ECONOMY

884 anarchy
one in which f (G i )=G m i
,wherem > 0 (and often, for technical reasons, 1 ≥ m),
so that:
G m 1
p 1 (G 1 , G 2 )=
G m 1 + G m 2
For this form the probability of winning depends on the ratio of guns expenditures
by the two parties. Some effects of different military technologies can be captured by
the parameter m; for example, the technology of seventeenth-century warfare during
the Thirty Years War, with its large armies and artillery units, can be thought of as
having a higher value for this parameter than the feudal levies of medieval Europe.
Such factors have consequences for the size of political units.
(4)
1.2 How Power is Determined
We can now examine how the two countries can be expected to distribute their
resources between guns and butter in this simple setting. We abstract away from
all the collective action problems as well as those of strategic interaction between
domestic political groups, and suppose that each country behaves as a unitary actor.
With both countries caring about how much butter they will consume and supposing
that they are risk neutral, 3 the expected payoffs intheeventofwar,inwhichthe
winner receives all the butter and the loser receives nothing, would be the following:
W 1 (G 1 , G 2 )= p 1 (G 1 , G 2 )(B 1 + B 2 )= p 1 (G 1 , G 2 )[‚ 1 (R 1 −G 1 )+‚ 2 (R 2 −G 2 )]
W 2 (G 1 ,G 2 )= p 2 (G 1 ,G 2 )(B 1 + B 2 )= p 2 (G 1 , G 2 )[‚ 1 (R 1 −G 1 )+‚ 2 (R 2 −G 2 )] (5)
Note that these two payoffs are thought of as depending on the expenditures on
guns by the two parties, since given (2) the choice of guns by each party determines
the quantity of butter as well. Furthermore, because these payoffs arederivedonthe
condition that the two countries are risk neutral, the probability of winning for each
country, p 1 (G 1 , G 2 )andp 2 (G 1 , G 2 ), can also be interpreted as the share of total
butter each country receives in the shadow of war.
We are interested in deriving Nash equilibrium strategies for guns—that is, a
combination (G ∗ 1 , G ∗ 2 ) such that W 1(G ∗ 1 , G ∗ 2 ) ≥ W 1(G 1 , G ∗ 2 ) for all G 1 and
W 2 (G ∗ 1 , G ∗ 2 ) ≥ W 2(G ∗ 1 , G 2) for all G 2 . At a Nash equilibrium there is no incentive
for any party to deviate in their strategies. At that equilibrium the marginal benefit of
each country’s choice of guns equals its marginal cost so that:
∂p 1 (G ∗ 1 , G ∗ 2 )
∂G 1
(B ∗ 1 + B ∗ 2 )= p 1(G ∗ 1 , G ∗ 2 )‚ 1
∂p 2 (G ∗ 1 , G ∗ 2 )
∂G 2
(B ∗ 1 + B ∗ 2 )= p 2(G ∗ 1 , G ∗ 2 )‚ 2 (6)
³ Under risk neutrality there is neither aversion towards risk nor love of risk. Clearly, this is a strong
assumption that is nevertheless analytically very convenient that is almost always adopted when a certain
problem is first modeled. We discuss the effects of risk aversion in Section 2.