Tuesday, March 8, 2011

Its All Fun And Games Until Someone Loses a Vote

"It is enough that the people know there was an election. The people who cast the votes decide nothing. The people who count the votes decide everything." - Joesph Stalin

Watching the recent elections here in Ireland was fascinating. The coverage by our national broadcaster was excellent as it provided hours of drama as old political careers were destroyed and new ones started all in the glare of live television. Here in Ireland we use proportional representation with a single transferable vote. It is considered a fair system but as often is the case it makes it very difficult to avoid coalition governments. Watching how it all played out as the votes were tallied I couldn't help think about a book I read last year "Prediction" by Bruce Bueno De Mesquita. In his book Bruce describes with the aid of game theory how we can forecast and even engineer events.

In this book he gives various examples from his work about how he used game theory to predict foreign political events for the U.S. Government. However the story that sticks out for me is a very simple one. He was hired to help a retiring CEO. The CEO did not have a favourite in mind to replace him but he did know who he didn't want. However this person was the favourite among the board. Bruce was hired in secret to engineer a result in the up coming vote of the board to choose a successor. He couldn't for obivious reasons rig the vote as it had to appear to the board to be a fair vote. So how do you change the system to enigneer the outcome but on the surface appear fair?

The board consisted of 15 members which could all vote. Of the 15 members 5 canidates were put forward, Anthony, Barry, Claire, David and Ellie. Anthony is the popular choice among the board members but the CEO doesn't want him. Instead he wants David to win. The key to solving this is information. First we must gauge how each member of the board ranks each canidate. Luckily in this case the CEO knew his board members well so could provide a good estimate of how each member ranked the canidates. Armed with this knowledge Bruce could divide the board in certain voting blocks.

There were five distinct blocks, each containing three members. There prefrences from most to least as follows
  • Anthony, Claire, Barry, David, Ellie
  • Anthony, Ellie, David, Barry, Claire
  • Ellie, Anthony, David, Barry, Claire
  • Claire, Ellie, David, Barry, Anthony
  • Barry, Claire, David, Ellie, Anthony
In a straight forward one person one votes scienero Anthony would get 6 votes, Ellie, Barry and Claire 3 votes and David  0 votes. This exactly what the CEO wanted to avoid. So maybe a different system might help. One system would be the Borda count system. In this system each voter ranks their preferences in order. The number one preference in the case above gets 4 votes, the second 3 votes, the third two votes, the fourth one vote and the last vote 0 votes. Using the above preferences for each voting block the canidates now rank as follows;

  • Anthony      33 votes
  • Ellie             33 votes
  • Claire          30 votes
  • Barry          27 votes
  • David          27 votes
With a tie for first there would be a run off in which going by preferences Ellie would win. Poor old David doesn't stand a chance in this system. Also if the CEO were to change the voting system to such a degree it would raise suspicions that something wasn't quite right. To solve this problem a slightly different approach is needed. 

If we inspect the preferences we can see that in fact there is a solution in which we can get David elected in what would appear to be a fair system. The system used will be a head-to-head elimination system. To the board the two main candidates were Anthony and Ellie. So it would seem fair to ask the board to choose either Anthony or Ellie in a straight vote. The winner of this vote would than be placed against one of the remaining candidates until there was only one person remaining. Seems fair right?

In the first vote Ellie would beat Anthony 9 votes to 6. This is due to the above preferences in block 1 and 2 Anthony is favoured while in blocks 3,4 and 5 Ellie is favoured. Now the the next step is very important, who do you put up against Ellie? The CEO was very happy at this stage as Anthony would not be the new CEO but he still wanted to see David take over. Next Ellie and Claire would face off against each other. In this contest Claire would win 9 votes to 6. Now at this stage both favourites had been eliminated the CEO was very happy. 

In the third contest Claire would face Barry. Again returning to the preferences we see that Barry would win by 9 votes to 6 as he had the support in blocks 2,3 and 5 over Claire. This leaves us with the finally contest Barry against David. Using the above preferences we see that David has the support of blocks 2,3 and 4 making him the winner over Barry by 9 votes to 6. David was elected as the new CEO to the delight of the retiring CEO. To the board members it seemed a fair and open contest, no secret ballots, all they were asked to do was to pick between two people until all were eliminated bar one. This system was anything but fair.

  • Anthony versus Ellie: Ellie Wins
  • Ellie versus Claire: Claire Wins
  • Claire versus Barry: Barry Wins
  • Barry versus David: David Wins!
This was only possible for two very important reasons. One, the retiring CEO could provide enough information to properly understand all the possible outcomes of this scenario. Two, analysis of the preferences indicted that the voting was circular in nature, it could be setup in such a way to make any of the candidates the winner. Lesson to be learned here just because it looks fair doesn't mean it is.


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