FIVB Senior World Rankings

Match result

More information We set the result ...

We set the result ${\displaystyle R=R_{n}}$, where ${\displaystyle n}$ is the index of the actual result (set score) ${\displaystyle n=1}$ – A win 3–0 ${\displaystyle n=2}$ – A win 3–1 ${\displaystyle n=3}$ – A win 3–2 ${\displaystyle n=4}$ – A lose 2–3 ${\displaystyle n=5}$ – A lose 1–3 ${\displaystyle n=6}$ – A lose 0–3

Match Result ${\displaystyle R_{n}}$ ${\displaystyle P_{n}}$ A win 3–0 +2 ${\displaystyle P{\text{1}}}$ A win 3–1 +1.5 ${\displaystyle P{\text{2}}}$ A win 3–2 +1 ${\displaystyle P{\text{3}}}$ A lose 2–3 -1 ${\displaystyle P{\text{4}}}$ A lose 1–3 -1.5 ${\displaystyle P{\text{5}}}$ A lose 0–3 -2 ${\displaystyle P{\text{6}}}$

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Expected match result

The expected results is then calculated as ${\displaystyle E=R_{1}P_{1}+R_{2}P_{2}+R_{3}P_{3}+R_{4}P_{4}+R_{5}P_{5}+R_{6}P_{6}}$ where ${\displaystyle P_{n}}$ is the probability of the outcome ${\displaystyle R_{n}}$ obtained using the following model (known as Ordered probit):

Team A win 3–0

${\displaystyle P_{\text{1}}=\Phi (C_{\text{1}}+\Delta )}$

Team A win 3–1

${\displaystyle P_{\text{2}}=\Phi (C_{\text{2}}+\Delta )-\Phi (C_{\text{1}}+\Delta )}$

Team A win 3–2

${\displaystyle P_{\text{3}}=\Phi (C_{\text{3}}+\Delta )-\Phi (C_{\text{2}}+\Delta )}$

Team A lose 2–3

${\displaystyle P_{\text{4}}=\Phi (C_{\text{4}}+\Delta )-\Phi (C_{\text{3}}+\Delta )}$

Team A lose 1–3

${\displaystyle P_{\text{5}}=\Phi (C_{\text{5}}+\Delta )-\Phi (C_{\text{4}}+\Delta )}$

Team A lose 0–3

${\displaystyle P_{\text{6}}=1-\Phi (C_{\text{5}}+\Delta )}$

where ${\displaystyle \Phi (z)}$ is the Cumulative distribution function of the Normal distribution, and ${\displaystyle C_{1},\ldots ,C_{5}}$ are the cut-points

• ${\displaystyle C_{1}=-1.06}$
• ${\displaystyle C_{2}=-0.394}$
• ${\displaystyle C_{3}=0}$
• ${\displaystyle C_{4}=0.394}$
• ${\displaystyle C_{5}=1.06}$

set so that ${\displaystyle P_{n}}$ is the probability of the outcome ${\displaystyle n}$ between two equal strength opponents (that is when ${\displaystyle \Delta =0}$), which is derived from the actual match results of the past decade.

The parameter ${\displaystyle \Delta }$ represents the scaled difference of the teams rankings

${\displaystyle \Delta ={8(S_{\text{teamA}}-S_{\text{teamB}}) \over 1000}}$

where:

• ${\displaystyle S_{\text{teamA}}}$ – the team A's number of World Ranking scores before the game
• ${\displaystyle S_{\text{teamB}}}$ – the team B's number of World Ranking scores before the game