RD - DM II

The DPCT states that when certain assumptions are met, two political parties will converge to a policy platform.

1. Two parties, A and B.
2. Parties announce a policy platform and commit to it.
3. Equal probabilities when voters randomize between parties.
4. Voters have single-peaked preferences.

Parties converge to the median voter's bliss point.

If a party deviates, the other party can win by announcing the median voter's bliss point

There is always an incentive to move in response to opponents' positions.

$U(p; \alpha_i) = J(p) + K(\alpha_i) H(p)$

where $K(\alpha_i)$ is monotonic and H(p) and J(p) are common to all voters

A function between ordered sets that preseves or reverses the given order

Assuming sincere voting also, a Condorcet winner always exists and coincides with the bliss point of the voter with the median value $alpha_i, p(alpha_m)$

- The party’s competition leads to the median voter’s choice. Thus, political
parties select the policy $p = (p_1, p_2)$ such that

$\max_{p_1, p_2} U(p_1, p_2; \alpha_m)$

- There is no longer a need to have convergence btwn the two parties
- The policy implemented is still that of the median-ranked voter's bliss point

Intermediate preferences and preferences with single crossing

If it's none of those then, in the case of cycling, there is not pure strategy equilibrium for the political parties (e.g Condorcet paradox)

They imply the existence of a concave indirect utility function.

Then a Condorcet winner always exists and coincides with the bliss point of the median voter.

Consider $p' > p_m, U(p_m;alpha_m) > U(p';alpha_m$. By definition of single-crossing preferences, for all $alpha_i > alpha_m, U(p_m;alpha_i) > U(p';alpha_i$