Voting and Markov Processes Andrew Nicholson Department of Mathematics The University of North Carolina at Asheville One University Heights Asheville, NC 884. USA Faculty Advisor: Dr. Sam Kaplan Abstract Proceeding of the National Conference On Undergraduate Research (NCUR) 5 Virginia Military institute Washington and Lee University Lexington, Virginia April 1-3, 5 This project involves the study of stability and paradoxes in voting methods. For example imagine an election with candidates A, B, and C that among voters A is preferred to B, B is preferred to C and C is preferred to A. When this occurs which candidate wins depends on which voting method is used (such as the Borda count, plurality method, antiplurality method, run off elections, etc). This project aims to identify paradoxes and to reduce the number of paradoxes in voting. When paradoxes arise Markov chains may be created to choose a winner. In the study of voting methods, Markov processes are an ideal model for elections. A Markov chain encodes the strength of preference for each candidate relative to the others. Standard methods from Linear Algebra are employed to analyze which candidate is most strongly preferred. A Markov voting process is not immune to paradoxes, however. We carry out a comparative study between this new voting method with more standard methods and identify under what circumstances the Markov process is superior. Keywords: Voting Theory, Markov Processes 1. Introduction The inquiry into the mathematics of voting began with J. C. Borda in 177. Borda wondered if the French Academy of Science (FAC) was actually electing new members whom the current members really desired. The French Academy of Science used the one-person one-vote method, or the plurality method, in their elections for new membership. Borda disliked the plurality method because there are situations where the plurality method does not seem to show the views of voters. For example, suppose three candidates A, B, C, -- are running for membership in the FAC collect the following votes from fifteen voters. Number of Votes Table 1: Example 1 Voter Preferences 6 A B C 5 C B A 4 B C A
Where denotes preferred to. In the election outcome above voters rank candidates according to preference with each preference called a voter profile. Note that for the three candidate case there are 3!, or 6, voter profiles. By the plurality method candidates A, B and C would receive 6, 4, and 5 points respectively. Then the election would have A C B. However, if we look more closely it seems that 9 out of the 15 voters preferred candidates B and C to A. If this is the case then is candidate A really the candidate whom voters prefer? 1.1 The Borda Count Because of his mistrust for the plurality method, Borda created his own method, the Borda Count. The Borda Count is a positional voting method or a method where voters rank candidates in order of preference. For example, the election above used a positional voting method. The Borda Count allocates, 1, points to a voter s first, second and third place candidates. In the election above candidate A, B, and C would receive 1, 19 and 14 points respectively. Thus, by the Borda Count, the election outcome would become B C A. Notice that the Borda Count reverses the election outcome of the plurality method! The point system used for the Borda Count above is just one example of a Borda Count. The general Borda Count is a normalized voting vector, denoted as w λ. w λ = (1, λ, ) where λ 1 (1) Notice that we can get the plurality by simply setting λ= in (1). Furthermore, after normalizing our original Borda Count, where we assigned points of, 1, and, we see that λ=1/. The startling result is that by changing our value for λ from to 1/ we reversed out election outcome in Table 1. An infinite number of values for λ exist and if the election outcome can change by choosing different values for λ an important question comes to mind. Which value of λ is best? 1. The Condorcet Method A contemporary of Borda, Marie-Jean-Antoine-Nicolas de Condorcet, disliked the Borda Count because it is uncertain on which value of λ is the best. In fact, it seems that the best value of λ is the one which gives victory to your favorite candidate. Instead of the Borda Count, Condorcet championed a method which pits candidates against each other in pairwise elections. Surprisingly, Condorcet s new method is called the Condorcet method. The candidate who defeats all other candidates in pairwise elections is called the Condorcet Winner. In the election depicted in Table 1 we can use the Condorcet method to decide upon a winner. First we pit the candidates against each other in pairwise elections. For the pairwise election between A and B, A defeats B nine votes to six. We say that B A in the pairwise vote A vs. B. In the pairwise elections A vs. C and B vs. C, C and B win the pairwise elections respectively. Because B defeats A and C in pairwise elections we say that B is the Condorcet winner. We can depict this election outcome with the diagraph below. Figure 1: Digraph of a Condorcet Winner The winner of a pairwise election between two candidates has an arrow pointing to it. For example, in the pairwise election between candidates A and B, B is the winner thus there is an arrow points from A to B. Notice that the Condorcet method disagrees with the plurality method on who won the election. In this election it seems reasonable to accept the Condorcet winner over the plurality winner because of the uncertainty on which value of λ is superior.
