This is a response to Ari’s awesome post on human-computer symbiosis. Ari and I were chatting about the equation he developed and I was wondering if there were some further refinements that are possible… let’s take a look:

We are attempting to understand the total analytic capability for a given task ** a** of a human-computer team. Analytic capability in this case probably means:

Where ** A** is the answer to the analytic problem in question and

**is the time needed to arrive at the answer based on the inputs available. In the case of chess,**

*t*_{A}**could be the optimum next move given all previous information and**

*A***would be how long it takes to decide on this move.**

*t*_{A}Read on for a look at how this generalizes in human-computer symbiotic systems.

In the case of the human-computer team, we know that ** a **is going to be a function of both the human’s analytical capability

**and the computer’s analytical capability**

*h***(where both**

*c***and**

*h***have units of answers/time). In the limit case we know that:**

*c*Or in plain English, if there is no human present, the total analytic capability is simply the analytic capability of the computer. So the naïve solution would be that:

(4) clearly meets the limiting cases described in (2) and (3). Kasparov noticed a mixing function where the ability of the human and computer to work together becomes the dominant term — we might call this the mixing capability for the given task or ** m**. Including this phenomenon, the total analytic capability (4) would be re-defined as:

where ** m** has the property that:

Thus maintaining the limits expressed in (2) and (3) and adhering to the observation that if there is no human or computer component then there will be no mixing advantage. A naïve solution to this constraint would be simple linear mixing:

where ** M** (units of time per answer) is the mixing efficiency and will be primarily based on the type of task being solved — some analytical tasks lend themselves to a combined process more than others (for example, multiplying 20 digit numbers does not really benefit from the intuition of a human so the ability of a human and computer to perform this task is merely their additive ability).

What Kasparov noticed is that the mixing was primarily based on the quality of the process rather than the analytical power of either the human or computer separately. This seems to imply that we must somehow account for the fact that the quality of the human-computer interface is responsible for the quality of the mixing. This can be modeled as a unitless friction of interaction ** f_{i}** that impedes the ability of the human and computer to work together.

Equation (7) can thus be re-written as:

In this case, the maximum value for the mixing capability is realized when the friction of interaction goes to zero. This mixing capability is the same as the equation Ari developed (less the coefficient which is necessary to maintain consistent units throughout).

We can now re-write our analytic capability in (5) as:

Below, see a plot of this function over a range of values for ** h**,

**and**

*c***:**

*f*_{i}As can clearly be seen from this functional plot (note the vertical scale), the effect of interface friction dominates over the other terms whenever both the human and computer can make important contributions to the task at hand. The conclusion can be drawn that the most effective way to solve analytical problems is to minimize the friction of the human-computer interface; or to put it another way: optimal analytical systems are those that are built specifically to maximize the ability of the human to leverage the ability of the computer.

I am certain there is still the possibility for further refinement, for example: