III. A Somewhat-Interesting Result

 

By now, you've begun your search for your (1, 1, 1). It goes without saying that roleplay and writing are hobbies, so the people who do them shouldn't have to settle for less than what they want. However, the frequency with which you want to enjoy your hobby and the number of "perfect partners" you have is going to determine two things: how often you have to accept a less-than-perfect roleplaying situation, and how much less-than-perfect those situations have to be.

 

The purpose of this discussion is not to address the frequency with which you write. On the face of it, it can help you make decisions about your writing partners in cases where your ideal partner isn't available. With that said, one approach to the problem may demonstrate exactly how subjective this business is.

 

Suppose that an assessment instrument of the sort previously described exists. The result for two potential roleplay partners is the 3-tuple (V, M, S).  Of course, the two players would like for (V, M, S) = (1, 1, 1), but what happens when it isn't?

 

A person might accept some deviation from an ideal partner. Mathematically, this is the same as saying that the person will accept any (V, M, S) whose distance from the point (1, 1, 1) is less than or equal to a constant, D. Using the distance formula, we can categorize all such points:

 

{(V, M, S): (1 - V)² + (1 - M)² + (1 - S)² < D²}

 

This condition describes a sphere having radius D centered at (1, 1, 1), but we are only interested in the section of the sphere for which V, M, and S are less than or equal to 1. If we further limit our study to those points on the section of the sphere's exterior, we can obtain a constraint.

 

( 1 )                   (1 - V)² + (1 - M)² + (1 - S)² - D² = 0

 

This gives us a set of candidate points, but how should we judge them? Let's also suppose that we can judge the "fitness" F of the roleplay relationship characterized by (V, M, S) by summing its components.

 

( 2 )                    F = V + M + S

 

Thus, we want to maximize (2) subject to the constraint expressed by (1). A calculus method called the Lagrange multiplier allows us to do exactly that.

 

Different people will feel like bowing out of the mathematics of this at different points. If you have come this far and want to continue, please feel free. This is especially useful in finding kindred spirits who can check my work. Otherwise, you may want to skip on to the result and the conclusions. I will make sure to mark them.

 

Per the method, we define the Lagrange function L = X + λY, where X is the function we seek to optimize, and Y is the function providing the constraint. This gives:

 

( 3 )                    L(λ,V,M,S) = V + M + S + λ[(1 - V)² + (1 - M)² + (1 - S)² - D²]

 

Next, we must find the gradient of L and set it equal to 0. This involves taking partial derivatives of L with respect to V, M, S, and λ.

 

( 4a )                   ∂L/∂V = 1 + 2λ(V - 1) = 0

( 4b )                   ∂L/∂M = 1 + 2λ(M - 1) = 0

( 4c )                   ∂L/∂S = 1 + 2λ(S - 1) = 0

( 4d )                   ∂L/∂λ = (1 - V)² + (1 - M)² + (1 - S)² - D² = 0

 

Here comes the especially roundabout part! (I'm kidding just a little, but if you're skipping ahead, then I hope you didn't peek too early!)

 

Using (4a), (4b), and (4c) we solve for V, M, and S in terms of λ. Then we substitute those solved values into (4d) to solve for λ in terms of D. (D is just a number; remember – it represents the maximum distance in space from your ideal writing partner that you are willing to accept.) Having found λ in terms of D, we can use that value to find the V, M, and S that optimize our fitness function.

 

No, I swear I am not making this up. Here goes!

 

( 5a )                   1 + 2λ(V - 1) = 0 → V = 1 - (1/2λ)

( 5b )                   1 + 2λ(M - 1) = 0 → M = 1 - (1/2λ)

( 5c )                   1 + 2λ(S - 1) = 0 → S = 1 - (1/2λ)

 

( 6 )                    { 1 - [1 - (1/2λ)] }² + { 1 - [1 - (1/2λ)] }² + { 1 - [1 - (1/2λ)] }² - D² = 0 → λ = ± √3/2D

 

( 7a )                   V = 1 - (1/2λ) = 1 ± D/√3

( 7b )                   M = 1 - (1/2λ) = 1 ± D/√3

( 7c )                   S = 1 - (1/2λ) = 1 ± D/√3

 

Now for the payoff before the payoff.

