The goats of two villagers graze on a common pasture. The number of goats that villager i = 1, 2 keeps is xi , assumed to be real non-negative number. The payoff of villager i depends directly on xi . It also depends on how much of the pasture is left unused for regeneration. Namely, if X = x1+x2 is the total number of goats, and K is some exogenous capacity of the pasture, then the payoff is increasing in the difference (K X). To be precise, the total payoff of agent i = 1, 2 is ui = xi + K X. (If villagers select the number of goats so that K < X, then the pasture is destroyed, all goats die and the payoff drops to zero)

1. Villagers simultaneously decide how many goats to keep. Find a Nash equilibrium of this game. Illustrate with a clearly annotated graph of best reply functions.

2. Is this Nash equilibrium efficient (in the utilitarian sense)? Demonstrate and explain.