The tennis serve problem

You are allowed to hit three serves. Object of the game is to see how fast your fastest can be (it has to go in). Assume there is a linear relationship between serve speed and %chance of it going on (for simplicity sake lets say that you have a 100% chance of making a 50mph serve and a 0% chance of making a 120mph serve, and it is linear between those 2 speeds). If you miss all 3 serves, your score is 0. Otherwise, your score is the same as your fastest serve that went in (in mph).

SO....

How do you maximize your expected score? Do you start off with an easy serve, then go for pregressively faster serves? Or perhaps go for a medium speed serve first?

Is it possible to fit an equation to this problem that would tell you how fast a serve you should go for on the first, 2nd, and 3rd try (obviously the 2nd and 3rd serves would depend on whether you made the serves before it)? To make the problem more difficult (for those that are really math buffs), how would the equation change if you were given 4 serves or 5 or 10? Also, how would the equation change if we recognize that it is not a linear relationship between speed and chances of going in?

Discuss.

Best,
-Grant

[alThor]

We need to know the benefit (payoff) of having a successful serve of X mph for any X, and also the payoff of having no successful serve.

After that, it's a simple backwards induction computation. Figure out how fast your last serve should be to maximize expected payoff, given the status of the previous attempts. Then analyze 2nd last serve, etc.

Also, do we get 3 attempts even if the first attempt is a successful serve? That's not how tennis works, of course, so be clear how the game actually works.

Regardless, with no further information about payoffs, it seems one cannot deduce anything.

alThor

What do you mean by benefit/payoff? Lets just assume you get a dollar for every mph of your fastest of the three serves. You get 3 serves, regardless of whether previous serves are successful. Disregard any real tennis rules. If you miss, it counts as 0mph (zero $). If your fastest serve is 100mph, you receive 100$.

Could you please elaborate on how you would work backwards?

The way to calculate this is to calculate what the best strategy is for 1 serve then use this result to calculate the best strategy for 2 serves and so on.

In general for x{i} where x is speed of serve with i serves left:
Probability of successuful serve = (120 - x{i}) / 70
Probability of unsuccessful serve = (x{i} - 50)/ 70
Expectation{i} = x{i} * (120 - x{i}) / 70 + Expectation{i-1} * (x{i} - 50) / 70
We have Expextation{0} = 0
Therefore Expectation{1} = x{1} * (120 - x{1}) / 70 + 0 * (x{1} - 50) / 70
To maximise Expectation{1}, we use calculus to find where gradient is equal to 0.
dExpectation{1}/dx{1} = (120 - (2 * x{1})) / 70
0 = (120 - (2 * x{1})) / 70
x{1} = 60

So with 1 serve you should try 60mph which gives the maximum expectation of 3600/70 = 51.42857143

So for 2 serves we calculate:
Expectation{2} = x{2} * (120 - x{2}) / 70 + 3600/70 * (x{2} - 50) / 70

Finding the maximum as above we obtain the best serve speed for 2 serves remaining is 12000/140 = 85.71428571 and a expectation of 68.22157434.

I haven't got time at thae moment to calculate further values, but the above gives the method of solving these types of problems. The approximate result for 3 serves is to serve at ~94 mph with an expectation of ~77.8

In general the best strategy is to serve fastest at the beginning and decrease the serve speed as you continue to fault. I suspect that this is true of any function, not just linear, where there is a negative correlation between serve speed and success.

[TomCollins]

You are most definitely wrong. I propose the following strategy that beats yours. Mine may not even be the best.

Try 60mph first. If you make it, try 90mph. If you miss, try 90mph again.

We make 90mph .367 of the time. We make 60mph .612 of the time, and we miss both .02 of the time.
This gives us an expectation of 69.79 points.

[TomCollins]

The optimum for 2 serves is to serve 75mph first, 97 if you make, and 60 if you miss.

This will give an EV of 71.226.

You should always choose 60mph if you miss the first serve. If you make the first serve, you should choose (x/2)+60 for your second serve, where x is the speed you had on the first serve.

[BruceZ]

[ QUOTE ]
In general for x{i} where x is speed of serve with i serves left:
Probability of successuful serve = (120 - x{i}) / 70
Probability of unsuccessful serve = (x{i} - 50)/ 70
Expectation{i} = x{i} * (120 - x{i}) / 70 + Expectation{i-1} * (x{i} - 50) / 70

[/ QUOTE ]

This ignores the fact that if we make it on the first serve, we can increase our speed for an even higher expectation. This leads to much more complicated multi-variable gradients.

yes I answered the wrong question. I assumed once you make a successful serve you get no more serves (ie as in a real tennis point).

[xorbie]

[ QUOTE ]
You are most definitely wrong. I propose the following strategy that beats yours. Mine may not even be the best.

Try 60mph first. If you make it, try 90mph. If you miss, try 90mph again.

We make 90mph .367 of the time. We make 60mph .612 of the time, and we miss both .02 of the time.
This gives us an expectation of 69.79 points.

[/ QUOTE ]

85-->102.5-->111.25 with 50 if you miss the first two has ex expectation of over 80.

[UprightCreature]

Toms solution is correct for the two serve case.

For the three serve case the solution is:

First try
82.64
Second try:
Pass then 102.09
Fail then 74.74
Third Try:
Pass Pass 111.05
Pass Fail 101.32
Fail Pass 97.37
Fail Fail 60

This results in an expectation of 82.11