The Dictator and the Parties A Study on Policy Co-operation in Mineral Economies

IU

off the incumbent expects to receive if she deviates (I ^1D ) and what she will receive if co-
operation is maintained (I^1C).

In order to obtain the gains of defection, it is assumed that if the incumbent deviates she will try
to secure as many elections as possible. Thus the punishment phase is characterised by the
implementation of a minimax or myopic strategy (s^m). During the period of deviation (not
necessarily equal to one electoral period) the incumbents will increase the pace of expenditure
reducing the stock of reserves in W^d equal to (W₀ - W ), assuming that deviation occurs at
t=0. By following a myopic strategy, the deviant incumbent will stay in office with certainty n

electoral periods.

Assuming that n is an integer, (i.e., that in the period t = n+1¹⁶ expenditure will be set at x=r)
the incumbent expects to obtain a pay-off after defecting equal to:¹⁷

(11) 1¹D = K[1 + (δ 1)p_e + ( δ2) p_e +...] + δn [ 1¹N ]

(1 - P_r δN ) K

^{(1 - δ}N ^{)[1 + δ}N ^{- 2δ}N . p_r ^]

I ^1N is the expected pay-off for the incumbent party in the stationary equilibrium and
P_e = P ( xe ) = 1 means that the incumbent's re-election is certain. The sequence of discount
factors during the deviation period reflects the fall of reserves. They are expressed as:
δ = 4Wo -(x∙ -r),Д] ; δn-, = δ,-2.4W0 - (n- 1)(x^e -r),μ] ; δ, = δ,-ɪ-δ[W^e,д]

¹⁶ The subscript (n+1) indicates the electoral period at which the system would rest again in stationary
Nash equilibrium if a deviation occurred in the current period.

¹⁷ For the deviant case in which xn+1 > r, the gains of defection will be greater; therefore, by assuming n as
an integer the temptation will be underestimated. However, a probability of re-election at t=n+1 different
than P(r) will complicate the algebra and clarity of exposition without changing the conclusion of the
analysis. The pay-off under deviation for this more general is obtained by developing eq. (1) :

I¹D = K[¹⁺...^δnPe ^{+ δ}n+1 p_d^{] + (1 -}p_d ^-p_r ^{+ 2}p_dp_r)μ^{n+1 δ}NI^{1 n +} (Pd ⁺ p_r ^-2p_dp_r)μⁿ⁺1^δ_NI⁰N
Where: p_d = p(Xd ) = p(r + Wld - n(Xe - r )]) refers to the probability that the current incumbent
will be in office at time t=n+1, given that defection has occurred at time t=0. X^d stands for the expenditure
during the period when reserves reached the equilibrium level.

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