Fortune's algorithm - Revision history

en>Nimur: /* Algorithm description */Avoid use of "complex": use "complicated". The functions is purely real, composed from a piecewise sequence of polynomial curves.

2015-01-03T10:27:45Z

Algorithm description: Avoid use of "complex": use "complicated". The functions is purely real, composed from a piecewise sequence of polynomial curves.

← Older revision		Revision as of 12:27, 3 January 2015
Line 1:		Line 1:
	~~Nice to meet you~~, ~~my car warranty title~~ [http://~~Www~~.~~Angieslist~~.com/~~articles/what-does-and-doesnt-your-auto-warranty-cover.htm extended~~] ~~car [http://Www.motortrend.com/new_cars/auto_warranty/ warranty] is Ling~~ and ~~I totally dig that title~~. ~~Managing people is how I make money [http://www.miketoerner.com/ActivityFeed/MyProfile/tabid/60/userId/35329/Default.aspx auto warranty]~~ and ~~it's some thing I truly enjoy. Years~~ in ~~the past we moved to Arizona but my wife desires us to move~~. One of the things I adore most is greeting card collecting but I don't have the time recently.<br><br>Also visit extended [http://www.sceltic.at/index.php?mod=users&action=view&id=51301 car warranty] my weblog - extended car warranty ([http://www.Lexhousing.com/ActivityFeed/MyProfile/tabid/56/userId/284/language/en-US/Default.aspx hop over to here])		Parole or Probation Officer Colton Jernigan from Carberry, has several pursuits including croquet, [http://comoficarrico.comoganhardinheiro101.com/ como ganhar dinheiro na internet] and cigar smoking. Is a travel enthusiast and recently made a vacation in Yin Xu.

93.220.17.131: The implementation doesn't work. I have tried it. Saving time and nerves of future readers.

2014-02-06T23:31:13Z

The implementation doesn't work. I have tried it. Saving time and nerves of future readers.

← Older revision		Revision as of 01:31, 7 February 2014
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	~~In [[game theory]], a game is said to be a '''potential game''' if the incentive of all players~~ to ~~change their [[strategy (game theory)\|strategy]] can be expressed using a single global function called the '''potential function'''. The concept was proposed in 1973 by [[Robert W. Rosenthal]].~~		Nice to meet you, my car warranty title [http://Www.Angieslist.com/articles/what-does-and-doesnt-your-auto-warranty-cover.htm extended] car [http://Www.motortrend.com/new_cars/auto_warranty/ warranty] is Ling and I totally dig that title. Managing people is how I make money [http://www.miketoerner.com/ActivityFeed/MyProfile/tabid/60/userId/35329/Default.aspx auto warranty] and it's some thing I truly enjoy. Years in the past we moved to Arizona but my wife desires us to move. One of the things I adore most is greeting card collecting but I don't have the time recently.<br><br>Also visit extended [http://www.sceltic.at/index.php?mod=users&action=view&id=51301 car warranty] my weblog - extended car warranty ([http://www.Lexhousing.com/ActivityFeed/MyProfile/tabid/56/userId/284/language/en-US/Default.aspx hop over to here])

	~~The properties of several types of potential games have since been studied. Games can be either ''ordinal'' or ''cardinal'' potential games. In cardinal games~~, ~~the difference in individual~~ [[payoff]]s for each player from individually changing one's strategy ''[[ceteris paribus]]'' has to have the same value as the difference in values for the potential function. In ordinal games, only the signs of the differences have to be the same.

	The potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure [[Nash equilibrium\|Nash equilibria]] can be found by locating the local optima of the potential function. Convergence and finite-time convergence of an iterated game towards a Nash equilibrium can also be understood by studying the potential function.

	~~==Definition==~~
	~~We will define some notation required for the definition. Let <math>N</math> be the number of players, <math>A<~~/~~math> the set of action profiles over the action sets <math>A_{i}<~~/~~math> of each player and <math>u</math> be the payoff function~~.

	~~A game <math>G=(N,A=A_{1}\times\ldots\times A_{N}, u: A \rightarrow \reals^N) <~~/~~math> is:~~

	* an '''exact potential game''' if there is a function <math>\Phi: A \rightarrow \reals</~~math> such that <math> \forall {a_{~~-~~i}\in A_{~~-~~i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,~~
	~~::<math> \Phi(a'_{i},a_{~~-~~i})~~-~~\Phi(a''_{i},a_{~~-~~i}) = u_{i}(a'_{i},a_{~~-~~i})~~-~~u_{i}(a''_{i},a_{-i})</math>~~

	~~::That is~~: ~~when player <math>i<~~/~~math> switches from action <math>a'<~~/~~math> to action <math>a''</math>, the change in the potential equals the change in the utility of that player~~.

