The Voting Model and Reality

The Paradox of Voting

The median voter model is a standard theoretical model that is used to determine public spending in a political process. However, there are several problems. An important one is concerned with the existence of stable unique equilibrium. This depends upon the shape of preference for public goods. As shown in Fig. 12.2, if a preference produces a peak and has a unimodal shape, we have stable unique equilibrium; if not, we may not attain stable equilibrium by majority voting.

Let us consider the following example. In Fig. 12.3, preferences for public goods show both peaks and troughs. Namely, until Y0, utility decreases with public goods. Then, utility increases with public spending to a certain level. Finally, utility decreases with public spending.

For example, consider public education spending. If public education is less than Y0, agents may prefer private education to public education. In this instance, they must pay taxes to support public education even if they choose private education. Thus, their utility decreases with regard to public education. If Y is greater than Y0, they now choose public education. Then, so long as the benefit is greater than the cost of public education, their utility increases with regard to public education.

Imagine that there are three agents: rich (R), middle income (A), and poor (P). The levels of public education are high (H), middle (M), and low (L). It is plausible

Fig. 12.3 Preference for public education

The paradox of voting

Fig. 12.4 The paradox of voting

to assume that the rich agent always prefers private education and is against an increase in public education.

Thus, as shown in Fig. 12.4, her or his utility decreases with regard to public education. However, the poor agent always prefers public education. Her or his utility is highest at M, and H is better than L. The middle-income agent prefers private education if public education is less than M, and prefers public education if it is higher than M. Thus, her or his utility is highest at H, and L is better than M.

In this example, majority voting does not produce stable equilibrium. The outcome depends upon how voting occurs. Consider the choice between L and M. L has the most votes. In the choice between L and H, H wins. However, in the choice between H and M, M wins. It follows that majority voting that wins with regard to any choice does not exist. This is called the paradox of voting.

 
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