(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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Definitions of the dynamics can be \ found in the closed group at the end of this section.\nDetails: If you use \ Logit[\[Eta]] or ILogit[\[Eta]], be sure to enter a value for the noise \ parameter \[Eta] in the closed group below. Making \[Eta] very small \ generates close approximations of the best response dynamic. Similarly, if \ you use SelMut[MM], enter the mutation matrix MM in its closed group. If you \ use Projection and if you are not using ", StyleBox["Mathematica", FontSlant->"Italic"], " version 5 or above, do not take initial condtions on the boundary. 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Note that the selection-mutation dynamic is only well defined for \ nonnegative payoffs, and that it is not invariant to the addition of a \ constant to all matrix entries.", FontFamily->"Palatino"] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"MM", "=", RowBox[{"(", GridBox[{ {"1", "0", "0"}, {"0", "1", "0"}, {"0", "0", "1"} }], ")"}]}], " ", ";"}]], "Input"] }, Closed]], Cell[BoxData[GridBox[{ { ButtonBox["Replicator"], ButtonBox[\(Logit[\[Eta]]\)], ButtonBox["BNN"], ButtonBox["PD"], ButtonBox["Projection"], ButtonBox["Combined"]}, { ButtonBox["MSReplicator"], ButtonBox[\(ILogit[\[Eta]]\)], ButtonBox["EP"], ButtonBox["PC"], ButtonBox[\(SelMut[MM]\)], ButtonBox["Other"]} }, RowSpacings->0, ColumnSpacings->0, GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]]], "Input", Active->True, Evaluatable->False], Cell[BoxData[ \(\(Dynamic = Replicator\ ;\)\)], "Input"], Cell[TextData[{ "If the dynamic you consider has a nontrivial component of rest points, the \ program will compute and plot a random subset of this component. If you set \ ", StyleBox["findrestpoints", FontWeight->"Bold"], " equal to 0, the computation and plotting is skipped." }], "Text", FontFamily->"Palatino"], Cell[BoxData[ \(\(findrestpoints = 1\ ;\)\)], "Input"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Definition of combination of two dynamics", FontSlant->"Italic"]], "SmallText"], Cell["\<\ To define a combination of two dynamics, specify the two dynamics \ to be combined, the weights on the dynamics, and the characterization of the \ rest points of combined dynamic. Nash is the default setting for the set or \ rest points. This is correct, for example, with combinations of the \ replicator dynamic and an excess payoff dynamic like the BNN dynamic. If the \ rest points aren't identical to the Nash equilibria for the dynamics you have \ specified, you can replace Nash with Automatic if the combined dynamic is \ smooth. (If the combined dynamic isn't smooth, you're on your own!) The \ last line of code formally defines the combined dynamic according to your \ specificcations.\ \>", "Text", FontFamily->"Palatino"], Cell[BoxData[{ \(\(dyn[1] = Replicator\ ;\)\), "\[IndentingNewLine]", \(\(dyn[2] = BNN\ ;\)\), "\[IndentingNewLine]", \(\(weight[1] = .9\ ;\)\), "\[IndentingNewLine]", \(\(weight[2] = .1\ ;\)\), "\[IndentingNewLine]", \(\(RPCharacterization[Combined] = Nash\ ;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Combined[x_, F_] := weight[1]\ \ \(dyn[1]\)[x, F] + \ weight[2]\ \(dyn[2]\)[x, F]\ ;\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Definitions of dynamics", FontSlant->"Italic"]], "SmallText", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[{ \(\(Clear[Replicator]\ ;\)\ \ \), "\[IndentingNewLine]", \(\(Clear[BNN]\ \ ;\)\), "\[IndentingNewLine]", \(\(Clear[Logit]\ ;\)\), "\[IndentingNewLine]", \(\(Clear[rho]\ ;\)\), "\[IndentingNewLine]", \(\(Clear[PD]\ ;\)\ \), "\[IndentingNewLine]", \(\(Clear[rho1]\ ;\)\), "\[IndentingNewLine]", \(\(Clear[SelMut]\ ;\)\)}], "Input", CellOpen->False, InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Replicator Dynamic", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(Replicator[x_, F_] := {x\[LeftDoubleBracket]1\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]1\[RightDoubleBracket], x\[LeftDoubleBracket]2\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]2\[RightDoubleBracket], x\[LeftDoubleBracket]3\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]3\[RightDoubleBracket]}\ ;\)\)], \ "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[Replicator] = Automatic\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Maynard Smith Replicator Dynamic", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(MSReplicator[x_, F_] := \(1\/Fbar[x, F]\) {x\[LeftDoubleBracket]1\[RightDoubleBracket]\ \ \(Fhat[ x, F]\)\[LeftDoubleBracket]1\[RightDoubleBracket], x\[LeftDoubleBracket]2\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]2\[RightDoubleBracket], x\[LeftDoubleBracket]3\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]3\[RightDoubleBracket]}\ ;\)\)], \ "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[MSReplicator] = Automatic\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Logit Dynamic", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(Logit[\[Eta]_]\)[x_, F_] := {\[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket]\/\[Eta]\)\/\(\[Sum]\+\(i = 1\)\%3 \ \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]i\[RightDoubleBracket]\/\[Eta]\ \)\) - x\[LeftDoubleBracket]1\[RightDoubleBracket], \[ExponentialE]\^\(\(F[x]\ \)\[LeftDoubleBracket]2\[RightDoubleBracket]\/\[Eta]\)\/\(\[Sum]\+\(i = \ 1\)\%3 \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]i\[RightDoubleBracket]\/\ \[Eta]\)\) - x\[LeftDoubleBracket]2\[RightDoubleBracket], \ \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]3\[RightDoubleBracket]\/\[Eta]\ \)\/\(\[Sum]\+\(i = 1\)\%3 \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]i\ \[RightDoubleBracket]\/\[Eta]\)\) - x\[LeftDoubleBracket]3\[RightDoubleBracket]}\ ;\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[Logit[_]] = Automatic\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["I-Logit Dynamic", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(ILogit[\[Eta]_]\)[x_, F_] := {\(x\[LeftDoubleBracket]1\[RightDoubleBracket]\ \ \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\/\[Eta]\ \)\)\/\(\[Sum]\+\(i = 1\)\%3 x\[LeftDoubleBracket]i\[RightDoubleBracket] \ \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]i\[RightDoubleBracket]\/\[Eta]\ \)\)\ - x\[LeftDoubleBracket]1\[RightDoubleBracket], \ \(x\[LeftDoubleBracket]2\[RightDoubleBracket]\ \[ExponentialE]\^\(\(F[x]\)\ \[LeftDoubleBracket]2\[RightDoubleBracket]\/\[Eta]\)\)\/\(\[Sum]\+\(i = \ 1\)\%3 x\[LeftDoubleBracket]i\[RightDoubleBracket] \ \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]i\[RightDoubleBracket]\/\[Eta]\ \)\) - x\[LeftDoubleBracket]2\[RightDoubleBracket], \(x\[LeftDoubleBracket]3\ \[RightDoubleBracket]\ \[ExponentialE]\^\(\(F[x]\)\[LeftDoubleBracket]3\ \[RightDoubleBracket]\/\[Eta]\)\)\/\(\[Sum]\+\(i = 1\)\%3 x\ \[LeftDoubleBracket]i\[RightDoubleBracket] \[ExponentialE]\^\(\(F[x]\)\ \[LeftDoubleBracket]i\[RightDoubleBracket]\/\[Eta]\)\) - x\[LeftDoubleBracket]3\[RightDoubleBracket]}\ ;\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[ILogit[_]] = Automatic\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["BNN dynamic", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(BNN[x_, F_] := {\(Fhatplus[x, F]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - x\[LeftDoubleBracket]1\[RightDoubleBracket]\ \(\[Sum]\+\(i = \ 1\)\%3\( Fhatplus[x, F]\)\[LeftDoubleBracket] i\[RightDoubleBracket]\), \(Fhatplus[x, F]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - x\[LeftDoubleBracket]2\[RightDoubleBracket]\ \(\[Sum]\+\(i = \ 1\)\%3\( Fhatplus[x, F]\)\[LeftDoubleBracket] i\[RightDoubleBracket]\), \(Fhatplus[x, F]\)\[LeftDoubleBracket]3\[RightDoubleBracket] - x\[LeftDoubleBracket]3\[RightDoubleBracket]\ \(\[Sum]\+\(i = \ 1\)\%3\( Fhatplus[x, F]\)\[LeftDoubleBracket] i\[RightDoubleBracket]\)}\ ;\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[BNN] = Nash\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Excess payoff dynamic", "Subsubsection", InitializationCell->True], Cell[TextData[{ "Here one specifies the vector field sigmatilde that defines an excess \ payoff dynamic. This definition should