Question

1. Consider an election where there are two political parties (Republicans vs Democrats). During the campaign, each party will choose a policy position which is represented as a point along a line (from zero to one). Each voter has a preferred policy and voters will vote for the candidate whose policy is closest to their own position. The voter’s preferences are distributed uniformly along this policy line. The party that gets majority of the votes wins the election. Each party prefers winning to tying and prefers tying to losing the election.

   1. Formulate this situation into a strategic game.

   2. Suppose that both parties are choosing the most extreme policiess. That is,R is choosing the policy pR = 1 and D is choosing the policy pD = 0. What proportion of the votes would each party get? What would be the outcome of the election?

   3. Suppose that the R party hires you to advise their campaign. Given D’s policy position in part (b), what policy position would you prescribe? What proportion of the votes would each party get? What would be the outcome of the election?

   4. Now suppose that R is choosing the policy pR = 0.8 and D is choosing the policy pD = 0.4. What proportion of the votes would each party get? What would be the outcome of the election?

   5. Suppose that the R party hires you to advise their campaign. Given D’s policy position in part (d), what policy position would you prescribe? What proportion of the votes would each party get? What would be the outcome of the election?

   6. Show that both parties choosing the median policy position pR = pD = 0.5 is a Nash Equilibrium of this game. (First, describe the outcome of the game then check whether either party has an incentive to deviate)

Answer 1

One of the cornerstones of a vibrant democracy is citizens’ ability to influence government through voting. In order for that influence, citizens must send clear signals to their leaders about they wish the government to do. Then It makes sense, then, a democracy will benefit if voters have several clearly differentiated options available to them at the polls on Election Day. Having these options means voters can select a candidate who more closely represents their own preferences and It also gives individuals who are considering voting a reason to participate, you are more likely to vote if you care about who wins and who loses. The existence of two major parties, especially in our present era of strong parties, leads to sharp distinctions between the candidates and between the party organizations.

The two-party system came into being because the structure of U.S. elections, with one seat tied to a geographic district, tends to lead to dominance by two major political parties. Even when there are other options on the ballot, most voters understand that minor parties have no real chance of winning even a single office. Hence, they vote for candidates of the two major parties in order to support a potential winner. Of the 535 members of the House and Senate, only a handful identify as something other than Republican or Democrat. Third parties have fared no better in presidential elections. No third-party candidate has ever won the presidency. Some historians or political scientists might consider Abraham Lincoln to have been such a candidate, but in 1860, the Republicans were a major party that had subsumed members of earlier parties, such as the Whig Party, and they were the only major party other than the Democratic Party. A simple version of this model is a strategic game in which the players are the candidates and a policy is a number, referred to as a “position”. (The compression of all policy differences into one dimension is a major abstraction, though political positions are often categorized on a left–right axis.) After the candidates have chosen positions, each of a set of citizens votes (nonstrategically) for the candidate whose position she likes best. The candidate who obtains the most votes wins. Each candidate cares only about winning; no candidate has an ideological attach. Eg; At one extreme, the parties may be completely passive in the determination of the election agenda. This could happen in a number of ways. Voters, for instance, may simply focus on those areas where current policy is out of tune with electoral preferences, that is, on those areas where the opposition offers a better alternative; as a result the incumbent is bound to lose and we get regular alternations between the parties. As another possibility, external forces (the press, for instance, or business interests) may have the power to impose an agenda of their own liking. 3 See Nannestad and Paldam (2002) for a survey of the literature on the cost of ruling. 3 We shall disregard these cases of passive parties and assume, instead, that the interaction between the strategic decisions of the two parties determines the agenda. This approach may lead to either a first- or a second-mover advantage. As a simple example of the former, consider a twodimensional policy space but assume that there is only room for one issue on the agenda. ‘Right’, for instance, may want the agenda to focus on taxes while ‘Left’ would like to focus on health care. In this case, whichever party gets to choose the first (and only) election issue will win. Schofield’s (2004) analysis of the US presidential election in 1860 describes a real-world example (see also Riker (1982, 1986), Dixon and Schofield (2001)). At the Freeport debate in 1858, Lincoln had forced the future Democrat candidate Stephen Douglas to support the Dred Scott4 decision of the U.S. Supreme Court which implied that no state could exclude slavery from its limits. Douglas’s position was a threat to the non-slave states of the North, especially to their free, scarce and well-paid labour, and Lincoln succeeded in making this threat understood during the two years before the election date. At the same time, Douglas appeared to be anti-slavery to Southern voters when he conceded that citizens of a Territory of the West could exclude slavery prior to the formation of a State constitution. His argument that local regulations could interfere with private property of slaves was understood by Southerners as an indication that he would not protect the slave institution. By focusing the election on the slave issue, Lincoln achieved two things. He gained support in the North and provoked a split among the Democrats who fielded two candidates, Douglas supported by the Northern Democrats and Breckinridge as the candidate of the Southern Democrats. At the election Lincoln gained 40 per cent of popular votes and 180 votes out of 303 of the Electoral College (none from the Border or the South); Douglas received 29 per cent of popular votes and 12 votes of the Electoral College; Breckinridge got 18 per cent of popular votes but 72