document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173267>Question 1173267</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Shown below is the solution to the linear program for finding Player A's optimal mixed strategy in a two-person, zero-sum game.\r<BR>\n" );
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document.write( "OBJECTIVE FUNCTION VALUE  =     3.500\r<BR>\n" );
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document.write( "VARIABLE\r<BR>\n" );
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document.write( "VALUE\r<BR>\n" );
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document.write( "REDUCED COSTS\r<BR>\n" );
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document.write( "PA1\r<BR>\n" );
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document.write( "0.050\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "PA2\r<BR>\n" );
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document.write( "0.600\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "PA3\r<BR>\n" );
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document.write( "0.350\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "GAINA\r<BR>\n" );
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document.write( "3.500\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "CONSTRAINT\r<BR>\n" );
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document.write( "SLACK/SURPLUS\r<BR>\n" );
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document.write( "DUAL PRICES\r<BR>\n" );
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document.write( "1\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "−0.500\r<BR>\n" );
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document.write( "2\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "−0.500\r<BR>\n" );
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document.write( "3\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "  0.000\r<BR>\n" );
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document.write( "4\r<BR>\n" );
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document.write( "0.000\r<BR>\n" );
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document.write( "  3.500\r<BR>\n" );
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document.write( "a.\r<BR>\n" );
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document.write( "What is Player A's optimal mixed strategy?\r<BR>\n" );
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document.write( "b.\r<BR>\n" );
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document.write( "What is Player B's optimal mixed strategy?\r<BR>\n" );
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document.write( "c.\r<BR>\n" );
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document.write( "What is Player A's expected gain?\r<BR>\n" );
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document.write( "d.\r<BR>\n" );
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document.write( "What is Player B's expected loss? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850745 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850745\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down the information from the linear program output to answer each question:\r<BR>\n" );
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document.write( "**a. Player A's Optimal Mixed Strategy**\r<BR>\n" );
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document.write( "* **PA1 = 0.050:** Player A should play strategy 1 with a probability of 0.05 (or 5%).<BR>\n" );
document.write( "* **PA2 = 0.600:** Player A should play strategy 2 with a probability of 0.60 (or 60%).<BR>\n" );
document.write( "* **PA3 = 0.350:** Player A should play strategy 3 with a probability of 0.35 (or 35%).\r<BR>\n" );
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document.write( "Therefore, Player A's optimal mixed strategy is (0.05, 0.60, 0.35).\r<BR>\n" );
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document.write( "**b. Player B's Optimal Mixed Strategy**\r<BR>\n" );
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document.write( "To find Player B's optimal mixed strategy, we look at the **DUAL PRICES** of the constraints. These represent the optimal probabilities for Player B's strategies.\r<BR>\n" );
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document.write( "* Constraint 1's Dual Price = -0.500<BR>\n" );
document.write( "* Constraint 2's Dual Price = -0.500<BR>\n" );
document.write( "* Constraint 3's Dual Price = 0.000\r<BR>\n" );
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document.write( "However, the Dual Prices must be converted into positive values, and then normalized, as they represent the negative of the player B's strategy.<BR>\n" );
document.write( "Also, the dual prices represent the negative of the player B's strategy, and the 4th constraint represents the gain.<BR>\n" );
document.write( "Therefore, the dual prices of constraints 1,2, and 3 are the negative of the probabilities of player B's strategies.\r<BR>\n" );
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document.write( "To normalize the dual prices, we must take the absolute value of the dual prices. Then add them together.\r<BR>\n" );
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document.write( "0.5 + 0.5 + 0 = 1\r<BR>\n" );
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document.write( "Then we can divide each dual price by the sum.\r<BR>\n" );
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document.write( "* Player B strategy 1: 0.5/1 = 0.5<BR>\n" );
document.write( "* Player B strategy 2: 0.5/1 = 0.5<BR>\n" );
document.write( "* Player B strategy 3: 0/1 = 0\r<BR>\n" );
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document.write( "Therefore, Player B's optimal mixed strategy is (0.5, 0.5, 0).\r<BR>\n" );
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document.write( "**c. Player A's Expected Gain**\r<BR>\n" );
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document.write( "* **OBJECTIVE FUNCTION VALUE = 3.500**<BR>\n" );
document.write( "* **GAINA = 3.500**\r<BR>\n" );
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document.write( "Player A's expected gain is 3.5.\r<BR>\n" );
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document.write( "**d. Player B's Expected Loss**\r<BR>\n" );
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document.write( "In a zero-sum game, Player B's expected loss is equal to Player A's expected gain.\r<BR>\n" );
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document.write( "Therefore, Player B's expected loss is also 3.5.<BR>\n" );
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