Combinatorial Game Theory
who wins, and by how much
Series

Approximation — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the temperature that bounds the loss. Temperature runs up the page and value across it. Each wall is where a player is willing to move once a tax of that much is charged per move; above the temperature at which they meet, neither wants to move and the position is worth its mean value. The height of the meeting point is what is at stake.

    A rule that is never right and cannot be far wrong

    Playing the hottest component is not optimal — over 440 measured lines it costs something on 17 of them. What makes it worth having is that the cost is bounded by the temperature, provably, and that the same test run with the ordering reversed breaks the bound on 54.

    part 1 · complexity
  2. The bound, and the cost, on four pools. Hottest-first play against optimal play on boards of two, three and four components drawn from four pools, 4,240 lines in all. The bound on the cost is the largest temperature on the board, which reaches 3; the worst cost measured anywhere is 1, and 100 lines meet the bound exactly.

    The bound names the hottest part and the cost does not

    Moving in the hottest component costs at most the largest temperature on the board, and that bound is attained: 100 lines of 4,240 pay exactly it. It is still the wrong quantity. Across four pools and boards of two, three and four parts the cost is nothing on 90.8% of lines and otherwise takes one of two values — half a point or one — on boards whose largest temperature runs to three, and it exceeds the coolest component on 13 lines and twice it on none.

    part 2 · complexity
  3. The more cheap fights there are, the cheaper the rule is. Hottest-first play against optimal play over 10,410 lines, grouped by how many components share the lowest temperature on the board. Lines costing more than that temperature occur only where one or two components share it; over the 3,230 lines with three or more, none does.

    The cheap fights make the rule cheaper

    A conjecture stands that playing the hottest part costs at most the coolest temperature times the number of parts sharing it — proposed on a range where that number never exceeds two. Swept to five-part boards over 10,410 lines it is false, and false the other way round: every line costing more than the coolest part has one or two parts at that temperature, and over the 3,230 lines with three or more, not one does.

    part 3 · complexity

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