Someone in the paradox studies course was planning to talk about some chess sophistry, saying that chess is a boring game, but his reasoning has serious problems. Let me reinforce it a bit:
I’ll first offer a chess variant with slightly altered rules, and start with a math problem:
**Two-move chess:** Modify the rules of chess so that in each round each player may make two consecutive moves, with everything else unchanged. Prove that the player to move first has a strategy of at least a draw.
The proof is very simple:
Proof by contradiction. Assume the first player has no strategy of at least a draw. That is, no matter how the first player moves, the second player can make him lose.
Then the first player’s two moves on the first turn can simply be to bring a knight out and then move it back, which is equivalent to not moving at all, so the position is effectively turned over to the second player, and by assumption the second player has a winning strategy. Contradiction. Therefore the first player has a strategy of at least a draw.
This line of thought is very similar to that classmate’s sophistry, but it makes the point much better! Because this is a rigorous mathematical proof rather than sophistry. The key is the nonconstructive proof by contradiction. Think about the claims of the intuitionists—doesn’t this make it easier to appreciate why intuitionists are so resistant to nonconstructive proofs? Is two-move chess a boring game? Similarly, all games, as long as one can choose to “pause,” give the first player a strategy of at least a draw.
In addition, if a game is played alternately by first player and second player, and each move has only finitely many possible choices, and the game will be decided within finitely many moves (not counting draws), that is to say, if all of its variations can be exhausted, then one can prove that one side must have a winning strategy. I didn’t find a rigorous proof, but I myself came up with a line of proof, and reasoning along these lines should be fine.
I used to have seen something like this—maybe someone invented some combinatorial method or whatever.
My line of proof is this:
Draw a tree diagram. The first player has several (finite) choices on the first move, so draw several branches with solid lines; then under each branch, draw all of the second player’s choices with dashed lines, and continue layer by layer like that. If a move ends the game in a win, then no further branching occurs. If the last line is solid, then the first player wins; if it is dashed, then the second player wins.
Since the total number of variations is finite, this tree diagram is finite.
If the game is finite, then this tree diagram can be drawn.
Then start erasing from the bottom, according to the following rule: if the final line is dashed, erase the solid line connected to the top of that dashed line, as well as the other lines branching from that top node. The reason is that this solid line is a choice made by the first player, and if the first player chooses such a line, then the second player has a winning move somewhere along the way. Since both sides are rational, the first player of course knows that if he goes down this path the opponent may have a chance to win in one move, so the first player will not choose that straight line. Likewise, if the terminal line is a solid line, then erase this solid line together with the preceding dashed line… In this way, each erasure removes at least two lines, and the total number of lines is finite, so this erasing process will definitely come to an end.
In the end, there are only two possible situations when nothing more can be erased: one is that only several solid lines of the first layer remain, and these cannot be erased; in that case, the first player has a winning strategy. The other is that all the lines have been erased; that means the second player has a winning strategy.
Thus, all games are boring: because if all possible moves are finite, then there must be an optimal strategy, so one only needs to mechanically follow the optimal strategy, which is boring. If there is no definite optimal strategy, then only when the game: 1. includes a random factor, then it is just like tossing a coin, leaving it to luck—boring! 2. has unexhaustible moves, that is to say, a game that can never be finished. What’s the point of a game that can never finish? So yes, they are all boring.
Translated from the Chinese original with AI assistance. The original text is authoritative.
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