Game theory: Why do top players play many openings?

Fact 1: Top chess players play many different openings - and it is advantageous for them to do so.

Fact 2: Chess is a sequential game with complete information.

These two facts almost contradict. The perfect chess player has no need to play a mixed strategy. A mixed strategy, in the chess context, involves playing different moves in the same board position. There is no need to do this to play perfect chess because information is complete. In a game like poker, where some information is hidden from your opponents, the perfect player does mix strategies (do different things in response to the same game state) for the purpose of concealing their private information otherwise their opponents might learn their hand from their actions, but there is no such incentive in chess.

While the perfect player has no need to mix over strategies, but there's a large advantage to playing many different strategies for very strong, but imperfect players. This is because, when imperfect players play each other, chess is no longer a game of complete information!

The game of chess is not just the sequential, complete information game played at the board. There is an initial stage of the game that happens before move 1 called preparation. Preparation is costly and the decision of how to prepare and for what is a decision players make under uncertainty. Preparation involves players studying the chess game tree either broadly or deeply. If a player only plays one opening, it is less costly to explore that opening deeply. If a player plays many openings, it is risky to prepare deeply for an opening they may not play. So, preparation will tend to be broad and not as deep.

Therefore, to reduce the depth at which your opponent prepares for you, it is necessary to credibly be willing to play many openings. This credibility is established by playing different openings in other games.

Chess is only a game of complete information for perfect players. For very strong but imperfect players, chess has private information. This is why chess is exciting to watch and why you are never quite sure what will happen next.