# Create the most 'stressful' tennis game ever!

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).

• This just sounds like a homework question. What are your thoughts? – Math1000 Feb 1 '15 at 14:29
• @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 1 '15 at 14:33
• @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
• I feel like the Total Stress of a match will always be 1. It should be properly demonstrated, though. – Masclins May 11 '15 at 6:16
• @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 9:26