be stated in terms of the excess \ payoff vector Fhat[x,F], which is defined as Fhat[x,F] = F[x] - Fbar[x,F]. \ (", StyleBox["Mathematica", FontSlant->"Italic"], " subtracts the scalar from the vector correctly.) When computing rest \ points, the program assumes that sigmatilde is acute, so that the rest points \ of the dynamic are the Nash equilibria of the underlying game." }], "Text", InitializationCell->True, FontFamily->"Palatino"], Cell[BoxData[ \(\(sigmatilde[x_, F_] := \ {\((Max[ 0, \(Fhat[x, F]\)\[LeftDoubleBracket]1\[RightDoubleBracket]])\)^2, \ \[IndentingNewLine]\((Max[ 0, \(Fhat[x, F]\)\[LeftDoubleBracket]2\[RightDoubleBracket]])\)^2, \ \((Max[0, \(Fhat[x, F]\)\[LeftDoubleBracket]3\[RightDoubleBracket]])\)^2};\)\)\ ], "Input", InitializationCell->True], Cell[BoxData[ \(\(EP[x_, F_] := {\(sigmatilde[x, F]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - x\[LeftDoubleBracket]1\[RightDoubleBracket]\ \(\[Sum]\+\(i = \ 1\)\%3\( sigmatilde[x, F]\)\[LeftDoubleBracket] i\[RightDoubleBracket]\), \(sigmatilde[x, F]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - x\[LeftDoubleBracket]2\[RightDoubleBracket]\ \(\[Sum]\+\(i = \ 1\)\%3\( sigmatilde[x, F]\)\[LeftDoubleBracket] i\[RightDoubleBracket]\), \(sigmatilde[x, F]\)\[LeftDoubleBracket]3\[RightDoubleBracket] - x\[LeftDoubleBracket]3\[RightDoubleBracket]\ \(\[Sum]\+\(i = \ 1\)\%3\( sigmatilde[x, F]\)\[LeftDoubleBracket] i\[RightDoubleBracket]\)}\ ;\)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(\(RPCharacterization[EP] = Nash\ \ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True], Cell[BoxData[""], "Input", InitializationCell->True], Cell["Pairwise Difference Dynamic", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(rho[x_, F_] := {{0, Max[\(F[x]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket], 0], Max[\(F[x]\)\[LeftDoubleBracket]3\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket], 0]}, {Max[\(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]2\[RightDoubleBracket], 0], 0, Max[\(F[x]\)\[LeftDoubleBracket]3\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]2\[RightDoubleBracket], 0]}, {Max[\(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]3\[RightDoubleBracket], 0], Max[\(F[x]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]3\[RightDoubleBracket], 0], 0}};\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(PD[x_, F_] := Transpose[rho[x, F]] . \ x - {{x\[LeftDoubleBracket]1\[RightDoubleBracket], 0, 0}, {0, x\[LeftDoubleBracket]2\[RightDoubleBracket], 0}, {0, 0, x\[LeftDoubleBracket]3\[RightDoubleBracket]}} . rho[x, F] . {1, 1, 1}\ ;\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[PD] = Nash\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Pairwise Comparison Dynamic", "Subsubsection", InitializationCell->True], Cell["\<\ Here one specifies the vector field rho that defines an excess \ payoff dynamic. When computing rest points, the program assumes that rho is \ separable and sign-preserving, so that the rest points of the dynamic are the \ Nash equilibria of the underlying game.\ \>", "Text", InitializationCell->True], Cell[BoxData[ \(\(rho1[x_, F_] := {{0, Max[\(F[x]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket], 0], Max[\(F[x]\)\[LeftDoubleBracket]3\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket], 0]}, {Max[\(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]2\[RightDoubleBracket], 0], 0, Max[\(F[x]\)\[LeftDoubleBracket]3\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]2\[RightDoubleBracket], 0]}, {Max[\(F[ x]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]3\[RightDoubleBracket], 0], Max[\(F[x]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - \(F[ x]\)\[LeftDoubleBracket]3\[RightDoubleBracket], 0], 0}};\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(PC[x_, F_] := Transpose[rho1[x, F]] . \ x - {{x\[LeftDoubleBracket]1\[RightDoubleBracket], 0, 0}, {0, x\[LeftDoubleBracket]2\[RightDoubleBracket], 0}, {0, 0, x\[LeftDoubleBracket]3\[RightDoubleBracket]}} . rho1[x, F] . {1, 1, 1}\ ;\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[PC] = Nash\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Projection