I think minesweeper makes a nice analogy with many of the ideas of epistemic rationality espoused in this community. At a basic level, it demonstrates how probabilities are subjectively objective—our state of information (the board state) is what determines the probability of a mine under an unknown square but that there really is only one correct set of mine probabilities. However, we also run quickly into the problem of bounded cognition. In this situation we resort to heuristics. Of course, heuristics are of varying quality, and it is possible with mathematics, to make better heuristics.
For example, if you find that the set of possible configurations of mines in a particular neighborhood is partitioned into, say, those that involve k mines and those that involve k+1 mines, then you can get a pretty good estimate of the probability that the true configuration will be in one partition or the other. It depends on the density of mines under the squares that aren’t known (something like a prior).
There are situations which come up in which one must decide just what one’s goals are—is it to survive the next click or to maximize the chance to win the game? Often these two goals result in the same decision, but sometimes, interestingly, they result in different decisions.
I like to play on a 24x30 board (maximum allowed on windows machines), with 200 mines. This makes the game rarely about deductive logic alone. Situations in which probability theory is necessary come up all the time with this density.
I think minesweeper makes a nice analogy with many of the ideas of epistemic rationality espoused in this community. At a basic level, it demonstrates how probabilities are subjectively objective—our state of information (the board state) is what determines the probability of a mine under an unknown square but that there really is only one correct set of mine probabilities. However, we also run quickly into the problem of bounded cognition. In this situation we resort to heuristics. Of course, heuristics are of varying quality, and it is possible with mathematics, to make better heuristics.
For example, if you find that the set of possible configurations of mines in a particular neighborhood is partitioned into, say, those that involve k mines and those that involve k+1 mines, then you can get a pretty good estimate of the probability that the true configuration will be in one partition or the other. It depends on the density of mines under the squares that aren’t known (something like a prior).
There are situations which come up in which one must decide just what one’s goals are—is it to survive the next click or to maximize the chance to win the game? Often these two goals result in the same decision, but sometimes, interestingly, they result in different decisions.
I like to play on a 24x30 board (maximum allowed on windows machines), with 200 mines. This makes the game rarely about deductive logic alone. Situations in which probability theory is necessary come up all the time with this density.