(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 371274, 9067] NotebookOptionsPosition[ 356037, 8579] NotebookOutlinePosition[ 357714, 8633] CellTagsIndexPosition[ 357326, 8621] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ The game of active SETI: Breaking the \[OpenCurlyDoubleQuote]Great Silence\[CloseCurlyDoubleQuote]\ \>", "Title", CellChangeTimes->{{3.518444847699396*^9, 3.518444862725532*^9}, { 3.540898563090246*^9, 3.5408985647587967`*^9}}, TextAlignment->Center, TextJustification->0.], Cell["Supplementary Material", "Subtitle", CellChangeTimes->{{3.518444869253996*^9, 3.51844488478071*^9}}, TextAlignment->Center, TextJustification->0.], Cell["Harold P. de Vladar", "Text", CellChangeTimes->{{3.518444889439906*^9, 3.51844491550877*^9}}, TextAlignment->Center, TextJustification->0.], Cell["IST Austria. Am Campus 1. A-3400 Klosterneuburg, Austria.", "Text", CellChangeTimes->{{3.518444922303594*^9, 3.5184449316637583`*^9}, 3.5408985993004923`*^9}, TextAlignment->Center, TextJustification->0., FontSlant->"Italic"], Cell["e-mail: Harold.Vladar@ist.ac.at", "Text", CellChangeTimes->{{3.518444933373866*^9, 3.5184449447139263`*^9}, { 3.540898607386298*^9, 3.5408986123884563`*^9}}, TextAlignment->Center, TextJustification->0., FontSlant->"Italic"], Cell[CellGroupData[{ Cell["Readme first", "Subsection", CellChangeTimes->{{3.5408986304114447`*^9, 3.54089865891899*^9}}], Cell["\<\ This mathematica notebook contains the step-by-step derivations, function \ implementations and simulations of the content in the main text. The notebook \ can be read as a supplementary material, but can also be used to explore the \ model and framework beyond some of the assumptions of the main text. \ \>", "Text", CellChangeTimes->{{3.544150744962956*^9, 3.544150865072877*^9}, { 3.544150936769473*^9, 3.544151009065749*^9}, {3.5441639741201887`*^9, 3.544163974621622*^9}}], Cell["\<\ The section \"Implemented functions\" contains a list of the functions that \ are implemented in this notebook.\ \>", "Text", CellChangeTimes->{{3.544164042314489*^9, 3.544164047976096*^9}}, FontWeight->"Plain"], Cell["\<\ The section \"Implementing the games\" is both, a basic introduction to \ simple games and a tutorial on how to use the commands implemented in this \ notebook.\ \>", "Text", CellChangeTimes->{{3.544150744962956*^9, 3.544150865072877*^9}, { 3.544150936769473*^9, 3.544150937465437*^9}, {3.5441510126181498`*^9, 3.544151034554468*^9}, {3.544151095722975*^9, 3.54415114198713*^9}, { 3.544164023267028*^9, 3.544164040020451*^9}}], Cell["\<\ The last section \"Definition of the functions\" contains just that. However \ the functions are initialized automatically, and can be used as a \"black box\ \".\ \>", "Text", CellChangeTimes->{{3.5441508832458677`*^9, 3.5441509244652557`*^9}, { 3.544151038523007*^9, 3.5441510904667463`*^9}, {3.544151148549839*^9, 3.5441511547074738`*^9}}], Cell[CellGroupData[{ Cell["Disclaimer", "Subsubsection", CellChangeTimes->{{3.5408986304114447`*^9, 3.540898634205468*^9}}], Cell[TextData[{ "This notebook is intended only as a guide on how to apply the \ game-theoretical framework to the selected problem of METI. However, the two \ scenarios \"communicative\" and \"belligerent\" societies is and ", StyleBox["ad hoc", FontSlant->"Italic"], " choice motivated by wide-spread and possibly unfounded ideas on the \ plausible nature of extra-terrestrial societies. Other choice of behaviours \ are theoretically possible, and upon the decision of messaging \ extra-terrestrial civilizations, the author of this document, H.P. de Vladar, \ takes absolutely no