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**Baseball**

Common in baseball and other sports, the

**log5**method provides an estimate for the probability \( p \) of participant 1 beating participant 2 given respective success probabilities \( p_1, p_2 \). Also let \( q_* = 1 -p_* \) in the following. The log5 estimate of the outcome is then:

\begin{align}

p &= \frac{p_1 q_2}{p_1 q_2+q_1 p_2}\\

&= \frac{p_1/q_1}{p_1/q_1+p_2/q_2}\\

\frac{p}{q} &= \frac{p_1}{q_1} \cdot \frac{q_2}{p_2}

\end{align}

The final form uses the

**odds ratio**, \( \frac{p}{q} \). Additional factors can be easily chained using this form to provide more complex estimates. For example, let \( p_e \) be an environmental factor, then:

\begin{align}

\frac{p}{q} &= \frac{p_1}{q_1} \cdot \frac{q_2}{p_2} \cdot \frac{q_e}{p_e}

\end{align}

**Chess**

The most common rating system in chess is the

**Elo rating system**. This has also been adopted for various other uses, e.g. ``hot or not'' websites. This system assigns ratings \( R_1, R_2 \) to players 1 and 2 such that the probability of player 1 beating player 2 is approximately:

\begin{align}

p &= \frac{e^{R_1/C}}{e^{R_1/C}+e^{R_2/C}}

\end{align}

Here \( C \) is just a scaling factor (typically \( 400/\ln{10} \) ). The Elo rating is connected to log5 via setting \( e^{R/C} = p/q \). We then recover:

\begin{align}

\frac{p}{q} &= e^{R/C}\\

p &= \frac{e^{R/C}}{1+e^{R/C}}\\

R &= C\cdot \ln(p/q)

\end{align}

Note that \( p \) is also the probability of this player beating another player with Elo rating 0. The Elo system generally includes enhancements accounting for ties, first-move advantage and also an online algorithm for updating ratings. We'll revisit these features later.

**Psychology**

The

**Bradley-Terry-Luce**(BTL) model is commonly used in psychology. Given two items, the probability \( p \) that item 1 is ranked over item 2 is approximately:

\begin{align}

p &= \frac{Q_1}{Q_1+Q_2}

\end{align}

In this context \( Q_* \) typically reflects the amount of a certain quality. That this model is equivalent to the previous models is immediate:

\begin{align}

Q &= e^{R/C} = p/q\\

R &= C\cdot \ln(Q) = C\cdot \ln(p/q)\\

p &= \frac{Q}{1+Q}

\end{align}

**Psychometrics**

The dichotomous (two-response)

**Rasch**and

**item response models**are commonly used in psychometrics. For the Rasch model, let \( r_1 \) represent a measurement of ability and \( r_2 \) the difficulty of the test item. The Rasch model estimates the probability of correct response \( p \) as:

\begin{align}

p &= \frac{e^{r_1-r_2}}{1+e^{r_1-r_2}}

\end{align}

The one-parameter item response model estimates:

\begin{align}

p &= \frac{1}{1+e^{r_2-r_1}}

\end{align}

These are clearly equivalent to each other and to the previous models.

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