please answer part 3

Consider the following game, where player 1 chooses a strategy U or M or D and player 2 chooses a strategy L or R. 1. (2 points) Under what conditions on the parameters is U a strictly dominant strategy for player 1 ? 2. (2 points) Under what conditions will R be a strictly dominant strategy for player 2 ? Under what conditions will L be a strictly dominant strategy for player 2 ? 3. (6 points) Let a=2,b=3,c=4,x=5,y=5,z=2, and w=3. Does any player have a strictly dominant strategy? Does any player have a strictly dominated strategy? Solve the game by iterated deletion of strictly dominated strategies. A concept related to strictly dominant strategies is that of weakly dominant strategies. A strategy s weakly dominates another strategy t for player i if s gives a weakly higher payoff to i for every possible choice of player j, and in addition, s gives a strictly higher payoff than t for at least one choice of player j. So, one strategy weakly dominates another if it is always at least as good as the dominated strategy, and is sometimes strictly better. Note that there may be choices of j for which i is indifferent between s and t. Similarly to strict dominance, we say that a strategy is weakly dominated if we can find a strategy that weakly dominates it. A strategy is weakly dominant if it weakly dominates all other strategies. 4. In part (3), we solved the game by iterated deletion of strictly dominated strategies. A relevant question is: does the order in which we delete the strategies matter? For strictly dominated strategies, the answer is no. However, if we iteratively delete weakly dominated strategies, the answer may be yes, as the following example shows. In particular, there can be many "reasonable" predictions for outcomes of games according to iterative weak dominance. Let a=3,x=4,b=4,c=5,y=3,z=3,w= 3. (a) (4 points) Show that M is a weakly dominated strategy for player 1. What strategy weakly dominates it? (b) (6 points) After deleting M, we are left with a 22 game. Show that in this smaller game, strategy R is weakly dominated for player 2 , and delete it. Now, there are only 2 strategy profiles left. What do you predict as the outcome of the game (i.e., strategy profile played in the game)? (c) (4 points) Return to the original game of part (4), but this time note first that U is a weakly dominated strategy for player 1 . What strategy weakly dominates it? (d) (6 points) After deleting U, note that L is weakly dominated for player 2 , and so can be deleted. Now what is your predicted outcome for the game (i.e., strategy profile played in the game)? Consider the following game, where player 1 chooses a strategy U or M or D and player 2 chooses a strategy L or R. 1. (2 points) Under what conditions on the parameters is U a strictly dominant strategy for player 1 ? 2. (2 points) Under what conditions will R be a strictly dominant strategy for player 2 ? Under what conditions will L be a strictly dominant strategy for player 2 ? 3. (6 points) Let a=2,b=3,c=4,x=5,y=5,z=2, and w=3. Does any player have a strictly dominant strategy? Does any player have a strictly dominated strategy? Solve the game by iterated deletion of strictly dominated strategies. A concept related to strictly dominant strategies is that of weakly dominant strategies. A strategy s weakly dominates another strategy t for player i if s gives a weakly higher payoff to i for every possible choice of player j, and in addition, s gives a strictly higher payoff than t for at least one choice of player j. So, one strategy weakly dominates another if it is always at least as good as the dominated strategy, and is sometimes strictly better. Note that there may be choices of j for which i is indifferent between s and t. Similarly to strict dominance, we say that a strategy is weakly dominated if we can find a strategy that weakly dominates it. A strategy is weakly dominant if it weakly dominates all other strategies. 4. In part (3), we solved the game by iterated deletion of strictly dominated strategies. A relevant question is: does the order in which we delete the strategies matter? For strictly dominated strategies, the answer is no. However, if we iteratively delete weakly dominated strategies, the answer may be yes, as the following example shows. In particular, there can be many "reasonable" predictions for outcomes of games according to iterative weak dominance. Let a=3,x=4,b=4,c=5,y=3,z=3,w= 3. (a) (4 points) Show that M is a weakly dominated strategy for player 1. What strategy weakly dominates it? (b) (6 points) After deleting M, we are left with a 22 game. Show that in this smaller game, strategy R is weakly dominated for player 2 , and delete it. Now, there are only 2 strategy profiles left. What do you predict as the outcome of the game (i.e., strategy profile played in the game)? (c) (4 points) Return to the original game of part (4), but this time note first that U is a weakly dominated strategy for player 1 . What strategy weakly dominates it? (d) (6 points) After deleting U, note that L is weakly dominated for player 2 , and so can be deleted. Now what is your predicted outcome for the game (i.e., strategy profile played in the game)