In the first stage of the game, the incumbent chooses a type for service that will be provided after the election, while the opposing party simultaneously chooses the quality of a challenger to run against the incumbent. There is then a subgame (the game's second stage) during which the incumbent and challenger each produce a flow of proposals for rates at which each will provide the service after the election, if victorious, while a financial contributor simultaneously produces a flow of proposals for money it will give to each candidate's campaign. In the subgame the three players act in continuous time, their movements being restricted to the dynamics of a system of ordinary differential equations. That system starts from specified kinds of initial conditions and runs for a finite time that is common knowledge. At the end of that time the voters learn the then-current parameter values and the election occurs. The probability that the incumbent wins the election is a function of the current values of the service rates and the contributions. The incumbent, the opposing party, the challenger and the contributor have the function as common knowledge and all fully anticipate the probability. Payoffs to the incumbent, opposing party, challenger and contributor are functions of the probability, evaluated at the time of the election. The players have complete information about their own and one another's payoff functions. Overall, the solution concept for the game is perfect Nash equilibrium. In each subgame the solution concept is Cournot-Nash equilibrium, with some refinements to choose initial conditions and specify what occurs when no Cournot-Nash equilibrium exists.
The model draws on several ideas developed in other formal work. As in many other models, the parties, candidates and contributor interact before the election based on rational expectations of voters' behavior. In Baron's (1989a; 1989b; 1994) models of interactions between candidates and contributors, the probability that each candidate wins the election is an explicitly specified function. Baron (1989a) showed that solutions with incumbent advantages in both contributions and reelection chances can be produced using a variety of exogenous differences between the candidates, including differences in recognition, valuation of the office, service effectiveness and interest-group support. The present model treats the incumbent and the challenger asymmetrically, but the asymmetries do not necessarily lead to an incumbent advantage. Austen-Smith (1987; 1995) has also analyzed games with policy selection, campaign contributions and elections.
The present model is also informed by the concept that contributors view their contributions as investments. In connection with electoral outcomes, this idea has been explored in formal work especially by Welch (1980), Denzau and Munger (1986) and Baron (1989a), both formally and empirically by Snyder (1990), Stratmann (1992) and Grier, Munger and Roberts (1994), and empirically by McCarty and Rothenberg (1996). The present model and Snyder's model imply the same behavior for an ``investor-contributor'' in the case where, in the present model, the type of service being provided makes voters indifferent to the amount of contributions being given to the candidates. Snyder does not model the concept of different types of service. In this sense the present model can be viewed as a generalization of his approach.
The type of service concept is defined in terms of the reaction service provokes among voters (Denzau and Munger 1986). Voters in the model respond to the difference between the amount of service the incumbent will provide, if she wins the election, and the amount that would be provided by the challenger should he win. The type of the service determines whether a larger amount of service from a candidate attracts or repels voters. For types of service that repel voters, a candidate's chances of winning the election increase if he or she promises to provide less service.
The concept of different types of service is motivated by the observation that service can be distributed in a variety of ways and produce a variety of externalities in a district. The taxes that must ultimately be collected to pay for spending are a negative externality associated with each service increase. Service that distributes benefits widely may offset the costs this externality imposes on most constituents. Staff assigned to the district to support casework may be an example of this kind of service: many may choose not to use the staff, but any constituent who wishes to do so can. Service that targets benefits more narrowly will do little to offset the costs most voters face, unless the benefits create significant positive externalities. For example, highway construction contracts go to individual firms, and so provide highly concentrated benefits, but when completed the highway itself will be a local public good. Situations in the model where voters are hostile to service are supposed to represent situations where the benefits from service, including externalities, do not exceed the costs in the particular district.
I interpret challenger quality as referring to the way the challenger computes his payoffs during the campaign. I assume that the incumbent cares about the negative effect an increase in voters' hostility to service would have on her reelection chances. The incumbent would like to minimize those effects. If the challenger's quality is high then he also cares about the effect an increase in voters' hostility to service would have on his chances of winning. The highest quality challenger weighs these concerns as strongly as the incumbent does; the highest quality challenger thinks just like the incumbent. A low quality challenger ignores the effects of potential changes in voters' responses to service.
Voters do not respond to the challenger's quality in any direct way in the model. This does not mean that the challenger's quality has no effect on the probability that the incumbent wins the election. Voters in the model respond to the rate at which each candidate turns contributions into service, and to the amount and type of service to be provided after the election. The amount of service is an increasing function of the contributions to each candidate and of each candidate's service rate. The service rates promised by the candidates, the contributions to the candidates, the service type and the quality of the challenger are all jointly determined, because of the strategic interactions among the parties, candidates and contributor. So the challenger's quality does affect voters, albeit indirectly.
In the system of differential equations that describes each subgame, the candidates adjust their intended service rates, while the contributor adjusts the contributions it will make to each candidate. The adjustment process is continuous time Cournot adjustment: the players all act noncooperatively, with each player making the adjustment in each parameter that would produce the largest improvement in its payoff if all the other parameters were to remain constant.