The Politics of Numbers: Is Majority Rule Always Fair?


Is the electorate always right? Mathematician George Szpiro shows how majority voting can throw up some unexpected results.

The electorate is always right, isn't it? Not necessarily. As George Szpiro shows, majority decisions can result in unintended cycles, a dunce may upset a sure thing, and dishonest electors are able to manipulate elections.

We have known for centuries that majority rule does not always reflect the electorate's true will

Imagine the following discussion that allegedly took place many years ago in a New York restaurant. A waiter informs the diner, ‘for dessert, we have apple pie and brownies’. The customer decides on apple pie. A few moments later the flustered waiter returns to inform the patron that he had forgotten to mention that the restaurant also offers ice cream. ‘In that case I will have a brownie’, the guest announces after some short reflection.

This anecdote, ascribed to the philosopher Sidney Morgenbesser from Columbia University, encapsulates a problem that has forever plagued decision-making procedures. Obviously, the diner did not care one way or another about ice cream, since he did not choose it, even when it was offered. But its sudden availability did reverse his choice between the first two alternatives. Something like this just should not happen. But things like that do happen, and more often than one thinks. Indeed they can have much graver consequences than the simple choice of a dessert.

In the 2000 US presidential elections, a clear, if slim, majority of the electorate preferred Al Gore to George Bush (50,999,897 votes for Gore, 50,456,002 for Bush). Bush nevertheless became president because – after intervention by the US Supreme Court – he garnered Florida's 25 electoral votes. It was the presence of a complete outsider on the ballot, the Green Party candidate Ralph Nader, that decided the election result. With only 2,882,995 votes across the United States, Nader never stood a chance of winning. But by attracting a measly 97,421 votes in Florida, Nader probably denied Al Gore the few hundred votes that he would have required in order to win the state and become president.

Knowing full well that their candidate stood no chance, rational Greens should have voted for their second most preferred candidate. By giving their vote to a spoiler, the Greens managed to get a candidate elected whom the majority clearly did not want. Did the democratic tradition let the American people down?

The Paradox of Majority Rule

The sad truth is that majority rule may not reflect the true will of the electorate. This has been known for centuries. In 1785, the nobleman Jean-Marie-Antoine Nicolas de Caritat, Marquis de Condorcet published a 200-page pamphlet titled Essay on the Application of Probability Analysis to Majority Decisions. He presented an astonishing paradox that can be illustrated by a simple example. Let us say Peter, Paul and Mary must decide what to buy for their after-dinner drinks. Their preferences are:

  • • 
    Peter: Amaretto > Grappa > Limoncello
  • • 
    Paul: Grappa > Limoncello > Amaretto
  • • 
    Mary: Limoncello > Amaretto > Grappa

(‘>’ indicates ‘preferred to’)

Committed as they are to democratic values, the three decide to go by the majority opinion. A majority prefers Amaretto to Grappa (Peter and Mary) and a majority prefers Grappa to Limoncello (Peter and Paul). Based on these two rounds they can make their decision: purchase a crate of Amaretto. But surprise, surprise: Paul and Mary protest. What happened? The most reasonable selection method was used – one person, one vote – and they are still not happy? Paul and Mary have a legitimate grumble. They point out that they would prefer even Limoncello, the lowest ranked option, over Amaretto. How come? Well, had the three campers had a third round of voting, between Limoncello and Amaretto, a majority would have preferred Limoncello (Paul and Mary). But if they bought Limoncello, Peter and Paul would protest just as vigorously. They prefer Grappa to Limoncello. So here we have it, a paradox. Try as you might, the final result is that Amaretto is preferred to Grappa, Grappa to Limoncello, Limoncello to Amaretto, Amaretto to Grappa…

Condorcet's Paradox, as the conundrum was henceforth called, can be the source of much abuse (see Box). For example, a person setting the agenda at a board meeting can subtly influence the outcome of decisions by manipulating the order in which votes are taken. So what is the solution? The depressing answer is that there is none. With majority rule there is no way out of Condorcet's Paradox. What a let-down for democracy.

The Borda Count

Enter Condorcet's contemporary and intellectual sparring partner, Jean-Marie de Borda, a naval officer and mathematician. Borda also suggests that electors list their preferences but then proposes that they assign points to each of the alternatives, or candidates. The lowest ranked candidate receives one point, the next lowest two points and so on. In a race with 10 candidates, the preferred candidate would receive 10 points. The candidate with the most points wins. By using Borda's method, the vexing Condorcet cycles can often be broken, and a clear winner can be declared. (Unfortunately, for the drinks example above, even Borda's method would not help.)

