An arms buildup is thought to have been a contributing factor to World War I. The naval arms race between Germany and Great Britain is particularly noteworthy. In 1889, the British adopted a policy for maintaining naval superiority whereby they required their navy to be at least twoand-a-half times as large as the next-largest navy. This aggressive stance induced Germany to increase the size of its navy, which, according to Britain’s policy, led to a yet bigger British navy, and so forth. In spite of attempts at disarmament in 1899 and 1907, this arms race fed on itself. By the start of World War I in 1914, the tonnage of Britain’s navy was 2,205,000 pounds, not quite 2.5 times that of Germany’s navy, which, as the second largest, weighed in at 1,019,000 pounds.10 With this scenario in mind, let us model the arms race between two countries, denoted 1 and 2. The arms expenditure of country i is denoted and is restricted to the interval [1,25]. The benefit to a country from investing in arms comes from security or war-making capability, both of which depend on relative arms expenditure. Thus, assume that the benefit to country 1 is so it increases with country 1’s expenditure relative to total expenditure. The cost is simply so country 1’s payoff function is
and there is an analogous payoff function for country 2:
These payoff functions are hill shaped.
a. Derive each country’s best-reply function.
b. Derive a symmetric Nash equilibrium.