Imperfect — when you can’t see their cards
In hide-and-seek games (and in hidden-army games like the ones on this site) the best plan is often a mix you choose at random on purpose. Try to beat a seeker that learns your habits, then work through four cases.
Exploit Me
A case · you decide
Aiyana spends a piece to learn a piece
Aiyana — a club player deciding whether to sacrifice a small piece to find out what a hidden one is
Two of her opponent's pieces haven't moved. One of them is probably the strongest piece on the board. Aiyana can send a 4 to strike one of them. If she learns where the strongest piece is, she estimates it's worth about 8 points of position. Losing the 4 costs her about 2.
If the chance that the struck piece is the strong one is about one in four, the probe's expected value is 0.25 × 8 − 2 = 0. If it's one in two, the value is 0.5 × 8 − 2 = 2.
Strong players do exactly this: they spend cheap pieces early, sometimes falling behind in material, because what they learn changes every later decision.
Aiyana thinks it's about even odds (one in two) that the piece is the strong one. Is the probe worth it?
Aiyana keeps a rough note of probe costs and what they bought. Over a season she spends fewer pieces on guesses and more on the questions that change her plan.
Devika scores her own guesses
A case · you decide
Devika scores her own guesses
Devika — a player who writes a probability next to each guess about a hidden piece, then checks them at the end
Devika marked three hidden pieces: 90% "bramble", 60% "big piece", 50% "the Moonstone". At the end, the first was a bramble, the second wasn't big, the third wasn't the Moonstone.
Her Brier score is ((0.9−1)² + (0.6−0)² + (0.5−0)²) ÷ 3 = (0.01 + 0.36 + 0.25) ÷ 3 ≈ 0.21.
Saying 50% about everything would score 0.25. She did a little better than that — on three guesses, which is far too few to be sure of anything.
What is the honest conclusion from three guesses and a Brier score of about 0.21?
Devika keeps marking guesses, and only reads anything into her score once she has a few dozen.
Joaquin and the slow-walking runner
A case · you decide
Joaquin and the slow-walking runner
Joaquin — a player deciding how far bluffing should go in a game with a younger sibling
Joaquin moves a weak runner one square at a time, the way a strong piece would move, and his younger brother backs away from it. Later his brother says it "felt like lying." The rules allow it. Joaquin wonders what he owes his brother as a teacher, not just as an opponent.
In a hidden-information game, pretending is part of the rules both players accept when they sit down. Outside that frame, the same move would be deception.
With a beginner, the frame may not be clear yet. Some players name it out loud first; some soften it; some play at full strength because that is what was agreed.
What does Joaquin do next game?
Joaquin settles on naming the frame before each game. Whatever he chooses, he can say why.
Aiyana and Devika audit the machine
A case · you decide
Aiyana and Devika audit the machine
Devika & Aiyana — two players reading about a program that beat strong humans at a hidden-rank game
A news story says a program "has mastered hidden information." The paper reports a record against a set of expert players over a few dozen games, and that it was trained with far less compute than an earlier system.
Aiyana notes it is a real and impressive result. Devika lists what the claim depends on: the format played, the opponents, the number of games, and whether the opponents had time to adapt to it.
A few dozen games against a handful of people is strong evidence about those matches. It is weaker evidence about every human, every format, or "hidden information" in general.
Which summary fits the evidence best?
Their rule for headlines: restate the claim in the size the evidence actually supports.