One criticism of the so-called Borda count is that only the ordering of preferences is taken into account; the intensity of preference is not being expressed. Giving one extra point for each extra rank implicitly assumes that the additional preference accorded the top candidate over the second is the same as that accorded the ninth over the tenth. Far more importantly, however, a paradoxical situation may arise through the sudden appearance of a clearly inferior candidate. Even though he or she would be ranked low on every voter's list, his or her addition to the roster might influence the election's outcome: let us assume that 60 electors prefer Ginger to Fred, and 40 prefer Fred to Ginger.

  • • 
    60 electors: Ginger > Fred
  • • 
    40 electors: Fred > Ginger

Number of points awarded:

  • • 
    Ginger: (60 × 2) + (40 × 1) = 160
  • • 
    Fred: (60 × 1) + (40 × 2) = 140

Obviously, the Borda count declares Ginger the winner with 160 points, against Fred's 140. Now Bozo appears on the scene. Nobody really likes Bozo but his entry persuades 30 of Fred's voters to rank Ginger even behind Bozo.

  • • 
    60 electors: Ginger > Fred > Bozo
  • • 
    10 electors: Fred > Ginger > Bozo
  • • 
    30 electors: Fred > Bozo > Ginger

Number of points awarded:

  • Ginger:

  • (60 × 3) + (10 × 2) + (30 × 1) = 230

  • Fred:

  • (60 × 2) + (10 × 3) + (30 × 3) = 240

  • Bozo:

  • (60 × 1) + (10 × 1) + (30 × 2) = 130

Now Ginger receives 230 points, Fred 240 and Bozo 130. Bozo's entry caused Fred to win. Hence, by adding a dunce to the roster (ice cream to the dessert menu, Nader to the ballot) the outcome of an election could be changed completely. Thus, the Borda count is open to manipulation through so-called strategic voting. By putting the most dangerous challenger dead last, a group of electors can ensure that a second or third choice is pushed to the fore.

When someone pointed out to Borda that his method could easily be manipulated by a group of electors who decide to deprive the front-runner of victory, he was indignant. ‘My scheme is only intended for honest men’, he snapped. Well, with all due respect to Borda's belief in the electors’ sincerity could Greens have been considered dishonest had they cast their ballots for Al Gore rather than for their true first choice, hoping at least to beat George W Bush?

Condorcet Cycles – Easy as Paper, Stone, Scissors

Condorcet cycles arise because preferences are intransitive, which is a technical term that means that preferences, unlike for example weights or lengths, do not carry through when choosing among three or more items. While one can infer from the observations that an elephant is heavier than a horse and a horse is heavier than a rabbit that the elephant is heavier than a rabbit, this does not hold for preferences among, say, different drinks, as expressed by majorities.

The situation can be illustrated with the popular children's game'paper, stone, scissors’: paper wraps stone, stone scratches scissors, scissors cut paper. The game's eternal appeal – or endless boredom – resides in intransitivity: the fact that paper beats stone and stone beats scissors does not mean that paper beats scissors. None of the three items is absolutely superior to the two others; preferences cycle around and around.

Arrow's Impossibility Theorem

Many adaptations have been suggested over the years, but no matter what variant was considered, all were fraught with some problem or another. Majority decisions can result in cycles, a dunce may upset a sure-thing choice, dishonest electors are able to manipulate an election. But maybe there is another ingenious way to elicit the people's true choice?

Enter Kenneth Arrow, who would win the Nobel Prize in economics in 1972. In his doctoral thesis in 1952, he answered the above question with a resounding ‘no’. The basic problem is that individual preferences cannot be simply aggregated into a ‘people's choice’. After positing a few reasonable-sounding requirements that a good elections system should have – for example, no ordering of candidates must be excluded, no choice should be imposed, there should be no dictator – Arrow proved mathematically that no electoral system exists that fulfils all these requirements (Arrow's Impossibility Theorem).

The only way to devise an electoral system that would have no cycles would be to relinquish one of the axioms. A dictatorship, for example, is a very efficient manner of governing: whatever the dictator opts for goes, irrespective of how the remaining electors may vote. But the most obvious requirement to be thrown overboard is the ‘independence of irrelevant alternatives’. Arrow required that the election of the preferred candidate be independent of whether an irrelevant candidate, without any chance of winning, appears on the scene. By dropping that assumption – thus allowing Bozo to throw an election into disarray or a restaurant patron to befuddle the waiter – our beloved majority voting system could be saved.

