2.3. Strategies and equilibrium concept
Suppose that the game begins in period 0, with the null history h0. For t > 1, let ht ={x0,
x1,..., xt-1} be the realised choices of actions at all electoral periods before t, and let Ht=(X)t be
the space of all possible period-t histories. I will restrict the attention to strategies in which the
past ht influences current decisions xi only through its effect on the state variable (Wt), which
summarises the direct effect of the past on the present environment. This class of strategies are
called Markov strategies. So a pure strategy si for player i (i=I,II) is a sequence of maps sit -
one for each electoral period- from values of the state variable at time t (Wt) into the action set
of spending decisions X (i.e., s: W → X). A Markov perfect equilibrium is a profile of
Markov strategies (s*1, s*π ) that yields a Nash equilibrium in every proper subgame. This
equilibrium concept has the property of being empty-threat proof because each player’s
strategy is the best response for every possible state.
In this game grim strategies will be used to explore the possibility of “early-stopping”
equilibrium, which will be associated with co-operative behaviour between both players. These
strategies are of the form: ‘Play xt = r in period t, and continue to play xt = r so long as the
realised action in the previous period xt-1 was r. If not, play the minimax strategy for the rest
of the game, that is to say, try to secure as many elections as possible.10
2.4. Payoff functions
The payoff function is ∑“0 μtg(Wt,x0...xt), which given the assumptions stated before
becomes ∑“0 μtp(x0...xt )c>t(Wt)K. The following probability tree depicts the structure of
the payoff during a sequence of electoral periods. For the sake of simplicity in the
representation, p and the discount factor are assumed to be constant, which is only true in a
particular situation when a stationary state has been reached.
10 By playing minimax the incumbent (Player I) plays the best response assuming that the opposition party
(Player II), once in office, will play that strategy that gives the worse result for her. That is, player II will try
to minimise player I's pay-off once he has the turn to play. Under this assumption, the best that Player I can
do while she has the turn to play is to minimise the chances of Player II getting into office (consequently,
minimising the opposition party's pay-off).