If voting cannot be perfect, let us make it at least simple

“Voting theory distinguishes between many different voting procedures, but in everyday life majority voting is the most common. We receive a list of candidates and we give a vote to the one we like the most. We use it in different situations, almost on a daily basis in our lives” – says Attila Tasnádi, talking about the most direct impact of voting theory on our daily lives. Together with fellow authors Dezső Bednay and Balázs Fleiner, they compared different voting methods and came up with some surprising results. The conclusions, summarized in a workshop paper entitled Which social choice rule is the most dictatorial?, were presented at the XXXV Hungarian Conference on Operational Research held at the Corvinus University of Budapest at the end of June. 

Borda-Condorcet debate 

There are written records of several different voting methods from as early as the 11th and 13th centuries. The birth of the theory of voting and the first analyses of the different methods are linked to the debate between an engineer, Jean-Charles de Borda, and a mathematician, Nicolas de Condorcet, when the former was  elected President and the latter Secretary of the French Royal Academy of Sciences at the end of the 18th century.  

Borda suggested that the obvious voting method would be for each voter to rank the five available candidates, with the candidates receiving points based on their rankings.  

Condorcet argued that this method could be manipulated, as the voter has no interest in drawing up the “true ranking”, but instead wants his/her own favourite to win and will play around with his/her list. “Borda responded by saying that “Borda responded by saying that his scheme was intended only for honest men”, notes Attila Tasnádi. 

The method proposed by Condorcet meant deciding pairwise matches by voting. Points are awarded for each victory in a pairwise contest, and at the end the points are added up and the candidate with the highest total number of points wins. “Condorcet’s method was also criticised, because in some cases it may not yield a winner (i. e. when participant A beats participant B, B beats C, but C beats A – ed.). Since then other methods have been developed”.  

God help Maine when mathematics reach for her! 

From the point of view of the history of science, the debate was finally settled and neither of the two proved to be entirely right, since another researcher, Kenneth Arrow’s thesis, stated that there is no such thing as a voting procedure that satisfies all the requirements. Rather than looking for the perfect voting method in every respect, voting theory aims to examine certain criteria. 

Arrow’s theorem approached voting procedures with axioms – it is a normative approach, and says that one should not differentiate on the basis of the outcome of a vote, but the characteristics of the vote should determine the method that is to be used. Examples of such characteristics are voter anonymity and neutrality, i.e. the order in which the candidates are listed (e.g. on the ballot paper). 

Apportionment is also a characteristic used in voting systems, which means that votes cast are converted to mandates using a certain ratio. In the 19th and 20th centuries, the very issue of apportionment was the subject of a long debate in the United States, as the size of the House of Representatives was constantly changed. In the case of the State of Maine, the number of their delegates to the House of Representatives varied from 3-4 due to apportionment, which resulted in the saying “God help Maine when mathematics reach for her!”  

Attila Tasnádi believes that the normative approach advocated by Arrow helps to filter out logical inconsistencies, moreover the impossibility theorems warn researchers not to develop excessive expectations of voting methods. 

When voting involves a large number of people, it should be done simply 

Following Arrow’s conclusions, Attila and the other authors were not comparing voting procedures to find the perfect method, but were more interested in their effectiveness. They created objective functions to measure quality, and found that for large numbers of voters, the different methods converge to the same outcome, i.e. there is no difference, whichever is applied among them. 

“There are often informational arguments in favour of simple voting procedures, and our study shows that their use is indeed justified for country-wide elections, as it is easier for voters to choose the best candidate than to rank or allocate scores,” explains Attila. 

The paper has mot been published yet, it is currently being considered for an international journal on economics, but Attila also notes that this is a field where the lead time is typically long. For the time being, they are generalising their results based on initial feedback and will present their study soon after making changes. 

What does science have to do with Google’s search algorithm? 

Voting is not only used for political votes in everyday life, sometimes we don’t even realise the many areas in which we encounter it. “Voting is one way of making decisions in the community. We also vote when we fill in a questionnaire to say what we think public money should be spent on. This is what happens when we decide at the residents’ meeting that the house needs a ramp or the roof needs repairing,” explains Attila Tasnádi.  

At the FIFA Ballon d’Or voting, sports journalists still use Borda’s method to rank the best players. “Voting is everywhere and sometimes we don’t even notice it”.  

Voting theory is also gaining ground in the IT sector, as the theory of search and recommendation algorithms has become more accurate owing to research in this area. The PageRank algorithm has enabled Google to produce much better results than previous internet search engines. Attila considers that the heart of the algorithm comes from scientometrics, which concerns itself with identifying the most cited and therefore the most successful journals. The algorithm started to look at the same thing, more specifically the websites that are most frequently linked to on the internet. In a similar way to the way we read and browse through citations to other journals in a journal, or links to other pages on a website.  

However, we may drop a completely new journal from the shelf, or move to a different webpage without following the links. According to this thought, a random factor was included in the algorithm, namely that there is a 15 percent probability of being transported from one website to another. This will result in a more complete hit list, as the user is not stuck in a local search range, and the algorithm allows the search to be performed over a slightly wider spectrum. Although Robin Li had introduced this in the RankDex algorithm before Google, Google secured the patent and thus the world success. The Chinese search engine Baidu is, however, based on the RankDex algorithm, notes Attila Tasnádi, professor at Corvinus University, on an important application of the results of voting theory.