Thinking Under Fog

Four short lessons on deciding when you cannot see everything: reading what is visible, what information is worth, pretending fairly, and staying steady when things go wrong.

Spark & Whetstone · sparkwhetstone.org/learn/thinking-under-fog · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Read what you can see

In a hidden-army game you never see the other side’s ranks, but you see plenty: which pieces have moved, how far, and which ones stayed still all game. Each of those is a fact you can reason from.

A piece that moved more than one square in a straight line can only be a scout. A piece that has never moved might be a bomb or the flag — or simply a piece that has not been needed yet. Facts narrow the possibilities; they rarely settle them.

The same game was invented more than once: a version in Paris before the First World War, and similar games in China and the Philippines where a referee says only who loses each battle. People keep reinventing it because deciding with half the picture is a common human situation.

Question. An opponent’s piece has not moved for the whole game. What is the most careful conclusion?

It is certainly the flag.
It could be a bomb or the flag, or a piece that has not been needed yet.
It must be a weak piece the opponent is hiding.

Say it back. In your own words: what can a piece that never moves tell you, and what can it not tell you?

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Lesson 2. Information has a price

Strong players often spend a small piece early, on purpose, just to learn what a hidden piece is. They fall behind in material and gain knowledge that shapes every later move.

A rough way to weigh it: how likely is the probe to tell you something useful, times how much that knowledge is worth, compared with what you give up. If the chance is one in two, the knowledge worth eight, and the cost two, the probe is worth about two in your favour.

The same reasoning applies off the board: a test, a second opinion, a small experiment. Paying for information is sensible when what you would learn changes what you do next.

Question. A probe costs you 2. There is a 1-in-4 chance it reveals something worth 8. What is it worth on average?

About 0 — it breaks even.
About 6 in your favour.
About 2 in your favour.

Say it back. In your own words: when is it worth paying something just to find out more?

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Lesson 3. Pretending fairly

Moving a small piece slowly, the way a strong one would move, is a bluff — and in this kind of game it is fair. Both players agreed to it by sitting down to play. Inside that agreement, pretending is part of the game rather than a trick on a person.

Outside it, the same move would be deception. That is why, when you play with someone new to the game, especially a child, it helps to say the frame out loud first: "pretending is allowed in this game, for both of us."

Younger children find this harder with people they know well than with strangers, so an explicit agreement matters more, not less, at home.

Question. You are about to play a hidden-army game with an eight-year-old for the first time. What helps the bluffing feel fair?

Never bluff, so the game stays honest.
Say first that pretending is allowed for both of you in this game.
Bluff without mentioning it; they will work it out.

Say it back. In your own words: what makes a bluff fair inside a game, and when would the same move not be fair?

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Lesson 4. Staying steady

In hidden-information games you will lose strong pieces to things you could not see. That is not a mistake; it is the game. The useful question afterwards is whether the decision was sensible given what you knew, not whether it worked out.

Players who do well over many games tend to think in games rather than single moves: one bad surprise does not change the plan unless it changes what you know.

A short pause after a setback — a breath, a look at the whole board — is often worth more than a quick reply.

Question. Your strongest piece is removed by a hidden bomb you had no way of knowing about. What is the most useful way to look at it?

It was a bad move, so play more cautiously from now on.
Ask whether the move was reasonable with what you knew, and what you have just learned.
The game is lost; resign.

Say it back. In your own words: how would you tell a bad decision apart from a good decision that met bad luck?

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At the table: The two-cup table

Players: 2 or more, any ages. You'll need: Two cups and a small object (a coin, a button).

One person hides the object under a cup while the other looks away.
Agree a price: finding it under the left cup scores 1, under the right cup scores 2 for the hider if missed.
Play ten rounds, then talk: did the hider fall into a pattern? Could the seeker read it?
Swap roles and try hiding at random, about two times out of three on the left.

Answers

It could be a bomb or the flag, or a piece that has not been needed yet. — Never moving is consistent with a bomb or the flag, which cannot move, but also with any piece that simply was not used. It shortens the list; it does not name the piece.
About 0 — it breaks even. — One quarter of 8 is 2; subtract the cost of 2 and the expected value is 0. At that point it is a judgement call; at better odds it pays.
Say first that pretending is allowed for both of you in this game. — Naming the frame makes the bluff part of a shared game. It costs a little surprise and keeps the game feeling fair.
Ask whether the move was reasonable with what you knew, and what you have just learned. — A reasonable decision can still meet bad luck. You have also learned where a bomb is, which is useful for the rest of the game.