It is natural to wonder if there are any problems with the Condorcet method. Unfortunately it turns out that there is one major problem with the Condorcet method. In some election outcomes there is no Condorcet winner! These election outcomes are called cyclic election outcomes. The winner of a cyclic election outcome is the candidate voted upon last using the Condorcet method. An example of an election with a cyclic election outcome is illustrated below. Table : Cyclic Election Outcome Number of Votes Voter Preferences 4 A C B 3 B A C C B A In the pairwise elections A vs. B, A vs. C, and B vs. C, the winners are B, A, and C respectively. Notice that no candidate beats all other candidates in pairwise elections. There is no Condorcet winner and we have a cyclic election outcome. The diagraph of the cyclic election outcome depicted in Table is shown below. Figure : Diagraph of no Condorcet Winner Furthermore, the candidate voted upon last in pairwise elections is the winner. For example, if the first pairwise vote were to be A vs. B, the second A vs. C and the third B vs. C, then C would be the winner. However if instead the first pairwise vote was to be B vs. C, the second A vs. C and the third A vs. B then B would become the winner. It seems that both the Borda Count and the Condorcet method have certain unavoidable flaws. There are an infinite number of values for λ to choose from, some which give victory to different candidates. Also there are certain election outcomes in which there is no Condorcet winner. The failure of both of these voting systems raises the question: Is there a better method for choosing a winner?. The New Method The Russian mathematician Andrei Markov developed a mathematical model called a Markov chain which can be used to predict the outcome of a dynamical system. A Markov chain is a sequence of probability vectors in a stochastic matrix, a matrix in which all of the columns sum to unity 1. If we raise this matrix to higher and higher powers a feel for the long term behavior of the system can be determined. The limit of the Markov matrix M is lim M n = L () n Where n exists in the natural numbers and L is the unique vector that is both the eigenvector for M with eigenvalue λ=1 and whose components add to unity. In other words, for a Markov matrix you can compute the long term behavior with the following procedure: 3
1. Compute the eigenvalues of the stochastic matrix M, call them λ.. Compute the eigenvector with eigenvalue λ=1, call it V. 3. Normalize V and call it L. (In this setting normalize means to divide V by the sum of its coordinates.) The vector L is the limit of the Markov matrix and gives the long term behavior of the system. When applying Markov chains to an election outcome it can be useful to use a digraph model. Returning to Figure 1, our diagraph model is reproduced below. Figure 1: Digraph of a Condorcet Winner One can interpret the digraph above as each candidate being in a room connected by corridors that can only be traveled through in one direction shown by the direction of the arrow. Then one can imagine votes starting in each room moving randomly from one room to the other as time progresses. For example, say a vote starts in candidate A s room. The vote can then move to either C or B. If the vote moves to C it will then move to B and stay there, for there are no corridors leaving from B. Obviously, if the vote moves to B then it will remain in B for all time as well. As time progresses all votes will end up in candidate B s room, thus B is called the Markov winner. 3. Main Theorem The fact that candidate B from example 1 is both the Condorcet winner and the Markov winner is not accidental. It turns out that if there is a Condorcet winner, then it is also a Markov winner. Proof: If there is a Condorcet winner, then it is also a Markov winner. Suppose there is an election with n candidates with a Condorcet winner. Call the Condorcet winner A. Then in the digraph of the election, there are n-1 arrows points to A from all other n-1 candidates. The probably of a vote to move to A is at least 1/n. Figure 3: Probability of a vote to move to candidate A 4
