Median voter theorem

The median voter theorem states that "a majority rule voting system will select the outcome most preferred by the median voter".

The median voter theorem makes two key assumptions. First, the theorem assumes that voters can place all election alternatives along a one-dimensional political spectrum. It seems plausible that voters could do this if they can clearly place political candidates on a left-to-right continuum, but this is often not the case as each party will have its own policy on each of many different issues. Similarly, in the case of a referendum, the alternatives on offer may cover more than one issue. Second, the theorem assumes that voters' preferences are single-peaked, which means that voters choose the alternative closest to their own view. This assumption predicts that the further away the outcome is from the voter's most preferred outcome, the less likely the voter is to select that alternative. It also assumes that voters always vote, regardless of how far the alternatives are from their own views. The median voter theorem implies that voters have an incentive to vote for their true preferences. Finally, the median voter theorem applies best to a majoritarian election system.

To appreciate the logic of the median voter model, consider a setting where three individuals, Al, Bob, and Charlie, are to choose a restaurant for lunch. Al prefers a restaurant where lunch costs $5, Bob favors somewhat better fare at a restaurant serving $10 lunches, and Charlie wants a gourmet restaurant where lunch will cost around $20. Bob can be said to be the median voter, because there are exactly the same number of people who prefer a more expensive restaurant than Bob as there are who prefer a less expensive restaurant than Bob: here one each. For convenience assume that, given any two options, each member of the lunch group prefers restaurants with prices closer to their preferred restaurant to those that are farther from it. Now consider some majority decisions over alternative restaurants:

The weak form of the median voter theorem says the median voter always casts his or her vote for the policy that is adopted. Note that Bob always votes in favor of the outcome that wins the election. Note also that Bob's preferred $10 restaurant will defeat any other. If there is a median voter, his or her preferred policy will beat any other alternative in a pairwise vote. (The median voter's ideal point is always a Condorcet winner.) Consequently, once the median voter's preferred outcome is reached, it cannot be defeated by another in a pairwise majoritarian election. The strong form of the median voter theorem says the median voter always gets his most preferred policy.

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