liability on the outcomes of such decision. The \ mathematical and statistical tools provided in this document are intended as \ a guidance, and any action regarding messaging extra-terrestrial \ civilizations is to be properly consulted with pertinent governmental and \ international organizations." }], "Text", CellChangeTimes->{{3.5441511595936823`*^9, 3.544151659448015*^9}}], Cell["\<\ The conclusions derived in this work and the tools provided in this document \ are not applicable for an intentional reply to a positive beacon coming from \ an extra-terrestrial source.\ \>", "Text", CellChangeTimes->{{3.5441515972315693`*^9, 3.544151600731518*^9}, { 3.5441516735327463`*^9, 3.544151751080686*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Implemented functions", "Subsection", CellChangeTimes->{{3.543125688578405*^9, 3.543125702340865*^9}, { 3.543125949348239*^9, 3.543125951538809*^9}, {3.544155972983675*^9, 3.544155997935512*^9}}], Cell["\<\ In order to have the description of a function FOO[], type ?FOO and shift + \ enter.\[LineSeparator]For example\ \>", "Text", CellChangeTimes->{{3.5441560207179193`*^9, 3.544156092052856*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Plot"}]], "Input", CellChangeTimes->{{3.544156095008562*^9, 3.544156096442803*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Plot\\\", \\\"[\\\", RowBox[{StyleBox[\\\ \"f\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"x\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) generates \ a plot of \\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\) as a function of \\!\\(\ \\*StyleBox[\\\"x\\\", \\\"TI\\\"]\\) from \ \\!\\(\\*SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]]\\) to \\!\\(\\*SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]\\). \\n\\!\\(\\*RowBox[{\\\"Plot\\\", \ \\\"[\\\", RowBox[{RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \ \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\ \"TR\\\"]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"x\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) plots \ several functions \\!\\(\\*SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"i\\\", \\\"TI\\\"]]\\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Plot"]}]], "Print", "PrintUsage", CellChangeTimes->{3.544156104827641*^9}, CellTags->"Info3544163304-8020089"] }, Open ]], Cell["The following functions are defined in this notebook :", "Text", CellChangeTimes->{{3.544164088514731*^9, 3.544164114326249*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "\[ScriptCapitalV]"}]], "Input", CellChangeTimes->{{3.544156206370306*^9, 3.5441562095949297`*^9}}], Cell[BoxData[ StyleBox["\<\"\[ScriptCapitalV][] Gives the generic payoff matrix of a game.\ \\n\[ScriptCapitalV][t] gives the payoff for the game of type t, where t is \ E,C, or B.\\nThe symbol \[ScriptCapitalV] is a script capital \ \[ScriptCapitalV], and is entered as \[EscapeKey]scV\[EscapeKey].\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.544157426402528*^9}, CellTags->"Info3544164626-8020089"] }, {2}]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "PlayGame"}]], "Input", CellChangeTimes->{{3.544156221997992*^9, 3.544156225254837*^9}}], Cell[BoxData[ StyleBox["\<\"PlayGame[\[Sigma]1,\[Sigma]2,\[ScriptCapitalV]] returns the \ payoff of \[ScriptCapitalV] when player 1 chooses strategy \[Sigma]1 and \ player 2 chooses strategy \[Sigma]2. If \[Sigma]1 or \[Sigma]2 are not \ strategies but symbols (probabilities), the function returns the mean payoff \ given the strategy of the other player.