	* a '''weighted potential game''' if there is a function <math>\Phi: A \rightarrow \reals</~~math> and a vector <math>w \in \reals_{++}^N<~~/~~math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}<~~/~~math>,~~
	~~::<math> \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) = w_{i}(u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i}))</math>~~

	* an '''ordinal potential game''' if there is ~~a function <math>\Phi: A \rightarrow \reals</math> such~~ that ~~<math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,~~
	~~::<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Leftrightarrow~~
	~~\Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i})>0</math>~~

	* a '''generalized ordinal potential game''' if there is ~~a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\~~ ~~\forall {a'_{i},\ a''_{i}\in A_{i}}</math>,~~
	::~~<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Rightarrow~~
	~~\Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) >0 <~~/~~math>~~

	*a '''best-response potential game''' if there is a function <math>\Phi: A \rightarrow \reals</~~math> such that <math>\forall i\in N,\ \forall {a_{-i}\in A_{-i}}</math>,~~
	~~::<math>b_i(a_{-i})=\arg\max_{a_i\in A_i} \Phi(a_i,a_{-i})</math>~~
	~~where <math>b_i(a_{-i})</math> is the best payoff for player <math>i</math> given <math>a_{-i}</math>~~.

	~~==A simple example==~~
	~~{{Payoff matrix \| Name = Fig~~. ~~1: Potential game example~~
	~~\| 2L = +1 \| 2R = –1 \|~~
	~~1U = +1 \| UL = <small>{{nowrap\|+b<sub>1<~~/~~sub>+w, +b<sub>2<~~/~~sub>+w}}<~~/~~small> \| UR = <small>{{nowrap\|+b<sub>1<~~/~~sub>–w, –b<sub>2<~~/~~sub>–w}}<~~/~~small> \|~~
	~~1D = –1 \| DL = <small>{{nowrap\|–b<sub>1<~~/~~sub>–w, +b<sub>2</sub>–w}}</small> \| DR = <small>{{nowrap\|–b<sub>1</sub>+w, –b<sub>2</sub>+w}}</small> }}~~
	~~In a 2-player, 2-strategy game with externalities, individual players~~' ~~payoffs are given by the function {{nowrap\|u<sub>i</sub>(~~s<sub>i</sub>, s<sub>j</sub>)}} = {{nowrap\|b<sub>i</sub> s<sub>i</sub> + w s<sub>i</sub> s<sub>j</sub>}}, where s<sub>i</sub> is players i's strategy, {{nowrap\|s<sub>j</sub>}} is the opponent's strategy, and w is a positive [[externality]] from choosing the same strategy. ~~The strategy choices are +1 and −1, as seen~~ in the ~~[[payoff matrix]] in Figure 1~~.

	~~This game has a potential function {{nowrap\|P(s<sub>1</sub>, s<sub>2</sub>)}} = {{nowrap\|b<sub>1</sub> s<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> + w s<sub>1</sub> s<sub>2</sub>}}.~~

	~~If player 1 moves from −1 to +1,~~ the ~~payoff difference~~ is ~~Δu<sub>1</sub> = {{nowrap\|u<sub>1</sub>(+1, s<sub>2</sub>) – u<sub>1</sub>(–1, s<sub>2</sub>)}} = {{nowrap\|2 b<sub>1</sub> + 2 w s<sub>2</sub>}}~~.

	~~The change in potential is ΔP = {{nowrap\|P(+1, s~~<~~sub~~>2<~~/sub~~>~~) – P(–1, s<sub>2<~~/~~sub>)}} = {{nowrap\|(b<sub>1<~~/~~sub> + b<sub>2</sub> s<sub>2</sub> + w s<sub>2</sub>) – (–b<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> – w s<sub>2</sub>)}} = {{nowrap\|2 b<sub>1</sub> + 2 w s<sub>2</sub>}} = Δu<sub>1</sub>~~.

