¹ See Donald Stokes, ‘Spatial Models of Party Competition’, American Political Science Review, 57 (1963), 368–77; and Donald Stokes, ‘Valence Politics’, in Dennis Kavanagh, ed., Electoral Politics (Oxford: Clarendon Press, 1992), pp. 141–62.

² Stokes, ‘Valence Politics’, p. 143.

³ Stokes, ‘Valence Politics’, p. 143.

⁴ See, for instance, Roy Pierce, ‘Candidate Evaluations and Presidential Electoral Choices in France’, and Richard Johnston, ‘Prime Ministerial Contenders in Canada’, both in Anthony King, ed., Leaders’ Personalities and the Outcomes of Democratic Elections (Oxford: Oxford University Press, 2000), pp. 96–126 and pp. 158–83 respectively; and Ivor Crewe and Anthony King, ‘Did Major Win It? Did Kinnock Lose? Leadership Effects in the 1992 Elections’, in Anthony Heath, Roger Jowell and John Curtice, eds, Labour’s Last Chance? The 1992 Election and Beyond (Aldershot, Hants.: Dartmouth, 1994), pp. 51–86.

⁵ On two-party elections, see Timothy Groseclose, ‘A Model of Candidate Location when One Candidate Has a Valence Advantage’, American Journal of Political Science, 45 (2001), 862–86; and John Londregan and Thomas Romer, ‘Polarization, Incumbency, and the Personal Vote’, in William A. Barnett, Melvin Hinich and Norman Schofield, eds, Political Economy: Institutions, Competition, and Representation (New York: Cambridge University Press, 1993), pp. 355–77. On multiparty elections, see Norman Schofield and Itai Sened, Multiparty Democracy: Elections and Legislative Politics (Cambridge: Cambridge University Press, 2006); and James Adams, Samuel Merrill III and Bernard Grofman, A Unified Theory of Party Competition: A Cross-National Analysis Integrating Spatial and Behavioral Factors (Cambridge: Cambridge University Press, 2005).

⁶ Londregan and Romer, ‘Polarization, Incumbency, and the Personal Vote’.

⁷ In the Londregan–Romer model this uncertainty relates to voters’ evaluations of the candidates’ abilities to perform constituent service, although we do not restrict the meaning of valence in this way. By contrast, in ‘A Model of Candidate Location when One Candidate Has a Valence Advantage’, Groseclose analyses a model in which the candidates’ uncertainty is over the location of the median voter.

⁸ Studies that emphasize the importance of the parties in the governing coalition include David Austen-Smith and Jeffery Banks, ‘Elections, Coalitions, and Legislative Outcomes’, American Political Science Review, 82 (1988), 405–22; and G. Bingham Powell, Elections as Instruments of Democracy: Majoritarian and Proportional Visions (New Haven, Conn.: Yale University Press, 2000). Studies on the importance of the formateur include David Baron, ‘Comparative Dynamics of Parliamentary Governments’, American Political Science Review, 92 (1998), 593–609. Michael Laver and Kenneth Shepsle, Making and Breaking Governments (Cambridge: Cambridge University Press, 1996), emphasize the importance of the party with jurisdiction over the relevant government ministry.

⁹ Michael McDonald and Ian Budge, Elections, Parties, and Democracy: Conferring the Median Mandate (Oxford University Press, 2005). We note that a related literature analyses the relationship between parties’ policy programmes and the published policy declarations of the government (see, e.g., Paul Warwick, ‘Coalition Policies in Parliamentary Democracies: Who Gets How Much and Why’, Comparative Political Studies, 34 (2001), 1212–36). Note, however, that these studies take as their dependent variable the government’s policy declarations rather than the actual government policy outputs that were observed. Policy-seeking parties are presumably concerned with actual policy outputs as opposed to policy promises.

¹⁰ See McDonald and Budge, Elections, Parties, and Democracy, Tables 12.3–12.5 on pp. 220–4. In addition to the McDonald–Budge results, we note that Seok-ju Cho and John Duggan (in ‘Bargaining Foundations of the Median Voter Theorem’, Journal of Economic Theory, forthcoming) present important theoretical results that, as legislators become arbitrarily patient, a large class of bargaining models of distributive politics collapse to the position of the median legislator. This result, which runs counter to the folk theorem for repeated games that any possible division of resources can be supported as a subgame perfect equilibrium outcome, also supports our assumption of the policy primacy of the MPP. Thus we have both theoretical and empirical reasons to believe that our model is relevant to policy making in real world democracies.

¹¹ Londregan and Romer, ‘Polarization, Incumbency, and the Personal Vote’. See also Theorem 4 in Donald Wittman, ‘Spatial Strategies When Candidates Have Policy Preferences’, in James Enelow and Melvin Hinich, eds, Advances in the Spatial Theory of Voting (Cambridge: Cambridge University Press, 1990), pp. 66–98.

¹² Schofield and Sened, Multiparty Democracy: Elections and Legislative Politics.

¹³ Groseclose, ‘A Model of Candidate Location when One Candidate Has a Valence Advantage’, and Londregan and Romer, ‘Polarization, Incumbency, and the Personal Vote’.

¹⁴ Thus, if the s_k's are ordered so that s ₁ ≤ s ₂ ≤…≤ s_K, then the MPP is that party k_M such that the set of parties 1,…, k_M and parties k_M, k_M + 1,…, K each include a majority of the seats in parliament.

¹⁵ The assumption that politicians who gain power are constrained to implement their pre-election policy promises is standard in spatial models with policy-seeking parties/candidates; for a review of this literature, see John Roemer, Political Competition: Theory and Applications (Cambridge, Mass.: Harvard University Press, 2001). This assumption is typically justified on the grounds that politicians project that, in future elections, retrospective voters will punish office-holders who violate their pre-election policy promises, and that these projected future electoral reverses will in turn generate policy losses that exceed the policy gains office-holders can achieve by reneging on their promises during the current inter-election period.

¹⁶ We say that a function U is concave and peaks at x ₀ if it is continuous, and if for all x in the domain of U for which x ≠ x ₀, and U(x ₀) > U(x). (This is a special case of strict single-peakedness.) Note that if U is concave and peaks at x ₀, then U is strictly increasing on the left of x ₀ and strictly decreasing on the right, i.e., if x ₁ < x ₂ ≤ x ₀, then U(x ₁) < U(x ₂) and if x ₀ ≤ x ₁ < x ₂, then U(x ₁) > U(x ₂).

¹⁷ Two striking recent examples of such phenomena occurred during the course of the German parliamentary election campaign in September 2002, and the Spanish election campaign in March 2004. In Germany the disciplined, forceful campaign waged by the SDP and its leader, Gerhard Schroeder, enhanced the party’s valence image and helped it achieve an unexpectedly strong election result. In Spain, the response of the governing party (the People’s party) to the Madrid train bombing – which occurred just days before the election – was widely believed to have harmed its reputation for competence and honesty, thereby contributing to its unexpectedly poor showing.

¹⁸ An anonymous reviewer notes that this assumption is consistent with a model of ‘expressive’ voting, i.e. one where citizens derive utilities from casting votes that express their sincere preferences, rather than deriving instrumental benefits from influencing the election outcome (on this point, see Geoffrey Brennan and Loren Lomansky, Democracy and Decision: The Pure Theory of Electoral Preference (Cambridge: Cambridge University Press, 1993). In Remark 2 below we address the issue of instrumental (i.e. outcome-oriented) voting.

¹⁹ Denote by j_M the party supported by the median voter. Because, for each party, the voter utility ag(s_j, x_i) declines as the voter position x_i recedes from the party position while the valence component (V_j + ε_j) is identical across voters, it follows that all voters located to the left of the median voter prefer party j_M to all parties whose policy positions lie to the right of j_M, and vice versa. Hence party j_M is the MPP under sincere voting.

²⁰ This equilibrium among voters is not to be confused with the equilibrium among parties that is the primary topic of this article. Denote by j_M the MPP under sincere (expressive) voting. Assume that all voters other than a focal voter are sincere. A focal voter who prefers j_M – which includes the median voter – cannot improve her utility by switching her vote. Secondly, a focal voter who prefers a party located to the left of j_M can only alter the identity of the MPP by switching her vote to a party located to the right of j_M. However, given that (1) all voters who prefer a party located to the left of j_M must themselves be located to the left of the median voter’s position m, and (2) all voters located to the left of the median voter prefer j_M to all parties located right of j_M (see fn. 19), it follows that no focal voter located to the left of m can have an instrumental incentive to switch her vote in order to alter the election outcome in an attempt to change the identity of the MPP. A similar argument applies to a focal voter located to the right of m.

²¹ For an extension of our model that relaxes this independence assumption, see James Adams, Lawrence Ezrow, Samuel Merrill III and Zeynep Somer-Topcu, ‘Policy-Seeking Parties in Proportional Systems with Valence-Related Uncertainty: Does Collective Responsibility for Performance Alter Party Strategies?’ Public Choice (forthcoming).

