Classroom games: candidate convergence.

1. Introduction

A critical part of teaching political economy is getting students to understand the link between voter preferences, candidate position taking, and vote aggregation rules. While in the abstract it is easy to understand that candidates will often converge to the median voter, the ease with which candidates respond is not often grasped. This exercise mimics multiple elections with candidates holding no information about the distribution of voter preferences. Parallels are also drawn to one-dimensional spatial location models.

The median voter model is one of the basic building blocks for political economy. The model assumes that voters have single-peaked preferences over a single policy dimension. This means that each voter has a single preferred position that can be mapped onto the policy dimension and the voter's utility decreases as a function of distance from that preferred position. Candidates seek election by proposing a policy position that will be implemented if elected. Voters then compare the utility they get from the policy position proposed by each candidate. In a two-candidate model the median voter's position is in equilibrium and candidates converge to the equilibrium. The simplicity of the model and its unique equilibrium allows those teaching various aspects of political economy to quickly cut through the complexity of political elections.

This classroom experiment has three goals. The first is to demonstrate that competitive elections quickly drive candidates to adopt winning positions. Second, the exercise points to the stability of an equilibrium. This is especially useful for students who feel that an equilibrium is a fuzzy concept with little explanatory power. In this exercise candidates who deviate from the equilibrium are quickly punished. Third, the exercise introduces a feature common to many political systems: redistricting. This can be thought of as a change in the preferences of some voters. Simple changes in student preferences can demonstrate the suddenness with which equilibrium can be changed and how quickly candidates will converge on the new equilibrium. Typically the exercise is run before introducing the concept of a median voter. Candidate behavior can be used as data to illustrate convergence to the median voter and to illustrate the power of equilibrium.

The exercise can be run with any size class and in less than 30 minutes. Classes with more than 100 students will require more preparation time. The exercise usually goes through 15 distinct elections. The entire exercise can be carded out by a single person, although with large classes appointing election monitors to pass out material and to count ballots is helpful.

2. Instructor Procedures

The materials needed for this exercise are limited and all of the preparations can be handled beforehand. All that is needed is

* A spreadsheet or some other device to generate voter distributions

* Index cards (enough so that everyone in the class has one index card).

* A smaller set of index cards of a different color (enough for 10-20% of the number in the class)

* Numbered poker chips (or some other randomizing device) numbered from 1 through the number in the class

* An experimenter record sheet (see the Appendix)

* Oral instructions to be read by the experimenter (see the Appendix)

* An overhead projector for the student instructions or enough sheets with the printed student instructions (see the Appendix)

* A student record sheet (optional)

Preexperiment Preparations

The elections take place on a single dimension made up of a whole number line ranging from 1 to 100. To begin preparations a distribution of voters needs to be constructed, index cards must be prepared, a random device must be designed and various forms should be assembled. The tasks prior to the experiment include the following:

(i) Generate a distribution of voters. The easiest way to do so is to build a spreadsheet with rows equal to the number of students in the class. One column has numbers ranging from 1 to the number of students in the class. The second column assigns each cell (student) a whole number on the number line ranging from 1 to 100. Rather than assigning numbers along a uniform distribution, I use a skewed distribution either to the left or to the right. This number will constitute a student's "ideal" position in the campaign space. People can have the same number (preference) if there are a lot of students. Typically I choose numbers so that the position of the median voter is close to 65. The third column should copy the numbers from the second column. In this column 10% to 20% of the "ideal" positions should be changed so that the equilibrium is shifted. This can easily be done by changing the values of the high valued students (if the equilibrium is above 50) and assigning them low values. I typically try to shift the equilibrium by 25 or 30 points to the left. Highlight the numbers that have been changed. To make life easier, keep the numbers that are changed in sequence with the ID numbers. So, for 100 students in a class and if 20 ideal positions are changed, change IDs that range between 71 and 90. (1)

(ii) Prepare the index cards. The index card needs two pieces of information: an 1D number and an ideal position. The ID is simply copied from the first column and placed in the upper-left corner of the index card. The ideal position is copied from the second column and put in the middle of the index card. The second, smaller set of index cards that differ in color should now have the ID copied from the first column and the new value from the third column of the spreadsheet. That set of cards needs to be generated only for those numbers that were changed (and highlighted).

(iii) Substitute the colored index cards for cards with the same ID. If the first deck of ID cards is white and the second, smaller deck is blue, the final deck will have almost all white cards and some blue. The remaining small stack will have all white cards. This is done so that during the exercise it will be easy to swap cards (change preferences) for a small subset of the students.

