Here’s a game that you can play for real against a human opponent. If the administrators don’t mind, you can play it right here in the comments.
Pract is played using finite sequences of integers, called “sequences.”
To start, each player chooses a well-defined infinite set of sequences, such that every sequence is either demonstrably in or demonstrably out of the set. The game ends when one player guesses the other’s set.1
Once they have picked their sets, the players take turns.
- A player specifies a sequence by writing each integer, in order, in decimal.
- A sequence’s classification relative to a set is a statement of whether the sequence is in or out of the set.
- A player classifies a sequence by writing the sequence’s classification relative to that player’s chosen set.
- The current player is the player whose turn it currently is.
- A player’s final score is the number of guesses made by that player plus the length of that player’s statement (at the end of the game) of the definition of his or her own set.
On each turn, the current player may either:
- Try to guess the other player’s set, in which case the other player indicates whether the guess was correct or not. Or,
- Specify a sequence, in which case both players classify that sequence.
- If, during the previous turn, the other player made an incorrect guess of the current player’s set, the specified sequence must have an opposite classification relative to the actual set than it does relative to the guessed set.
- If the other player instead specified a sequence, the two sequences (the one specified by the current player this turn and the one specified the other player on the previous turn) must have opposite classifications relative to the current player’s set.
When a correct guess is made, each player states the definition of his or her own set. The player who guessed correctly makes this statement second. Then the game ends.
Winning and Losing
If the player who guessed correctly has a lower final score than the other player, the player who guessed correctly wins and the other player loses. Otherwise, both players lose.
If a player withdraws from the game during his or her own turn, both players lose.2
Statement lengths affect final scores, so language, notation, and encoding are relevant. In most cases, common sense, context, and the medium itself should suffice. (Pract is meant to be played over the Internet in a forum, chat room, mailing list, or other similar medium.)
Pract is meant as way to practice both reasoning and being reasonable. Those who would rather argue well than play well should spectate.
Here’s an example of the beginning of a game between Alice and Bob:
alice> 2, 4, 6 alice> in bob> out bob> 7, 2, 4 bob> in alice> out
The above example shows two turns: the first is Alice’s and the second is Bob’s. Both players have used their turn to specify a sequence rather than make a guess, and each player classifies each sequence as required by the rules.
Bob is being redundant when he classifies the sequence he specified. The rules prevent him from selecting a sequence that is out of his own set because the sequence specified by Alice is out of his set.
Here’s an example showing the end of a game:
cathy> 5, 7, 8 cathy> out dave> out dave> increasing cathy> no cathy> 3, 2 cathy> in dave> out dave> 4, 4, 2, 2 dave> in cathy> out cathy> even numbers dave> yes dave> all even cathy> odd sum dave> length 8 + 5 guesses = 13 cathy> length 7 + 4 guesses = 11
The first turn shown is Cathy’s, and she uses it to specify a sequence. On the next turn, Dave makes an incorrect guess. Then there are two more turns on which a sequence is specified. Finally, Cathy ends the game by making a correct guess. The players state their sets and calculate their scores. From the calculations, we can infer that there were several previous guesses by each player.
Cathy is being redundant when she classifies
3, 2. The rules prohibit her from selecting a sequence that is both non-increasing and out of her set because Dave incorrectly guessed on the previous turn that her set was the set of all increasing sequences. (In a typical game, nearly half the classifications are redundant. The redundancy makes it much easier to verify that players are following the rules.)
The players are not using full sentences or punctuation to describe the sets, and Dave’s description of his own set is shorter than Cathy’s equivalent description when she guesses it. This is reasonable, as long as it does not introduce ambiguity or involve silly things like languages invented on the fly.
There may be situations where it is not clear how the rules apply. For example, the correctness of a guess may depend on some open question in mathematics. So long as players try to avoid such situations, rather than cause them, they should be rare and resolvable.
ETA: New rule for withdrawing. My main concern from the beginning was that both players would be stumped by their own biases and unable to finish the game. The discussion on how to play in bad faith gave me an idea for how to deal with that. ↩