Totally symmetric non-random games?

I´m not sure if there are symmetric games with no randomness. But there are some with "symmertic randomness", where the random choices effect both, for example Ricochet Robots.

I remember a WW1 dogfighting game I used to play with my dad when I was younger where you each had a book with an image of your opponents relative position and a choice of aerial manoeuvres. Both players chose their move at the same time and told each other the corresponding page number where they then had to check the result of both moves combined.

Sorry for the hideous description but it's the only game I can think of that comes close.

I'm not really a games theorist (in the sense of the branch of maths called 'game theory') but the following observations seem applicable here:

Note: mathematical game theorists mean very precise things by their terms, which are sometimes different from how board gamers use those words. In particular, a "strategy" in game theory is a complete plan for every eventuality in a game - something which tells you what decision to make whenever you get to make a move.

When there is no luck or hidden information in a game (i.e. when you know the results of your moves in advance), then, for perfect players, the result of the entire game is already decided before you start. Chess is a game like this - either there's a single winning "strategy" (in the game-theoretic sense) for White, or for Black, or both players have a non-losing strategy leading to a draw. What makes chess interesting as a game is that we haven't figured out what the perfectly optimal strategy/ies are (in game theoretic terms, chess is not "solved").

But in the abstract, chess has an optimal "pure" strategy. Suppose chess is solved, and it turns out that White is the winner. Then there is (at least one) correct move White can make at every possible turn which will result in eventually checkmating Black.

However, in games like scissors-paper-stone, players don't know in advance what the result of their move will be (because it depends on what the other players choose). It's known in game theory that you can guarantee (on average) minimal loss in these games by using a "mixed strategy": one in which you take random decisions. In scissors-paper-stone, if you play completely at random, nobody can get an advantage over you by strategic play. In other games, the randomness of the "unbeatable" (in game theory, this means "unbeatable except by pure luck") mixed strategy might be much more complicated, but the principle is the same.

Of course, a game could have both types of complexity: chess-like (where it's effectively impossible to determine the best pure strategies), and mixed-strategy (where the unbeatable strategy requires random choices), without even having any randomness in the game rules. Imagine playing Race For The Galaxy with the deck of cards shuffled and turned face-up at the start of the game (and every player's hand face up). You'd know exactly which cards were coming up, but because of the simultaneous role selection, you still wouldn't know who would get them.

You could also have a totally symmetric game in which players took turns: imagine a game which involved playing two games of chess at once, one as White and one as Black. The turn sequence would run as follows: Player A takes a move as White (on the first board), then Player B takes a move as White (on the second board), then Player B takes a move as Black (on the first board), then Player A takes a move as Black (on the second board).
This is effectively what happens when you play a 2-game match of chess. (My girlfriend tells me that in some chess tournaments, they play simultaneously.)

Games like "Revolution!" has zero-randomness determined by cards or dice. It is solely trying to outguess your opponents bids on influential people. This makes it more like Rock-paper-scissors in essence.

For games where all the randomness starts before the game, try "Endeavor." This one does not have simultaneous turn resolution, but the "starting player" rotates around amongst the players, so it is something you plan for. You're better off getting a description of play from board-game-geek than me, though.