2. The Bradley-Terry Model

The Bradley-Terry Model

声明：本文为本人毕业研究报告《The Exploration of Pairwise Comparison in Football Application》中的部分内容摘录与整理，仅用于学习与交流。

Introduction

The core Bradley-Terry model was proposed by Ralph Allan Bradley [1], and it was further redefined by transitioning from the normal distribution used in Thurstone’s model [2] to a squared hyperbolic (Logistic Density) secant distribution. The following model formulation provided here primarily focuses on the Bradley-Terry model, intentionally omitting Thurstone’s model which was detailed earlier.

Model Formulation 2

Based on the context of Thurstone’s model for sensory difference testing, where stimuli $O_1,O_2,...,O_n$ evoke sensations $X_i$ quantitatively compared on a sensation scale $S$, the probability of preference $p_{ij}$ between stimuli is explored.

Building upon Thurstone’s theoretical foundation to achieve a new model of preference probabilities via the logistic density (Squared Hyperbolic Secant Density), the probability of preference $p_{ij}$ is redefined as:

$$\begin{array}{}\text{(9)}& p_{ij}=P(X_i>X_j)=\frac{1}{4}{\int }_{-(S_i-S_j)}^{\mathrm{\infty }}{\mathrm{sech}}^2\left(\frac{y}{2}\right)dy,\end{array}$$

where $S_i={\log }_e({\pi }_i)$ and $S_j={\log }_e({\pi }_j)$ denote the log-transformed strengths of stimuli $i$ and $j$, thus measuring stimulus strength on Thurstone’s perceptual scale.

Simplifying this to the Bradley-Terry model:

$$\begin{array}{}\text{(10)}& p_{ij}=\frac{{\pi }_i}{{\pi }_i+{\pi }_j}.\end{array}$$

The probability $p_{ij}$ that stimulus $i$ is preferred over stimulus $j$ can be articulated in terms of their relative strengths or perceived intensities, ${\pi }_i$ and ${\pi }_j$.

Example 2

Let’s apply this model to assess which of two football players, Player A and Player B, is more likely to be perceived as the better player based on their performances in a season. Assume the strengths of the players are quantified based on their contribution scores, such as goals, assists, and defensive actions, over the season.

Player A’s strength, ${\pi }_A$, is 20 (goals + assists + key defensive actions).
Player B’s strength, ${\pi }_B$, is 15.

Calculate the probability of preference $p_{ij}$:

$$p_{ij}=\frac{{\pi }_A}{{\pi }_A+{\pi }_B}=\frac{20}{20+15}=\frac{20}{35}\approx 0.571$$

This means there is a $57.1\mathrm{\%}$ chance that Player A is perceived as the better player compared to Player B based on their performance metrics.

Model Derivation 2 (Method 1)

The logistic probability density function (PDF) with location parameter $\mu$ and scale parameter $s$ is given by:

$$f(x;\mu ,s)=\frac{e^{-\frac{x-\mu }{s}}}{s{(1+e^{-\frac{x-\mu }{s}})}^2}$$

Simplifying for $\mu =0$ and $s=1$:

$$\begin{array}{}\text{(11)}& f(x;0,1)=\frac{e^{-x}}{{(1+e^{-x})}^2}.\end{array}$$

The $\mathrm{sech}$ function is defined as the reciprocal of the hyperbolic cosine ($\cosh$) function:

$$\mathrm{sech}(x)=\frac{1}{\cosh (x)}=\frac{2}{e^x+e^{-x}}.$$

Now, observe that the denominator of equation (11) can be related to the $\cosh$ function:

$$1+e^{-x}=2\cdot e^{-x/2}\cdot \left(\frac{e^{x/2}+e^{-x/2}}{2}\right)=2\cdot e^{-x/2}\cdot \cosh \left(\frac{x}{2}\right)$$

Then, rewrite equation (11):

$$\begin{array}{rl}f(x;0,1)& =\frac{e^{-x}}{(2\cdot e^{-x/2}\cdot \cosh (\frac{x}{2}){)}^2}\\ & =\frac{1}{4}\cdot \frac{1}{{\cosh }^2(\frac{x}{2})}\\ & =\frac{1}{4}\cdot {\mathrm{sech}}^2\left(\frac{x}{2}\right),\end{array}$$

therefore, the redefined Thurstone’s model (equation 9) of the core Bradley-Terry model follows the logistic distribution (${\mathrm{sech}}^2$).

