Greetings to the kind tutor who would mind helping me regarding question (c) of this exercise about pure and mixed strategies in static games.

Below the exercise and my answer:

The municipality of Madrid is organizing an operation called "Madrid Verde". In a street of Chamartn, each family that has a house there will get two trees. Only two neighbors live on this street (only two houses). Each of the two neighbors must decide how many of these trees he will plant in his garden (in which case you cannot see the trees from the street) and how many he will plant in the entrance of his house (in which case you can see the trees from the street). The trees that can be seen from the street contribute to the revaluation/improvement of the neighborhood. Neighbor 2 values more than neighbor 1 the trees that can be seen from the street because he intends to sell his house soon. Suppose that the decision of each neighbor is private and that once they have planted the trees they cannot change their position. Let ?₁ and ?₂ denote the number of trees that Neighbor 1 and Neighbor 2, respectively, has decided to plant in his garden. Let x_c denote the number of trees that can be seen from the street. The utility function of Neighbor 1 is given by ?₁ (?₁ , ?_? ) = ?₁ (1,5 + ?_? ) and that of Neighbor 2 by ?₂ (?₂ , ?_? ) = ?₂ (1,5 + ??_? ), where ? > 1.

(a) Represent the above game in normal form and find all its Nash Equilibria in pure strategies.

(b) Do the equilibriums that you have found in question 1 above maximize the total social utility (the total social utility is given by ?_? (?₁ , ?₂ ) = ?₁ (?₁ ,?_? ) + ?₂ (?₂ , ?_? )? Justify your answer.

(c) Find all the mixed strategy Nash equilibria.

My answer:

(a) N = {Neighbor 1, Neighbor 2}

S₁ = S₂ = {0 , 1 , 2} ? Each of the two neighbors gets two trees then must decide how many of these trees he will plant in his garden and how many he will plant in the entrance of his house.

X_c = number of trees planted in the entrance.

X_c = 4 - X₁ - X₂ where X₁ & X₂ denote the number of trees that neighbor 1 and neighbor 2 ,respectively, has decided to plant in his garden.

U₁(X₁, X_c) = X1(1.5 + X_c) = X₁(1.5 + 4 - X₁ - X₂ ))

U2(X₂, Xc) = X2(1.5 + aXc) = X₂(1.5 + a(4 - X₁ - X₂ )) where a > 1.

Below the game in the normal form: