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$CellContext`NETR$, $CellContext`NEBL$, $CellContext`NEBR$, $CellContext`NE$, \ $CellContext`AEF$, $CellContext`BEF$, $CellContext`qO$, $CellContext`pO$, \ $CellContext`AMS$, $CellContext`BMS$}, $CellContext`ADTL$ = If[$CellContext`ATL >= $CellContext`ABL, 1, 0]; $CellContext`ADTR$ = If[$CellContext`ATR >= $CellContext`ABR, 1, 0]; $CellContext`AD$ = If[ And[$CellContext`ADTL$ == 1, $CellContext`ADTR$ == 1], "Top", If[ And[ Or[$CellContext`ADTL$ == 0, $CellContext`ATL == $CellContext`ABL], Or[$CellContext`ADTR$ == 0, $CellContext`ATR == $CellContext`ABR]], "Bottom", "None"]]; $CellContext`BDTL$ = If[$CellContext`BTL >= $CellContext`BTR, 1, 0]; $CellContext`BDBL$ = If[$CellContext`BBL >= $CellContext`BBR, 1, 0]; $CellContext`BD$ = If[ And[$CellContext`BDTL$ == 1, $CellContext`BDBL$ == 1], "Left", If[ And[ Or[$CellContext`BDTL$ == 0, $CellContext`BTL == $CellContext`BTR], Or[$CellContext`BDBL$ == 0, $CellContext`BBL == $CellContext`BBR]], "Right", "None"]]; $CellContext`NETL$ = If[ And[$CellContext`ADTL$ == 1, $CellContext`BDTL$ == 1], 1, 0]; $CellContext`NETR$ = If[ And[$CellContext`ADTR$ == 1, Or[$CellContext`BDTL$ == 0, $CellContext`BTL == $CellContext`BTR]], 1, 0]; $CellContext`NEBL$ = If[ And[ Or[$CellContext`ADTL$ == 0, $CellContext`ATL == $CellContext`ABL], $CellContext`BDBL$ == 1], 1, 0]; $CellContext`NEBR$ = If[ And[ Or[$CellContext`ADTR$ == 0, $CellContext`ATR == $CellContext`ABR], Or[$CellContext`BDBL$ == 0, $CellContext`BBL == $CellContext`BBR]], 1, 0]; $CellContext`AEF$ = If[$CellContext`ATL $CellContext`BDTL$ + $CellContext`ATR ( 1 - $CellContext`BDTL$) >= $CellContext`ABL $CellContext`BDBL$ + \ $CellContext`ABR (1 - $CellContext`BDBL$), "Top", "Bottom"]; $CellContext`BEF$ = If[$CellContext`AEF$ == "Top", $CellContext`BDTL$ "Left" + (1 - $CellContext`BDTL$) "Right", $CellContext`BDBL$ "Left" + (1 - $CellContext`BDBL$) "Right"]; $CellContext`qO$ = If[$CellContext`BD$ == "Left", 1, If[$CellContext`BD$ == "Right", 0, N[ ReplaceAll[$CellContext`q, Last[ Solve[$CellContext`ATL $CellContext`q + $CellContext`ATR ( 1 - $CellContext`q) == $CellContext`ABL $CellContext`q + \ $CellContext`ABR (1 - $CellContext`q), $CellContext`q]]]]]]; $CellContext`pO$ = If[$CellContext`AD$ == "Top", 1, If[$CellContext`AD$ == "Bottom", 0, N[ ReplaceAll[$CellContext`p, Last[ Solve[$CellContext`BTL $CellContext`p + $CellContext`BBL ( 1 - $CellContext`p) == $CellContext`BTR $CellContext`p + \ $CellContext`BBR ( 1 - $CellContext`p), $CellContext`p]]]]]]; \ $CellContext`AMS$ = If[$CellContext`ATL (1.01 $CellContext`qO$) + $CellContext`ATR (1 - 1.01 $CellContext`qO$) > ($CellContext`ABL 1.01) $CellContext`qO$ + $CellContext`ABR (1 - 1.01 $CellContext`qO$), 1, 0]; $CellContext`BMS$ = If[$CellContext`BTL (1.01 $CellContext`pO$) + $CellContext`BBL (1 - 1.01 $CellContext`pO$) > ($CellContext`BTR 1.01) $CellContext`pO$ + $CellContext`BBR (1 - 1.01 $CellContext`pO$), 1, 0]; $CellContext`NE$ = DeleteCases[ If[ And[$CellContext`AD$ == "None", $CellContext`BD$ == "None"], {$CellContext`NETL$ { "Top", "Left"}, $CellContext`NETR$ { "Top", "Right"}, $CellContext`NEBL$ { "Bottom", "Left"}, $CellContext`NEBR$ {"Bottom", "Right"}}, If[$CellContext`AD$ == "Top", {$CellContext`NETL$ {"Top", "Left"}, $CellContext`NETR$ { "Top", "Right"}}, If[$CellContext`AD$ == "Bottom", {$CellContext`NEBL$ { "Bottom", "Left"}, $CellContext`NEBR$ {"Bottom", "Right"}}, If[$CellContext`BD$ == "Left", {$CellContext`NETL$ { "Top", "Left"}, $CellContext`NEBL$ {"Bottom", "Left"}}, If[$CellContext`BD$ == "Right", {$CellContext`NETR$ { "Top", "Right"}, $CellContext`NEBR$ { "Bottom", "Right"}}]]]]], {0, 0}]; Grid[{{ Style["Simultaneous Game: Pure Strategy", Bold, Large], SpanFromLeft}, { Grid[{{"", "", Style["Player B", ColorData[97][1]], SpanFromLeft, SpanFromLeft, SpanFromLeft}, { "", "", "Left", SpanFromLeft, "Right", SpanFromLeft}, { Style["Player A", ColorData[97][2]], "Top", Style[$CellContext`ATL, ColorData[97][2], If[ And[$CellContext`BR$$ == 1, $CellContext`ADTL$ == 1], { Underlined, Bold}, ColorData[97][2]]], Style[$CellContext`BTL, ColorData[97][1], If[ And[$CellContext`BR$$ == 1, $CellContext`BDTL$ == 1], { Underlined, Bold}, ColorData[97][1]]], Style[$CellContext`ATR, ColorData[97][2], If[ And[$CellContext`BR$$ == 1, $CellContext`ADTR$ == 1], { Underlined, Bold}, ColorData[97][2]]], Style[$CellContext`BTR, ColorData[97][1], If[ And[$CellContext`BR$$ == 1, Or[$CellContext`BDTL$ == 0, $CellContext`BTL == $CellContext`BTR]], { Underlined, Bold}, ColorData[97][1]]]}, {SpanFromAbove, "Bottom", Style[$CellContext`ABL, ColorData[97][2], If[ And[$CellContext`BR$$ == 1, Or[$CellContext`ADTL$ == 0, $CellContext`ATL == $CellContext`ABL]], { Underlined, Bold}, ColorData[97][2]]], Style[$CellContext`BBL, ColorData[97][1], If[ And[$CellContext`BR$$ == 1, $CellContext`BDBL$ == 1], { Underlined, Bold}, ColorData[97][1]]], Style[$CellContext`ABR, ColorData[97][2], If[ And[$CellContext`BR$$ == 1, Or[$CellContext`ADTR$ == 0, $CellContext`ATR == $CellContext`ABR]], { Underlined, Bold}, ColorData[97][2]]], Style[$CellContext`BBR, ColorData[97][1], If[ And[$CellContext`BR$$ == 1, Or[$CellContext`BDBL$ == 0, $CellContext`BBL == $CellContext`BBR]], { Underlined, Bold}, ColorData[97][1]]]}}, Frame -> { None, None, {{{3, 3}, {3, 4}} -> Thick, {{4, 4}, {3, 4}} -> Thick, {{3, 3}, {5, 6}} -> Thick, {{4, 4}, {5, 6}} -> Thick}}, Spacings -> {1, 1}], Grid[{{ Style["Dominant Strategies:", Bold, Medium], Style["Pure Strategy Nash Equilibria:", Bold, Medium]}, { If[$CellContext`DS$$ == 1, TextGrid[{{"Player A:", $CellContext`AD$}, { "Player B:", $CellContext`BD$}}], ""], If[$CellContext`NEI$$ == 1, $CellContext`NE$, ""]}}, Frame -> { None, None, {{{1, 2}, {1, 1}} -> True, {{1, 2}, {2, 2}} -> True}}]}, {}, {}, { Style["Simultaneous Game: Mixed Strategy", Bold, Large], SpanFromLeft}, If[$CellContext`MS$$ == 1, If[ Or[$CellContext`AD$ == "Top", $CellContext`AD$ == "Bottom"], { TextGrid[{{((( "Because player A has a dominant strategy, they will play \ " <> $CellContext`AD$) <> " with certainty (p=") <> ToString[ If[$CellContext`AD$ == "Top", 1, 0]]) <> ")."}, {((("This implies that player B will play " <> If[$CellContext`AD$ == "Top", $CellContext`BDTL$ "Left" + (1 - $CellContext`BDTL$) "Right", $CellContext`BDBL$ "Left" + (1 - $CellContext`BDBL$) "Right"]) <> " with certainty (q=") <> ToString[ If[$CellContext`AD$ == "Top", $CellContext`BDTL$ 1 + (1 - $CellContext`BDTL$) 0, $CellContext`BDBL$ 1 + (1 - $CellContext`BDBL$) 0]]) <> ")"}}]}, If[ Or[$CellContext`BD$ == "Left", $CellContext`BD$ == "Right"], { TextGrid[{{((( "Because player B has a dominant strategy, they will play \ " <> $CellContext`BD$) <> " with certainty (q=") <> ToString[ If[$CellContext`BD$ == "Left", 1, 0]]) <> ")."}, {((("This implies that player A will play " <> If[$CellContext`BD$ == "Left", $CellContext`ADTL$ "Top" + (1 - $CellContext`ADTL$) "Bottom", $CellContext`ADTR$ "Top" + (1 - $CellContext`ADTR$) "Bottom"]) <> " with certainty (p=") <> ToString[ If[$CellContext`BD$ == "Left", $CellContext`ADTL$ 1 + (1 - $CellContext`ADTL$) 0, $CellContext`ADTR$ 1 + (1 - $CellContext`ADTR$) 0]]) <> ")"}}]}, { TextGrid[{{ "Player A's expected pay-off from playing top:", \ $CellContext`ATL $CellContext`q + $CellContext`ATR (1 - $CellContext`q)}, { "Player A's expected pay-off from playing bottom:", \ $CellContext`ABL $CellContext`q + $CellContext`ABR (1 - $CellContext`q)}, { "Player A must be indifferent between playing top and \ bottom:", $CellContext`ATL $CellContext`q + $CellContext`ATR ( 1 - $CellContext`q) == $CellContext`ABL $CellContext`q + \ $CellContext`ABR (1 - $CellContext`q)}, { "Therefore the optimal q for player B to choose is:", "q'=" <> ToString[$CellContext`qO$]}, { If[$CellContext`ATL ( 1.01 $CellContext`qO$) + $CellContext`ATR (1 - 1.01 $CellContext`qO$) > ($CellContext`ABL 1.01) $CellContext`qO$ + $CellContext`ABR (1 - 1.01 $CellContext`qO$), "If q>q', player A should always choose top because it \ yields a higher payoff", "If q>q', player A should always choose bottom because it \ yields a higher payoff"], SpanFromLeft}}], TextGrid[{{ "Player B's expected pay-off from playing left:", \ $CellContext`BTL $CellContext`p + $CellContext`BBL (1 - $CellContext`p)}, { "Player B's expected pay-off from