Donald Saari from the University of California at Irvine points out that Arrow's famous theorem should not be seen solely in a negative light: it simply reflects the fact that the goal of creating fair voting systems fails if we introduce rules that demand paired comparisons of candidates, separate from the whole set of candidates. Doing so dismisses crucial information about voters’ preferences. But reinserting the lost information into the system leads to Borda's method. As we saw, however, this method has its own drawbacks.

A Third Way?

So is there an election procedure that would somehow avoid cycles in spite of Arrow's Impossibility Theorem? Michel Balinski and Rida Laraki from the École Polytechnique in Paris think there is. They came up with a new proposal that was able, they claimed, to avoid all obstacles, like Condorcet's Paradox or problems surrounding Borda's scheme, as well as Arrow's Impossibility Theorem.


[ Outsider Ralph Nader unwittingly decided the 2000 presidential election in George W Bush's favour ]

Despite its flaws majority rule is still the most robust electoral system

The two mathematicians suggested an enhanced election procedure. Electors would no longer just pop a slip of paper with the name of their preferred candidate in the ballot box and they would not simply rank them as Borda required. Instead, they would fill out an ‘evaluation form’ in which they would assess all candidates, by assigning them grades ranging from ‘very good’ to ‘good’ to ‘satisfactory’ and so on, all the way down to ‘reject’. The percentage of votes each candidate receives in each category would be noted. Then each candidate's so-called median would be determined: beginning with ‘very good’ the percentages are added until the candidate has obtained at least half the electors. The candidate with the best median or, if more than one candidate have the same median, the one with the higher percentage at the median would be declared winner.

The two mathematicians claim that their method allows a more differentiated picture since voters’ opinions about all candidates are taken into account via the evaluation forms. Thus it would not suffice to get the nod from a plurality of voters; candidates must strive to get the best grades from all citizens. Ostensibly this method avoids all traps. Condorcet's Paradox is circumvented because the assignment of grades does not depend on the candidates’ ranking. It does not fall into the Borda trap because the addition or removal of candidates does not change the evaluation accorded the initial candidates. And it steers clear of Arrow's depressing conclusion, they claim, because there is no attempt to aggregate different people's preferences. Instead, voters express, in commonly used words, the intensity of their preferences.

This raises a host of questions about the use and interpretation of expressions. Do all voters give the same meaning to words like ‘good’ or ‘acceptable’? Balinski claims that in practice it is valid to postulate the existence of common language. In judging figure skaters, for example, or wines, he believes that judges do in fact have a common language of evaluation. However, the periodic scandals at Olympic Games and other sporting events would let us believe otherwise.

Be that as it may, Balinski and Laraki got a chance to test their method at three polling stations during the French presidential elections in 2007. Neither of the two favourites, Nicolas Sarkozy on the right or Ségolène Royal on the left, received the best result. It was the third-ranked centrist François Bayrou who would have won. Sixty-nine per cent of the voters assessed him as being ‘satisfactory’ or better. Only 58 per cent said the same of Royal, and Sarkozy trailed far behind with only 53 per cent of the voters giving him at least a satisfactory grade. Similarly, Bayrou was rejected by only 7 per cent of the voters, while the rejection rate for Royal was 13 per cent and a whopping 28 per cent for Sarkozy.

Meanwhile, Eric Maskin from the Institute for Advanced Study in Princeton (Nobel Prize in economics 2007) and Partha Dasgupta from Cambridge University take a more practical approach. Given that there is no election method that works well under all circumstances, they ask which voting rule works well under most circumstances? Their answer is … our beloved majority rule. Specifically, they prove mathematically that if any voting rule works well for some set of candidates and electors’ preferences, then majority rule will work well, too. Conversely, for any voting method that is different from majority rule, there exists some set of candidates and electors’ preferences on which it does not work well but majority rule does. Thus, majority rule is essentially the unique voting rule that works well in most cases. It is, in this sense, the most robust voting rule.


  • Balinski, M. and Laraki, R. (2007‘A Theory of Measuring, Electing and Ranking’Proceedings of the National Academy of Sciences USA104 (21)87205
  • Dasgupta, P. and Maskin, E. (2008‘On the Robustness of Majority Rule’Journal of the European Economic Association6 (5),94973.
  • Saari, D. G. (2008‘Mathematics and Voting’Notices of the American Mathematical Society55 (April), 44855.