Electoral System and Number of Candidates: Candidate Entry Under Plurality and Majority Runoff

Abstract

We know that electoral systems have an effect on the number of competing candidates. However, a mystery remains concerning the impact of majority runoff. According to theory, the number of competing candidates should be equal (or only marginally larger) under majority runoff than under plurality. However, in real-life elections, this number is much higher under majority runoff. To provide new insights on this puzzle, we report the results of a laboratory experiment where subjects play the role of candidates in plurality and majority runoff elections. We use a candidate-only and sincere-voting model to isolate the effect of the electoral system on the decision of candidates to enter the election. We find very little difference between the two electoral systems. We thus re-affirm the mystery of the number of competing candidates under majority runoff.

Appendix: Game-Theoretical Equilibria

In this Appendix, we compute the equilibria of a nine-player game, which mimics the laboratory experiment presented in this chapter. The game is identical to the experiment except that it is supposed to be one-shot, whereas in the experimental sessions, we repeat the game three times with fixed positions for all players. We first consider a streamlined version of the game (full information game), neglecting the uncertainty due to the random choice of 50 out of 90 voters. Then, we study the game with uncertainty, taking into account this random draw of voters. Below, we also discuss, in view of the results, the pertinence of the equilibrium approach.

In the full information game, all nine players know the payoffs with certainty. If we consider that all 90 voters are turning out, the situation in which only the median player (i.e., the candidate located at position E) enters is a Nash equilibrium under both plurality and majority runoff. This player would indeed defeat any other player in a pair-wise competition. None of them should thus enter.

With a cost of entry of 1/5, as in the experiment presented in the chapter (to simplify the analysis, the gain from winning an election is normalized to one), there exist six pure Nash equilibria under plurality, and three pure Nash equilibria under majority runoff.

The three equilibria under majority runoff are: (such as in the chapter, the players are denoted A, B, C, … I)

• {E}: Only player E (the median player) enters. Her payoff is of 1 – (1/5) = 4/5.

• {D,F}: Players 4 and 6 enter with a payoff of (1/2) – (1/5) = 3/10.

• {C,G}: Players 3 and 7 enter with a payoff of (1/2) – (1/5) = 3/10.

It is easy to check that these three situations are equilibria. It is more tedious to make sure that there is no other equilibrium; we achieved this with the help of a computer.

The three situations above are also pure strategy Nash equilibria under plurality. However, there are three more equilibria under plurality, which involve three entering players with a payoff of (1/3) – (1/5) = 2/15:

• {B,E,H}: A symmetric situation with the centrist and two rather extreme players.

• {A,F,G}: A non-symmetric situation involving one extreme player.

• {C,D,I}: The mirror situation of the previous equilibrium.

It is interesting to observe that the reasoning that if there are five entering players or less my probability of winning is 1/5 or more, and thus that my entering cost is covered, is not sufficient. It is true that if six players enter, then at least one of them has a chance of 1/6 or less to win, and should thus not enter. If there are exactly five entering players, these players have a probability of winning of 1/5 only if they have equal chances. As soon as they do not have equal chances (and they never have), the probability of winning of at least one of them goes below 1/5. This(-ese) player(s) should thus not enter. The situation where there are five entering subjects is thus really an upper bound (and a crude one) for rational entry.

Let us now consider the game with uncertainty on turnout. If there were 89 participating voters instead of 90 (as in the full information game), the payoffs of the candidates would arguably be extremely close to those of the full information game, and therefore the equilibria would be the same. Now, with 50 voters turning out, as in the experiment, the noise introduced in the game is more important. To compute the payoffs in that case, we used computer simulations: with n independent random draws of 50 out of 90 voters, one can compute the average payoff of each candidate over these n draws. By the law of large numbers, these payoffs converge to the exact payoffs when n becomes large. We observed empirically that the payoffs do not vary by more than 1 % for n = 10,000 draws.

