A Solvable Game is a game that whose outcome can hypothetically be predetermined with absolute certainty, if we assume that all players involved are playing in the most optimal manner possible for them.
A game is guaranteed to be solvable if it has no elements of luck present, meaning the game is entirely deterministic. These requirements are not absolute, however, as a game involving luck may still be solvable as long as the result of the game remains identical no matter the result of the random element.
A game being solved implies that there exists some flowchart that a given player can follow when the game starts, and guarantee victory for themselves, regardless of the actions of the other players. Even when a game is solved, this flowchart may not be discovered itself, but rather it was merely proven that the given player must win (or that the game must end in a tie). This is known as an “Ultra-Weak Solution”, while the flowchart described above is known as a “Weak Solution.” A “Strong Solution” to a game is when we have an algorithm that can produce perfect play regardless of the state of the game. All cases are still considered solving the game.
Note that a game whose outcome is heavily stacked against a player is not necessarily solvable; it is impossible to know the outcome of a slot machine despite it typically being, on the absolute majority of times, a loss to the player.
Some examples of solved games include:
Tic-Tac-Toe: the game ends in a draw. In fact, I would wager most of the readers of this page solved tic-tac-toe on their own.
Connect Four: the first player wins.
Checkers: the game ends in a draw.
Some games are hypothetically solvable yet no solution has been found yet, typically due to the game having a large search space. (Meaning - the amount of different ways a game may proceed in is so large, it is unreasonable to fully traverse all of these options even with modern computing power.) An example of such a game is, at the time of writing, Chess.
It has been proven that the amount of possible board states for chess is at least 10120. To put that number into perspective - if we assume a computer can fully analyze a billion (109) chess states per second, and we get one million (106) computers to do nothing but try out chess states until the end of time, it would take us 10105 seconds. Or,
roughly 31688764600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 years.
The universe would explode a few times before you finished half of it.