Leaving the world of annoying theory and getting into some of the coolest stuff in the whole class, chapter seven discusses the way modern democracies generally make collective choices, some form of voting.

People choose goods based on the utility, budget (B), and relative prices (p_C, p_G) they face.

B = p_C * C^* + p_G * G^*

However, in this case C represents general consumption and G represents a government good. The superscripted asterisk indicates the consumer's optimal level of these goods. A problem which has be mentioned earlier is that the level of government spending is chosen collectively. So it is quite likely that this person will not get to be at the optimal level of consumption given preference, budget, and relative prices.

How are these levels of government spending and associated taxation set? Pareto, Hicks-Kaldor, Lindahl or some other scheme? In the US, many of these decisions are set by some sort of majority rule voting.

Assume the government must run a balanced budget so total tax collections (T) must equal spending on the publicly provided good (p_G * G). Total tax collections would be equal to the sum of what all the individuals pay (Sum_i T_i).

p_g * G = T = Sum_i (T_i)

Individual's share of cost = T_i/(Sum_i (T_i)) = T_i/(p_G * G) = tau_i

B_i = p_c * C + tau_i * (p_G * G) = p_c * C + (tau_i * p_G) * G

(tau_i * p_G) = "tax price" of the publicly provided good.

If the publicly provided good was just like a private good and could be purchased like an ordinary good, then this would be point (G^*, C^*) in Figure 1. At G^* (and C^*), the agent gets utility of 7. If the level of the publicly provided good differs from G^*, then the agent will get lower utility, like a utility of 5 from levels of the public good like G₂ or G₃. G₂ corresponds to too little of the publicly provided good and G₃ corresponds to too much. G₁ and G₅ correspond to utility of 2. The happiness of this agent as a function of the level of the publicly provided good and its associated tax burden is show in Figure 2.

This approach was developed by Ken Arrow (Nobel Prize in 1972) and Duncan Black. Figure 2 depicts a single peaked public sector preference. If this person got to vote for what level of the publicly provided good, proposals close to G^* would be preferred to those further from it. Different people have different preferences and budgets and therefore different G^*'s. Consider Figure 3, with three consumers with differing optimal levels of G. In an election with two alternatives G_α^* and G_ß^*, G_ß^* would win with two votes (from ß and γ) against one vote (from α) for G_α^*. Actually, any proposal to the right of G_α^* and to the left of G_γ^*, would defeat any proposal outside that range by at least a two vote to one margin. The level of publicly provided good G_ß^* would defeat any other proposal in this set up. It is what is known as a Majority Rule Equilibrium.

Majority Rule Equilibrium (MRE)

Any point, G^E, such that no alternative proposal, G', can beat G^E by a strict majority. It is assumed that everyone votes sincerely.

Sincere voting implies that people will always vote for the proposal they prefer. Why wouldn't somebody do that? A person might vote for a less preferred proposal now in the hope that it would be easily defeated later by a proposal which the voter preferred to either of the current choices. This will make more sense soon. Voting now for a less preferred choice in order to get to vote for some future more preferred choice is known as no "sophisticated" voting. It is what some of the opponents of NAFTA claimed to be doing when they said they believed in trade but "Not this NAFTA." The claimed that while they favored NAFTA to no agreement, they were voting against NAFTA in the hopes of getting some better "NAFTA" agreement later. This is a particularly unsavory example of sophisticated voting because most of the people who said this were lying about wanting an agreement at all.

In a simple majority election, the decisive voter is the median voter or the person who pushes the winning side from getting exactly half the votes to getting half the votes plus one. You might say that in such a situation, every voter is the median voter but if policy choices can be expressed along one dimension as in Figures 3 and 4, the concept of the median voter is clear. Sometimes the median voter is not the middle voter as with votes that require a two-thirds majority or a plurality (the largest set of voters received by any alternative but less than half the votes cast).

Consider Figure 4. This depicts a choice of the public good "Ideology." There are seven voters. The three left most voters we will call Democrats and the three right most voters we will call Republicans. The middle voter is the decisive median voter. Suppose that this reflects the electorate for the presidential election and that before a candidate can be nominated to run for president, that candidate must participate in primaries. The median Democrat is voter 2, the median Republican is voter 6. This model predicts that in the primaries the Democratic candidates would play liberal and pursue voter 2, the Republican candidates would play conservative and pursue voter 6. This would help them win their parties' nominations. In the general election, they would both pursue voter 4, the decisive median voter in the general election, and so would moderate the stands they took in the primaries. Cases in point: Clinton on tax increases and Bush on abortion. This model also predicts that with three candidates, there is a strong incentive for both parties to become very similar on issues in an attempt to woo the same median voter. With three major candidates, this model predicts behavior much more like locational models of firms as studied in Industrial Economics where firms will try to put as much distance between each other as possible so as to monopolize a range of the market (or voters).

