The tragedy of democratic voting systems

A common issue in participatory/democratic infrastructures is the problem of aggregating individual preferences into a global one. Voting is often the most public and visible example of mass collective decision-making. Such problem of aggregation occurs in many different disciplines: in decision theory , in social choice theory & welfare economics and in artificial intelligence. In the case of voting systems, the problem is how votes are counted and aggregated to get an acceptable final result, how choosing such aggregation function? Understanding the advantages and drawbacks of each proposed system is thus crucial in democracy and has justified a quantity of theories, called Voting theory, leading to many discovered paradoxes and surprises.

What is a good voting system?

Before investigating this topic, I didn’t imagine the diversity of voting methods available (did you know Lewis Caroll,also mathematician, was the inventor of a “unconventional” voting system called asset voting, in which both the voters and the candidates participate?). Whatever is the purpose of the vote, a stupid online poll, the Oscar for the best movie or the presidential election, you have to be aware that all these systems are not perfect. But let’s start by identifying what could be an ideal voting system. To define potentially desirable properties of voting systems, several ‘fair’ criteria have been proposed. Here are some traditional ones:

  1. Non-dictatorship: no single voter possesses the power to determine the group’s preference.
  2. Independence of irrelevant alternatives (IIA): The introduction of a candidate should not should not change the existing relative order among other candidates.
  3. Unanimity (or Pareto efficiency): If everyone prefers A to B (A>B), then society should prefer A to B.
  4. Universality (or Unrestricted domain): requires that a social choice function always produce a ranking. no ties is admitted (If the system doesn’t produce a single winner, it isn’t very useful).
  5. Condorcet criterion: if a candidate beats every other candidate in pairwise comparison, this candidate should win.
  6. Consistency: if  voters are separated in n parts and, in each part, the result is the same choice being selected, an election of the entire electorate also selects this choice.

You can find other criteria in Wikipedia, such as simplicity, speed of vote-counting, robustness to fraud or tactical voting. We will also see later that new criteria like the “bayesian regret” i.e. the expected avoidable human unhappiness is, for some, the “gold standard” to compare single winner methods in real world situations.

The (fake) lesson of the Arrow’s Impossibility & Co

Taking individually, these properties seem to be intuitively compelling but integrating all of them in a voting system is mathematically impossible due logical incompatibilities. Economist Kenneth Arrow proved an impossibility theorem [2] in the 50’s, which demonstrates it is impossible to design a voting system that satisfies all the first 4 criteria (universality, non-dictatorship, unanimity, IIA) in the case of strict ordering preferences by individuals (e.g. voter’s preference is candidate A> candidate C> candidate B). The Gibbard–Satterthwaite theorem provides similar results for the case of a single winner method. Furthermore Young and Levenglick (1978) have established the incompatibility of the Condorcet and consistency properties. Participation (none of the voters is disillusioned by submitting his true ballot) has also been shown in Moulin (1988) to be incompatible with the Condorcet property.

Even if people will vote in a honest and rational way (which is not the case), aggregating their choices in a rank order always lead to the possibility of paradox. No method is totally free from paradox. If these previous theorems don’t borrow people trusting dictatorship regime (those who are ready to trust one “strong and visionary” man), it is quite an issue for partisans of democracy. So a more substantive way to argue for or against a particular election method would be to compare how frequently failures occur, under what conditions they occur, and how severe they are.

Furthermore we will see later that this popularized Arrow impossiblity theorem, overused to demonstrate the failure of democracy processes over the last 50 years, is limited to ordinal voting system (ranking based system) but irrelevant in the case of cardinal voting system (i.e. scoring based system)

The two party domination and the Duverger Law

The problem of not ideal system and emerging paradox is that it could lead to tactical voting i.e. voting not according to your preferences but for someone else to finally get a better result.

