1. **State the problem:** We need to find the rational outcome of the given game using Iterated Elimination of Strictly Dominated Strategies (IESDS). 2. **Explain IESDS:** IESDS involves repeatedly removing strategies that are strictly dominated, meaning there is another strategy that always gives a strictly better payoff regardless of the opponent's choice. 3. **Analyze Player 1's strategies:** - Compare W, X, Y, Z across Player 2's strategies A, B, C, D. - Check if any strategy is strictly dominated by another. 4. **Check Player 1:** - W vs X: For A, W=5 vs X=4 (W better); B: W=3 vs X=5 (X better); C: W=4 vs X=6 (X better); D: W=3 vs X=4 (X better). X is better in 3 out of 4 cases but not strictly dominating W. - W vs Y: A:5 vs 6 (Y better); B:3 vs 4 (Y better); C:4 vs 7 (Y better); D:3 vs 5 (Y better). Y strictly dominates W. 5. **Eliminate W for Player 1.** 6. **Check X vs Y:** - A:4 vs 6 (Y better); B:5 vs 4 (X better); C:6 vs 7 (Y better); D:4 vs 5 (Y better). Y better in 3 out of 4 but not strictly dominating X. 7. **Check Z vs X:** - A:4 vs 4 (equal); B:3 vs 5 (X better); C:5 vs 6 (X better); D:3 vs 4 (X better). X strictly dominates Z. 8. **Eliminate Z for Player 1.** 9. **Analyze Player 2's strategies:** - Compare A, B, C, D across Player 1's remaining strategies Y and X. 10. **Check A vs B:** - Y:2 vs 10 (B better); X:1 vs 9 (B better). B strictly dominates A. 11. **Eliminate A for Player 2.** 12. **Check C vs D:** - Y:1 vs -1 (C better); X:8 vs 0 (C better). C strictly dominates D. 13. **Eliminate D for Player 2.** 14. **Check B vs C:** - Y:10 vs 1 (B better); X:9 vs 8 (B better). B strictly dominates C. 15. **Eliminate C for Player 2.** 16. **Remaining strategies:** - Player 1: X, Y - Player 2: B 17. **Check if any Player 1 strategy is dominated given Player 2 chooses B:** - X payoff: 5 - Y payoff: 4 - X strictly dominates Y against B. 18. **Eliminate Y for Player 1.** 19. **Final rational outcome:** Player 1 chooses X, Player 2 chooses B with payoff (5,9). **Answer:** The rational outcome by IESDS is $(X, B)$ with payoffs $(5, 9)$.