036&fromuid=4313]</li>
 
</ul>


Often, [[symmetric game]]s (where the payoffs do not depend on which player chooses each action) are represented with only one payoff. This is the payoff for the row player. For example, the payoff matrices on the right and left below represent the same game.
== five hundred and sixtieth chapters joint seal ==


{| align="center"
Chapter 65 joint seal!<br><br>five hundred and sixtieth chapters joint seal!<br><br>Hu dry and others appear, no doubt like a force-like, so too have some of the elders of [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys] the people into the ultimate state of spiritual suddenly boosted.<br><br>'Men of orders, replaced several elders!' large silhouette instantly flashed in the sky beyond the energy shield, tiger stem a wave, suddenly, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html gps 腕時計 カシオ] a [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html casio 腕時計 ゴールド] dozen strong came from outside the hospital, is flying leap, Under that with the tacit understanding, will cover the maintenance of energy and had a smooth handover.<br><br>hands of the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html カシオ腕時計 メンズ] burden of being received, and that several elders also a big sigh of relief, directed at Hu et al thankful dry arch of the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計 カシオ] hand, and then plunged sky, hunkered down, the rapid recovery of the body consumes vindictiveness.<br>Although Hu dry<br>brought strong outside the hospital, in general than in the strength of the president who is weak on some old, but it is a lot of wins in personnel, so even gorgeous
|
相关的主题文章：
{| border="1" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 1em; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
  <ul>
|+ align=bottom|''Both players''
 
|
  <li>[http://www.battle-ball.com/bbs/aska.cgi http://www.battle-ball.com/bbs/aska.cgi]</li>
! ''Stag''
 
! ''Hare''
  <li>[http://www.danpayneactor.com/cgi/gbdan/guestbook.cgi http://www.danpayneactor.com/cgi/gbdan/guestbook.cgi]</li>
|-
 
! ''Stag''
  <li>[http://www.blogharbor.com/cgi-bin/page.cgi http://www.blogharbor.com/cgi-bin/page.cgi]</li>
| 3, 3
 
| 0, 2
</ul>
|-
! ''Hare''
| 2, 0
| 2, 2
|}
|
{| border="1" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 1em; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
|+ align=bottom|''Just row''
|
! ''Stag''
! ''Hare''
|-
! ''Stag''
| 3
| 0
|-
! ''Hare''
| 2
| 2
|}
|}
 
== Uses of normal form ==
 
=== Dominated strategies ===
{| border="1" align=right cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 1em; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
|+ align=bottom|''The Prisoner's Dilemma''
|
! ''Cooperate''
! ''Defect''
|-
! ''Cooperate''
| −1, −1
| −5, 0
|-
! ''Defect''
| 0, −5
| −2, −2
|}
 
The payoff matrix facilitates elimination of [[dominated strategy|dominated strategies]], and it is usually used to illustrate this concept. For example, in the [[prisoner's dilemma]] (to the right), we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for good. However, if they both defect, they will both be locked up for longer. One can determine that ''Cooperate'' is strictly dominated by ''Defect''. One must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. This shows that no matter what the column player chooses, the row player does better by choosing ''Defect''. Similarly, one compares the second payoff in each row; again 0 > −1 and −2 > −5. This shows that no matter what row does, column does better by choosing ''Defect''. This demonstrates the unique [[Nash equilibrium]] of this game is (''Defect'', ''Defect'').
 
=== Sequential games in normal form ===
[[File:SGPNEandPlainNE explainingexample.svg|300px|thumb|Both extensive and normal form illustration of a sequential form game with subgame imperfect and perfect Nash equilibriium marked with red and blue respectively.]]
 