	~~The solution for player 2 is equivalent~~. ~~Using numerical values b<sub>1<~~/~~sub> ~~=&~~nbsp;2, b<sub>2</sub> ~~= −1, w = 3, this example transforms into a simple [[battle of the sexes (game theory)\|battle of the sexes]], as shown in Figure 2. The game has two pure Nash equilibria, (+1, +1) and (−1, −1). These are also the local maxima of the potential function (Figure 3). The only [[stochastically stable equilibrium]] is (+1, +1), the global maximum of the potential function.
	~~<center>~~
	{\|
	\|-
	~~\| width=50% \|~~
	~~{{Payoff matrix \| Name = Fig. 2: Battle of the sexes (payoffs)~~
	~~\| 2L = +1 \| 2R = –1 \|~~
	~~1U = +1 \| UL = 5, 2 \| UR = –1, –2 \|~~
	~~1D = –1 \| DL = –5, –4 \| DR = 1, 4 }}~~
	~~\| width=50% \|~~
	~~{{Payoff matrix \| Name = Fig. 3: Battle of the sexes (potentials)~~
	~~\| 2L = +1 \| 2R = –1 \|~~
	~~1U = +1 \| UL = '''4''' \| UR~~ = ~~'''0''' \|~~
	~~1D = –1 \| DL = '''–6''' \| DR = '''2''' }}~~
	\|}
	~~</center>~~
	~~A 2-player, 2-strategy game cannot be a potential game unless~~
	~~:<math>~~
	~~[u_{1}(+1,-1)+u_1(-1,+1)]-[u_1(+1,+1)+u_1(-1,-1)~~] =
	~~[u_{2}(+1,~~-~~1)+u_2~~(~~-1,+1)]-[u_2(+1,+1)+u_2(-1,-1)]~~
	~~</math>~~

	~~==References==~~
	* Dov Monderer and Lloyd S. Shapley: "Potential Games", ''Games and Economic Behavior'' 14, pp. 124–143 (1996).
	* Emile Aarts and Jan Korst: ''Simulated Annealing and Boltzmann Machines'', John Wiley & Sons (1989) ISBN 0-471-92146-7

	~~==External links==~~
	* Lecture notes of Yishay Mansour about [http://www.~~math~~.~~tau.ac.il~~/~~~mansour~~/~~course_games~~/~~scribe~~/~~lecture6~~.~~pdf Potential and congestion games]~~

	~~[[Category:Game theory]~~]

131.155.204.198: Removed a line that was outdated in Pseudocode.

2013-11-27T13:27:45Z

Removed a line that was outdated in Pseudocode.

← Older revision		Revision as of 15:27, 27 November 2013
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	~~Hello friend~~. ~~Allow me introduce myself~~. ~~I am Luther Aubrey~~. The ~~preferred pastime~~ for ~~my children~~ and me is ~~dancing~~ and ~~now I~~'~~m attempting to earn cash with it~~. ~~Arizona has always been my living place but my spouse wants us to move~~. ~~Interviewing~~ is ~~what she does.~~<br><br>~~Here~~ is ~~my page~~ [~~http:~~//~~eric~~.~~jandjsitebuilders~~.~~com~~/~~UserProfile~~/~~tabid~~/~~109~~/~~userId~~/~~106643~~/~~Default~~.~~aspx~~ http://~~eric~~.~~jandjsitebuilders~~.~~com~~/]		In [[game theory]], a game is said to be a '''potential game''' if the incentive of all players to change their [[strategy (game theory)\|strategy]] can be expressed using a single global function called the '''potential function'''. The concept was proposed in 1973 by [[Robert W. Rosenthal]].

			The properties of several types of potential games have since been studied. Games can be either ''ordinal'' or ''cardinal'' potential games. In cardinal games, the difference in individual [[payoff]]s for each player from individually changing one's strategy ''[[ceteris paribus]]'' has to have the same value as the difference in values for the potential function. In ordinal games, only the signs of the differences have to be the same.

			The potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure [[Nash equilibrium\|Nash equilibria]] can be found by locating the local optima of the potential function. Convergence and finite-time convergence of an iterated game towards a Nash equilibrium can also be understood by studying the potential function.

			==Definition==
			We will define some notation required for the definition. Let <math>N</math> be the number of players, <math>A</math> the set of action profiles over the action sets <math>A_{i}</math> of each player and <math>u</math> be the payoff function.

			A game <math>G=(N,A=A_{1}\times\ldots\times A_{N}, u: A \rightarrow \reals^N) </math> is:

			* an '''exact potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
			::<math> \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) = u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})</math>

			::That is: when player <math>i</math> switches from action <math>a'</math> to action <math>a''</math>, the change in the potential equals the change in the utility of that player.

			* a '''weighted potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> and a vector <math>w \in \reals_{++}^N</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
			::<math> \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) = w_{i}(u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i}))</math>

			* an '''ordinal potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
			::<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Leftrightarrow
			\Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i})>0</math>

			* a '''generalized ordinal potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
			::<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Rightarrow
			\Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) >0 </math>

			*a '''best-response potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math>\forall i\in N,\ \forall {a_{-i}\in A_{-i}}</math>,
			::<math>b_i(a_{-i})=\arg\max_{a_i\in A_i} \Phi(a_i,a_{-i})</math>
			where <math>b_i(a_{-i})</math> is the best payoff for player <math>i</math> given <math>a_{-i}</math>.

			==A simple example==
			{{Payoff matrix \| Name = Fig. 1: Potential game example
			\| 2L = +1 \| 2R = –1 \|
			1U = +1 \| UL = <small>{{nowrap\|+b<sub>1</sub>+w, +b<sub>2</sub>+w}}</small> \| UR = <small>{{nowrap\|+b<sub>1</sub>–w, –b<sub>2</sub>–w}}</small> \|
			1D = –1 \| DL = <small>{{nowrap\|–b<sub>1</sub>–w, +b<sub>2</sub>–w}}</small> \| DR = <small>{{nowrap\|–b<sub>1</sub>+w, –b<sub>2</sub>+w}}</small> }}
			In a 2-player, 2-strategy game with externalities, individual players' payoffs are given by the function {{nowrap\|u<sub>i</sub>(s<sub>i</sub>, s<sub>j</sub>)}} = {{nowrap\|b<sub>i</sub> s<sub>i</sub> + w s<sub>i</sub> s<sub>j</sub>}}, where s<sub>i</sub> is players i's strategy, {{nowrap\|s<sub>j</sub>}} is the opponent's strategy, and w is a positive [[externality]] from choosing the same strategy. The strategy choices are +1 and −1, as seen in the [[payoff matrix]] in Figure 1.

			This game has a potential function {{nowrap\|P(s<sub>1</sub>, s<sub>2</sub>)}} = {{nowrap\|b<sub>1</sub> s<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> + w s<sub>1</sub> s<sub>2</sub>}}.

			If player 1 moves from −1 to +1, the payoff difference is Δu<sub>1</sub> = {{nowrap\|u<sub>1</sub>(+1, s<sub>2</sub>) – u<sub>1</sub>(–1, s<sub>2</sub>)}} = {{nowrap\|2 b<sub>1</sub> + 2 w s<sub>2</sub>}}.

			The change in potential is ΔP = {{nowrap\|P(+1, s<sub>2</sub>) – P(–1, s<sub>2</sub>)}} = {{nowrap\|(b<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> + w s<sub>2</sub>) – (–b<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> – w s<sub>2</sub>)}} = {{nowrap\|2 b<sub>1</sub> + 2 w s<sub>2</sub>}} = Δu<sub>1</sub>.

			The solution for player 2 is equivalent. Using numerical values b<sub>1</sub> = 2, b<sub>2</sub> = −1, w = 3, this example transforms into a simple [[battle of the sexes (game theory)\|battle of the sexes]], as shown in Figure 2. The game has two pure Nash equilibria, (+1, +1) and (−1, −1). These are also the local maxima of the potential function (Figure 3). The only [[stochastically stable equilibrium]] is (+1, +1), the global maximum of the potential function.
			<center>
			{\|
			\|-
			\| width=50% \|
			{{Payoff matrix \| Name = Fig. 2: Battle of the sexes (payoffs)
			\| 2L = +1 \| 2R = –1 \|
			1U = +1 \| UL = 5, 2 \| UR = –1, –2 \|
			1D = –1 \| DL = –5, –4 \| DR = 1, 4 }}
			\| width=50% \|
			{{Payoff matrix \| Name = Fig. 3: Battle of the sexes (potentials)
			\| 2L = +1 \| 2R = –1 \|
			1U = +1 \| UL = '''4''' \| UR = '''0''' \|
			1D = –1 \| DL = '''–6''' \| DR = '''2''' }}
			\|}
			</center>
			A 2-player, 2-strategy game cannot be a potential game unless
			:<math>
			[u_{1}(+1,-1)+u_1(-1,+1)]-[u_1(+1,+1)+u_1(-1,-1)] =
			[u_{2}(+1,-1)+u_2(-1,+1)]-[u_2(+1,+1)+u_2(-1,-1)]
			</math>

			==References==
			* Dov Monderer and Lloyd S. Shapley: "Potential Games", ''Games and Economic Behavior'' 14, pp. 124–143 (1996).
			* Emile Aarts and Jan Korst: ''Simulated Annealing and Boltzmann Machines'', John Wiley & Sons (1989) ISBN 0-471-92146-7

			==External links==
			* Lecture notes of Yishay Mansour about [http://www.math.tau.ac.il/~mansour/course_games/scribe/lecture6.pdf Potential and congestion games]

			[[Category:Game theory]]

en>Mtthll: add link to GitHub JavaScript library

2012-05-22T19:18:14Z

add link to GitHub JavaScript library

New page

Hello friend. Allow me introduce myself. I am Luther Aubrey. The preferred pastime for my children and me is dancing and now I'm attempting to earn cash with it. Arizona has always been my living place but my spouse wants us to move. Interviewing is what she does.<br><br>Here is my page [http://eric.jandjsitebuilders.com/UserProfile/tabid/109/userId/106643/Default.aspx http://eric.jandjsitebuilders.com/]