Elegance

I have no idea how you would measure elegance, but if you could increase elegance without changing anything else then I'd say that would qualify as improving the game.

How could you do improve elegance without changing "anything"? One way would be unifying the art/iconography of the components. Besides making the game look better, it just plain makes everything easier to comprehend. Another way would be to make the rules more internally consistent by using the same terms/metaphors to refer to equivalent mechanics.

An example of something that could have been made more elegant without changing any probabilities would be the "saving throw" or "resistance roll" from old AD&D and Rolemaster, respectively. In both cases they used the same dice as a combat roll, and the same beat-this-target-number mechanic, but were pointlessly made into discrete rules to give players another thing they needed to learn.

I agree with polyobsessive about less rules/more strategy. Which is hard to achieve.

I think elegance is my top priority in game design. I want my rules to fit on a playing card!

I don't think elegance = improvement. Sometimes too much can be taken out.

I try to combine rules and reduce unneeded rules. Also, I try to make rules that are multifunctional. I have a game in which you can lay cards on top of each other. One rule is that you can't lay a card on a person. Some cards have people on them. With this, I can restrict placement of cards on multiple types of other cards very easily.

I would agree with polyobsessive.

For an out of board-games example example, look into flocking behavior.

Such complex shapes and moves, all bases on three simple rules.

It's a big term we use in software development. The rawest definition being: Maximum value from minimum effort. But it really falls more into a butterfly-ee sort of artsy thing.

In the design process, I think it largely lives in that aha moment where you start adding rules or mechanisms to control an issue and then suddenly you see a solution by removing all the extra and making a small adjustment. Or when playtesting reveals a bonus undesigned dimension of play.

I often wander into the realm of terseness in pursuit of what I think will be clarity or elegance.

As a gamer, I experience elegance both in mechanics that solve issues I've seen in other games or deliver a superior experience with less effort.

I particularly appreciate game designs that yield big emergent gameplay. It's most spectacular in gateway games like Codenames.

I'm a huge fan of writing out objectives for a game (a laundry list of what it is supposed to allow players to do/experience) as soon as you start formulating a rule system and then combining/tossing/simplifying rules until I cannot see how to slim it down anymore without compromising my core 2 or 3 objectives.

A lot of the time the question of whether to cut or keep a rule you've had in your game since its beginning can be answered right away by seeing if it contributes to the core goals. If it does not, you can ALWAYS come up with a mechanic/tweak an existing rule to keep the game playable but in a more thematic or efficient way. Even if its not immediately apparent, seeing that the current solution is outside the scope of the game is the first step to finding a solution within scope.