{| 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%;"
|+ align=bottom |''A sequential game''
|-
|
! ''Left, Left''
! ''Left, Right''
! ''Right, Left''
! ''Right, Right''
|-
! ''Top''
|align=center|4, 3
|align=center|4, 3
|align=center|−1, −1
|align=center|−1, −1
|-
! ''Bottom''
|align=center|0, 0
|align=center|3, 4
|align=center|0, 0
|align=center|3, 4
|}
 
These matrices only represent games in which moves are simultaneous (or, more generally, information is [[perfect information|imperfect]]). The above matrix does not represent the game in which player 1 moves first, observed by player 2, and then player 2 moves, because it does not specify each of player 2's strategies in this case. In order to represent this [[sequential game]] we must specify all of player 2's actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before, ''Left'' and ''Right''. Unlike before he has four strategies, contingent on player 1's actions. The strategies are:
# ''Left if player 1 plays Top and Left otherwise''
# ''Left if player 1 plays Top and Right otherwise''
# ''Right if player 1 plays Top and Left otherwise''
# ''Right if player 1 plays Top and Right otherwise''
On the right is the normal-form representation of this game.
 
== General formulation ==
 
In order for a game to be in normal form, we are provided with the following data:
 
* There is a finite set ''P'' of players, which we label {1, 2, ..., ''m''}
 
* Each player ''k'' in ''P'' has a finite number of [[pure strategy|pure strategies]]
 
::<math> S_k = \{1, 2, \ldots, n_k\}. </math>
 
A ''pure strategy profile'' is an association of strategies to players, that is an ''m''-[[tuple]]
 
:<math> \vec{s} = (s_1, s_2, \ldots,s_m) </math>
 
such that
 
:<math> s_1 \in S_1, s_2 \in S_2, \ldots, s_m \in S_m </math>
 
A ''payoff function'' is a function
 
:<math> F: S_1 \times S_2 \times \ldots \times S_m \rightarrow \mathbb{R}. </math>
 
whose intended interpretation is the award given to a single player at the outcome of the game. Accordingly, to completely specify a game, the payoff function has to be specified for each player in the player set ''P''= {1, 2, ..., ''m''}.
 
'''Definition''': A ''game in normal form'' is a structure
 
:<math> G=\langle P, \mathbf{S}, \mathbf{F}\rangle </math>
 
where:
 
:<math>P=\{1,2, \ldots , m\}</math>
 
is a set of players,
 
:<math>\mathbf{S}= \{S_1, S_2, \ldots, S_m\} </math>
 
is an ''m''-tuple of pure strategy sets, one for each player, and
 
:<math> \mathbf{F} = \{F_1, F_2, \ldots, F_m\} </math>
 
is an ''m''-tuple of payoff functions.
 
== References ==
* D. Fudenberg and [[Jean Tirole|J. Tirole]], ''Game Theory'', MIT Press, 1991.
 
* {{Citation | 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; [http://www.morganclaypool.com/doi/abs/10.2200/S00108ED1V01Y200802AIM003 free online] at many universities.
 
* [[R. Duncan Luce|R. D. Luce]] and [[Howard Raiffa|H. Raiffa]], ''Games and Decisions'', Dover Publications, 1989.
 
* {{Citation | 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 Chapter 3. [http://www.masfoundations.org/download.html Downloadable free online].
 
* J. Weibull, ''Evolutionary Game Theory'', MIT Press, 1996
 
* [[John von Neumann|J. von Neumann]] and [[Oskar Morgenstern|O. Morgenstern]], ''Theory of games and Economic Behavior'',  John Wiley Science Editions, 1964. Which was originally  published in 1944 by Princeton University Press.
 
== External links ==
* http://www.whalens.org/Sofia/choice/matrix.htm
 
{{game theory}}
 
[[Category:Game theory]]

Latest revision as of 20:38, 5 April 2014

the moment somewhat suddenly nodded.

Of the body is the Bodhi saliva? But the door to the heritage Hsiao, how could draw Jean Monnet day trip to take heart or something? Mind flashed wipe suspect this 'confusion', Xiao Yan's eyes suddenly catch カシオ 掛け時計 a glimpse of the Soviet Union one thousand large elders, the moment somewhat suddenly nodded.