In order to obtain the set of all equilibria in the game with uncertainty, we computed the payoff of each candidate for each configuration, and checked for each configuration if it was an equilibrium (this is the case if each candidate wins with a probability higher than 20 % and if once another candidate enters, she wins with a probability lower than 20 %). It appears that all the equilibria of the game with full information are still equilibria. Moreover, there is an extra equilibrium in the game with uncertainty under majority runoff: {B,D,F,H}. Besides, there exists no other equilibrium. In the following paragraphs, we provide some insights on these results.

First, it is easy to show that the single-player (median player) equilibrium is still an equilibrium (under both plurality and majority runoff). If the other candidates do not enter, player E is obviously right to pay the entry cost, as she will win. In contrast, other players should not enter. Consider player D, if she enters, she obtains the ‘turning out’ voters located at positions between 1 and 40 (on the 90-point scale), while player E obtains those located at positions between 41 and 90. Player D wins if the number of her ‘turning out’ voters is strictly larger than the number of ‘turning out’ voters of player E, wins with probability of 1/2 if these numbers are equal, and loses otherwise. Player D wins with an approximate probability of 12 %. This probability is less than 1/5, which means that she should not enter. The situation is similar (or even worse) for other players. The single candidate equilibrium is an equilibrium of the game with uncertainty.

The equilibrium {D,F} is also still an equilibrium of the game with uncertainty (under both plurality and majority runoff). The expected payoff of player D and F is (by symmetry) (1–2) – (1/5) > 0, just like in the situation where all 90 voters are counted. If player C enters, she obtains the ‘turning out’ voters located at positions between 1 and 30, player D obtains those located at positions between 31 and 45, and player F obtains those located at positions between 46 and 90. Under plurality, the chances of player F are larger than 97 %. Therefore, player C should not enter (under both plurality and majority runoff). Here again, the situation is even worse for the other players. Similarly, {C,G} is an equilibrium of the game with uncertainty under both electoral systems. In that case, the most dangerous challenger is player E. If E enters, she wins with a probability smaller than 1 % under plurality, and with probability 2 % under majority runoff.

With three candidates, {B,E,H} is an equilibrium of the game with uncertainty under plurality, but not under majority runoff. In this configuration, player B obtains the ‘turning out’ voters located at positions between 1 and 30, player E obtains those located at positions between 31 and 60, and player H obtains those located at positions between 61 and 90. As a result, each candidate wins with probability 1/3 under plurality, and this configuration is an equilibrium. However, player E is much more likely to win under majority runoff, as she almost surely wins when she reaches the second round. Under this electoral system, E wins with probability 67 %, whereas B and H win with probability 17 % each. Hence, {B,E,H} is not an equilibrium, but it is not far from being one.

With four candidates, {B,D,F,H} is not an equilibrium of the game with uncertainty under plurality, but it is under majority runoff. In this configuration, player B obtains the ‘turning out’ voters located at positions between 1 and 25, player D obtains those located at positions between 26 and 45, player F obtains those located at positions between 46 and 65, and player H obtains those located at positions between 66 and 90. Under plurality, the two central candidates (D and F) win with probability 7 % only, which explains that {B,D,F,H} is not an equilibrium. However, under majority runoff, these central candidates are advantaged when they reach the second round, as they almost surely win. It appears that players D and F reach the second round with probability 25 % and win with probability 24 %, whereas player B and H win with probability 26 %. Hence, {B,D,F,H} is an equilibrium. Note that this result is related to the noise introduced in the game: in the full information game for instance, players D and F receive a strictly lower number of votes than players B and H, and they never reach the second round.

Finally, with five candidates, one can wonder whether {A,C,E,G,I} is close to being an equilibrium of the game with uncertainty. The computation yields a probability of winning for extreme candidates (A and I) of 3 % under plurality and lower than 1 % under majority runoff. This configuration is thus far from being an equilibrium.

The equilibrium analysis leads us to conclude that there should be a low or moderate number of entering players: at most three under plurality, and at most four under majority runoff. However, one should note that this is really a typical equilibrium reasoning. If I think that few other players are running, it is not unreasonable to take a chance. Suppose for instance that I am the median candidate and that I observe that in the past elections, four or five players were entering, it becomes very reasonable for me to enter under majority runoff.