International Security
Arma virumque cano

This paper investigates the impact of information concerning stocks of weapons on the possible outcomes of an arms race. Several attempts have been made to find a behavioural foundation for Richardson's model (Brito (1972), Simaan and Cruz (1975)) and to estimate the coefficients of the model (McGuire, Desai and Blake). This paper tries to provide a more satisfactory strategic underpinning of the Richardson model using a feedback model where information on the development of the arms race is incorporated into the strategies of the countries involved. This means that the appropriate solution concept of this game is the so-called subgame-perfect equilibrium. This concept not only allows for the exploitation of information, but also implies that there are no commitments. The question that is asked in the paper is whether the so-called open-loop solution with commitments and without the possibility to exploit information leads to more or less weapon accumulation than the subgame- perfect solution. In other words, does verification lead to fewer or more weapons. Second, the concepts are compared regarding the resulting welfare for the countries involved in the arms race. Finally, the results of different expectations patterns and matters of credibility are evaluated. The microeconomic foundation of the model is kept very simple. The indirect utility functions of the two governments involved in the arms conflict can be characterized as a "guns versus butter" dilemma. The more a government invests in arms, the less its constituents can consume and the more they will feel safe (which obviously also depends on the investment in arms of the other country).

The most widely used model of arms races is due to Richardson, who analysed a dynamic model in which the rate of change of each country's weapon stocks depends on three factors: "defence", "fatigue" and "grievance or hatred". The defence term depends on the weapon stock of the other country, while the fatigue term is a negative function of the country's own arms stock. The grievance term does not depend on the levels of the weapon stocks, but rather reflects the country's fixed desire to increase its stock of weapons regardless of the behaviour of the other country. Several attempts have been made to find a behavioural or strategic micro foundation for the equations in Richardson's model. Brito (1972) reformulated the Richardson model as a differential game, but restricted himself to arms accumulation in which the countries are committed to their announced policies. Simaan and Cruz (1975) obtained a feedback (or subgame-perfect) Nash equilibrium solution, which relies on nations being able to monitor their rival's weapon stock. The objectives of this paper are to provide a more satisfactory strategic foundation for the Richardson equations and to show that the subgame-perfect Nash equilibrium leads to less weapon accumulation than the open-loop Nash equilibrium. The paper focuses on the impact of information concerning the rival country's weapon stock, and the results suggest that countries should be encouraged to observe and verify each other's weapon stocks.
The model considers two countries involved in arms conflict. "The West" is a decentralized market economy comprised of a representative household, a representative firm and the government. There is no private capital accumulation, although the government does invest in weapon stocks. There is only one domestically produced commodity, which can be used for both consumption and investment purposes. The government demands goods for investment purposes, the household supplies labour and demands goods for consumption purposes, and the firm demands labour and supplies goods. The real wage adjusts in order to ensure labour market equilibrium. The government finances the provision of public goods, i.e. weapons, by means of non- distortionary taxes and chooses its expenditure on weapons so as to maximize the utility of the representative household. The household's utility depends on consumption, leisure and defence; defence is a characteristic which depends on the difference between home and foreign weapon stocks. When consumption and leisure are normal goods, there is a "guns versus butter" dilemma; the more a government invests in arms, the less its constituents can consume and the more they will feel safe (which obviously depends on the other country's investment in arms).

The decentralized market economy is engaged in competitive arms accumulation with a centrally planned economy, called the East. The East has the same technology and preferences, but its government plans consumption, leisure and arms spending to maximize utility, subject to the production function constraint.

The decentralized market economy of the West and the centrally planned economy of the East are identical, because no distortions or market imperfections have been considered. One possible form of asymmetry between the two economies occurs if wages and prices are rigid in the short run, because then the West is likely to be in a regime of Keynesian unemployment and the East in a regime of repressed inflation. These extensions have implications for the conflict over arms accumulation which are discussed in the paper. We first consider a "cooperative" equilibrium; since the game between the two countries is zero sum (at the margin), the cooperative outcome involves a moratorium on investment in weapons, so that stocks fall to zero.

In the long run this is a consequence of defence depending on the difference in arms levels rather than their sum: consumption in both countries can rise with no loss in security if no arms are accumulated.

The cooperative outcome is not sustainable, however, as each country has an incentive to deviate from a multilateral arms treaty by increasing its security at the expense of its rival, if the desired lead in weapons is greater than zero. Furthermore, even if countries do not care about the sum of stocks of weapons, a moratorium on investment in arms would seldom be observed in the real world. We therefore consider non-cooperative outcomes as well.

If West and East do not cooperate and neither dominates the arms race, a Nash equilibrium is appropriate. The most common concept of Nash equilibrium is perhaps the open-loop Nash equilibrium solution, which has been used by Brito. It presumes that the optimal investments in arms at each point of time are only conditioned on the initial weapon stocks and therefore the expected investments of the rival do not depend on past or current weapon stocks or on past or current investments of the country under consideration. It follows that the open-loop Nash solution requires that each country pre-commits itself to a path of investment in arms and that the expectations of each other's paths of investment are correct in equilibrium.

Since the marginal value of Eastern (or Western) weapons to the West (or East) does not affect the open-loop Nash equilibrium solution, it does not matter whether the countries can observe their own weapon stocks. This means that the open-loop Nash equilibrium solution also describes the situation where each country can monitor its own weapon stocks but not those of the other nation.

We also consider a closed-loop Nash equilibrium solution in which each country can condition its investment in weapons on the current and, possibly, past stocks of weapons, so that each country should be able to monitor its own as well as foreign weapon stocks. This type of information structure does not give rise to a unique equilibrium. We therefore introduce the additional requirement of subgame perfectness. An equilibrium solution is subgame-perfect if for each subgame over a remainder of the planning period, the relevant part of the solution is also a Nash equilibrium. Each player (country) expects the other player to react rationally at any point in time to the information available at that time and in equilibrium these expectations are correct. Subgame perfectness rules out threat equilibria, which rely on information patterns with memory, and equilibria which imply future investments that are not rational to carry out if called upon to do so in the future.

The main result of the paper is that the subgame-perfect solution leads to less arms accumulation and more welfare than the open- loop solution. Monitoring of foreign weapon stocks, i.e. verification, leads to lower weapon stocks than in the absence of monitoring, since monitoring decreases the grievance coefficients. The intuition behind this result is that, when one country considers the purchase of one additional unit of weapons, it considers not only the direct marginal contribution to security and welfare, but also the reaction of its rival. In other words, it takes account of the fact that its rival's security is worsened and therefore it will purchase more weapons. This therefore reduces the direct marginal contribution to security and welfare so that there is less incentive to invest in weapons than when countries cannot observe their rival's weapon stock. The obvious policy implication is that countries should be encouraged to monitor each other's weapon stocks as this will lead to some unilateral disarmament. Another effect of monitoring is that the defence coefficient can easily be shown to be larger than without monitoring. Hence, the adaptation to the (lower) steady-state levels of arms is faster than in the absence of monitoring. It is also true, however, that the cooperative solution has an even faster speed of adjustment than the non- cooperative solution with monitoring. The outcome of the game with pre-commitment and with one dominant player is that both countries are better off than in the game without dominant players and that the dominant country is worse off than the follower country.

Our analysis also reveals that the desired lead of weapons over the rival country and the relative priority of "guns" rather than "butter" increase the "grievance or hatred" coefficients in Richardson's model at which future utility is discounted and therefore also increase the steady-state levels of weapon stocks. The rate of discount, the depreciation rate of weapon stocks, and the relative priority of "butter" rather than "guns" diminish the "defence" coefficients and the speed of adjustment to these steady-state levels.