New Oscar Voting system [3] has been criticized due to possible tactical voting. To really understand the impact of a bad voting system on politics a good example is its effect on the two-party system. Why do you think lots of countries using plurality voting (each voter votes for only one candidate. The candidate with the most votes wins) like the US seem incapable of escaping a two-party political system? the culture? No…

French sociologist Duverger observed that the plurality voting method tends to favor a two-party system, whereas top-two runoff (It holds a runoff election between the two highest polling options in the first round if there is no absolute majority (50% plus one) and proportional representation tend to multipartism. This tendency for plurality voting to maintain two-party domination is so reliable that it has become known as Duverger’s Law [4]. A top-two runoff system (e.g. French presidential election system) differs considerably. To echo Duverger, most of the approximately 30 countries which use this system have escaped two-party domination [5]. because it is the result of voter psychology and the different tactical incentives at play. But Even the top two runoff has side effect. Which criteria failed? The IIA criterion, but also the Concordet criterion.

So Choosing the right voting system has deep impact: changing the voting system could lead to a new layout of the political landscape. And For lots of expert the plurality system is considered as the most problematic of all voting methods. Here are common side effects.

  • Concordex paradox [6]: Back in 1770, the Marquis de Condorcet discussed a method for the aggregation of preferences which led to the first aggregation problem. the transivity property (i.e. if A>B and B>C then A>C) at the individual level could fail at a collective level (called intransitivity of majority preferences) i.e. collective preferences can be cyclic, even if the preferences of individual voters are not.
  • No Show Paradox [7] arising when a voter is better off not voting than to get a better result.
  • The spoiler effect [8]: in close election, when a minority party candidate with no chance to win “takes” some votes from the similar big party leading to the winning of the dissimilar party.

Here a short video about a common strategy used for election for instance.

Sustainable method: Score voting

So the prospects of finding a best voting method have been disappointing, due to the negative results obtained through the systematic axiomatic analysis (Arrow Impossiblity) employed during the last half of this century. but some quite robust methods exist and have been advocated.

From Bees and to Sparta

İn term of sustainable method, Nature may have a thing or two to teach people about collective decision-making. For instance, Honeybees have been “voting” in single winner “elections” for 20-50 million years [9,10]. They’ve held far more elections than humans. Whıch election method do bees use? Each spring, half of worker bees leave with their queen to start a new hive (2000 and 20000 bees). The important decision is where to build the new hive? Among the bees that depart are scouts (5%) that search for potential sites within about 100 square kilometers. They usually find about 20 different options. To report back, each use a waggle dance to advertise the site and indicate the orientation of the location. The longer the dance, the more enthusiasm they are for this site (quality of the evaluation).

Other bees observe these reports and fly out to check the alleged sites for themselves. some bees who already have been scouts can choose to re-explore their own sites or the sites advertised by others. So after some time (max = 1 week), multiple “camps” of bee scouts emerge, each camp advertising different potential nest sites. The site with the highest average score i.e. average total dance-length among all the dancers for a site is the winner. Another interesting detail, the bees refuse to terminate an election until a “quorum” of at least 10-15 scout bees have explored a site, certainly due to preserve a minimal quality in each site-evaluation. About 90% of the time, the bee swarm succeeds in selecting (what appears to entomologists to be) the best one.

Such system is known in the human society as the “score voting” (aka range voting). Such voting system is based on an entirely different paradigm: ratings rather than rankings. Voters rate each candidate between 0-9 for instance. The scores for each candidate are summed, and the candidate with the highest sum is the winner. Sparta, the longest-lasting substantially democratic government in history, voted in a similar way from about 700 b.c.e. until at least 220 b.c.e. Spartans elected Gerontes and Ephors (council members who had the power to dethrone kings) by means of a shouting system. The candidate with the loudest support won.

