In a **pairwise vote**, voters are asked to repeatedly pick between pairs of options, selecting the one they favor of the two. The procedure then combines all these pairwise choices in order to establish a global ranking for all options. A pairwise vote is statistical in nature, it must infer preference data that voters have not explicitly stated in order to obtain a result.

This statistical property allows obtaining an approximate preference ordering over a large list of options without overwhelming the voter with too much work. For example, if each voter was asked to establish a preference over 50 items, they would be exhausted and participation would suffer.

The **pairwise-beta** is a simple bayesian method used to rank items in pairwise votes. It is based on the beta-binomial model [1], which is composed of a beta prior and a binomial likelihood. This model is very tractable: because the beta distribution is conjugate to the binomial, the posterior also has beta form and is easily obtained:

I will not present a formal justification of the pairwise-beta model for pairwise comparisons, rather I will present some intuitions that should convey how and why the model works.

The key bridge to interpret pairwise comparisons in terms of the beta-binomial model is to realise that the better-worse relation between options maps directly into the success/failure outcomes of a bernoulli trial. We can thus establish a correspondence[4] between pairwise comparisons and bernoulli trials:

- each item
*i*corresponds to a sequence of bernoulli trials,*B*_{i} - a comparison in which i wins corresponds to a success in
*B*_{i} - a comparison in which i loses corresponds to a failure in
*B*_{i}

The question we are trying to answer is

*Given the proportion of comparisons in which i wins, what is the proportion of items that are globally better than i?*

that reformulated in terms of our correspondences becomes

*Given a sequence of bernoulli trials B _{i}, what is the proportion of successes/losses for i?*

Which is a case of standard binomial proportion estimation[2]. As we noted before, the posterior of the beta binomial is also a beta distribution, given by

If we want a point estimate we just use the mean for this distribution which is

This gives us, for each item *i*, an estimation for the number of items that are better/worse than itself. This leads directly to a global ranking: the best ranked items will be those which are estimated to be better than most other items.

** **In summary, the procedure is

For each item*i*, obtain the corresponding sequence of bernoulli trials*B*_{i}- For each item i, calculate the posterior beta distribution mean given the data from 1)
- Create a global ranking based on the proportions for each item, as calculated in 2)

The pairwise-beta model is simple but not perfect. In particular it does not exploit information about the strength of the opposing item in a pairwise comparison. However, despite this drawback it performs well in practice. Please refer to [3] for details.

[1] http://www.cs.cmu.edu/~10701/lecture/technote2_betabinomial.pdf

[2] http://homepage.ntu.edu.tw/~ntucbsc/%A5%CD%AA%AB%C2%E5%BE%C7%B2%CE%ADp%BF%D4%B8%DF%A4@971008/[3]%20Chapter%208.pdf

[3] http://arxiv.org/pdf/1202.0500v2.pdf

[4] The correspondence is two to one, as each comparison yields two bernoulli trials