Now call the set of n-1 candidates Ω. Then our diagraph simplifies to: Figure 4: Simplified figure Where P is the probability of a vote moving to A. The stochastic matrix of the digraph, M is: M From To = A Ω A Ω 1 p (3) 1 p Next, we compute the eigenvalues of M. MV = λv (4) Where λthe eigenvalue of M and V is is the eigenvector. Continuing with our analysis: 1 λ p = 1 p λ (5) + λ( p ) + (1 p) = λ (6) λ = 1, 1 p (7) Using our eigenvalue λ=1 to compute our eigenvector V: 1 p v 1 p v 1 v = 1 v 1 (8) This implies that the components of our eigenvector V: v = 1 pv v1 + (9) This implies that: V 1 A = Ω (1) 5
Notice that V is already normalized implying that V is indeed L. Furthermore, L shows that candidate A is the Markov winner. Therefore if there exists a Condorcet winner, then it is also a Markov winner. QED. 4. Draw Backs with the Markov Method The Condorcet winner is also a Markov winner, but what about when there is no Condorcet winner? For this analysis return to the diagraph expressed in Figure. Figure : Diagraph of no Condorcet Winner Notice that as time progresses votes will never become trapped in any candidates room. However, perhaps a sense of how much time votes spend with each candidate can be attained through Markov chains. By computing L one can sense how long votes are spending with each candidate. In this case, the candidate whom votes spend the most time is called the Markov winner. However, in the three candidate case our unique vector L gives a null result. Votes spend the same amount of time with each candidate! Thus it seems that the Markov method is not very useful for the three candidate cyclic case, but perhaps with a cyclic case involving more than three candidates the Markov method will produce a winner. 5. Detecting Subtle Differences In cases with more than three candidates the Markov method has a good chance of producing a winner. Below is a digraph of a cyclic election outcome for a five candidate election. Figure 5: Five candidate cyclic election outcome 6
The Markov matrix for this case is shown below. From To A M = B 1/ (11) C 1 1/ D E A B 1/ 3 1/ 3 1/ 3 C 1/ D 1/ E 1/ 1/ It follows from the procedure for calculating L: M n... A.146... B = L.93... C.146... D.195... E (1) From this analysis is appears that votes spend the most time with candidate C. We call candidate C the Markov winner. 6. Summary Borda began the investigation of the mathematics behind voting and successfully showed that the plurality method, used by the FAC and the majority of American elections, was a special case of the Borda Count. The Borda Count is a positional voting method where points are assigned to candidates according to rank. The problem with the Borda Count is the uncertainty on which value of λ is the best. By varying λ the election outcome can change dramatically, as can be seen in the first example. Because of this uncertainty, Condorcet created the Condorcet method. The Condorcet method pits candidates against each other in pairwise elections. The candidate who beats all other candidates in pairwise elections is the Condorcet winner. Because there is no question about which point value each rank in the voter preference the Condorcet method is superior to the Borda Count. However, there does not always exist a Condorcet winner. These conclusions are called cyclic election outcomes and render the Condorcet method useless in analyzing the election result. The new method, the Markov method, aims to rectify the null result of a cyclic election outcome. By the proof above, when there is a Condorcet winner the winning candidate is also a Markov winner. Furthermore, when faced with a cyclic election outcome the Markov method may still be able to choose a winner. The candidate whom votes spend the most time with is the Markov winner. Unfortunately, the Markov method fails in the three candidate cyclic election outcome because all votes spend an equal amount of time with each candidate. However, for elections with more than three candidates the Markov winner can choose a winner as can be seen in the five candidate election example above. 7. References 1. David Lay, Linear Algebra and its Applications (Boston: Addison Wesley, 3), 88.. Donald Saari and Fabrice Valongnes, Geometry, Voting and Paradoxes, Mathematics Magazine 71, no. 4 (1998): 44. 7