(iv) Prepare a randomizing device. I typically use numbered poker chips, but anything can be used. Plastic poker chips are cheap and can be numbered with a permanent ink marker. These chips can be drawn from a box, a hat, or anything else that is handy.

(v) Prepare overhead slides with instructions for the students and the experimenter's record sheet. In lieu of this, pages of the instructions can be copied for students and distributed.

(vi) Copy student record sheets. This is optional since students can keep track of their own earnings using their own paper and an example of a record-keeping sheet can be posted on the board or on an overhead.

This is the extent of preclass preparation. Because there is some fluctuation in attendance I always prepare a full set of materials for everyone in the class, ff the index cards are thoroughly shuffled and randomly distributed, those cards that are left over will not affect the lesson.

Class turnout is very high on experiment day. In part this is because I require students to pay a lab fee at the outset of class. This is a fee of $5.00 per student and that amount goes into a central pool of money. I match the central pool with my own money (obviously my classes are well below 100 people). Students understand that throughout the course they will be in many experiments and in each experiment they will earn points. The points are normalized as a share of their winnings in each experiment. At the end of the semester students are paid their normalized share from the central pool. So, each experiment is incentivized and counts toward their earnings at the end of the semester. This certainly is not needed for this particular experiment to work. However, even a nominal prize of a candy bar for the individual accumulating the most points (and a reprint of the faculty member's latest publication for the individual with the fewest points) can be useful.

Conducting the Experiment

The experiment requires approximately 30 minutes from handing out materials to the final election cycle. Because I post the data and ask students to write up the results, I ordinarily run the exercise in the final 30 minutes of class. At the next class the results are discussed and the exercise is integrated into the lecture. However, there is no reason that the experiment could not be run in the first part of the class with the data plotted and a short lesson drawn in the last portion of the class.

The following steps are taken to conduct the experiment. Throughout I assume that an overhead projector is used, but similar sequences can be used for materials that are distributed by hand.

(i) A short set of instructions are read (see Experimenter Oral Instructions in the Appendix). These provide minimal information to the students and give them an overview of the experiment.

(ii) The large stack of index cards should be thoroughly shuffled and passed out to students. If student record sheets are used, they too should be passed out at the same time. If the class is large, then randomly choose two students to help with all the tasks. They will help with passing out material and with counting votes. If there are any remaining cards, set them aside. Do not confuse them with the smaller, second, stack of index cards that will be used to change preferences.

(iii) Read the remaining instructions and put up the student instructions on the overhead. There are three primary points that students need to understand. The first is that, except for the candidates, everyone votes. The second is that candidates can choose any policy they wish. The incumbent always enjoys a first-mover advantage in that the challenger may not adopt the same policy as the incumbent. Only the winning candidate is paid. The loser earns nothing. Third, voters are paid on the basis of the position adopted by the winning candidate. In this world a candidate's announced position is immediately implemented. Voters have to solve for their earnings in between election cycles. It is worth spending some time on this so that everyone knows how to calculate their earnings. Finally, there is a simple set of rules designed to keep the experiment flowing and to minimize distractions.

(iv) The experiment begins by randomly drawing two candidates from the numbered poker chips. Individuals whose ID numbers match the numbers drawn become the candidates. Those candidates stand and a coin is flipped to determine who will be the incumbent. The incumbent is asked to state a position (a number between 1 and 100). That position is recorded on the Experimenter Record Sheet (see the Appendix). The challenger is asked to state a position that is also recorded on the experimenter record sheet.

(v) Voters are now asked to vote. Voters are cautioned that they may only vote once and a show of hands is asked for all those in favor of the incumbent's position as recorded on the record sheet. With a large class two vote counters may come in handy--with each counting a section of the class. When there is agreement over the number of votes, it should be recorded on the record sheet. A show of hands should then be asked for those voting for the challenger's position. Again the votes should be counted and posted. The winner should then be declared and the policy position adopted should be circled on the experimenter record sheet and the winning proposal written in the last column. Students should be told that this is the policy adopted for this election cycle and they should calculate their earnings. In the event of a tie a coin should be flipped to determine which policy is adopted.

(vi) The winning candidate is the incumbent (or remains the incumbent). The losing candidate rejoins the voters. A new challenger is drawn from the poker chips (this can be done with or without replacement) and that candidate is asked to stand. The incumbent then states a policy position that is recorded. The challenger is then asked to state a different policy position and it is recorded. Step (v) above is then repeated.

(vii) Steps (v) and (vi) are repeated for 8 to 12 periods until candidates converge to the equilibrium. With small classes and with students randomly drawn as candidates there can be some fluctuation around the equilibrium. Even so, candidates typically converge very quickly.

