Solutions to Chapter 4 Exercises

S1. (a) For Rowena, Up strictly dominates Down, so Down may be eliminated. For Colin, Right strictly dominates Left, so Left may be eliminated. These actions leave the pure-strategy Nash equilibrium (Up, Right). (b) Down is dominant for Rowena and Left is dominant for Colin. Equilibrium: (Down, Left) with payoffs of (6, 5). (c) There are no dominated strategies for Rowena. For Colin, Left dominates Middle and Right. Thus these two strategies may be eliminated, leaving only Left. With only Left remaining, for Rowena, Straight dominates both Up and Down, so they are eliminated, making the pure-strategy Nash equilibrium (Straight, Left). (d) Beginning with Rowena, Straight dominates Down, so Down is eliminated. Then for Colin, Middle dominates both Right and Left, so both are eliminated, leaving only Middle. Because Straight and Up both give a payoff of 1, neither may be eliminated for Rowena, so there are two, pure- strategy Nash equilibria: (Up, Middle) and (Straight, Middle). S2. (a) Zero-sum or constant-sum game (payoffs in all cells sum to 4) (b) Non-zero-sum (c) Zero-sum or constant-sum (payoffs in all cells sum to 6) (d) Zero-sum or constant-sum (payoffs in all cells sum to 7) S3. (a) (i) The minima for Rowena’s strategies are 3 for Up and 1 for Down. The minima for Colin’s strategies are 0 for Left and 1 for Right. (ii) Rowena wants to receive the maximum of the minima, so she chooses Up. Colin wants to receive the maximum of the minima, so he chooses Right. Again, the pure-strategy (minimax) Nash equilibrium is (Up, Right). (b) Not a zero-sum game, so minimax solution is not possible. Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company (c) (i) The minima for Rowena’s three strategies are 1 for Up, 2 for Straight, and 1 for Down. The minima for Colin’s strategies are 4 for Left, 2 for Straight, and 1 for Right. (ii) Rowena wants the strategy that gets her the maximum of her minima, or 2, which she gets from playing Straight. Colin’s maximum of the minima is 4, so he plays Left. This yields the pure-strategy (minimax) Nash equilibrium of (Straight, Left). (d) (i) The minima for Rowena’s strategies are 1 for Up, 1 for Straight, and 0 for Down. The minima for Colin’s strategies are 1 for Left, 6 for Middle, and 4 for Right. (ii) Rowena wants to receive the maximum of the minima, so she chooses Up or Straight. Colin wants to receive the maximum of the minima, so he chooses Middle. Again, the two pure-strategy (minimax) Nash equilibria are (Up, Middle) and (Straight, Middle). S4. (a) Rowena has no dominant strategy, but Right dominates Left for Colin. After eliminating Left for Colin, Up dominates Down for Rowena, so Down is eliminated, leaving the pure-strategy Nash equilibrium (Up, Right). (b) Down and Right are weakly dominant for Rowena and Colin, respectively, leading to a Nash equilibrium at (Down, Right). Best-response analysis also shows another Nash equilibrium at (Up, Left). (c) Down is dominant for Rowena; Colin will then play Middle. Equilibrium is (Down, Middle). (d) There are no dominant or dominated strategies. Use best-response analysis to find the equilibrium at (North, East) with payoffs of (7, 4). (The equilibrium is not in dominant strategies— another interesting point to convey to students.) S5. (a) Neither Rowena nor Colin has a dominant strategy, because neither has one action that is its best response, regardless of its opponent’s action. (b) For Colin, East dominates South, so South may be eliminated. Then, for Rowena, Fire dominates Earth, so Earth may be eliminated. Doing so then allows East to dominate North for Colin, so North may be eliminated. Finally, for Rowena, Water dominates Wind, so Wind may be eliminated. Elimination of dominated strategies reduces the strategic-form game to the following