Question

We are presented with land use choices.  Each individual is free to choose their own development strategy based on the profit potential of the development.

a) Does either landowner have a dominant strategy? A dominant strategy is what you are going to do, knowing what they are going to do—this leads to the Nash Equilibrium  Explain how and what the dominant strategy is (hint, there does not need to be two dominant strategies).

b) Is there a Nash equilibrium? Explain.

c) Develop a Profit Payoff for this choice where the alternative Nash Equilibrium would occur.

d)  What is the socially optimal solution?

e)  How can we arrive at this point (consider the Coase Solution).  Explain the necessary payoff.

Payoff Matrix

                                               Smith

                                                Apple Orchard                Pig Farm

            Rental Housing            J:  Profit =$700               J:  Profit = $400

                                                S:  Profit =$200               S:  Profit = $450  

Jones

            Bee Keeping                J:  Profit =$650               J:  Profit = $450

                                                S:  Profit =$400               S:  Profit = $500

Answer 1

a) A dominant strategy is a strategy that gives the player the highest payoff regardless of the other players' decisions (i.e. it gives the highest payoffs in each of the strategies the other player chooses)

In this case Smith has a dominant strategy of selecting a Pig farm. Smith's profit with Pig farm is higher than Apple Orchard if Jones selected Rental Housing (As 450> 200) and also Smith's profit with Pig farm is higher than Apple Orchard if Jones selected Bee keeping (As 500>400).

Thus in either case the payoff from Smith choosing Pig farm is higher and Choosing Pig farm is his dominant strategy. (Jones does not have a dominant strategy as Rental housing gives higher profit in some cases to Jones and Bee keeping gives a higher profit in other cases)

b) A nash equilibrium is a case where no player can increase his profits or payoff by changing his decision assuming the other players stick to their decision. In this case we have a Nash equilibrium of Smith choosing Pig farm and Jones choosing Bee keeping. Smith cannot increase his profit of 500 by selecting Apple orchard as his profit will then decrease to 400, and similarly Jones cannot increase his profit of 450 by selecting Rental housing as then his profit will reduce to 400.

Thus the Nash equilibrium is Smith selects Pig farm and has a profit of 500$ and Jones selects Bee keeping and has a profit of 450$

c) The alternative Nash equilibrium would be of Jones selecting Rental housing and Smith selecting Apple Orchard. There can be many profit payoff matrices where that would be the case however for simplicity we can just exchange the values of the profits of Smith for each decision so that the ne nash equilibrium becomes Rental housing and Apple Orchard

Jones\Smith	Apple Orchard	Pig farm
Rental Housing	700 , 450	400, 200
Bee Keeping	650 , 500	450 , 400

In this case the dominant strategy for Smith is Apple Orchard and so he will always select Apple Orchard and Now Jones selects rental housing for highest profit and now neither has any incentive to change their strategy and it becomes a nash equilibrium.

d) The socially optimal solution in the original question will be the set of strategies that leads to the highest combined profit which is Bee keeping and Apple Orchard where the combined profit is 650+400=1050$

e) Since the nash equilibrium of the original question was Bee keeping and Pig farm where the combined profit was 950. To go from that to Bee keeping and Apple Orchard, the profit of Jones will increase by 200 but the profit of Smith will reduce by 100. They can arrive at this new point by bargaining. Since Smith has his profits reduced by 100, Smith will be willing to change to Apple Orchard if he is paid 100 or more. Similarly Jones' profit increase by 200 if Smith moves to Apple Orchard, therefore Jones will be to pay upto 200 (200 or less). Therefore according to Coase theorem, if there is no transaction costs, they will bargain such that Jones pays smith somewhere between 100 to 200 and we arrive at the new point of socially optimal solution of Bee keeping and Apple Orchard.

Hope it helps. Do ask for any clarifications if required.