\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.54415622610087*^9}, CellTags->"Info3544163425-8020089"] }, {2}]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "S"}]], "Input", CellChangeTimes->{{3.544156500122431*^9, 3.5441565011627607`*^9}}], Cell[BoxData[ StyleBox["\<\"S is a strategy for the games: search only (defect).\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.5441565017835073`*^9}, CellTags->"Info3544163701-8020089"] }, {2}]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "SB"}]], "Input", CellChangeTimes->{{3.544156503949567*^9, 3.544156507777008*^9}}], Cell[BoxData[ StyleBox["\<\"SB is a strategy for the games: search & broadcast \ (cooperate).\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.544156508517974*^9}, CellTags->"Info3544163708-8020089"] }, {2}]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "\[DoubleStruckCapitalE]U"}]], "Input", CellChangeTimes->{{3.544157044011629*^9, 3.544157046415399*^9}}], Cell[BoxData[ StyleBox["\<\"\[DoubleStruckCapitalE]U[\!\(\*SubscriptBox[\(\[ScriptCapitalV]\ \), \(1\)]\),\!\(\*SubscriptBox[\(\[ScriptCapitalV]\), \(2\)]\)] returns the \ matrix of the expected mean payoff, in terms of the entries of the payoffs \!\ \(\*SubscriptBox[\(\[ScriptCapitalV]\), \(1\)]\) and \!\(\*SubscriptBox[\(\ \[ScriptCapitalV]\), \(2\)]\). It is necessary that \!\(\*SubscriptBox[\(\ \[ScriptCapitalV]\), \ \(1\)]\)\[NotEqual]\!\(\*SubscriptBox[\(\[ScriptCapitalV]\), \(2\)]\).\\nThe \ symbol \[DoubleStruckCapitalE] is a double-strike E, and is entered as \ \[EscapeKey]dsE\[EscapeKey].\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.544157460313653*^9}, CellTags->"Info3544164660-8020089"] }, {2}]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "\[DoubleStruckCapitalE]U2"}]], "Input", CellChangeTimes->{{3.544157411579155*^9, 3.544157414751199*^9}}], Cell[BoxData[ StyleBox["\<\"\[DoubleStruckCapitalE]U2[c,\[Rho],r] returns the matrix of \ the expected mean payoff for a compound game involvinf communicative and \ belligerent societies. It is necessary that \[Rho]>1. It is equivalent to \ \[DoubleStruckCapitalE]U[\[ScriptCapitalV][C],\[ScriptCapitalV][B]] \\nThe \ symbol \[DoubleStruckCapitalE] is a double-strike E, and is entered as \ \[EscapeKey]dsE\[EscapeKey].\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.544157415478039*^9}, CellTags->"Info3544164615-8020089"] }, {2}]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "mixStrategy"}]], "Input", CellChangeTimes->{{3.5441575690964003`*^9, 3.54415757208251*^9}}], Cell[BoxData[ StyleBox["\<\"mixStrategy[c,\[Rho],e] returns the mix strategy for the \ compound game \[ScriptCapitalV][C] \[ScriptCapitalV][B]. It requires that \ \[Rho]>1.\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.54415757295463*^9}, CellTags->"Info3544164772-8020089"] }, {2}]] }, Closed]], Cell[CellGroupData[{ Cell["Implementing the games", "Subsection", CellChangeTimes->{{3.5408986304114447`*^9, 3.540898634205468*^9}, { 3.5413985606552553`*^9, 3.5413985617992077`*^9}, {3.543126418541778*^9, 3.5431264216877823`*^9}}], Cell[CellGroupData[{ Cell["The basic structure of games", "Subsubsection", CellChangeTimes->{{3.543125688578405*^9, 3.543125702340865*^9}}], Cell["\<\ The payoff matrix of a simple 2x2 row-game is implemented in the function \ \[ScriptCapitalV]\ \>", "Text", CellChangeTimes->{{3.5431194527955723`*^9, 3.543119457190847*^9}, { 3.543119498121748*^9, 3.5431195055404587`*^9}, {3.543119860278668*^9, 3.543119887860489*^9}, {3.543125710070012*^9, 3.5431257377110023`*^9}}], Cell["For example, the generic payoff matrix is", "Text", CellChangeTimes->{{3.5431250684394827`*^9, 