# How to distribute points to players based on their effort?

Assume that a HOME team and an AWAY team play each other in a game. The HOME team defeats the AWAY team, so the HOME team gains 30 points and the AWAY team loses 30 points. Each team has two players.

The winning team distributes the points that it won to its players based on the number of strokes each player had in the game.

For example, Player 1 had 15 strokes and Player 2 had 5 strokes. There was a total of 20 strokes.

The formula for distribution points to the winning team members would be:

$$\text{POINTS} \times \frac{\text{PLAYER_STROKES}}{\text{TOTAL_STROKES}} = \text{PLAYER_POINTS}$$

Out of 30 points, Player 1 would receive 22.5 points and Player 2 would receive 7.5 points.

My question is, how can this same type of distribution be applied to the losing team in the inverse order. I want the losing team's player with the least strokes to lose the most points and the player with the most strokes should lose the least points.

The AWAY team loses 30 points. Player 1 has 17 strokes. Player 2 has 10 strokes. Total Strokes = 27

It should also be mentioned that the formula should work if the teams have more players.

Any ideas?

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You'd like to penalize each player for each stroke which is not his. The penalty per stroke should be divided among all players who did not make that stroke. I assume only one player is responsible for the stroke. Therefore:

$$\mathrm{SP} = \mathrm{STROKE\_PENALTY} = \frac{1}{\mathrm{TEAM\_SIZE}-1}\\ \mathrm{MS} = \mathrm{MISSED\_STROKES} = \mathrm{TOTAL\_STROKES}-\mathrm{PLAYER\_STROKES}\\ \mathrm{PPS} = \mathrm{POINTS\_PER\_STROKE} = \frac{\mathrm{TOTAL\_POINTS}}{\mathrm{TOTAL\_STROKES}}\\ \mathrm{PLAYER\_PENALTY} = \mathrm{SP} \times \mathrm{MS}\times \mathrm{PPS}$$

For your example: $$\mathrm{SP} = \frac{1}{2-1} = 1\\ \mathrm{PPS} = \frac{30}{27}\\ \mathrm{PLAYER\_PENALTY}_\mathrm{PLAYER\_1} = 1 \times (27 - 17) \times \frac{30}{27} = 10 \times \frac{30}{27} \approx 11.111...\\ \mathrm{PLAYER\_PENALTY}_\mathrm{PLAYER\_2} = 1 \times (27 - 10) \times \frac{30}{27} = 17 \times \frac{30}{27} \approx 18.888...\\$$

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Your solution does not add up to 30 in the example provided. – Jonathan Rich Mar 20 '13 at 17:33
Sorry, changed that. Now it does – Roelof van den Berg Mar 20 '13 at 17:47
Thanks, this was easy to understand. It seems to have solved my problem! – JV- Mar 20 '13 at 18:19

Use the ratio of the reciprocal to the summation of the reciprocals:

S = $\sum s^{-1}$

$r = \frac{s^{-1}}{S}$

$p_p = rp_t$

Where $s$ is the number of strokes, $S$ is the summation of the reciprocals, $r$ is the ratio of points lost, and $p_t$ is the total points lost.

So, in your example, $S = \frac{27}{170}$, $r_1 = \frac{170}{459}$, $r_2 = \frac{170}{270}$, and:

$p_1 = \frac{5100}{459} = 11.1$

$p_2 = \frac{5100}{270} = 18.9$

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