Part 4

Consider a researcher named J, who is interested in the relations between congressional campaign finance and benefits provided to special interests. J likes formal models, and is in particular a fan of game theory. So J would like to know if there is a fairly simple game that captures some interesting essence of the interactions between a member of the U.S. Congress and special interests who might contribute money to the member's campaign.

J decides to study a game with three players: an incumbent running for reelection; a challenger; and a political action committee (PAC) representing some special interest. The incumbent moves first, by promising to deliver some amount of benefits to the special interest if he wins reelection. Having heard the incumbent's promise, the challenger then announces a benefit amount she will give to the special interest if she wins. Having heard both candidates' promises, and believing they are credible, the PAC then decides how it is going to allocate its support between the incumbent and the challenger. The PAC can decide to favor the incumbent, to be neutral, or to favor the challenger.

Each candidate wants to maximize the probability that he or she wins the election. The PAC wants to maximize the amount of benefit the special interest can be expected to receive after the election. The expected benefit depends on the probability that the incumbent is reelected. Let p denote this probability. Then the probability that the challenger wins, defeating the incumbent, is q = 1 - p. Let denote the benefit amount promised by the incumbent, and let denote the benefit amount promised by the challenger. The expected benefit amount is computed using the expected value formula, .

The incumbent's probability of winning reflects three kinds of reactions by voters. First, because of his reputation for effectiveness over many years, the incumbent begins with a slight but significant advantage. Second, because voters do not share in the benefit to be given to the special interest, they tend to prefer the candidate who makes the smallest promise. But, third, the PAC launches a negative campaign against the candidate who promises the smallest amount, as well as against the candidate it is not supporting. The campaign is effective: the more the PAC spends on the campaign, the less likely voters are to support the candidate the PAC is attacking. In formal terms,

where represents the incumbent's baseline advantage, denotes the size of the negative campaign directed against the challenger, and denotes the size of the negative campaign directed against the incumbent. The PAC uses the following rules to determine the sizes of the negative campaigns.

incumbent's is bigger same size challenger's is bigger PAC's stance

pro-incumbent 0 1 0 1

neutral 0 0 0 0

pro-challenger 1 0 1 0

When the PAC is supporting the incumbent and the incumbent promises an amount smaller than the challenger, the PAC mounts an especially severe negative campaign against the incumbent due to feelings of betrayal. On the other hand, a pro-incumbent PAC's negative campaign against a challenger whose promises less than the incumbent is not that severe.

The cost of the negative campaign is subtracted from the PAC's payoff; i.e., the PAC's payoff is . Researcher J assumes that the challenger cannot match the incumbent's highest promised benefit levels, due to the inferior seniority the challenger will have if she wins. So J lets the incumbent choose from the set , while the challenger chooses from the set . Finally, J assumes the incumbent's baseline advantage is . The payoffs associated with the possible combinations of candidate and PAC choices are shown in Table 2. That table would be the normal form of the game if the three players moved simultaneously instead of sequentially.

Table 2: Payoffs for game of Part 4

1 4

incumbent's promise 1 @c@ .71, .29, 0 @c@ .44, .56, -4.32 c@ pro-incumbent PAC

5 @c@ .41, .59, .65 @c@ .53, .47, 3.28 c@ challenger's promise

1 4

incumbent's promise 1 @c@ .6, .4, 1.0 @c@ .64, .36, .57 c@ neutral PAC

5 @c@ .36, .64, 1.11 @c@ .49, .50, 4.16 c@ challenger's promise

1 4

incumbent's promise 1 @c@ .43, .57, 0 @c@ .47, .53, -1.42 c@ pro-challenger PAC

5 @c@ .61, .39, -5.55 @c@ .52, .48, 1.52 c@

Identify the Nash equilibrium (or equilibria) in this game (supply a demonstration). Does the assumption that the players move sequentially rather than simultaneously make any difference? What would happen if the special interest made its decision about how to allocate its support before either of the candidates were assumed to make their promises about benefits? What conclusions should researcher J draw from this modeling effort? What are the major strengths and weaknesses of this particular exercise?