1. **Problem 1: Use Iterated Elimination of Strictly Dominated Strategies (IESDS) to find the rational outcome of the game.** 2. The game matrix is: | | A | B | C | D | |-------|-------|-------|-------|-------| | W | (5,0) | (3,1) | (4,4) | (3,5) | | X | (4,1) | (5,9) | (6,8) | (4,0) | | Y | (6,2) | (4,10)| (7,1) | (5,-1)| | Z | (4,3) | (3,2) | (5,3) | (3,2) | 3. **Step 1: Eliminate strictly dominated strategies for Player 1.** - Compare rows W, X, Y, Z: - W vs Z: W is better than Z in A (5>4), B (3>3 equal), C (4>5 no), D (3>3 equal) so no strict domination. - X vs Z: X better than Z in A (4>4 equal), B (5>3), C (6>5), D (4>3) so Z is strictly dominated by X. - Eliminate Z. 4. **Step 2: Eliminate strictly dominated strategies for Player 2.** - Compare columns A, B, C, D for Player 2's payoffs: - Column A payoffs: 0 (W),1 (X),2 (Y) - Column B payoffs:1 (W),9 (X),10 (Y) - Column C payoffs:4 (W),8 (X),1 (Y) - Column D payoffs:5 (W),0 (X),-1 (Y) - Check if any column is strictly dominated: - D is strictly dominated by B because for Player 2: 5<1 no, so no. - C is not dominated. - A is dominated by B? 0<1 no. - No columns strictly dominated. 5. **Step 3: Eliminate strictly dominated strategies for Player 1 again with reduced matrix (no Z).** - Compare W, X, Y: - W vs X: W better in A (5>4), B (3<5 no), C (4<6 no), D (3<4 no) no domination. - X vs Y: X better in A (4<6 no), B (5<4 no), C (6<7 no), D (4<5 no) no domination. - Y vs W: Y better in A (6>5), B (4>3), C (7>4), D (5>3) so W is strictly dominated by Y. - Eliminate W. 6. **Step 4: Check Player 2's strategies again with reduced matrix (X,Y only).** - Payoffs for Player 2: - A: 1 (X), 2 (Y) - B: 9 (X), 10 (Y) - C: 8 (X), 1 (Y) - D: 0 (X), -1 (Y) - D is strictly dominated by C because 0<8 and -1<1 for Player 2. - Eliminate D. 7. **Step 5: Final reduced matrix:** | | A | B | C | |-------|-------|-------|-------| | X | (4,1) | (5,9) | (6,8) | | Y | (6,2) | (4,10)| (7,1) | 8. **Step 6: Check for further domination:** - For Player 1: X vs Y no strict domination. - For Player 2: A vs B vs C no strict domination. 9. **Rational outcome after IESDS is the reduced game above.** --- 2. **Problem 2: Use IESDS to shorten the game and find Pure Strategy Nash Equilibrium (PSNE).** Game matrix: | | A | B | C | D | |-------|--------|--------|--------|--------| | V | (5,-2) | (-2,5) | (-1,6) | (5,4) | | W | (12,1) | (3,3) | (0,2) | (6,1) | | X | (6,-1) | (2,0) | (1,1) | (-2,0) | | Y | (2,8) | (1,9) | (-5,4) | (3,2) | 10. **Step 1: Eliminate strictly dominated strategies for Player 1.** - Compare rows: - V vs X: V better in A (5<6 no), B (-2<2 no), C (-1<1 no), D (5>-2 yes) no domination. - W vs X: W better in A (12>6), B (3>2), C (0<1 no), D (6> -2) no domination. - Y vs X: Y better in A (2<6 no), B (1<2 no), C (-5<1 no), D (3> -2 yes) no domination. - No strict domination for Player 1. 11. **Step 2: Eliminate strictly dominated strategies for Player 2.** - Compare columns: - A vs B: A payoffs (-2,1,-1,8), B payoffs (5,3,0,9) - A is strictly dominated by B for Player 2? -2<5 yes, 1<3 yes, -1<0 yes, 8<9 yes, so A is strictly dominated by B. - Eliminate A. 12. **Step 3: New matrix without A:** | | B | C | D | |-------|--------|--------|--------| | V | (-2,5) | (-1,6) | (5,4) | | W | (3,3) | (0,2) | (6,1) | | X | (2,0) | (1,1) | (-2,0) | | Y | (1,9) | (-5,4) | (3,2) | 13. **Step 4: Check for further domination for Player 2.