(4) In this game, each of two players can volunteer some of their spare time planting...

(4) In this game, each of two players can volunteer some of their spare time planting and cleaning up the community garden. They both like a nicer garden and the garden is nicer if they volunteer more time to work on it. However, each would rather that the other person do the volunteering. Suppose that each player can volunteer 0, 1, 2, 3, or4 hours. If player 1 volunteers x hours and 2 volunteers y hours, then the resultant garden gives each of them a utility payoff equal to √ x + y. Each player also gets disutility from the work involved in gardening. Suppose that player 1 gets a disutility equal to x (and player 2 likewise gets a disutility equal to y). Hence, the total utility (payoff) of player 1 is Π1(x, y) = √ x + y−x, and that of player 2 is Π2(x, y) = √ x + y−y. Write down the best response set of each player to every strategy of the other players 1 .

(6) Consider again the situation of strategic interaction in problem (4) Suppose that an outside observer asked the following question: How much total time should be volunteered in order to make the sum of their utilities the highest? How would you answer this observer?

SOLVE 6 PLEASE

Solutions

Expert Solution

(4) Total Utility of player 1:

Total Utility of player 2:

Player 2 utility is completely dependent on player 1's choice of number of hours working on garden. Hence, any choice of number of hours utilized on working in garden is best response of player 2 to every choice of player 1.

Therefore, best response set of player 2 (BR₂) = {0,1,2,3,4} for all members in BR₁ (best response set of player1).

Player 1's best response is also independent of player 2's choice of hours. He will spend either 0 or 1 hour working on garden because in that case his utility is and in all other cases his utility is less than y.

Therefore, best response set of player 1 (BR₁) = {0,1} for all members in BR₂.

(6) Sum of utilities:

Our objective is to maximize (1) subject to .

The optimal y will be 4 and optimal x will be 1.

Rahul Sunny answered 1 year ago

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