Dynamic", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(closetozero = 10^\((\(-8\))\)\ ;\)\)], "Input", InitializationCell->True], Cell[BoxData[{ \(\(\[Pi]bir[{aa_, bb_, cc_}, F_] := Evaluate[ Which[aa > closetozero && bb > closetozero\ && \ cc > closetozero\ , \ F[{aa, bb, cc}], \ aa > closetozero\ && \ bb > closetozero\ && \ cc \[LessEqual] \ closetozero\ , \ {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket]}, aa > closetozero\ && \ bb \[LessEqual] closetozero\ && \ cc > \ closetozero, {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket]}, aa \[LessEqual] \ closetozero\ && bb > closetozero\ && \ cc > \ closetozero, {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}, aa > closetozero\ && bb \[LessEqual] \ closetozero\ && \ cc \[LessEqual] \ closetozero\ \ && \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket] \ \[GreaterEqual] \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], \ {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket]}, aa > closetozero\ && bb \[LessEqual] \ closetozero\ && \ cc \[LessEqual] \ closetozero\ && \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket] < \ \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket]}, aa \[LessEqual] \ closetozero\ && bb \[LessEqual] \ closetozero\ && \ cc\ > closetozero\ \ && \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket] \ \[GreaterEqual] \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \ {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket]}, aa \[LessEqual] \ closetozero\ && bb \[LessEqual] \ closetozero\ && \ cc > \ closetozero\ \ && \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket] < \ \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}, aa \[LessEqual] \ closetozero\ && bb > \ closetozero\ && \ cc\ \[LessEqual] closetozero\ && \ \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket] \ \[GreaterEqual] \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], \ {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}, aa \[LessEqual] \ closetozero\ && bb > \ closetozero\ && \ cc\ \[LessEqual] closetozero\ && \ \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket] < \ \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], {\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \ \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket], \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket]}]\ ]\ \ ;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(InUse[{aa_, bb_, cc_}] := \ Evaluate[ If[aa > closetozero, 1, 0] + If[bb > closetozero, 1, 0] + If[cc > closetozero, 1, 0]]\ ;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(sstar[{aa_, bb_, cc_}, F_] := \ \((\ s = InUse[{aa, bb, cc}]\ ; \ Do[\ Evaluate[If[s \[GreaterEqual] 3, s = 3; \ Break[], s = s]]; Evaluate[ If[1\/s\ \(\[Sum]\+\(j = 1\)\%s\( \[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] j\[RightDoubleBracket]\)\ \[LessEqual] \ \(\[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] s + 1\[RightDoubleBracket]\ , s = s + 1]], \ {i, 3}]; Return[s])\)\ ;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Projection[{aa_, bb_, cc_}, F_] := {Evaluate[ If[\((aa > closetozero\ || \ \((\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket] \ \[GreaterEqual] \((\(1\/sstar[{aa, bb, cc}, F]\) \((\[Sum]\+\(j = 1\)\%\(sstar[{aa, bb, \ cc}, F]\)\(\[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] j\[RightDoubleBracket])\))\))\))\), \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - \ \((\(1\/sstar[{aa, bb, cc}, F]\) \((\[Sum]\+\(j = 1\)\%\(sstar[{aa, bb, cc}, F]\ \)\(\[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] j\[RightDoubleBracket])\))\), 0]], Evaluate[ If[\((bb > closetozero\ || \ \((\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket] \ \[GreaterEqual] \ \((\(1\/sstar[{aa, bb, cc}, F]\) \((\[Sum]\+\(j = 1\)\%\(sstar[{aa, bb, \ cc}, F]\)\(\[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] j\[RightDoubleBracket])\))\))\))\), \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - \ \((\(1\/sstar[{aa, bb, cc}, F]\) \((\[Sum]\+\(j = 1\)\%\(sstar[{aa, bb, cc}, F]\ \)\(\[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] j\[RightDoubleBracket])\))\), 0]], Evaluate[ If[\((cc > closetozero\ || \ \((\(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket] \ \[GreaterEqual] \ \((\(1\/sstar[{aa, bb, cc}, F]\) \((\[Sum]\+\(j = 1\)\%\(sstar[{aa, bb, \ cc}, F]\)\(\[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] j\[RightDoubleBracket])\))\))\))\), \(F[{aa, bb, cc}]\)\[LeftDoubleBracket]3\[RightDoubleBracket] - \ \((\(1\/sstar[{aa, bb, cc}, F]\) \((\[Sum]\+\(j = 1\)\%\(sstar[{aa, bb, cc}, F]\ \)\(\[Pi]bir[{aa, bb, cc}, F]\)\[LeftDoubleBracket] j\[RightDoubleBracket])\))\), 0]]\ }\ ;\)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(\(RPCharacterization[Projection] = Nash\ ;\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Selection-mutation", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(SelMut[MM_]\)[x_, F_] := \ Transpose[ MM]\ . \ {{x[\([1]\)], 0, 0}, {0, x[\([2]\)], 0}, {0, 0, x[\([3]\)]}} . F[x] - \((x . F[x])\)\ x\ ;\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(\(RPCharacterization[SelMut[___]] = Automatic\ ;\)\(\[IndentingNewLine]\)\)\)], "Input", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["Other", "Subsubsection", InitializationCell->True, CellTags->"dynamicslibrary"], Cell["\<\ Define any other dynamic you want to use. When defining be sure you \ follow same style with previously defined dynamics. Otherwise, you may need \ to do major changes in the program.\ \>", "Text", InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(Other[x_, F_] := {x\[LeftDoubleBracket]1\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]1\[RightDoubleBracket], x\[LeftDoubleBracket]2\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]2\[RightDoubleBracket], x\[LeftDoubleBracket]3\[RightDoubleBracket]\ \ \(Fhat[x, F]\)\[LeftDoubleBracket]3\[RightDoubleBracket]};\)\)], "Input",\ InitializationCell->True, CellTags->"dynamicslibrary"], Cell[BoxData[ \(\(RPCharacterization[Other] = Automatic\ \ ;\)\)], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Choice of contour function", "Section", Evaluatable->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Choose a contour function here. To use one of the ", ButtonBox["built-in contour functions"], ", select the rhs of \"", StyleBox["SimplexContourFunction", FontWeight->"Bold"], " = xxx\" (excluding the semicolon) and then press the button corresponding \ to the contour function you desire. If you choose \ StableGameLyapunov[Dynamic], the program will use the Lyapunov function that \ is appropriate for the dynamic you selected in the previous section. ", StyleBox["SphereContourFunction", FontWeight->"Bold"], " is used for phase diagrams drawn on the surface of the sphere (see the \ next section)." }], "Text", FontFamily->"Palatino"], Cell[BoxData[ RowBox[{GridBox[{ { ButtonBox["Speed"]}, { ButtonBox["L1Speed"]}, { ButtonBox["RSSpeed"]} }, RowSpacings->0, ColumnSpacings->0, GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]], " ", GridBox[{ { ButtonBox["NormalFormPotential"]}, { ButtonBox["CongestionPotential"]}, { ButtonBox["NormalFormLogitPotential"]}, { ButtonBox["CongestionLogitPotential"]} }, RowSpacings->0, ColumnSpacings->0, GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]], " ", GridBox[{ { ButtonBox[\(StableGameLyapunov[Dynamic]\)]} }, RowSpacings->0, ColumnSpacings->0, GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]]}]], "Input", Active->True, Evaluatable->False], Cell[BoxData[ \(\(SimplexContourFunction = Speed\ ;\)\)], "Input"], Cell[BoxData[ \(\(SphereContourFunction = Speed\ ;\)\)], "Input"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Definitions of contour functions", FontSlant->"Italic"]], "SmallText", InitializationCell->True, CellTags->"contourfunctions"], Cell[CellGroupData[{ Cell["Speed functions", "Subsubsection", CellTags->"contourfunctions"], Cell["\<\ Speed is the standard (Euclidean) speed on the simplex. L1Speed is \ self-explanatory. 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