Examine the following game tree RentA (16, x) Rent S Buy B (8, 12) Rent C (6, 6)

Question

Examine the following game tree RentA (16, x) Rent S Buy B (8, 12) Rent C (6, 6) Buy S2 Buy D (9,7 Fred and Sally are planning on running competing restaurants. Each must decide whether to rent space or buy space. Fred goes first at decision node F. Sally goes second at either decision node or decision node S2 (depending on what Fred chose to do at decision node F. Note that the payoff to Sally at terminal node A is X a. if X 12, what terminal node will the subgame perfect Nash equilibrium path lead to? Click to select) v C.Suppose that X-11 but that it is now possible for Fred to make a side payment of value Vto Sally that will boost her payout at decision node A from X= 11 to X= 11 + V. what is the minimum amount that V can be such that the subgame perfect Nash equilibrium path will lead to terminal node A? Assume that Vcan take on only discrete units (0,1,2,3, .). Minimum payment of V

Explanation / Answer

a) We solve using Backward Induction. We see what happens given Sally will go first. If X<12, for S1, Sally will go for Buying since payoff 12 will be greater than X. For S2, Sally will again go for Buying since 7>6. So the option to Rent is eliminated. Now coming to Fred, he can go for (Rent, Buy) or (Buy, Buy). Here, (Buy, Buy) will give him a higher payoff of 9 at node D compared to 8 at node B. Thus, (Buy,Buy) is the Sub-game perfect Nash Equilibria.

b) If X>12, for S1, Sally will go for Renting since payoff wil be >12 as compared to payoff of 12 in Buying. For S2, Sally will go for Buying since 7>6. So the option to Buy is eliminated for S1 and the option to Rent is eliminated for S2. Now coming to Fred, he can go for (Rent, Rent) or (Buy, Buy). Here, (Rent, Rent) will give him a higher payoff of 16 at node A compared to 9 at node D. Thus, (Rent,Buy) is the Sub-game perfect Nash Equilibria.

c) For the sub-game Nash Equilibria path to lead to node A, first Sally must select node A to node B for S1 itself. This will happen only when X>12. Thus, (11+V)>12 which gives V>1. Since V can take only discrete values, the minimum value for V so that node A is SPNE is 2.

Therefore, minimum payment of V = 2.

Thank You and Best of Luck :)

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