Considering the difference in the log-transformed strengths:

$$\begin{array}{}\text{(12)}& \mathrm{\Delta }S=S_i-S_j={\log }_e({\pi }_i)-{\log }_e({\pi }_j),\end{array}$$

we have the cumulative probability that $X_i>X_j$ under the transformed logistic distribution:

$$p_{ij}={\int }_{-\mathrm{\Delta }S}^{\mathrm{\infty }}\frac{1}{4}{\mathrm{sech}}^2\left(\frac{x}{2}\right)dx.$$

The cumulative distribution function (CDF) of the logistic distribution is given by:

$$\begin{array}{}\text{(13)}& F(x)=\frac{1}{1+e^{-x}}\end{array}$$

where $\mu =0$ and $s=1$.

Substitute equation (12) into the logistic CDF (13) to calculate the probability $p_{ij}$:

$$\begin{array}{}\text{(14)}& p_{ij}=F(\mathrm{\Delta }S)=\frac{1}{1+e^{-({\log }_e({\pi }_i)-{\log }_e({\pi }_j))}}\end{array}$$

To simplify using properties of logarithms:

$$p_{ij}=\frac{1}{1+e^{-{\log }_e({\pi }_i/{\pi }_j)}}$$

Since $e^{-{\log }_e(x)}=\frac{1}{x}$, apply this to the formula:

$$p_{ij}=\frac{1}{1+\frac{1}{{\pi }_i/{\pi }_j}}=\frac{1}{1+\frac{{\pi }_j}{{\pi }_i}}=\frac{{\pi }_i}{{\pi }_i+{\pi }_j}$$

Method 2

Lecture Notes 24 [3] provide a detailed derivation of the core Bradley-Terry model. Let $p_{ij}$ denote the probability that item $i$ is preferred over item $j$, modeled as an independent Bernoulli random variable. The log-odds of $p_{ij}$ is given by

$$\log \left(\frac{p_{ij}}{1-p_{ij}}\right)=S_i-S_j,$$

where $S_i=\log ({\pi }_i)$ and $S_j=\log ({\pi }_j)$ represent the log-transformed strengths of items $i$ and $j$, respectively.

Eliminate the logarithm, yielding

$$\frac{p_{ij}}{1-p_{ij}}=e^{S_i-S_j}.$$

Rearrange the equation to solve for $p_{ij}$:

$$\begin{array}{rl}{p}_{ij}& =(1-p_{ij})e^{S_i-S_j}\\ p_{ij}+p_{ij}{e}^{S_i-S_j}& =e^{S_i-S_j}\end{array}$$

which simplifies to

$$p_{ij}=\frac{e^{S_i-S_j}}{1+e^{S_i-S_j}}=\frac{e^{\log ({\pi }_i)-\log ({\pi }_j)}}{1+e^{\log ({\pi }_i)-\log ({\pi }_j)}}$$

Using the property that $e^{\log (x)}=x$,

$$p_{ij}=\frac{{\pi }_i/{\pi }_j}{1+{\pi }_i/{\pi }_j}=\frac{{\pi }_i}{{\pi }_i+{\pi }_j}=\frac{e^{S_i}}{e^{S_i}+e^{S_j}}$$

which matches the stated form.

Conclusion

To sum up, when transformed by the logarithm ($S_i=\log ({\pi }_i)$), it is used primarily to simplify multiplicative relationships into additive ones. Representation of the model as a ratio manages the original multiplicative relationship in a way that improves computational efficiency and numerical stability.

However, the Bradley-Terry model [1] has certain limitations, as it requires making a clear preference between any two items, which does not align with real-world application scenarios. Next, we will introduce the Rao-Kupper model [4] to address this issue by considering a third situation: considering the unsure case of A and B (either is possible).

References

• [1] Ralph Allan Bradley. Some statistical methods in taste testing and quality evaluation. Biometrics, 9(1):22–38, 1953.
• [2] Frederick Mosteller. Remarks on the method of paired comparisons: I. the least squares solution assuming equal standard deviations and equal correlations. Psychometrika, 16(1):3–9, 1951.
• [3] STATS 200: Introduction to Statistical Inference. Lecture 24 — the bradley-terry model. https://web.stanford.edu/class/archive/stats/stats200/stats200.1172/Lecture24.pdf, 2016. [Online; accessed 9-April-2024]
• [4] P. V. Rao and L. L. Kupper. Ties in paired-comparison experiments: A generalization of the bradley-terry model. Journal of the American Statistical Association, 62(317):194–204, Mar 1967. Available: https://www.jstor.org/stable/2282923