(viii) Once candidates have converged read the next section of the instructions (see Appendix). Take the small stack of index cards and hand them out to each person with a colored index card. These cards should match on the student IDs. This can be thought of as equivalent to redistricting, generational replacement, or some general shift in preferences. Retain the current incumbent and randomly choose a new challenger. Repeat steps (v) and (vi) for 8 to 12 new election cycles.

(ix) When students have converged to the new equilibrium and repeated the process several times the exercise can be halted. Either collect the student record sheets to record how they did or ask for the highest score(s). Subject payments can be made at this time or later if you are so inclined.

The end of the class period usually determines the conclusion of the exercise, but as noted above it need not. I then post the experimenter record sheet data on the web and ask students to graphically represent the data in order to show me what happened during the course of the experiment. I also provide them with several leading questions asking them to explain what they thought was the lesson.

3. Class Discussion

The class discussion can focus on several points. The first has to do with candidate convergence and the power of the median voter. Figure 1 is taken from one class exercise. Although the incumbent won the first election, challengers then won the next series of elections. There was a steady convergence to the equilibrium (at policy position 34) until election 5 in which the recent incumbent overshot. By period 6 the candidates were at or near the equilibrium and both remained side-by-side. Between periods 9 and 10 a handful of voters were given new preferences and candidates immediately adjusted, although they did not know the new equilibrium. Again there was some fluctuation until candidates converged to equilibrium located at 57. Class discussion pertained to the fact that both candidates offered similar positions and that at some point the challenger could do no better than to offer positions around those adopted by the incumbent.

[FIGURE 1 OMITTED]

The second point concerns the power of the median voter, if need be the instructor can put up the distribution of voters' preferences and tall about the predictive value of the median voter. Figure 1 can be referenced in this discussion because it illustrates the strategies used by candidates in an effort to locate the median. Moreover, this point can demonstrate how it is that simplifying voter interests to the median voter is not much different from choosing a representative agent from a market.

The third point illustrates the impact of redistricting (or some other change in voter preference) and the responsiveness of candidates to voters. At this point it is always nice to demonstrate to students that candidates are catering to voters and their interests and not ignoring voter concerns. Candidates quickly accommodate changing voter preferences.

There are obviously a number of extensions to take. One direction is what happens if candidates care about more than simply winning the election? Will this lead them to adopt policies that differ from the median voter? While possible, it is always interesting to get students to think about what effect failed (unelected) politicians have--after all, they cannot vote on policy. The lesson for candidates is that winning ensures that they get a voice at the policy table, whereas losing provides no such guarantee. A second direction involves alternative aggregation rules. An easy addition might consider the impact of primary elections that pick candidates who then stand for general election. If the primaries involve people who are clustered together at different ends of the policy space (suppose the policy space involves an ideological continuum rather than a number line) then it is unlikely that candidates will converge, but the winner will still be responsive to the median voter. With sufficiently foresighted voters, candidates would converge to the median, regardless of the distribution in the primaries. A third direction involves candidates not being bound to their stated policy decision (they get to announce a new position once they are elected). Here it might be interesting to ask whether there will be reputational effects and whether an incumbent, in a repeated game, has an incentive to change that policy even if something different is preferred. A fourth direction is to imagine what would happen with more than two candidates and whether the equilibrium would change. This can lead to discussions of rules governing entry and whether a winner-take-all system leads to strict two-party competition over time. A final direction might be to ask the stability of these results when voters have preferences across many dimensions and candidates stake out positions on many dimensions. Unfortunately, this problem is not very tractable. Here the standard results from social choice theory can be applied.

An interesting feature to this exercise is that it need not be only about candidate convergence and the median voter. Indeed the original variation of this model derives from Hotelling (1929) and spatial location theory. The classic question is "Where should a pair of ice cream vendors locate on a beach in order to maximize sales and minimize the distance that beach goers have to wall?" In this sense the exercise can be related to purely economic conceptions of equilibrium. Rather than voting, students could choose the vendor from whom they wished to buy. The shift in people's preferences in the second half of the periods could be explained as a change in the attractiveness of the beach (or an oil spill at one end of the beach).

A second interesting feature of this exercise is to present the idea that voting is little more than information pooling. None of the voters know anything about the underlying distribution of voter preference. The candidates know little more about the distribution. Yet the median position is revealed very quickly and efficiently. Moreover, like price, the equilibrium is resilient to change.

