The following game between the EU and UK begins in 2015 when the EU must decide whether to change the terms of the UK's membership.

- If the EU says "Yes" to altered membership for the UK, the game ends and both players get a payoff of 0.
- If the EU says "No," the UK must decide whether to Remain or Leave.
- If the UK Remains, the game ends and the UK gets a payoff of -5 while the EU gets a payoff of 5.
- If the UK chooses to Leave, the EU must choose either a Weak or Tough negotiating position. After this choice, the UK must decide whether to pursue a Hard or Soft Brexit.
- If the EU chooses Weak and the UK chooses Soft, the EU gets a payoff of -10 while the UK gets a payoff of 5.
- Any other combination of Weak/Tough and Hard/Soft will lead to a breakdown in negotiations, with the EU getting a payoff of -40 and the UK getting a payoff of -80.

(i)
(a) Draw the game tree of this game. [3 Marks]

(b) Represent this as a normal form game (i.e., a matrix) and find all the pure-strategy Nash equilibria. [4 Marks]

(c) Find all the Subgame Perfect Nash Equilibria (SPNE) in this game. How many are there? Why should we expect this to be the number of SPNE? [4 Marks]

(d) Is it a credible strategy for the EU to play Tough? Is it a credible strategy for the UK to play Hard? Discuss why or why not in each case. [4 Marks]

To draw the game tree, we start with the initial decision of the EU to change the terms of the UK's membership or not. If the EU says Yes, the game ends with a payoff of 0 for both players. If the EU says No, we move to the next decision point where the UK chooses to Remain or Leave.

If the UK Remains, the game ends with a payoff of -5 for the UK and 5 for the EU. If the UK Leaves, the EU must choose between a Weak or Tough negotiating position, and the UK must decide between a Hard or Soft Brexit. There are four possible outcomes in this case: Weak-Soft, Weak-Hard, Tough-Soft, and Tough-Hard, each with different payoffs. To represent this game as a normal form game or a matrix, we list the possible strategies for each player and their corresponding payoffs.

The strategies for the EU are to say Yes or No initially and to choose Weak or Tough in the second stage. The strategies for the UK are to Remain or Leave initially and to choose Hard or Soft in the second stage. The matrix will have 8 cells representing the payoffs for each combination of strategies.To find all the Subgame Perfect Nash Equilibria (SPNE), we need to identify strategies that are optimal at each decision point, given the strategies of the other player.

To know more about Game tree visit:-

https://brainly.com/question/30500053

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