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Some sports, such as tennis, use a complicated points system (point, game, set, match; with deuces and tie-breaks) for what would otherwise be an extremely simple and monotonous sport. The main reason, I suppose, is to make the sport more stressful, by making some parts of the match more crucial than others. It should be possible to make it a lot more 'stressful' mathematically.

Mathematical scenario:
There is an event, played by 2 players, worth 1 point. The probability of winning the entire match, if they (a) win (b) lose the next point is calculated. The difference between these values is the 'stress' value of that point. The sum of all the stress values is called the 'total stress' value of the match, and this number divided by the number of points played is the 'average stress'. For players of equal skill, and stress-handling, the probability of either player winning any given point is $\frac12$. The 'skill' of a player is a randomly generated number from $0\%$ to $100\%$, and the probability of a player winning a point is given by $\frac{Skill(own)}{Skill(own)+Skill(opponent)}$

Your job is to create a points system, however simple or complicated, using any kind of conditions, limits, etc, that produces the maximum possible:
(a) Total stress
(b) Average stress
for a match. You are allowed to create your own definitions of 'point', 'set' and so on, however, the 'points' I am referring to is a single valid serve (double faults to be ignored) with a single winner.

The only constraint is that the total number of points played should be between $1000$ and $1500$ for at least $90\%$ of the scenarios (9 in every 10 matches should not have the players play less than $1000$ or more than $1500$ points).

ghosts_in_the_code
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    This just sounds like a homework question. What are your thoughts? – Math1000 Feb 01 '15 at 14:29
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    @Math1000 No, it is not a homework question. In fact, I thought it maybe was too broad to be asked on SE. I myself am confused as to how it is to be solved, as every possible system I think of leads to very similar results when I actually calculate the probabilities. I only have a feeling that it should be better to maximimize the number of points played, even for (b). – ghosts_in_the_code Feb 01 '15 at 14:33
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    @ghosts_in_the_code Just having a points system that allows for high stress cannot guarantee it, can it? Say for example, if one player runs away with the match from the beginning, then there will be no stress, regardless of points-systems, right? Do we need to assume that a match is played as evenly as possible for this question? – JLee May 10 '15 at 18:57
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    I feel like the Total Stress of a match will always be 1. It should be properly demonstrated, though. – Masclins May 11 '15 at 06:16
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    @JLee We simply assume that both players want to win, and are aware of the points system used. – ghosts_in_the_code May 11 '15 at 09:26
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    @AlbertMasclans There's a 50 rep bounty ending today, if you want to do so. – ghosts_in_the_code Dec 26 '15 at 03:46
  • Can the system penalize the win of a point? For example, can it insist that the winner must NOT score than so many points above the loser? What about parity? – Aravind Jun 03 '16 at 04:12
  • @Aravind The system can't force a player to give away a point, but it can penalise the player in such a manner that the player himself wants to give away the point. – ghosts_in_the_code Jun 03 '16 at 05:35
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    @ghosts_in_the_code I'll re-bounty this if no answers are given. I put the bounty because this interests me and honestly I'm not active on Math.SE anyways, I don't need the rep :-) – Insane Jun 04 '16 at 01:50
  • @Insane Ok thanks – ghosts_in_the_code Jun 04 '16 at 09:32
  • @ghosts_in_the_code I think your question is cool, but please, satisfy this need: the word "game" has 3 meanings in this context, game from "game theory", a tennis full match to 3 or 5 sets, and a single game inside a set 15-30-40-game. You should use different words in your post for each one of them. If not, reading and answering is a pain! – Santropedro Dec 16 '16 at 20:23
  • @Santropedro I guess the question is clear enough only. Which part do you find confusing? – ghosts_in_the_code Dec 17 '16 at 12:17
  • @ghosts_in_the_code Every time it says "game" I have to deduce from context, wich is something can be avoided if you use unique names. – Santropedro Dec 17 '16 at 15:28
  • @Santropedro Fine now? – ghosts_in_the_code Dec 17 '16 at 15:52
  • @ghosts_in_the_code Yes, much better, great job! – Santropedro Dec 19 '16 at 00:32

1 Answers1

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What you are missing is called "competitive balance" in the economic literature. You need a set of rules to assure it, otherwise, you will have a major leaguer play a t-ball player.

Since players are randomly generated, you need a rule system that begins to systematically sort players of similar rank.

There are two distinct issues here. The first surrounds the mean, the second the variance. Two t-ball players are unlikely to meet a stress condition because their natural variance is high. Two professional players are likely to face the stress conditions you are seeking because the mean rank will be highly similar and the variability will be very small.

Your rules do not seem to allow learning. Random systems will have grave difficulties in creating a stress system because it would be like playing competitive Roulette. While 90% of the scenarios have to have between 1000 and 1500 points, you could play low-level qualifying matches with few points. You could use something similar to a Swiss System tournament with an Elo system to rank. This is prebuilt so you can adapt it to your needs.

If sorting is not allowed, then handicapping is your alternative. This takes away any advantage. If you want to maximize stress, then you want to include rules to minimize the probability of a statistical run. The handicapping will normalize the odds, you will want to create some favoritism against the leader to minimize the variability.

Dave Harris
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