4. Further Reading

There are a number of readings that supply the basics for the median voter theorem. Riker (1982) has an accessible version of the median voter and its usefulness. Original statements of the median voter stem from Hotelling (1929), Black (1948), and Downs (1957). The classic statement by a political scientist concerning reelection incentives is found in Mayhew (1974). A general discussion of the intersection between politics and economics, touching on the median voter, can be found in Miller (1997). An experimental application of the median voter and elections is in Morton (1993). Interesting examples of the application of the median voter to policy and politics can be found in Boylan and McKelvey (1995), Duggan (2000), and Besley and Case (2003). Discussions of the concept are by no means limited to any of these articles.

Appendix

Instructions for Subjects

In this experiment you will sometimes be a voter and sometimes you will be a candidate. Voters and candidates earn points in different ways. Candidates are trying to win. Voters are trying to get the winning candidate to pick a policy close to their preferred point.

Each voter has a card with two pieces of information

* First, there is an ID number at the top left of the card. It will be used to determine candidates.

* Second, you have a preferred point. This is your ideal policy. It is private information and you should not reveal it to anyone.

Payment to Candidates

* If you win, you earn 50 points for that vote.

* If you lose, you earn 0 points for that vote.

You will have a decision sheet that looks like the following:

 A        B         C          D          E          F           G

Vote    Type:      Pre-     Winning    Absolute   Earnings   Cumulative
 #     Voter or   ferred   Candidate    Diffe-     (50-E)     Earnings
        Candi-    Point     Policy      rence
         date

 1        C                                          50          50

In this example suppose that you are a candidate. If so, you would mark "C" under column B. If you win the vote, then you earn 50 points. You would enter this under column F and your cumulative earnings for the first vote would simply be 50. On the other hand, if you lost the election, you would record 0.

Payment to Voters

* Once the election is held, calculate the absolute value for the difference in the winning candidate's policy and your own preferred policy.

[absolute value of Candidate Policy - Preferred Policy]

* Subtract that amount from 50. That gives you your earnings.

You will have a decision sheet that looks like the following:

You will have a decision sheet that looks like the following:

 A        B         C          D          E          F           G

Vote    Type:      Pre-     Winning    Absolute   Earnings   Cumulative
 #     Voter or   ferred   Candidate    Diffe-     (50-E)     Earnings
        Candi-    Point     Policy      rence
         date

 1        V         50        63          13         37          37

In this example, suppose it was the first vote, you were a voter (you would write in "V" in column B) and your preferred point was 50 (write that in column C). If the winning candidate proposed 63, you would write that in column D, you would find the absolute difference between the two and enter that under column E. You earnings would then be 50 minus the amount you entered under E. Enter this number in column F. Your cumulative earnings for the first vote would be 37.

Rules for Voters

* There is no talking. If you have a question you may address it to the experimenter.

* Voting will be by a show of hands. A tie will be decided by a flip of a coin.

* You must keep track of your decisions on the decision sheet. If you have any questions, please ask.

Rules for Candidates

* If you are the incumbent you get to make the first policy proposal.

* If you are the challenger you must pick a policy different from that announced by the incumbent.

* You may not talk, except to announce your policy proposal to the experimenter.

Experimenter Oral Instructions

(Read before passing out the materials)

In this experiment you will face a number votes. Sometimes you will be a voter and sometimes you will be a candidate. You need to pay careful attention to what is happening in the experiment.

In this experiment there will be two candidates and the remainder will be voters. Candidates are trying to win. Voters are trying to get the winning candidate to pick a policy close to their preferred point.

The policy space consists of a line that ranges from 1 to 100. A candidate can propose any policy on that space. An incumbent always makes the first announcement. The challenger makes a second announcement and can never choose the same policy as the incumbent.

Voters then vote for a candidate based on the announced policy position. The winning candidate's policy is implemented and voters are paid based on the difference between that policy and their own preferred policy.

Cards are now going to be passed out. They have been shuffled and are in random order. You should take one card and hide it so that others cannot see what is on the card. Once everyone has a card I will explain the information that it contains.

Read Once the Materials Have Been Passed Out

Cards have been passed out to each individual. On the card are two pieces of information. The first is the ID. This will be used to locate new candidates. The second is your preferred policy.

The experiment will be conducted as follows:

First, two candidates will be randomly selected. Selection will be made from a bag of poker chips, each with a number that matches an ID noted on the card. At the first vote, to determine which candidate will be the incumbent, a coin will be flipped.

Second, the incumbent will be asked to state a policy that will be implemented if that person is elected. The challenger will then propose a different policy. All policies must be integers.

Third, the voters will then be asked to vote for one candidate or the other based on the announced policy position. Votes will be taken by a show of hands and a simple majority will determine the outcome. If there is a tie, then a coin will be flipped.

