Spatial models of elections

1
Lecture 6

• Spatial models of elections
• Readings Shepsle, Analyzing politics, chapter 5

2
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

3
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?

4
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

5
Downsian model of party competition

• Median voter theorem If all voters 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

6
Which platform will parties choose?
A
B
C
x
y
1
2
3
4
5
0
7
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

8
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

9
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

10
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

11
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)

12
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

13
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

14
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)gtUi(B), then the
  probability that i votes for A is piA1 and hence
  piB0

15
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)ejgt0
  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
• piAf(Ui(A), Ui(B), ej)
• Let F(.) be the probability distribution function
  of the random shocks
• Remember that F(x)Probejltx and
  1-F(x)Probejgtx
• piAProbUi(A)-Ui(B)ejgt0
• piAProbejgtUi(A)-Ui(B)F(Ui(A)-Ui(B))

16
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)

17
Questions

• Assume that individuals have different
  preferences for percentage x of healthcare
  services that should be provided by the
  government. Let f(x) with 0ltxlt1 be the density
  function describing the distribution of
  preferences and assume that distribution is the
  following

f(x)
x
1
18
Questions

• 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?