There is a general feeling that Americans don't make much of an effort to vote and that voter turn-out is too low. It is a bit low by world standards. However, given the low probability of the election coming down to one vote and getting to be the median voter, it is a puzzlement for economists to explain why so many people vote. Perhaps it is to avoid being embarrassed like Michael Huffington was; he ran for the Senate in California but was shown not to have voted in many previous elections. (Things only got stranger from there.)

Is it silly to think of issues as one-dimensional? It can be since losers in elections try to break up winning coalitions. Case in point: Catholics generally voted Democratic until Republicans began heavily working the abortion issue. Another interesting group which made a similar switch in voting patterns are white southerners.

What would a voter's preferences look like in two dimensions? Consider two publicly provided goods, education and defense. A voter probably has optimal levels of provision of these goods, (E^*, D^*), with utility decreasing as the actual level of provisions of these goods moves away from these optimal levels. Consider Figure 5. Lower and lower levels of utility can be depicted as rings which are further and further out from the optimal point.

What would an election look like in this case with two goods? Let there be three voters, MF, BC, and RR. They each have differing optimal levels of both goods. BC likes some defense and lots of education. RR likes little public spending on education but lots of defense. MF likes little public spending in general. Their choices are shown in Figure 6. It is assumed again that given the choice of two proposals, a voter will vote for the proposal that comes closest to his or her optimal levels of the goods in question. Sadly, there is no longer a stable MRE. Instead we get vote cycling. Can you see how any choice of defense and education can get beaten by at least on alternative?

The triangular region which connects the three optimal points is known as the Pareto Region. A proposal which lies within this region can lose to an alternative outside the region but never by a unanimous vote. Starting from some point in this region, any movement of provision of public goods out of this region or to some other point in this region will make some voter or voters worse off.

 [Notes: I think Niskanen's model of Bureaucracy is very interesting. I also think Rosen's discussion of logrolling is quite well done. How do you suppose that would interact with the idea of a line item veto?]

This idea that with (at least) three voting blocks and (at least) three proposals, there would be no MRE comes from the work of a Frenchman named Condorcet. In 1765 he came up with the Majority Rule Paradox. Consider the following cases ....

Suppose that there are three voters, Moe, Larry and Curly. They are considering three projects: Bowling, Mud Wrestling, and Roller Derby. Here are their preferences for the first case.

If each just voted for which project they should pursue, each might vote his first choice and it would be a three way tie and no decisions would have been reached. Suppose instead that the vote was carried out in a series of binary elections.

So mud wrestling beats either of the other alternatives.

Mud wrestling > Roller Derby > Bowling. (here > means "is preferred to.")

Now consider this case with slightly different preferences.

Round 1: Bowling vs Mud Wrestling Bowling > Mud Wrestling

Round 2: Bowling vs Roller Derby Roller Derby > Bowling

Round 3: Mud Wrestling vs Roller Derby Mud Wrestling > Roller Derby

So here the is no majority rule equilibrium (MRE) because

Bowling > Mud wrestling > Roller Derby > Bowling.

This is a contradiction of how preferences are expected to go.

These guys being stooges, this cycling could go on forever. Suppose however that Curly decides to be a wise guy. Suppose he volunteers to run the votes and that there will just be two rounds of voting. In the second round he will have his favorite, Roller Derby go up against the winner from the first round. He knows that Roller Derby can beat Bowling in a straight vote and he knows that Bowling can beat Mud Wrestling in a straight vote. So by having Bowling beat Mud Wrestling in the first round and then Roller Derby beat Bowling the second round, Curly gets what he wants most. This is the advantage of agenda control. The ability to structure the vote can often determine the outcome in situations where cycling is possible.

Curly's scheme is very bad for Moe since it sticks Moe with his least preferred option. What could Moe do about this? He could poke Curly in the eyes and then slap him. Also, he could engage in "sophisticated voting." In the first round, even though he truly prefers Bowling to Mud Wrestling, Moe will vote for Mud Wrestling so that it can beat Roller Derby in the second round. Now, even though Moe does not get his most preferred outcome, at least he is not stuck with his last choice. Ah, but what if Curly expects Moe to do this and what if Larry starts to play the sophisticated voting game as well? The person who came up with this approach went crazy and he was neither the first nor the last game theorist to do so. I think it's best that we move on.

