Complementary mechanics are a coupling of mechanics where each reinforces the other in order to result in a desirable player behaviors. The best complementary mechanics reinforce each other as well as encourage player behaviors that relate to each mechanic; I'm calling this type 'completely complementary mechanics'.
Complementary mechanics are a very effective way to silently pressure a player into strategic actions with them not necessarily being aware of the greater importance of those actions. They must be carefully designed, however, so that the resultant behavior is fairly obvious given the mechanics involved.
Examples
Chess
If you are familiar with chess strategy, then you know that building a pawn wall is very important. The pawns, though weak individually, become a powerful tool that thwarts opponent movement when used together.
In order for it to work, a pawn wall must be zig-zag shaped; pawns capture diagonally, so for one to protect another, they must all be adjacent diagonally. To complement the fact that they capture diagonally, pawns can move one or two spaces on their first move, making the most basic pawn wall, a two space wide zig-zag one space in front of all of a players more powerful pieces, mostly trivial to accomplish.
Because most chess pieces cannot hop over allied pieces, the complementary mechanics of pawns naturally force you to move them into a defensive structure that is porous enough to let your pieces through but strong enough to repel opponent pieces until the mid-game.
Mechanics: Pawn capturing, one-or-two first move
Resultant Behaviors: building a pawn wall, opening maneuvering room for more powerful pieces
Sins of a Solar Empire
To achieve the next tech tier, you need a certain number of research labs. You can also only build a certain number of structures around each planet. Since there are numerous structures you could build and limited slots to build them in, the only choices you have as a player are to colonize more worlds or upgrade more worlds to get more slots to build more labs to research more techs to protect all those worlds and structures you colonized or built. Planets also incur a cost to operate until they are completely upgraded.
This is an example of completely complementary mechanics.
Mechanics: tech tiers need more research structures, only so many structure slots per planet
Resultant Behavior: colonize more planets, upgrade planets, and later, research more techs to protect your investments
Battlefleet Gothic
A game of space battles in the Warhammer 40,000 universe, whose table-top wargame is probably Games Workshop's most widely known offering. I don't know much about the history of naval type war games, but I get a strong sense from BG that there is a lot of influence from past games going on.
In BG, you must always move your ships (unless a ship is under special orders that involve some tradeoffs) because of the huge amount of momentum they have. This forces play to be dynamic and constantly changing. Ships also are harder to hit from the front and ever harder to hit from the rear, so positioning is a vital component of the game.
By forcing the player to always move their ships, the game is forcing you to make tactical decisions every turn which it rewards you with by making ships have a varying degree of difficulty to attack based on relative position.
Mechanics: forcing the player to move, variable armor based on relative position
Resultant Behavior: fast-paced and tactical play
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