I have been using some terms idiosyncratically, so it seemed worth defining what I mean when I refer to “variation” and “uncontrolled randomization” as things apart from “randomness.” None of these are fixed dichotomies with hard edges, but I think they work as radial categories.
While I might argue that the general public misunderstands “randomization,” in a gaming context, it seems relatively clear. You roll a die, draw a card, shuffle tiles, pull something from a bag, etc.
What I characterize as “variation” may include randomization, but the randomization happens and is known before the game, not within it. (Whether you randomize within the game or just reveal the results of that randomization during the game is functionally irrelevant.) For example, the board for Settlers of Catan, Kingdom Builder, or Tiny Epic Kingdoms changes each game but is a known quantity when the game starts. You do not shuffle the board mid-game. Many games randomize or let you pick your hero or special ability, so those can vary from game to game, but you can plan and play on that basis. Chess has no such variation, and there have been many proposals over the years to introduce variation on starting positions to reduce the solvability of the game, the importance of opening books, and the advantage that computers have in the fixed game.
In that sense, “variation” is known randomization. It can vary in degree. A Settlers board rearranges a fixed set of hexes, while someone with all 206 (including promos) Dominion kingdom cards has 30,377,090,094,628,145 possible sets of 10. Or you can make it non-random by picking options (or more controlled by vetoing options/sets (“no central desert” in Settlers)).
Randomization also has degrees of control. Deck-building games are games of probability management, so I would characterize the order of your cards as controlled randomization. The DC comics deck-building game introduces more because the cards available to buy each turn are pulled from a deck, a source of uncontrolled randomization (subject to variation by pre-game customization of which cards go into the deck). Settlers falls somewhere in between: you manipulate probability by which numbers you build on, but the dice themselves are a source of uncontrolled randomization. All Settlers players have played the game with 3 10s in a row or with no 6s in the first 20 rolls. (If you are even more opposed to uncontrolled randomization than I am, you can buy a 36-card “dice deck” so that you get all possible rolls at a controlled frequency.)
Randomization does not simply mean “unknown.” Hidden information is often randomized, but they are not the same. Diplomacy is a game where much information is unknown but nothing is random. Bluffing games usually have random factors, but the important parts of play are what you can infer about other players’ pieces and what you do with your own. Random information is not just unknown but unknowable; at most, when it is a random pick from a fixed pool, you can estimate the probability of certain outcomes.
When I say that a game has a high degree of uncontrolled randomization, it implies that when you make a decision, the random component of the result is more important than the decision you made, either due to the amount of random variation in deciding the result or the relative importance of the random component. Some people love and enjoy those kinds of games, and there is certainly a time and a place for not wanting to be in control. I will tend to use it as a condemnation of a game, because I consider meaningful decisions to be the core of playing games.
When I say that a game has a high degree of variation or variability, it implies that the game is less solvable and has more replayability due to differences in game conditions that provide a broadly similar game experience with variations in the details. Higher ranges of variation can produce very different experiences from a base rule set. Because in most systems the number of bad random combinations is greater than the number of good random combinations, the best sources of variation have measures built in to make sure the resulting variation is viable.