It is possible, although usually difficult, to compute how efficient a user interface is relative to its theoretical optimum. Information theory shows how to compute how much information is present in a series of numbers or other symbols, given the current situation, the history, and knowledge of what symbols are likely. If you can estimate the probabilities of all of the possible player moves, you can estimate the information I that is present, according to information theory, in a move. The best your UI can possibly do is for W to be of the same order of computational complexity as I, meaning that O(W) = O(I).
I/W is a measure of the efficiency of your interface; it can be at most O(1). For every variable in the expression for I, it will have the same or a larger exponent in W, so it is easier to think in terms of W/I, which is a measure of how badly you abuse your players. Estimating I is much harder than estimating W; you can often obtain only an upper bound on I. In order for W/I to be meaningful, then, it should be a lower bound.
What you find, in a game with many units, is that for unit-based user interfaces, your lower bound on W grows much faster with the number of units than your upper bound on I does. This is because, most of the time, a lot of the units do pretty much the same thing, and knowing what a small fraction of a player’s units are doing would enable a skilled player to predict with good accuracy what the rest of their units are doing. This is exactly analogous to our user interface for sculpting clay: Most of the points on the surface of the sculpture have a surface tangent almost the same as do the points near them. An interface that requires you to move every unit individually is the equivalent of a sculpting interface that makes you move every voxel. The use of information theory allows you to calculate I, and learn the true dimensionality required of your UI, even when you don’t have a simple way to visualize the connection between the units in play.
To explain how to compute W and I in general would take another article. You can figure it out from a book on probability and information theory. A good primer is the book version of the 1949 paper that defined information theory, The Mathematical Theory of Communication, reprinted in 1998 by the University of Illinois Press. In difficult cases, I can be estimated by a combination of theory, and statistics gathered during playtesting.
Computers can now animate more units than any player could reasonably want to control, and the number will continue to increase exponentially. This leads to player frustration rather than fun. In a good user interface design, no player should control more than seven game entities. To enable this, the UI may let the player control something more abstract than an on-screen unit. This requires object-oriented developers to think of code objects as abstractions beyond the mere units on the screen. The UI may also give the player a chance to specify behaviors off-line in order to reduce the amount of on-line supervision needed.
Game developers can evaluate their user interfaces using a user-interface profiling tool, and by computing the work involved in different interfaces. They can even estimate their theoretical efficiencies, to know for sure whether there’s room for improvement. The ultimate goal of game design is to increase the game’s FPS - fun per second. The easiest way to do that is to pack the same action into fewer seconds, and the easiest way to do that is usually to improve the user interface.