Best method using the “Bayesian regret” criteria

In 2000, The mathematician Warren D. Smith performed an extensive set of computer calculations which showed the system working extremely well, even with high rates of tactical voting [11]. This is based on an objective “economic” indicator of voter satisfaction with (or “representativeness of”) election outcomes, called Bayesian regret [12] a statistical concept from economics that he transferred for voting methods, the “gold standard” for comparing single-winner election methods according to him. Range voting was the winner. Surprisingly this method also satisfies the 4 criteria (non-imposition, non-dictatorship, monotonicity, and IIA) and evade from Arrow’s theorem because it is a cardinal voting system (i.e. weight based system), while the “universality” criterion of Arrow’s theorem effectively restricts that result to ordinal voting systems (i.e. rank based).

Why do we vote? the paradox of voting

Another common mystery is: why do we vote?, why do we participate in large decision making? This question has been studied and debated with Turnout. Tracing the connections between participation in small groups and large-scale participation causes a difficulty for rational choice theory.

The failure of rational voting theory?

Much of the literature about The logic of collective action focuses on the explanation of varied social actions and outcomes, including spontaneous actions, social norms, and large institutions. Studies using game theory, which takes into account the ability of voters to interact, have also found that the expected turnout for any large election should be zero. The larger the population is, the lesser weight one individual will have as voter on the global result. This is one of the most notorious failures of the rational choice literature.

Rational ≠ from selfish

However, the literature on voter turnout has tended to associate rationality with selfishness.Traditionally, rationality is understood as a sort of cost-benefit analysis and utility maximization but implicitly refer to direct benefits to the voter. For a selfish voter, the expected benefits from being pivotal during the election vanish as n grows. But Fowler [13] broke the link between selfish preferences and rationality—two assumptions that have usually been linked but that have no logical connection— and shown that voting in large populations is perfectly rational. He found that voters are more likely than nonvoters to behave altruistically (as is consistent with the social-benefit utility model) and to display delayed-gratification behavior. As a result, as population size grows, an individual may change his or her vote and begin to vote for the social good instead of the individual good. These experimental findings linking turnout to altruism, patience, and party identification have the potential to unify psychological and political explanations of participation.

Genetic influence?

Another ınterestiıng study from Fowler [14] finds that genes significantly affect variation in voter turnout, shedding new light on the reasons why people vote and participate in the political system.In conducting their study, the authors examine the turnout patterns of identical and non-identical twins—including 396 twins in Los AngelesCounty and 806 twins in the National Longitudinal Study of Adolescent Health. Their findings suggest that 53% of the variation in turnout can be accounted for by genetic effects in the former, with similar outcomes in the latter. According to Fowler, “we expected to find that genes played some role in political behavior, but we were quite surprised by the size of the effect and how widely it applies to all kinds of participation.”

Conclusıon: The fog of war & saved lifes

The world is facing to several crises (e.g. species collapse, overpopulationand climate change). Democracies is an engine of national wealth and prosperity.Hardly known, however, is that freedom also saves millions of lives from famine, disease, war, collective violence, and democide (genocide and mass murder). Summarizing his lifetime of award-winning research into war, democracy, and genocide Rudolph J. Rummel [15] concludes with these two quotes:

  • To eliminate war, to restrain violence, to nurture universal peace and justice, is to foster freedom (liberal democracy).
  • The more democratic two regimes, the less likely violence between them; the more democratic a regime, the less its overall foreign violence; and the more democratic a regime, the less its genocide and mass murder (which in this century has killed about four times the battle dead of all its foreign and domestic wars).

In view of that, the world needs to make good decisions and have real democracy voting system. As we saw before the voting system could change completely the political landscape (e.g. dominated by two party system for countries like USA using plurality voting system). Warren Smith calculated that by adopting score voting system in US and thus making better decisions, it will lower the risk of a 2-billion population crash in 50 years, by 5%. That is, in expectation, 100,000,000 lives saved in 50 years.


(why usıng old school citation index instead of using hypertext in the text? to avoid serendipity behavior and stay focus on the article)

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