Xiao Li into the light beam lasts thirty-four minutes or so, that is out of line, watching its appearance, though not bitterly of 'color', but did not smile much, surely they have come up カシオ 時計 with something black emperor were not give their assessment is too high now.

mind flashed the idea of ​​this road, Xiao Yan is slowly stood up.

the time being as an object of 腕時計 メンズ casio attention, Xiao Yan so move naturally Lima will be a road around the eyes contains a variety of emotions pulled over.

For these eyes, Xiao Yan did not bother physique of a move, floating カシオ 掛け時計 across the air, then it is to fall into a light beam into.

stature casio 腕時計 phys swept into the beam of light, glare 相关的主题文章：

little girl in white this time

, This display Ziyun wing, rest assured that back into the hospital.

With the days 'color' bleak for some reason, Xiao Yan again smooth out those who want to challenge him busybodies, coming into the pan straight into the door, swaggering back to the pavilion, but when it pushed the door Pops, in front of white shadow suddenly Shanlue, so too he hurried Police took カシオ 時計 a step back, take a closer カシオ 時計 電波 look, could not help a bit of shock, the original character asks shadow in front of it, who actually is the so-called 'strong list' first, Warcraft purple research incarnation.

little girl in カシオ腕時計 メンズ white this time, his face looked at the door are looking forward to Xiao Yan, black eyes filled with mouth-watering flavor.

'Well, purple research, how do you come ah?' will bring to the heart カシオ 腕時計 ソーラー to put it down, Xiao Yan cough loudly, laughing, 'knead' the 'rub' Purple research little カシオ腕時計 メンズ head, then that is the flash sideways into the room, into the room, it was found that the children smoked, Hu Jia, Wu Hao, who are in them, and this 相关的主题文章：

some of the strength of incompetence

Are all could not http://nrcil.net/sitemap.xml help but burst of raspberry, is indeed the order of the bucket were other strong, カシオ gps 時計 just the hand, the presence of strong imperial struggle is no one who can cast out.

huge green 'color' spin shrouded wind Yunshan whole body, roared high-speed rotation, gales, many people will be on the plaza beneath the rickety stature are blown, some of the casio 腕時計 phys strength of incompetence, and even more are back again and again, finally and sat on the floor, his face looked up at the horror that almost casio 腕時計 スタンダード across the vast world of the wind spin, this is カシオ 腕時計 チタン like the terrible force of nature like, Yunshan single-handedly that it would be the call out, when really terrible.

wutherin winds wreak havoc in the cloud Arashiyama sky and over, along with the mountains in the woodlands also Kuangmeng surging waves from a tree at the center of a ** green 'color' appeared, and finally into a rounded, swept the four parties .

suddenly looked so big action Yunshan, Xiao Yan is Yi Zheng, immediately surfaced on the pale face 相关的主题文章：

five hundred and sixtieth chapters joint seal

Chapter 65 joint seal!

five hundred and sixtieth chapters joint seal!

Hu dry and others appear, no doubt like a force-like, so too have some of the elders of casio 腕時計 phys the people into the ultimate state of spiritual suddenly boosted.

'Men of orders, replaced several elders!' large silhouette instantly flashed in the sky beyond the energy shield, tiger stem a wave, suddenly, gps 腕時計 カシオ a casio 腕時計 ゴールド dozen strong came from outside the hospital, is flying leap, Under that with the tacit understanding, will cover the maintenance of energy and had a smooth handover.

hands of the カシオ腕時計 メンズ burden of being received, and that several elders also a big sigh of relief, directed at Hu et al thankful dry arch of the 時計 カシオ hand, and then plunged sky, hunkered down, the rapid recovery of the body consumes vindictiveness.
Although Hu dry
brought strong outside the hospital, in general than in the strength of the president who is weak on some old, but it is a lot of wins in personnel, so even gorgeous 相关的主题文章：

Retrieved from "https://en.formulasearchengine.com/index.php?title=Ambipolar_diffusion&oldid=240276"