²² Logit analysis has been employed extensively both in empirical studies of voting behaviour (see Schofield and Sened, Multiparty Democracy: Elections and Legislative Politics), and in spatial models of multiparty competition, see, e.g., Samuel Merrill III and James Adams, ‘Computing Nash Equilibria in Probabilistic, Multiparty Spatial Models with Non-Policy Components’, Political Analysis, 9 (2001), 347–61.

²³ See chap. 3 in Kenneth Train, Qualitative Choice Analysis (Cambridge, Mass: MIT Press, 1986) for a proof that the logit model implies choice probabilities of the functional form given by Equation 2.

²⁴ For example, under quadratic-loss utility for parties and voters, , where .

²⁵ The details of the proof of Theorem 1 are available on the website, http://course.wilkes.edu/merrill/.

²⁶ If U is continuous on a closed bounded interval I, then U is single-peaked (or, equivalently, strictly quasi-concave) if U has a unique local maximum on I (see p. 18 in Roemer, Political Competition: Theory and Applications). In particular, if U is single-peaked, there exists x ₀ ∈ I such that U(x ₀) > U(x) for all x ∈ I, x ≠ x ₀. Note that if a continuous function is concave and peaks at x ₀, then it is single-peaked.

²⁷ Nolan McCarty and Adam Meirowitz, Political Game Theory: An Introduction (Cambridge: Cambridge University Press, 2007).

²⁸ Londregan and Romer, ‘Polarization, Incumbency, and the Personal Vote’.

²⁹ In every scenario we have tested using quadratic voter utility, all parties have moved in the same direction as the focal party; using linear utility, some parties remain fixed while others move in the same direction as that of the focal party. See examples below.

³⁰ Norman Schofield, ‘Valence Competition in the Spatial Stochastic Model’, Journal of Theoretical Politics, 15 (2003), 371–83; and Norman Schofield, ‘The Median Voter Theorem under Proportional and Plurality Rule’ (paper presented at the Annual Meeting of the Public Choice Society, New Orleans, 2004). We note that Schofield’s work on multiparty elections also considers parties’ policy-seeking objectives, as well as their expectations about post-election coalition negotiations. However, here we consider only Schofield’s conclusions about vote-maximizing parties.

³¹ Schofield shows that the degree of policy moderation by high-valence parties depends on the specifics of the election context. However, if there is more than one high-valence party – as is typically the case in competitive multiparty systems – then the competing high-valence parties will typically not converge all the way to the centre of the voter distribution.

³² Norman Schofield and Itai Sened, ‘Multiparty Competition in Israel, 1988–1996’, British Journal of Political Science, 35 (2005), 635–63.

³³ An exception is a party with centrist policy preferences, which will invariably adopt a centrist policy strategy.

³⁴ Donald Wittman, ‘Candidate Motivation: A Synthesis of Alternatives’, American Political Science Review, 77 (1983), 142–57; Londregan and Romer, ‘Polarization, Incumbency, and the Personal Vote’; Groseclose, ‘A Model of Candidate Location when One Candidate has a Valence Advantage’.

³⁵ The parameter a = 0.25 is suggested by empirical voting analyses reported in chaps 4 and 6 in Adams, Merrill and Grofman, A Unified Theory of Party Competition. Substantively, this value implies that if the median voter M is located three units closer to Party A than to Party B along the 1–7 left–right scale, and these parties have equal valence images (i.e. V_A = V_B), then the probability that M will prefer A to B on election day is approximately 90 per cent. We note that realistic variations in the specified value of a did not substantially affect the parties’ equilibrium positions (decreasing a resulted in somewhat more dispersed positions and increasing a somewhat depressed party dispersion). With respect to variations in the other model parameters used for our examples, we found that: (1) results for linear loss utility for parties were similar to those for quadratic losses, but somewhat more dispersed; (2) results for larger party systems (i.e. more than four parties) were somewhat more dispersed. Results for alternative sets of assumptions about the parties’ valence images are reported below.

³⁶ Substantively, the settings (V_A = 0, V_B = 2, V_C = 2, V_D = 0) imply that if the median voter M is indifferent between a high-valence party (B or C) and a low-valence party (A or D) on policy grounds, the probability that M will prefer the high-valence party on election day is roughly 88 per cent.

