In order to answer this, you need to specify what you mean by "no luck". I assume you mean "no randomness". But here, we need to separate randomness of the game and randomness of the players. For the purposes here, I will assume that setup is part of the game, so an Abstract Strategy Game that is deterministic except for the choice of player (like Chess) isn't deterministic enough.
Absolutely No Randomness
Rock Paper Scissors is a complex example. While the mechanisms of the game are deterministic, the players may not be. Game theory says that the optimal strategy is to play at random, making each move with 1/3 probability. I heard a story of a player who placed 4th in a Rock Paper Scissors world championship by memorizing random sequences and playing them [citation needed] (I have not been able to find any corroborating evidence of this, so this is just hearsay).
Fortunately, the game-theory category that contains Rock Paper Scissors, namely Simultaneous Games, also includes games that are both deterministic and have deterministic optimal strategies. The Prisoner's Dilemma is the most famous example.
Another way to get to Randomness-Zero is to take an almost deterministic game like Chess and make a deterministic mechanism for choosing who goes first. A trivial example of this is for two players to agree to play not just to play Chess but also agree as which player will play which side; now we have a deterministic game. Or, you can add in bidding to determine who plays which side. One way to play Armageddon Chess (Chess played with very short time totals) is for players bid an amount of time they are willing to take if playing white, and whichever player bids lower plays white with that amount of time on their clock (black plays with some preset amount of time, like 5 minutes). If you do a simultaneous bid, this becomes a totally deterministic game.
The problem with games of this type is that there exists an optimal strategy and once found, the game is done. Most Checkers tournaments since the mid 1900s have implemented "three-move restriction", where the first three moves of each side are made randomly, because otherwise too many games are draws. At this point, this is essential, as Checkers has been solved as a game, and the base game (without a randomized opening) is in fact a draw.
No Structural Randomness
If we relax the constraint to just "the game itself can't do anything random" and ignore the players, we have a lot more options. Under this construction, Rock Paper Scissors is a deterministic game. We can then use this to bootstrap a game that is deterministic aside from setup into a totally deterministic game. Here are some examples:
- Play Rock Paper Scissors to determine who plays white in a game of Chess
- Play Rock Paper Scissors to determine who plays black in a game of Go
- Have a Rock Paper Scissors tournament to do country assignment in Diplomacy
This is a dangerous path, because under this model, we can actually remove the randomness from any game. If you're willing to go through sufficiently many iterations of Odds and Evens, you could replace die rolls and deck shuffling with "deterministic" processes.
Also, I'll add the obligatory caveat that dice rolls, deck shuffling, and digital random number generators are all only pseudorandom. There is even peripheral arguments that people can become skilled at rolling dice.
Games of Physical Skill
One other bit that drifts dangerously into another realm of philosophy are games of physical skill. Here are some examples that don't use a coin-toss to decide which player goes first:
The problem here is that most physical games depend on tiny differences in how objects interact with their environment to be interesting. Pro NBA players don't make 100% of foul shots even though that is supposed to be a totally controlled environment. These tiny differences are what people have attributed as "random" and are also what allows us to use dice or shuffling cards as effective randomness for other games.
The other problem is we don't know if the universe is deterministic or random on a quantum level. But, if the universe is deterministic, then there is no such thing as randomness, and this whole conversation is moot. This is where I'm going to stop and leave it to the physicists and philosophers.
