Question: ชื่อ-นามสกุล______________________________รหัสนักศึกษา________________\n\nข้อสอบครั้งที่ 4 วิชา1302332 Operations Research สำหรับตอนเรียนที่ 1\nคะแนนเต็ม 10 คะแนน ผู้ตรวจในการสอบ จะถูกปรับตกและได้เกรด F\nวันที่สอบ 22 ส.ค. 2551 เวลา 8.00-9.00 น. ให้ทำทุกข้อในกระดาษคำถาม\n\n1.\n\nb1 b2 b3 b4 b5\na1 0 4 2 -3 3\na2 3 1 6 2 4\na3 1 1 5 2 3\na4 1 6 3 1 -4\na5 1 1 4 2 3\n\n1.1 จากตารางเกม ให้ใช้วิธี Dominance Method เพื่อกำจัดกลยุทธ์ที่ไม่ดีออกไป\n1.2 ปัญหาเกมส์นี้เป็นแบบ Mixed Strategy หรือ Pure Strategy จงแสดงให้เห็น\n\n2. Player 1 writes an integer between 1 and 20 on a slip of paper. Without showing this slip of paper to player 2, player 1 tells player 2 what he has written. Player 1 may lie or tell the truth. Player 2 must then guess whether or no player 1 has told the truth. If caught in a lie, player 1 must pay player 2 10; if falsely accused of lying, player 1 collects 5 from player 2. If player 1 tells the truth and player 2 guesses that player 1 has told the truth, then player 1 must pay 1 to player 2. If player 1 lies and player 2 does not guess that player 1 has lied, player 1 wins 5 from player 2. Determine the value of the game and each player’s optimal strategy.
1. **Problem Statement:** Use the Dominance Method to eliminate dominated strategies from the given payoff matrix and determine if the game is a Mixed or Pure Strategy game.\n\nThe payoff matrix for Player 1 (rows) vs Player 2 (columns) is:\n$$\begin{array}{c|ccccc} & b1 & b2 & b3 & b4 & b5 \\ \hline a1 & 0 & 4 & 2 & -3 & 3 \\ a2 & 3 & 1 & 6 & 2 & 4 \\ a3 & 1 & 1 & 5 & 2 & 3 \\ a4 & 1 & 6 & 3 & 1 & -4 \\ a5 & 1 & 1 & 4 & 2 & 3 \\ \end{array}$$\n\n2. **Dominance Method:** A strategy is dominated if there exists another strategy that yields a better or equal payoff in every column and strictly better in at least one. We check rows and columns for dominance.\n\n3. **Check for dominated rows:**\n- Compare $a1$ and $a3$: $a3$ has payoffs $(1,1,5,2,3)$ which are all greater or equal to $a1$'s $(0,4,2,-3,3)$ except at $b2$ where $a1=4 > 1$. So no dominance here.\n- Compare $a2$ and $a3$: $a2=(3,1,6,2,4)$ vs $a3=(1,1,5,2,3)$. $a2$ is better or equal in all columns and strictly better in $b1, b3, b5$. So $a3$ is dominated by $a2$.\n- Compare $a4$ and $a5$: $a4=(1,6,3,1,-4)$ vs $a5=(1,1,4,2,3)$. $a5$ is better in $b3, b4, b5$ but worse in $b2$. No dominance.\n\n**Eliminate $a3$** as it is dominated by $a2$.\n\n4. **Check for dominated columns:**\n- Compare $b4$ and $b5$: For Player 1, payoffs in $b4$ are $(-3,2,2,1,2)$ and in $b5$ are $(3,4,3,-4,3)$. Since $b5$ yields better payoffs for Player 1 in most rows, $b4$ is dominated by $b5$.\n\n**Eliminate $b4$.**\n\n5. **Reduced matrix after elimination:**\n$$\begin{array}{c|cccc} & b1 & b2 & b3 & b5 \\ \hline a1 & 0 & 4 & 2 & 3 \\ a2 & 3 & 1 & 6 & 4 \\ a4 & 1 & 6 & 3 & -4 \\ a5 & 1 & 1 & 4 & 3 \\ \end{array}$$\n\n6. **Check again for dominance:**\n- Compare $a1$ and $a5$: $a5$ is better or equal in $b3$ and $b5$, but $a1$ is better in $b2$. No dominance.\n- Compare $a2$ and $a5$: $a2$ better in $b1, b3, b5$, $a5$ better in $b2$. No dominance.\n- Compare columns $b1$ and $b3$: $b3$ payoffs $(2,6,3,4)$ vs $b1$ $(0,3,1,1)$. $b3$ dominates $b1$.\n\n**Eliminate $b1$.**\n\n7. **Final reduced matrix:**\n$$\begin{array}{c|ccc} & b2 & b3 & b5 \\ \hline a1 & 4 & 2 & 3 \\ a2 & 1 & 6 & 4 \\ a4 & 6 & 3 & -4 \\ a5 & 1 & 4 & 3 \\ \end{array}$$\n\n8. **Determine if game is Mixed or Pure Strategy:**\n- Since no single pure strategy dominates and no saddle point (no cell is minimum in its row and maximum in its column), the game requires Mixed Strategies.\n\n**Answer:** The game is a Mixed Strategy game after eliminating dominated strategies $a3$, $b4$, and $b1$.\n\n---\n\n**Summary:**\n- Dominated strategies eliminated: $a3$, $b4$, $b1$.\n- The game requires Mixed Strategies to solve optimally.\n\n**q_count:** 2