Since I recently finished reading Donald G. Saari's wonderful book "Chaotic Elections - A Mathematician Looks at Voting" (published by the American Mathematical Society), I decided to give a short example of what goes wrong in elections, so you'll know how voting paradoxes influence our lives and why you should know something about it. This is about Germany, but I tried to design the example such that you don't have to know anything about Germany to understand it.

Look at the data from 2009's national election in Germany for the Bundestag, which is the lower house of parliament in Germany:

Party	% votes	(biased categorisation of parties)
CDU/CSU	33.8	christian-democratic
SPD	23.0	social-democratic
GRÜNE	10.7	sustainable/social
FDP	14.6	liberal
LINKE	11.9	socialist
other	6.0	(*)

(*) in Germany, only parties that get at least 5% are considered for a seat in the Bundestag (with very few exceptions). There are numerous parties that didn't reach this limit and they are subsumed under "other" and take 6% of all votes together.

As you can see, no party has more than 50%, not even closely. That's why parties have to join together in a coalition, that reaches the 50% limit together. Technically, it's possible that three parties join together in a coalition (then it would be effectively impossible to change politics via voting). It has happened just before the 2009 elections that the two biggest parties (CDU/CSU and SPD) formed a coalition. But that has not happened again, and as it's usually the case, a small party joined a big party in coalition (FDP and CDU/CSU).

Maybe you notice that FDP and CDU/CSU don't get 50% but only 14.6+33.8=48.4% of all votes. Well, this is correct (although you have to keep in mind that we are only talking about votes, not about people who are allowed to vote or inhabitants. I'm providing more numbers for this issue below). As I noted in the table above with a (*), the votes for parties that don't reach 5% are excluded from the election counting. Yes, that's true: If you vote for a party that's unlikely to reach 5%, your vote is lost and your opinion who should be in the parliament doesn't count any longer. But there are even more issues like that, so let's wait for a moment and look at the table of votes given, excluding the "other" parties:

Party	% votes
CDU/CSU	36.0
SPD	24.5
GRÜNE	11.4
FDP	15.5
LINKE	12.6

Now FDP and CDU/CSU have 15.5+36.0=51.5% of all votes. What about other coalition options? Just looking at the numbers, not politics, CDU/CSU+SPD would have been possible, too, with 60.5% of all votes. A coalition of three parties, FDP+SPD+GRÜNE would have collected 51.4% of all votes, so this would have been a possibility, too.

(comic licensed from Randall Munroe under a Creative Commons Attribution-NonCommercial 2.5 License)

Voting systems are designed (or at least, some people pretend they are) to give those power who are in favour of the masses. But what do the people really want? One (silly) way to look at the numbers would be: the more % a coalition has, the more people will be happy about it. Take another look at the numbers: in a coalition of CDU/CSU and FDP, there are 36.0 votes for CDU/CSU, thus 64.0 votes against CDU/CSU, and there are 15.5 votes for FDP, thus 84.5 votes against FDP. This means, strictly thinking about the numbers, that 64% of all votes are against this coalition.

You might say now: "maybe the votes for FDP have had CDU/CSU as their second choice?" and this is the right direction to look at. We're not counting second or third choices now and this poses several problems. If we'd ask the 6% of voters who were ignored because they've voted a small party, what their second choice would be - it may change the outcome of the elections. Imagine that half of the other-party-voters would have chosen the LINKE as second choice (this is an estimate which might be close to reality) and the other half is distributed like the other votes. Look at the changed vote table then:

Party	% votes
CDU/CSU	34.8
SPD	23.7
GRÜNE	11.0
FDP	15.1
LINKE	15.4

You see that FDP+CDU/CSU have now 15.1+34.8=49.9% of all votes. This is close, but not equal to 50%, so they could not form a coalition. This is just a little example to show what's odd in voting systems.