3.543125088501211*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[ScriptCapitalV]", "[", "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.543125091020813*^9, 3.543125096587161*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\[ScriptCapitalR]", "\[ScriptCapitalS]"}, {"\[ScriptCapitalT]", "\[ScriptCapitalP]"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.543125097115912*^9}] }, Open ]], Cell["And the speciffic games considered in this article are :", "Text", CellChangeTimes->{{3.5431251029376583`*^9, 3.543125118716172*^9}, { 3.543125506819153*^9, 3.543125510541991*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\[ScriptCapitalV]", "[", "\"\\"", "]"}], "//", "MatrixForm"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[ScriptCapitalV]", "[", "\"\\"", "]"}], "//", "MatrixForm"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[ScriptCapitalV]", "[", "\"\\"", "]"}], "//", "MatrixForm"}]}], "Input", CellChangeTimes->{{3.543125126007847*^9, 3.543125140722026*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", RowBox[{"-", "c"}]}, { RowBox[{"1", "+", "c"}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.543125141151284*^9, 3.54330466755761*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\[Rho]", RowBox[{"-", "c"}]}, { RowBox[{"1", "+", "c"}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.543125141151284*^9, 3.543304667564721*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\[Rho]", RowBox[{ RowBox[{"-", "c"}], "-", "e"}]}, { RowBox[{"1", "+", "c"}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.543125141151284*^9, 3.543304667567501*^9}] }, Open ]], Cell["\<\ To \"play\" a game use the function PlayGame. We need the strategy space, \ which is given by the two symbols S and SB. For example if both defect in the \ generic game we get the \[OpenCurlyDoubleQuote]suckers payoff\ \[CloseCurlyDoubleQuote]\ \>", "Text", CellChangeTimes->{{3.5431251569964046`*^9, 3.543125196464262*^9}, { 3.544157616281925*^9, 3.54415763315583*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PlayGame", "[", RowBox[{"S", ",", "S", ",", RowBox[{"\[ScriptCapitalV]", "[", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.54312531455997*^9, 3.5431253229147253`*^9}, 3.5431254432812157`*^9, {3.543125818522915*^9, 3.54312582269685*^9}}], Cell[BoxData["\[ScriptCapitalT]"], "Output", CellChangeTimes->{ 3.543125323942411*^9, {3.543125441918051*^9, 3.543125443655532*^9}, 3.543125819184607*^9}] }, Open ]], Cell["\<\ Similarly, for mixed strategies we may define a probability of cooperation P \ is the probability instead of a pure strategy. In the following example, we \ cooperate, and the oponent has a mixed strategy P:\ \>", "Text", CellChangeTimes->{{3.543125329546324*^9, 3.543125375778886*^9}, { 3.5431258288476963`*^9, 3.543125892094397*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"PlayGame", "[", RowBox[{"P", ",", "S", " ", ",", RowBox[{"\[ScriptCapitalV]", "[", "]"}]}], " ", "]"}], "\[IndentingNewLine]", RowBox[{"PlayGame", "[", RowBox[{"P", ",", "SB", ",", RowBox[{"\[ScriptCapitalV]", "[", "]"}]}], "]"}]}], "Input", CellChangeTimes->{{3.543125379257667*^9, 3.543125388131624*^9}, { 3.54312547845805*^9, 3.543125478854597*^9}, {3.543227085026655*^9, 3.543227096075274*^9}, {3.543227135745644*^9, 3.543227153439493*^9}, { 3.5432273669645967`*^9, 3.543227372465785*^9}, {3.543227474874728*^9, 3.543227479981881*^9}, {3.543227523544896*^9, 3.543227547976448*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "P"}], ")"}], " ", "\[ScriptCapitalP]"}], "+", RowBox[{"P", " ", "\[ScriptCapitalS]"}]}]], 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