U reaction function. At low levels of S weapons, in particular, for w _S
less than w + ̄ _S the war arms-investment strategy is best, as U can
obtain the redistribution of resources in the second period from in-
vesting in weapons and fighting. At high levels of W _s, however, in
particular above w + ̄ _S, the deterrence arms-investment strategy is
best, as U can avoid a costly investment in weapons in the first period.
Thus there is a switch at w + ̄ _S. The reaction function w _U (w _S ), rises with
w _S, showing that U will buy more weapons than S does, using a war
arms-investment strategy. At w + ̄ _S, however, the reaction function
shifts to the upper segment w _U (w _S ), again increasing with w _S, but
starting from a lower base, using a deterrence arms-investment
strategy. Similarly w _S (w _U ), the reaction function for S, rises with
w _U, but makes a discontinous jump down at the critical value w + ̄ _U for
which S switches from a war arms-investment strategy to a deterrence
arms-investment strategy.

The discontinuity of each reaction function implies that to achieve
an equilibrium it is necessary for both countries to use a mixed strat-
egy. With appropriate probabilities assigned to the use of a war arms-
investment strategy (consistent with the other side having low levels
of weapons) and remaining probabilities assigned to the use of a de-
terrence arms-investment strategy (consistent with the other side
having high levels of weapons), an equilibrium exists in figure 3. With-
out such mixed strategies there is no equilibrium. For example, if
each side believes that the other has a high level of weapons, then
each will adopt a deterrence arms-investment strategy. The result
will be an endless upward spiral in armament--an arms race without
end. ¹²

There are two possible ways to avoid the instability implied by
figure 3. One is to allow for voluntary transfers. This approach avoids
the discontinuity in the reaction functions, but it is clearly politically
unacceptable, involving one country voluntarily giving part of its en-
dowment to the other side. The other way is to avoid the discontinuity
by relying on weapons that do not have extreme effects on utility, in
particular, on nonnuclear rather than nuclear weapons. With such
weapons the war investment strategy is always optimal, leading to
continuous reaction functions, an equilibrium in weapons levels, the
avoidance of the use of mixed strategies, and the avoidance of an end-
less upward spiral in armaments, a never-ending arms race.

The theoretical two-period model of a potential arms race followed
by a potential conflict yields a dilemma. The outcome is either a neces-
sary transfer payment of resources from one country to the other or a

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