This simulation uses the following randomly drawn coordinates in 8 dimensions:
Given these coordinates, we can expect these two outcomes:
|Governor's total utility gain||Legislature's total utility gain|
If each issue were in a separate bill, how much total utility would each player gain?
If all issues were in a single bill, how much utility would each player gain?
This page illustrates arguments that I present in my paper on multidimensional veto bargaining. If you haven't read the paper, this page may not make much sense. But in brief...
Suppose that the legislature is considering passing one or more bills concerning 8 separate issues. The governor will then decide whether to veto or accept the legislature's proposal. The legislature might handle each issue separately, or it might bundle all those issues into a single bill.
We can plot the location of the legislature's "ideal policy" along a left-right liberal-conservative line. We can also plot the governor's ideal point and the location of the status quo policy. The numbers along the line are meaningless. They are drawn randomly from a normal distribution (mean 0, standard deviation 100). The numbers don't matter, only the relative location of G (the governor's ideal point), L (the legislature's ideal point), and SQ (the status quo). Each time you reload you'll see a new image, with G, L, and SQ drawn randomly from a normal distribution (mean 0, s.d. 100).
If the issues shown above were dealt with in separate bills, the legislature would maximize its utility (while avoiding a veto) by proposing SQu (i.e. "SQ unidimensional"). But if all these issues were all put into a single omnibus bill, the result would be SQm ("SQ multidimensional").
Each player's preferences are symmetrical and single-peaked. That means each player's utility ("happiness") rises linearly as the new status quo (SQu or SQm) moves closer to his ideal point. For example, suppose we are looking at a two-dimensional game. To find the legislature's overall utility from unidimensional bargaining, we first solve each part of the game (issue 1 and issue 2) separately to locate SQu. We then plot L on a graph, using L's ideal point in Issue 1 as the x-axis value and L's ideal point on Issue 2 as the y-axis value. We plot SQ, SQu, and SQm on the same graph in the same manner. The legislature's utility under the old status quo is the distance between L and SQ, which we can find using the Pythagorean Theorem. The legislature's utility under the new status quo (from sequential/unidimensional bargaining) is the distance between L and SQu. To see how much the legislature's utility increased under unidimensional bargaining, we subtract the distance L-SQu from the distance L-SQ. We can use the same procedure to calculate the legislature's utility gain from multidimensional (simultaneous) bargaining.
Note that this procedure will yield a different (and more accurate) measure of utility gains than you would get if you simply added up the apparent gain in utility from each of the "issues" shown in the figure.
This page uses the following model parameters. If you want to see how the model changes when these parameters are adjusted, or if you want to learn more about what these parameters are, try the veto bargaining simulator.
What is the utility cost to the legislature of making a proposal?
If the legislature proposes a new policy that is exactly as far from the governor's ideal point as the status quo, then the governor has no particular reason not to veto. Thus, the legislature will propose something marginally better than the status quo. To avoid a veto, the legislature will never propose something that doesn't grant the governor (and the legislature) at least this much utility. Stated differently, the governor will veto anything that does not give at least this much utility: