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Lecture 6. Spatial models of elections Readings: Shepsle, Analyzing politics, chapter 5. Direct versus indirect democracy. In the previous lecture we asked how individuals vote on policies and what is the outcome of the voting process

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Lecture 6

Lecture 6

Spatial models of elections

Readings: Shepsle, Analyzing politics, chapter 5

Direct versus indirect democracy
Direct versus indirect democracy

  • In the previous lecture we asked how individuals vote on policies and what is the outcome of the voting process

  • The outcome of the voting process should be a policy that is the Condorcet winner in the set of feasible policies

  • However, in reality citizens rarely vote directly on policies (example of direct vote on policy: referendum)

  • In representative democracies citizens vote on candidates who choose then policies on their behalf

Political parties and electoral competition
Political parties and electoral competition

  • We can still assume that citizens indirectly choose policies through candidates’ platforms according to the same criteria outlined in previous lectures

  • But in this case the crucial question becomes: how do parties choose platforms? And more generally, how do parties enter in the political competition?

Downsian model of party competition
Downsian model of party competition

  • The basic assumption of the Downsian model is that parties formulate policies in order to win elections

  • In other words, every party tries to maximise its votes

  • The second important step in the model is the assumption that the median voter theorem holds

Downsian model of party competition1
Downsian model of party competition

  • Median voter theorem: If all voter’s preferences are single peaked on a single dimension, then the most preferred point of the median voter is a Condorcet winner

    • Two parties compete for election

    • Parties compete on a single dimension

    • We can order the single dimension policies on a line

    • Citizens have single peaked preferences

Optimal party location
Optimal party location

  • When

    • parties care about winning elections

    • the median voter theorem holds

  • The Downsian model of political competition with two parties predicts a Nash Equilibrium where the party platforms converge to the median voter

Party convergence
Party convergence

  • Do we observe parties’ convergence in real world elections?

  • While the tendency to convergence is plausible, parties clearly do not always converge

  • In fact, divergence is common and party polarization is typical of most political races in modern democracies

Causes of party divergence
Causes of party divergence

  • Politicians’ policy motivation (lack of commitment technology)

  • Uncertainty about voters’ preferences

    • if the two previous assumptions hold

  • Alesina (AER,1988) shows that in a one shot game the two parties will diverge and in particular they will run with their most preferred platforms

Causes of party divergence1
Causes of party divergence

  • Although parties may have an incentive to announce platforms that are closer to the median voter, those announcements would not be credible

  • On the other hand, if the game is repeated, in some cases parties may credibly commit to choose a compromise policy that would be implemented whether they are in power or the other party is in power

Condorcet paradox and party competition
Condorcet Paradox and party competition

  • As we have seen in previous lectures, it is not very difficult to construct examples where the Condorcet winner does not exists

  • Remember: divide the dollar game

  • Any problem of sharing a fixed amount of resources among a given number if individuals (ex: budget allocation) would be affected by the so called Condorcet paradox (cycling majority)

Condorcet paradox and party competition1
Condorcet Paradox and party competition

  • If we apply this paradox to the political competition, we would obtain that each party can get the majority of votes through a marginal change of his platform so that we cannot observe convergence to a Condorcet winner

  • A party can switch from the state of loosing the election to the state of winning by marginally changing its platform

  • In other words, the pay-offs of parties are not continuous in platforms and for this reason we cannot obtain an equilibrium party platform

Condorcet paradox and party competition2
Condorcet Paradox and party competition

  • If parties pay-offs were continuous it would be possible to obtain equilibrium party platforms

  • If by marginally changing platforms parties would not be in the situation of winning or loosing for sure elections, than we would be able to overcome the Condorcet paradox

  • If there is some uncertainty on the preference of voters, than the status of parties would not which from losers to winner when platforms are altered since they would be winning/loosing with some probability, given any possible platform announcement

Deterministic voting
Deterministic voting

  • Let pix be the probability that individual i votes in favour of alternative x

  • Let Ui(x) be the utility i derives from x

  • In a deterministic voting setting preferences of people are known and given any two possible alternatives A and B, if Ui(A)>Ui(B), then the probability that i votes for A is piA=1 and hence piB=0

Probabilistic voting
Probabilistic voting

  • In a probabilistic voting setting we assume that because of a random shock, the voting rule of individual i is as follows:

  • i votes A over B if and only if Ui(A)-Ui(B)+ej>0 where ej is a random shock i.i.d. across individuals

  • Therefore, the probability that i votes A over B is a continuous function depending on Ui(A), Ui(B) and ej

    piA=f(Ui(A), Ui(B), ej)

  • Let F(.) be the probability distribution function of the random shocks

  • Remember that F(x)=Prob{ej<x} and 1-F(x)=Prob{ej>x}

  • piA=Prob{Ui(A)-Ui(B)+ej>0 }

  • piA=Prob{ej>Ui(A)-Ui(B)}=F(Ui(A)-Ui(B))

Probabilistic voting1
Probabilistic voting

  • Since candidates are not sure to win elections just proposing a given platform, their payoffs become a function of the platform and the probability of winning for every platform

  • It is possible to show that if the probability function is strictly concave, than the equilibrium of the game is unique and both candidates offer the same platform (Coughlin-Nitzan)


  • Assume that individuals have different preferences for percentage x of healthcare services that should be provided by the government. Let f(x) with 0<x<1 be the density function describing the distribution of preferences and assume that distribution is the following:





  • Show on the diagram the median of the distribution

  • Suppose that, given the above distribution, two parties have to choose their political platform specifying the percentage of healthcare that they would provide if they win the elections. Which platform will the two parties choose in equilibrium?