Question

1. (20 points) Consider the following game: Left 7,17 10,5 4,4 Top Middle Bottom Player B Middle 21,21 14,4 7,3 Right 14,11 4,3 10,25 Player A a. Does either player have a dominant strategy? What about a dominated strategy? b. What are the Nash equilibria of this game? C. Is there one Nash equilibrium that you think is a more likely outcome than the others? If so, why? If not, why not? d. Now suppose the game looks like this: 1 Left 7,17 10,5 4,4 Top Middle Bottom Player B Middle 15,21 14,4 7,3 Right 14,11 4,3 13,25 Player A Have the Nash equilibria changed from before? e. Which outcome would Player B prefer to happen? Why wouldn't it happen? Is there anything (outside the game) she could do to achieve it? Can you think of any (legal or not) real-world analogs to this scenario?

Answer 1

First 4 parts

a. If player 1 plays Top, player 2 is strictly better off by playing Middle because she gets the highest payoff which 21.

Similarly, if player 1 plays Middle, player 2 is strictly better off by playing Middle than left or right.

Similarly, if player 1 plays Bottom, player 2's best strategy is to play Right.

It means player 2 plays each strategy in each of the cases. No single strategy is a dominant strategy. Each strategy is player at least once by player 2. So, there is no dominated strategy.

Hence , there is neither a dominant strategy nor a dominanted strategy for player 2.

If player 2 plays Left, player 1 gets highest payoff by playing Middle. So, Player 1's best strategy is playing Middle.

Similarly, if player 2 plays Middle, player 1's best strategy is to play Top.

Similarly, if player 2 plays Right, player 1's best strategy is to play Top.

Here, player 1 doesn't play any single strategy in all cases. So, there is no dominant strategy.

But player 1 never plays Bottom. Because in each case, player is better off by playing Top or Middle rather than Bottom. So, in no case is the Bottom the best strategy for player 1. So, the strategy Bottom is dominated by top and middle for player 1. Both players would strictly prefer (Top,Middle) to (Middle,Left).

Hence, there is no dominant strategy for player 1 and the strategy Bottom is a dominanted strategy for player 1.

b. A strategy is called the Nash equilibrium strategy if each it is the best the strategy of each player given the strategy of the other player.

Here, from part a, we can see that

Player 1's best strategy given that player 2 plays Middle is Top. And player 2's best strategy given that player 1 plays Top Middle. So, it is the best strategy for each player given the other player's strategy. So, (Top,Middle) is a Nash equilibrium.

Similarly, player 1's best strategy given that player 2 plays Left is Middle. And player 2's best strategy given that player 1 plays Middle is Left. So, (Middle,Left) is a Nash equilibrium.

There is no such case apart from the above mentioned cases.

Therefore, (Top,Middle) and (Middle,Left) are the Nash equilibria of the game.

c. (Top,Middle) pays a payoff of 21 to each player and (Middle,Left) pays 10 to player 1 and 5 to player 2. So, each player gets higher payoff at (Top,Middle). So, (Top,Middle) is Pareto superior to (Middle,Left).

Therefore, (Top,Middle) is more likely to happen than (Middle,Left). (

d. No, the Nash equilibria will remain unchanged.

(Top, Middle) and (Middle, Left) will still be the Nash equilibria and no other Nash equilibrium is there is the modified game.