** - Compare B vs C: - B payoffs: 5,3,0,9 - C payoffs: 6,2,1,4 - B is not strictly dominated by C because 5<6 yes, 3>2 no. - Compare B vs D: - B payoffs: 5,3,0,9 - D payoffs: 4,1,0,2 - B not strictly dominated by D. - Compare C vs D: - C payoffs: 6,2,1,4 - D payoffs: 4,1,0,2 - C not strictly dominated by D. 14. **Step 5: Check for Player 1 domination again.** - V vs W: - B: -2<3 no - C: -1<0 no - D: 5<6 no - No domination. - W vs X: - B: 3>2 yes - C: 0<1 no - D: 6> -2 yes - No domination. - Y vs X: - B: 1<2 no - C: -5<1 no - D: 3> -2 yes - No domination. 15. **Step 6: Find Pure Strategy Nash Equilibria (PSNE) in reduced game.** - For each Player 1 strategy, find Player 2 best response: - V: Player 2 payoffs B=5, C=6, D=4 best is C. - W: B=3, C=2, D=1 best is B. - X: B=0, C=1, D=0 best is C. - Y: B=9, C=4, D=2 best is B. - For each Player 2 strategy, find Player 1 best response: - B: Player 1 payoffs V=-2, W=3, X=2, Y=1 best is W. - C: V=-1, W=0, X=1, Y=-5 best is X. - D: V=5, W=6, X=-2, Y=3 best is W. - PSNE candidates are where strategies are mutual best responses: - (W,B) since W best for B and B best for W. - (X,C) since X best for C and C best for X. - (W,D) since W best for D but D best for V not W, so no. 16. **PSNE are (W,B) and (X,C).** --- 3. **Problem 3: Use backward induction to find all Subgame Perfect Nash Equilibria (SPNE).** Game tree: - P1 chooses A or B. - If A, P2 chooses C or D with payoffs (9,2) or (5,2). - If B, P2 chooses E or F. - E leads to (5,1). - F leads to P1 choosing G or H with payoffs (1,1) or (2,0). 17. **Step 1: Solve last decision node (P1 chooses G or H).** - P1 compares payoffs: G=1, H=2. - P1 chooses H for payoff 2. 18. **Step 2: P2 chooses between E and F.** - P2 payoffs: E=1, F leads to P1 choosing H with payoff 0 for P2. - So P2 prefers E (payoff 1) over F (payoff 0). - P2 chooses E. 19. **Step 3: P1 chooses between A and B.** - If A, payoff is 9. - If B, payoff is 5 (from E). - P1 chooses A. 20. **SPNE strategies:** - P1: choose A at root, choose H at last node. - P2: choose C after A, choose E after B. --- 4. **Problem 4: Monopoly profit maximization.** Given: - Price function: $$P = 100 - 10Q$$ - Cost per unit: 5 21. **Step 1: Write revenue function:** $$R = P \times Q = (100 - 10Q)Q = 100Q - 10Q^2$$ 22. **Step 2: Write cost function:** $$C = 5Q$$ 23. **Step 3: Write profit function:** $$\pi = R - C = 100Q - 10Q^2 - 5Q = 95Q - 10Q^2$$ 24. **Step 4: Find quantity that maximizes profit by setting derivative to zero:** $$\frac{d\pi}{dQ} = 95 - 20Q = 0$$ 25. **Step 5: Solve for Q:** $$20Q = 95 \Rightarrow Q = \frac{95}{20} = 4.75$$ 26. **Step 6: Calculate maximum profit:** $$\pi(4.75) = 95(4.75) - 10(4.75)^2 = 451.25 - 225.625 = 225.625$$ **Final answers:** - Problem 1: Rational outcome after IESDS is the reduced game with Player 1 strategies X, Y and Player 2 strategies A, B, C. - Problem 2: PSNE are (W,B) and (X,C). - Problem 3: SPNE strategies are P1 chooses A at root, P2 chooses C after A, P2 chooses E after B, and P1 chooses H at last node. - Problem 4: Quantity for maximum profit is $4.75$ units and maximum profit is $225.625$.