EXAMPLE 6 Two-Dimensional Spatial Voting
Five board members of a company must decide between two other board members ( and ) for the vaunted position of chairman of the board. The voters are concerned with two issues: overseas expansion and the timing of the initial public offering. The voters’ ideal points and the candidates’ policy positions are ordered pairs , where indicates the position on overseas expansion (a larger value of means a more vast expansion) and where indicates the time until the initial public offering (a larger value of represents a longer time until the initial public offering will be held).
Algebra Review Appendix
Plotting Points in the Plane
Assume that and announce policy positions of (4, 2) and (8, 4), respectively. The ideal points of board members 1 through 5 are (5, 1), (4, 4), (6, 4), (9, 6), and (8, 2), respectively. The policy and ideal positions are graphed in Figure 12.3a.
Algebra Review Appendix
Distance and Midpoint Between Two Points in the Plane
The midpoint of (4, 2) and (8, 4) is on the line that connects these points; this is the dashed line in Figure 12.3b. The midpoint is calculated by taking the average of each coordinate. The -coordinate of the midpoint is and the -coordinate of the midpoint is . Equivalently, the midpoint satisfies
Just as the midpoint of two policy positions divides the left-right continuum into three regions in which (1) voters prefer one policy position, (2) voters prefer the other policy position, and (3) voters are indifferent to the choice between the two policy positions, there is also a way to use the midpoint to divide the two-dimensional policy space into three similar regions. In high school geometry, we learn how to draw a line through a point that is perpendicular to another line, using either a compass or dynamic geometry software.
The solid line in Figure 12.3c is through the midpoint and is perpendicular to the dashed line between the policy positions of and . It divides the two-dimensional space so that voters with ideal points below the solid line prefer over , voters with ideal points above the solid line prefer over , and voters with ideal points on the solid line are indifferent between and . Board members 3, 4, and 5 have ideal points above the solid line and vote for Candidate , while board members 1 and 2 have ideal points below the solid line and vote for Candidate . Board member is elected as chairman of the board.