playing right:", \ $CellContext`BTR $CellContext`p + $CellContext`BBR (1 - $CellContext`p)}, { "Player B must be indifferent between playing left and \ right:", $CellContext`BTL $CellContext`p + $CellContext`BBL ( 1 - $CellContext`p) == $CellContext`BTR $CellContext`p + \ $CellContext`BBR (1 - $CellContext`p)}, { "Therefore the optimal p for player A to choose is:", "p'=" <> ToString[$CellContext`pO$]}, { If[$CellContext`BTL ( 1.01 $CellContext`pO$) + $CellContext`BBL (1 - 1.01 $CellContext`pO$) > ($CellContext`BTR 1.01) $CellContext`pO$ + $CellContext`BBR (1 - 1.01 $CellContext`pO$), "If p>p', player B should always choose left because it \ yields a higher payoff", "If p>p', player B should always choose right because it \ yields a higher payoff"], SpanFromLeft}}]}]], {}], If[ And[$CellContext`MS$$ == 1, $CellContext`AD$ == "None", $CellContext`BD$ == "None"], { Show[{ Plot[-1, {$CellContext`x, 0, 1}, PlotRange -> {{-0.025, 1.1}, {-0.025, 1.1}}, AxesLabel -> {"p", "q"}], Graphics[{ ColorData[97][2], Thick, Line[{{0, $CellContext`qO$}, {1, $CellContext`qO$}}], Line[{{$CellContext`AMS$, $CellContext`qO$}, \ {$CellContext`AMS$, 1}}], Line[{{1 - $CellContext`AMS$, $CellContext`qO$}, { 1 - $CellContext`AMS$, 0}}], ColorData[97][1], Dashed, Line[{{$CellContext`pO$, 0}, {$CellContext`pO$, 1}}], Line[{{$CellContext`pO$, $CellContext`BMS$}, { 1, $CellContext`BMS$}}], Line[{{0, 1 - $CellContext`BMS$}, {$CellContext`pO$, 1 - $CellContext`BMS$}}]}]}, ImageSize -> Medium], SpanFromLeft}, {}], {}, {}, { Style["Sequential Game: Player A is Leader", Bold, Large], SpanFromLeft}, { If[$CellContext`EF$$ == 1, TreeGraph[{ Labeled[1, Placed[ Style["Player A", ColorData[97][2]], Above]], Labeled[2, Placed[ Style["Player B", ColorData[97][1]], Before]], Labeled[3, Placed[ Style["Player B", ColorData[97][1]], After]], Labeled[5, Placed[ Style[{ Style[$CellContext`ATL, ColorData[97][2]], Style[$CellContext`BTL, ColorData[97][1]]}, Bold, Medium], Below]], Labeled[4, Placed[ Style[{ Style[$CellContext`ATR, ColorData[97][2]], Style[$CellContext`BTR, ColorData[97][1]]}, Bold, Medium], Below]], Labeled[6, Placed[ Style[{ Style[$CellContext`ABL, ColorData[97][2]], Style[$CellContext`BBL, ColorData[97][1]]}, Bold, Medium], Below]], Labeled[7, Placed[ Style[{ Style[$CellContext`ABR, ColorData[97][2]], Style[$CellContext`BBR, ColorData[97][1]]}, Bold, Medium], Below]]}, { Labeled[ DirectedEdge[1, 2], Style["Top", Bold, Medium]], Labeled[ DirectedEdge[1, 3], Style["Bottom", Bold, Medium]], Labeled[ DirectedEdge[2, 4], Style["Right", Bold, Medium]], Labeled[ DirectedEdge[2, 5], Style["Left", Bold, Medium]], Labeled[ DirectedEdge[3, 6], Style["Left", Bold, Medium]], Labeled[ DirectedEdge[3, 7], 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