 

If (V, M, S) = (1 + D/√3, 1 + D/√3, 1 + D/√3), then F = 3 + D√3. Unfortunately, this value for (V, M, S) falls outside of the unit cube. Remember: V, M, and S must all be less than or equal to 1. Since D is positive, 1 + D/√3 > 1.

 

If (V, M, S) = (1 - D/√3, 1 - D/√3, 1 - D/√3), then F = 3 - D√3. The good news is that this value for (V, M, S) does fall inside of our unit cube. The bad news is that since the previous (invalid) value for (V, M, S) produced a larger value for F than did this value for (V, M, S), this value of F must be a minimum and not a maximum! What a gyp!

 

In a sense, the answer to the original question has been hiding in plain sight the entire time. When an optimization method like this one fails to find its optimum on its interior, we have to look at what are called boundary conditions. The maximum value is somewhere on the edges of the cube.

 

If we set any of V, M, or S equal to (1 - D) while allowing the other two to equal 1, we obtain F = 3 - D. Thus, the points (V, M, S) that maximize F are any of the following: (1 - D, 1, 1) ; (1, 1 - D, 1) ; (1, 1, 1 - D).

 

 

Those of you who were hiding until now can come back!

 

 

What exactly have we figured out?

 

Let's suppose that for you, D = 0.1 – in the most basic sense, it means that you would never settle for a writing partner with whom your views on volume, on mechanics, or on style don't align under at least 90% of circumstances.

 

Imagine then, that you are standing in the center of a room. At each of the room's four corners stands a potential writing partner. As long as we're imagining this situation, imagine further that all five of you have already taken our non-existent survey of roleplay compatibility. Your survey results have been correlated with each of the other four.

 

In Corner 1, we have someone who produced a respectable (0.942265, 0.942265, 0.942265). (The problem with irrational numbers like the square root of 3 is that they go on forever. I chose what I thought to be a "convenient" place to round the number.)

 

In Corner 2, we have someone who produced (1, 1, 0.9).

 

In Corner 3, we have someone who produced (1, 0.9, 1).

 

In Corner 4, we have someone who produced (0.9, 1, 1).

 

With the results tabulated, our fitness function says that you should choose to write with . . . any one of the four, except for the person in Corner 1!

 

In truth, that's not really what the results say. Given your choice of D, any of the four would be acceptable partners (for you). But according to the fitness function, the writers in Corners 2, 3, 4 all score higher (2.9 out of 3) than does the writer in Corner 1 (2.8268 out of 3).

 

But what's more: if the writer in Corner 1 suddenly and magically found his compatibility values to be (0.96, 0.96, 0.96), he would be closer in space to your ideal - his D would only be about 0.07 - but the fitness function would still reject him in favor of any of the other three, as his value for F would only be 2.88 out of 3.

 

 

IV. Tidying Up

 

If you have read this far, then you are certainly entitled to make a judgment about how much this result surprises you.

 

Perhaps it surprises you that a writer who is closer to one's ideal partner could somehow be judged to be less of a fit than are writers who are farther away from said ideal.

 

Perhaps it doesn't surprise you that there is something about a writer whose perfect alignment with a partner along two out of three measures that somehow makes up for the deviation in the third – at least, as compared to a writer who is close to ideal in all three measures but not perfectly aligned in any of them.

 

Having come this far myself, I have decided that I am both surprised and not surprised. When it comes down to it, mathematics and science are full of seeming-paradoxes such as this one. Further, I am reasonably certain that what I have found does not stem from an error in the method, but it could stem from an error in my application of the method. Thus, while I am satisfied that I have done all of the math correctly, I welcome any observations to the contrary.

 

This does leave the question of the fitness of the fitness function. Throughout this discussion, I have assumed that all three measures – volume, mechanics, and style – carry equal importance. Of course, this is going to vary from writer to writer, and in a way that can be quantified through study. It would represent a refinement to the as-yet nonexistent survey. In the sequel to this treatise that I intend never to write, perhaps I can address the premise of weighting the measures for their importance (or lack thereof) to a particular writer.

 

In the meantime, I hope that you've enjoyed reading this as much as I have enjoyed writing it. From the point where I started, it took me a long time to get everything the way that I wanted it. But in doing so, I've had to think extensively about what is important and what should be important to writers and roleplayers like me, who want to entertain themselves and others. I don't see how it can be a bad thing if more people are doing this, or if they have some framework for doing it.

 

I welcome your constructive comments and questions!