Olympics host election method

In the past October 2nd we witnessed the choice of Rio de Janeiro as the host of the forthcoming 2016 Olympic game, after defeating Chicago, Tokyo and Madrid. The loser cities were eliminated in that order in a poll system that deserves to be analysed.

In this system, the election is made in up to 3 stages until a city gets more than half of the votes (which is usually the last 1 on 1 round).

Among 94 valid votes in the first round, Chicago was the first to be out with only 18 votes. Madrid had the majority of the votes with 28, followed by Rio with 26 and Tokio with 22.

In the second round, the members of the COI representing the country of the losing city are entitled to vote, so there were two more voters.

The valid votes were 95 and the result was Rio 46 (+20), Madrid 29(+1) and Tokyo 20 (-2), the latter being eliminated. As we dont' know what happened with the invalid votes we cannot be sure of what exactly happened in this round, but it seems that all Chicago votes moved to Rio.

In the last round, with 98 valid votes Rio's candidature overwhelmed Madrid's by 66 to 32, again showing that almost all the votes of the losing city (Tokyo this time) moved to Rio.

The "political" reasons for this are already been argued in mass media across the world (at least in the candidate's countries), but we would like to have a further look on the election method.

For what we saw in this election Madrid was 'least preferred' among the 4 cities by roughly more than two thirds of the voters, and 'most preferred' among almost the other third. We can know that for the evolution of votes. In the first round, each voter chose its most prefered city. If that city passes that stage, it's supposed that he will keep that vote (not always true as seen in the real election, but mostly true). If the city is eliminated, he will vote in the next round for his second alternative. If that second alternative advances to the final, he will keep the vote, but if the city fails, the voter would chose its third option for the final.

When we have a situation like this, with a city (let's call it a cult city) with so many voters entirely against (4th preferred), but enough voters with total support, the election is condemned to have a final between the cult city and the survivor of the other two rounds. It could be argued that this is ok, as that city is the preferred among the three, but this resulted from an election where the voters of the cult city, not knowing the preference of the other voters, didn't express their opinion about the other three, because they were supporting their city.

Argentineans can remember a similar situation that arouse in the presidential elections of 2003, in which 4 candidates fought in the first round with almost equal votes (19%-24% each), but in the ballotage for defining between the two most voted candidates, the candidate who had obtained the majority of votes in the first round, had to give up and didn't even contended because polls were revealing that he wouldn't get no more votes than in the first round, giving his adversary more than 70% of votes.

Coming back to the Olympics election, an inmediate system that comes to our minds could be a negative or censorship method, in which voters choose which city wants to discard in each round. In the case of the last election, if this method was applied, Madrid would have been pulled out in the first round, and their unconditional voters would have helped to decide among the other three, with possibly a different outcome.

In the last election, it doesn't seem that the winner would have changed with the method, but at least, the final decision would be a much closer one. Let's play a little (not being rigorous with the figures) :

First round:
Madrid 66. We cannot know which were the preferences of the 33 voters for Madrid, and it was said that Madrid supported Rio as second chance, but let's asume 1/3 for each city.

Chicago 11, Tokyo 11 and Rio 11. Madrid is eliminated

Second round:

Now the game turns a more strategic one, and some voters could chose to eliminate not their least preferred, but also their second, as they fear that in an eventual final, this would be a stronger candidate. But let's keep that game theory out now, and assuming that voters don't have much information about the other voters, they will just vote for their least preferred.

In the election we saw that Chicago voters decided for Rio before Tokyo, so, all the 18 original Chicago votes would go to eliminate Tokyo. We don't know the preferences of Tokyo and Rio, so we will split their votes. (in parenthesis is the city that the voters chose in the first round of the real election)

Tokio gets: 11 (Madrid) + 18 (Chicago) + 13 (Rio) = 42

Chicago: 11 (Madrid) + 13 (Rio) + 11 (Tokio)= 35

Rio: 11 (Madrid) + 11 (Tokio) = 22

Tokyo is eliminated. (of course, taking into account the strategic view, normally Chicago would have tried to vote Rio, and Rio would have tried to get rid of Chicago)

Final Round:

We can now put things positive.

We have 22 votes for Chicago (those that already chose to kick out Rio) and 35 for Rio (those who voted to kick out Chicago). The remaining 42 votes are originated with voters that chose in the actual election Madrid, Chicago and Rio. We don't know about Madrid's people, so we assume again equal split.

Result:
Positive for chicago= 22 + 5 +18=45

Positive for Rio=35+5+13=53

Again, Rio wins, but it's a much closer election, and the preferences of more voters are represented.

This should be more deeply analysed, and of course it could be possible to find a different example of preferences of the voters in which the current method gives better results. This is because in each round, voters are asked to express only a piece of their preference order. The election could be sophisticated by asking each voter to initially express his full ordering and having a method to choose the city that """"""""" (multiple quotes) maximizes 'social' utility """"""