³⁷ Similar equilibrium strategies are also found in scenario 3A, in which parties B and D (the centre-left and far right parties) have high valence scores V_B = V_D = 2, whereas parties A and C (the extreme left and centre-right parties) have low valence scores V_A = V_C = 0. This scenario plausibly captures the strategic situation in Germany, which features four major parties – the Greens, the Social Democratic Party (SDP), the Free Democrats (FDP), and the CDU/CSU – of which the two smallest are the Greens who espouse sharply left-wing policies, and the FDP which currently espouses centre-right positions. As the valence of either left-of-centre party is increased (see scenarios 3B–3E), the optimum strategies at equilibrium for all parties shift to the left.

³⁸ See chap. 11 in Adams, Merrill and Grofman, A Unified Theory of Party Competition.

³⁹ See fn. 37 for the effects of varying other parameters.

⁴⁰ For instance, when Party A is the focal party, if the preferred positions for the other parties are equally-spaced at R_B = 3.25, R_C = 4.75, and R_D = 6.25, V_B = V_C = V_D = 0, and if V_A = 1, the parameters given by the regression equation predict that the (unnormalized) optimal strategy for party A is s_A*≅2.97 when (the unnormalized location) R_A = 2.50; that s_A* = 2.80 when R_A = 1.75; and that s_A* = 2.72 when R_A = 1.00. A similar example in which Party B is the focal party suggests that its optimal strategy becomes more extreme by 0.94 units as R_B moves from 4.00 to 2.50.

⁴¹ For example, when Party A is the focal party, if the (unnormalized) party preferred positions are equally-spaced at 1.75, 3.25, 4.75 and 6.25, and V_B = V_C = V_D = 0, then the regression equation projects that the (unnormalized) optimal strategies for party A are s_A* = 2.92 when V_A = 0; that s_A* = 2.80 when V_A = 1; and that s_A* = 2.69 when V_A = 2, i.e., Party A’s optimal strategy becomes more extreme by 0.23 units as V_A increases from 0.0 to 2.0. A similar example, in which Party B is the focal party, projects that its optimal strategy becomes more extreme by only 0.07 units as V_B increases from 0.0 to 2.0.

⁴² See pages 191–5 in Adams, Merrill and Grofman, A Unified Theory of Party Competition. Note that this effect is opposite to that found by Groseclose (‘A Model of Candidate Location when One Candidate Has a Valence Advantage’) in the two-party case when uncertainty pertains instead to the voter distribution.

⁴³ Zeynep Somer-Topcu, ‘Party Policy Strategies and Valence Issues: An Empirical Study of Ten Post-Communist European Party Systems’ (presented at the Conference on the Dynamics of Party Position-Taking, Binghamton, 2007).

⁴⁴ See Ian Budge, Hans-Dieter Klingemann, Andrea Volkens, Eric Tannenbaum and Judith Bara, eds, Mapping Policy Preferences: Estimates for Parties, Electors, and Governments 1945–1998 (Oxford: Oxford University Press, 2001).

⁴⁵ James Adams and Zeynep Somer-Topcu, ‘Party Policy Adjustment in Response to Rival Parties’ Policy Shifts: Spatial Theory and the Dynamics of Party Competition in Twenty-Five Post-War Democracies’, British Journal of Political Science, forthcoming 2009.

⁴⁶ Peter Van Roozendaal, ‘The Effect of Dominant and Central Parties on Cabinet Composition and Durability’, Legislative Studies Quarterly, 17 (2002), 5–36.

⁴⁷ Londregan and Romer, ‘Polarization, Incumbency, and the Personal Vote’.

⁴⁸ Numerical calculations for a valence-uncertainty model in which parties have mixed vote-maximizing and policy-seeking motivations yielded equilibrium strategies intermediate between dispersed strategies under policy-seeking and centrist strategies under vote-maximization. With respect to the issue of variable turnout in the electorate, see Jane Green, ‘Party Shifts and Electoral Penalties: Testing Theories of Party Competition in Britain’, Journal of Politics (forthcoming).

⁴⁹ On this point, see James Adams and Samuel Merrill III, ‘Why Small, Centrist Third Parties Motivate Policy Divergence by Major Parties’, American Political Science Review, 100 (2006), 403–17; and Lawrence Ezrow, ‘Parties’ Policy Programmes and the Dog that Didn’t Bark: No Evidence that Proportional Systems Promote Extreme Party Positioning’, British Journal of Political Science, 38 (2008), 470–97.

⁵⁰ If s_k = m or s_k = R_k, then an increase in V_k will not change the location of Party k’s optimal position.