Candidates are paid simply on the basis of whether or not they win election. A winning candidate gets 50 points following the election. A losing candidate gets 0 points.

Voters are paid based on how close they are to the winning policy. If you look at your instruction sheet, you will see an example of a decision sheet. In this example, suppose it was the first vote, you were a voter (you would write in "V" in column B) and your preferred point was 50 (write that in column C). If the winning candidate proposed 63, you would write that in column D, you would find the absolute difference between the two (columns C and D) and enter that under column E. You earnings are then calculated as 50 minus the amount you entered under column E. Enter this number in column F. Your cumulative earnings for the first vote would be 37. Please check this to make certain you know how to calculate your earnings. Does anyone have any questions?

On the sheet you will have to fill in a good deal of information following each vote. Please fill in all of the information. If you happen to be a candidate, you will have very little information to fill in.

Fourth, once voting has ended a new challenger will be selected from the bag of poker chips. The incumbent will remain the same and will again get to choose the policy first. The challenger will pick a different policy. The voting process will start over.

Are there any questions at this point?

Once the experiment is done, you will hand in your completed decision sheet.

Additional rules and examples are on the subject instruction handout.

Read Once Candidates Have Converged for Several Periods

At this point I want all of you who have [insert color of card here] cards to raise your hands. You are going to be given new cards that have a new ID on them and a new preferred policy. Once everyone gets their cards, we are going to continue.

Subject Decision Sheet

Decision Sheet      NAME: --(print)

 A        B         C          D          E          F           G

Vote    Type:      Pre-     Winning    Absolute   Earnings   Cumulative
 #     Voter or   ferred   Candidate    Diffe-     (50-E)     Earnings
        Candi-    Point     Policy      rence
         date

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25

                                                   Total:

Experimenter Record Sheet

         Incumbent   Incumbent   Challenger   Challenger   Winning
Vote #   Position      Vote       Position       Vote      Position

   1
   2
   3
   4
   5
   6
   7
   8
   9
  10
  11
  12
  13
  14
  15
  16
  17
  18
  19
  20
  21
  22
  23
  24
  25

(1) In extremely large classes the last two digits of the student's social security number could be used and the campaign space could run from 0 through 99. The problem with this technique is that it is likely to produce a uniform distribution with convergence at 50. Candidates could then be drawn at random from the class lists, with the instructor choosing a new challenger each time. To effect a change in preferences a number between zero and nine could be drawn from numbered chips and students with the last digit of their social security number matching the number drawn could now be instructed to use the first two digits of their social security number. Of course, this is of little help for universities and colleges in the upper plains states and to the west in which values would be either at 50 or nearly there. Alternative mechanisms could be used, but close attention needs to be paid to the aim of shifting preferences sufficiently to induce a new equilibrium.

References

Besley, T., and A. Case. 2003. Political institutions and policy choices: Evidence from the United States. Journal of Economic Literature 41:7-73.

Black, D. 1948. On the rationale of group decision making. Journal of Political Economy 56:22-34.

Boylan, R. T., and R. D. McKelvey. 1995. Voting over economic plans. American Economic Review 85:860-71.

Downs, A. 1957. An economic theory of democracy. New York: Harper.

Duggan, J. 2000. Repeated elections with asymmetric information. Economics and Politics 12:109-35.

Hotelling, H. 1929. Stability in competition. Economic Journal 39:41-57.

Mayhew, D. R. 1974. Congress. The electoral connection. New Haven: Yale University Press.

Miller, G. J. 1997. The impact of economics on contemporary political science. Journal of Economic Literature 35:1173-204.

Morton, R. B. 1993. Incomplete information and ideological explanations of platform divergence. American Political Science Review 87:382-92.

Riker, W. H. 1982. Liberalism against populism. San Francisco: W. H. Freeman and Co.

Rick K. Wilson, Department of Political Science, MS 24, P.O. Box 1892, Rice University, Houston, TX 77251-1892, USA; E-mail [e-mail].

Thanks go to students in my Congress classes for being guinea pigs for classroom experiments over the years. The work was completed under an NSF Infrastructure grant (SES 00-94800), although that agency bears no responsibility for the content. Time spent at the Russell Sage Foundation gave me the luxury of writing up this class experiment. Thanks go to the 2003 NSF Workshop on Classroom Experiments in Economics held in Tucson, Arizona, and to two anonymous reviewers of this journal. Special thanks go to Catherine Eckel, Charlie Holt, David Reiley, and Cathleen Johnson for comments.

Received April 2004; accepted June 2004.