4. The tragedy of the commons There is a sense that people tend not to care as fully as they should for environmental assets that are shared equally among large numbers of people. Even when each individual rationally maximizes his or her personal utility, u a situation develops creating a decrease in overall utility for the total number of those using the resource. The images of trash-strewn parks or over-used camp sites may spring to mind. The vast common of the earth's atmosphere may also be another important example. We would like to understand the essence of this paradox using game theory. Consider a highly simplified 2-player game, played over 2 periods - now (period 1), and in the future (period 2). Each player is consuming a resource, say water from a well, of an initial amount, y where there is no replenishing over time. If a situation arises where the consumptions of players 1 (Jill) and 2 (Jack) in a given period is greater than the amount ofwater in the well. they must simply split it. We can imagine an authority of some kind the water police standing near the well taking initial bids for a certain number of buckets. If the total number of buckets is less than the remaining yolume, the police allows them to extract the amount reo uested, it not they must split what is left between them. Giyen this simplified situation, we would like to understand the consumption strategy of each player given they will maximise individual utility, u. We assume the consumption of players 1 and 2 in period 1 is giyen by c: and c2. a] Show that for the last period [the future}, each player will get only x[y- {c1+c2}]. bl We will assume the utility function u has the form. ur=|r1l:c'r:l. Does this function exhibit a diminishing marginal utility as we expect it should? c] Now consider the 1st period. Player 1 must now choose the amount to consume in period 1. She knows that the utility of her choice will depend on player 2's choice, since their collectiye consumption determines how much is left over at the start of the final period. If she calls player 2's chosen consumption cf. show that her best response to c;* is: clicf'l: [y {1*}!2 dl Assuming player 2's response function is perfectly symmetric to player 1's. find Cal's1*}- e} With these response functions, a Mosh equilibrium obtains when neither side can do better in light of the other player's best response. Find the Nash equilibrium consumption leyels in period 1 for both players