What about other voting schemes? Lani Guinier wrote a book about alternative voting schemes entitled The Tyranny of the Majority (the book focuses mostly on "at large voting" and weighted voting where voters are given multiple votes to cast). In 1993, she was nominated by Bill Clinton for the head of the EEOC but the Republicans ran a seemingly effective counter-offensive against her, branding her a quota queen. They did this with no discernable sense of irony over their use of minority voting tactics such as filibusters or Constitutional protections of minority voting blocks through super-majorities for Constitutional amendments or impeachment, and the historic compromise of giving each state, no matter how small, two senators. It is ironic also because Guinier would join with many conservatives in opposing these oddly drawn Congressional districts which are designed to win ethnic minorities seats. Clinton, displaying more speed than he shows while jogging, dumped her nomination with a quickness that mirrored his ability to move a Big Mac into his mouth. What this because he was afraid to fight for her ideas? Or was it, as Chicago Tribune Columnist Clarence Page suggested, because when Clinton finally read her ideas and they made sense to him, maybe too much sense. Consider the following hypothetical example.

Three candidates are running for office: GHWB, WJC, and HRP. In an earlier example, Moe, Larry, and Curly each voted for his favorite activity and it was a three way tie with no decision. That did not happen with GHWB, WJC, and HRP. WJC got 43% of the vote, GHWB got 37% of the vote, and HRP got 19%. So WJC won in a plurality. Now consider voting method suggested by the work of Lani Guinier. It is likely that while the backers of HRP did not much like GHWB, it is likely that more than two thirds of them preferred GHWB to WJC. Had the voting been conducted according to some system where second choices of voters for third place candidates were given some weight, it is not implausible that GHWB would have won in an outright majority. It is not unbelievable that more than half the electorate would have preferred GHWB to WJC in a straight binary vote.

Would a scheme that takes into account voters second choices make any sense? In a March 18, 1994 editorial in The Chicago Tribune by Don Saari from Northwestern's math department, he suggests what is known as a "Borda Count" for taking a vote over several choices.

  A Borda Count method of voting is familiar to anyone who ran cross-country because this is the method used for scoring teams. The simplest way to do a Borda Count is for each voter to rank his or her preferences over the available choices, with a one for first choice, a two for second, a three for third, etc.

In the first example with Moe, Larry, and Curly (where Curly prefers Mud Wrestling to Bowling), the Borda Count would work out as follows.

Bowling: 1 vote from Moe, 3 from Larry, and 3 from Curly = 7 votes

Mud Wrestling: 2 votes from Moe, 1 from Larry, 2 from Curly = 5 votes

Roller Derby: 3 votes from Moe, 2 from Larry, 1 from Curly = 6 votes.

So just like before, Mud wrestling > Roller Derby > Bowling.

In the second example with Moe, Larry, and Curly (where Curly prefers Bowling to Mud Wrestling), the Borda Count would work out as follows.

Bowling: 1 vote from Moe, 3 from Larry, and 2 from Curly = 6 votes

Mud Wrestling: 2 votes from Moe, 1 from Larry, 3 from Curly = 6 votes

Roller Derby: 3 votes from Moe, 2 from Larry, 1 from Curly = 6 votes.

So just like before, we get no answer in a voting process. This is not necessarily bad, here the voters really have no majority preference so none should appear.

What about in the example with GHWB, WJC, and HRP? This is extreme guesswork. For the sake of making a point, suppose all of GHWB's voters would vote 1 for GHWB, 2 for HRP and 3 for WJC. Let WJC's voters go with 1 for WJC, 2 for HRP, and 3 for GHWB. Let HRP's voters go with 1 for HRP, one third of them go with 2 for WJC and 3 for GHWB, and two thirds of HRP's voters go with 2 for GHWB and 3 for WJC.

Recall that WJC got 43% of the vote, GHWB got 37% of the vote, and HRP got 19%. The Borda Count (normalizing the electorate to 100 voters) would come out as follows.

WJC: 43*1 + 37*3 + 1/2*19*2 + 1/2*19*3 = 43 + 111 + 12.7 + 38 = 204.7

GHWB: 43*3 + 37*1 + 1/2*19*3 + 1/2*19*2 = 129 + 37 + 19 + 25.3 = 210.3

HRP = 43*2 + 37*2 + 19*1 = 86 + 74 + 19 = 179

So WJC still beats GHWB by a whisker. However, Notice that HRP is now the winner. (Hmm, maybe the Electoral College system is looking better and better.)

What if we add a candidate that no one likes? For Minnesota, such a candidate might be Dan Reaves.

It is costly in time and travel to physically vote in elections. It is even more costly to stay informed about the issues. Given that in any but the very smallest election, a voter's chance of being the median (having the election decided by a margin of just one vote) is very close to zero, why do people vote? It's an economic paradox. Perhaps the attempts to build up an aura of civic duty or shame at not voting actually work.

However, is the median voter a good person to be making choices for society? The mean income in the US at least twice the median income. This is because the mean is bid up by some really rich people. So what can we guess about the preferences of the median voter? Would the median voter be in favor of high levels of social services (available to everyone) financed by high taxes on the rich? Could be. Given the choice, would the median voter prefer better care for the poor or better treatment of the middle class? Hmm, tough one.

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Questions, comments, typos? mwitte@nwu.edu