THIS IS STRATEGIC THINKING GAME THEORY QUESTION

Question 4. [30 total points] \"Mutual assurance.\" Andrew and Brianna grow and eat bananas. Their adjacent properties are hit by exactly one cyclone each year. Each storm destroys exactly one of their crops. With probability 1 ,/ 2 it destroys Andrew's crop but not Brianna's and with probability 1 / 2 it destroys Brianna's crop but not Andrew's. When a crop is not destroyed it yields 4 tonnes of bananas. Bananas cannot be stored between years. Each person's payoff in a given year from consuming bananas in that year is given by Bananas consumed (in [metric] tonnes) Payoff Notice that each incremental tonne of bananas consumed gives less incremental payoff than the previous tonne. Each person aims to maximize his or her present discounted expected payoff, where each has the same discount factor 6 = 3/4. If each person just consumes his or her own crop, each gets with probability 1/2 a payoff of 20 and with probability 1 f 2 a payoff of zero, for an expected payoff of 10. The (continuation) value of getting this expected payoff per year forever is 10/(15) = 10/(1/4) = 40. Alternatively, if in each period, the total crop of the two properties is shared equally between Andrew and Brianna, they each get a payoff of 14 in each period, and the continuation value in this case is 14/ (1/ 4) = 56. Now let's treat this as an innitely repeated game. In each period, after the cyclone has destroyed one of their two crops, the person whose crop is not destroyed can choose to give 0, 1, 2, 3 or 4 tonnes of bananas to the person whose crop was destroyed. This choice of gift is made c'er observing which crop was hit, and the corresponding tonnes of bananas are consumed that year. (a) (10 points) Show that both Andrew and Brianna adopting the following analog of the grim strategy is a subgame perfect equilibrium (SPE) of this repeated game, in which along the equilibrium path, after every storm, the person with bananas Page 4 of 5 Strategic Thinking: an introduction to game theory,r {ECON2141) gives one half of his or her crop (that is, two tonnes) to the person whose crop was destroyed