Maybe you're sitting in front of you computer, muttering "of course different voting systems lead to different outcomes, what's wrong about it?". In my personal opinion, a voting system should be designed in a way that the common wishes of the group are expressed in the result. Interestingly, mathematics tells us that this is impossible in many cases, especially if there are many options for a vote with a close outcome. The current voting system in Germany encourages strategic voting, which is voting a party (more generally an option) you don't prefer most, knowing that the mathematics add up so your real preference will win more likely because of your vote. It's even possible to weaken a candidate by voting for him. In the interesting Bush/Gore presidential election in the US, it's pretty sure that Gore would have won if Nader (the third candidate) wouldn't have been an option to vote for.

From the book "Chaotic Elections" you can learn that amongst all positional voting procedures, the one the least manipulatable by strategic voting is the Borda Count. For this procedure, you have to ask the voter about his second choice, and even his third, fourth (and so on). This sounds complicated and I don't think it's feasible yet for national votes. Maybe it suffices if we require to name a second choice. It does not suffice to open the possibility of naming a second choice - people don't use this opportunity very often (this has been seen in experiments). That's because almost nobody knows about voting paradoxes and how to create and avoid them. So we all are very manipulatable if we don't learn how elections work, mathematically. However, manipulating elections by choosing a specific voting procedure or by strategic voting is not the only method of manipulation in democratic settings: imagine wrong statistics (or even no statistics) provided by bureaucracy or manipulative nomination, which includes strategies like "I propose something very close to what everybody likes, but I bundle it with something everybody hates - then nobody will vote for it", which work if similar proposals are gathered to facilitate voting (in particular, the german e-petitioning system is vulnerable to this attack).

Some more numbers of general interest:

• total population, in millions: 81.8
• people allowed to vote, in millions: 62.2
• people who actually voted, in millions: 44.9
• people who did not vote but were allowed to, in millions: 17.3
• people who did not vote, including those not allowed to: 36.9
• percentage of actual voters in total population: 54.9
• percentage of people who voted for either CDU/CSU or FDP
  in total population: 26.6
• percentage of people who voted either CDU/CSU or FDP
  in population allowed to vote: 34.9
• percentage of people who didn't vote either CDU/CSU or FDP
  in population allowed to vote: 65.0

the most interesting number, for me, is the last one. It indicates that 65% of all people that are allowed to vote could have changed the outcome of the vote if they would have had been asked for their second choice. Maybe asking them (and counting with Borda Count) would have shown that CDU/CSU+FDP was a common choice - maybe we would have had a completely different parliament in Germany now. It's always questionable why 17.3 millions of Germans didn't vote although they've had the right to do so. Maybe they knew that they couldn't vote for their favourite party and change politics at the same time.

(comic licensed from Randall Munroe under a Creative Commons Attribution-NonCommercial 2.5 License)

I looked up the numbers used here from different statistical services and newspapers. All numbers are from 2009. Every number is either checked twice or calculated by myself. But beware: those are statistics, you're not supposed to trust them!

There is a review about the book here, if you're interested now.

There are many, many other voting paradoxes which can happen and affect our life, in every situation involving a democratic process with more than 2 options (**), which happens quite often. I recommend you to read something about this subject, maybe something shorter or better digestible than Saari's book - but if you're interested in politics, social phenomena or mathematics (or anything in-between), I guess it's best to take a look at Saari's book. It has parts which require you to understand some basic calculus, but in Germany you learn all this in the last years of highschool. Even if you ignore all calculations, the basic problems are well explained. I myself took some calculations for granted, since I began to trust the author after some examples turned out to be correct :-)

The most provocative quote from Saari, from page 100 of the book:

For a price, I will serve as a consultant for your group for your next election. Tell me who you want to win. After talking to the members of your organization, I will design a 'democratic procedure' which ensures the election of your candidate.

and after reading the book I have to admit that it's not a big challenge. Once you know the voter's preferences with a certain probability, you're able to design some voting system that "chooses" your favourite option. That's why most democracies tend to stick to one voting procedure instead of changing it for each election (and gerrymandering is a popular example of such manipulations).

(**): Little exercise: why aren't there any voting problems or paradoxes when there are only two options?