Question

1. Runnistan, Soggia and Wet shoreline on Overflowing Bay. The stan, Soggia and Wetland are three countries, each of which has a e on Overflowing Bay. They all have military and naval forces stationed bay, and depending on how they deploy their forces, two or more of them ay be able to control the bay effectively, and use the control to increase their trade and prosperity at the expense of the third. The strategies for the three countries are the positions at which they station their forces. Runnistan can position its forces in the north or in the south; Soggia can position its forces in the east or in the west; and Wetland can position its forces offshore on Swampy Island, which Wetland controls, or onshore. Payoffs are as follows: Runnistan Soggia Wetland East East North North North North South Payoffs (R, S, W) 7,7,1 0,0,0 6,6,6 7, 1,7 4,4,4 1,7,7 0,0,0 4,4,4 Onshore Offshore Onshore Offshore Onshore Offshore Onshore Offshore West East South South South West West a. Suppose this is a sequential game with Runnistan moving first, Soggia moving second and Wetland moving last. Draw the game in extensive form (as a tree) b. Find the rollback equilibrium. C. Which player would like to change the expected outcome of the game? d. What strategic move could that player make to improve their payoff? Explain clearly what the move is and how the others players would be expected to react to it. e. What could that player do to increase the credibility of their strategic move? f. Is there a combination of strategic moves that would yield the outcome of (6, 6, 6)? Explain clearly what those moves are.

Answer 1

Runnistan soggia Soggia Wetland HAC Wetland. Welland share/ hou Onshow off shore on- shou, show / . (77,1) (0,0,0) (,6,6) (7,1,7) (4,4,4) (1,7,2) (0,0,0) (4,4,4)

b) In this game, for roll back equilibrium, we first consider the subgame of length 1, in which player 3, that is, Wetland moves.

In the first subgame, out of the four, following history E, player 3's optimal action is choosing outcome On-shore, with payoff 1.

In the second subgame, following history W, player 3's optimal action is choosing outcome Off-shore, with payoff 7.

In the third subgame, player 3's optimal action is choosing outcome off-shore, with payoff 7.

In the fourth subgame, player 3's optimal action is choosing outcome off-shore with payoff 4.

Now, moving on to subgame of length 2, in which player 2 moves, given the optimal actions of player 3.

In the first subgame of length 2, optimal action for player 2, given player 3's action is to choose outcome East, that is, with payoff 7.

In the second subgame of length 2, optimal action for player 2, given player 3's action is to choose outcome East with payoff 7.

Now, moving on to subgame of length 3, (which is also the entire game), in which player 1 moves, given player 2 and 3's optimal actions. The optimal action for player 1 will be to choose outcome North with payoff 7.

Therefore, Roll-Back equilibrium is (North, East, On-shore) with payoffs (7,7,1).

c) Player 3 would like to change the expected outcome of the game.

d) Player 3 might like to choose offshore, which would give him higher payoff. The outcome of the game would change as follows:

In the first subgame of length 2, if player 3 chooses outcome off-shore, then player 2 would choose outcome West.

In the second subgame of length 2, if player 3 chooses outcome off-shore, then player 2 would choose outcome East.

In the subgame of length 3, player 1 would choose outcome North, given player 2 and 3's optimal choices.

Roll-Back equilibrium would change to (North, West, Off-Shore).