Bounding Game Temperature Using Confusion Intervals

Carlos Pereira dos Santos; Richard J. Nowakowski; Svenja Huntemann

REVIEW 2 major objections 6 minor 20 references

Bounding Game Temperature Using Confusion Intervals

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any class of short games, temperature is bounded by K/2 + J, where K and J are the maximum confusion-interval lengths of the games and their options.

desk verdict First general boiling-point bound, but the thermic-version lemma has a real gap that needs fixing. read the letter →

arxiv 1908.08471 v1 pith:XNS52KMQ submitted 2019-08-22 math.CO

Svenja Huntemann , Richard J. Nowakowski , Carlos Pereira dos Santos This is my paper · ORCID

classification math.CO MSC 91A46

keywords combinatorialgametheorytemperatureboilingpointconfusionintervalthermographDomineeringSnortshortgames

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves the first general upper bound on the temperature of any class of short combinatorial games. It shows that if every game in a class has confusion interval of length at most $K$ and every option has confusion interval of length at most $J$, then every game in the class has temperature at most $K/2+J$. Temperature measures how much a position's advantage can shift, and the boiling point of a class is the largest temperature in it, something previously bounded only for individual games. The paper also proves a way to estimate the confusion interval itself: the number of passing moves Right needs to win in $G^L-G$ gives an upper bound on $\ell(G)$. It applies both tools to prove that Domineering snakes on a $2\times n$ board have temperature at most $3$ and that Snort on a path has temperature at most $7.5$.

What carries the argument

The argument is carried by the thermograph of a game, the plot of its Left and Right stops as the game is cooled, together with the thermic version $\tilde G$, a game with a single Left option and a single Right option that has the same temperature as $G$. Decomposing the left wall of the thermograph into vertical and oblique segments below the mast, the paper observes that the vertical segments are paid for by an option's confusion interval while the oblique segments are paid for by the confusion interval of $G$ itself, and the temperature is the sum of both. The length of a confusion interval is $\ell(G)=LS(G)-RS(G)$, and Proposition 27 turns bounding it into a strategic question: if Right wins $G^L-G-K+\varepsilon$ moving second, then $\ell(G)\le K$. This converts temperature control into checking a difference game, which is how the Domineering and Snort examples are obtained.

What would settle it

A brute-force search over all short games of small game-tree depth looking for a game $G$ with $\ell(G)\le 6$, $\ell(G^L),\ell(G^R)\le 6$, and $t(G)>9$ would settle the tightness claim; the paper's own sequence approaches $9$ from below, so a single exceeding value would refute Theorem 25.

Watch

Extended reading notes

Core claim

The central discovery is that the temperature of a short game $G$ is controlled by the lengths of its confusion interval and those of its options. Writing $\ell(G)$ for the distance between Left and Right stops and $\tilde G$ for a thermic version of $G$ (a game with one Left option and one Right option whose temperature equals that of $G$), the paper proves $t(G)\le \ell(H)+\ell(G)/2$, where $H$ is the option on the side whose thermograph wall has more vertical segment. Consequently, for any class $\mathscr S$ with uniform bounds $\ell(G)\le K$ and $\ell(G^L),\ell(G^R)\le J$, the boiling point satisfies $BP(\mathscr S)\le K/2+J$. An explicit sequence of games with $K=J=6$ shows the bound is tight, with temperatures approaching $9$. This is the first boiling-point bound that applies to every class of short games.

Load-bearing premise

The proof assumes that for every hot game one can choose a single Left option and a single Right option so that the two-option game has exactly the same temperature as the original; the paper's justification is brief, and the main bound would not cover all short games if this failed.

Editorial extensions

If this is right

For any class of short games whose confusion intervals and option confusion intervals are uniformly bounded, the boiling point is finite and at most $K/2+J$, with no per-game computation needed.
The bound is sharp in general: for the class of games with $\ell(G),\ell(G^L),\ell(G^R)\le 6$, the temperatures of the displayed sequence approach $9$, so the boiling point is exactly $9$.
Domineering snakes fitting in a $2\times n$ grid have temperature at most $3$, proved by showing Right wins $H^L-H-2$ moving second and then applying Theorem 25.
Snort on a path of any length has temperature at most $7.5$, proved by showing Right wins $H^L-H-5$ moving second for paths of length at least $5$.
Since $t(G+H)\le\max\{t(G),t(H)\}$, these bounds apply to disjunctive sums of such positions, not just to connected boards.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's explicit claims, the same strategy should yield sharper constants for classes with few threats: when thermic options can be identified without building the whole thermograph, only their confusion intervals enter the bound, and the paper notes that replacing the constant $5$ by $4+\uparrow$ in the Snort proof would lower the path bound to $6$.
For games dominated by threats, like $\{\{10|1\}|-1\}$, the confusion interval can stay small while the temperature is large, so the $K/2+J$ bound is conservative exactly where threats matter; classes of games with mild threats may admit much better bounds.
A testable extension is to apply Proposition 27 to other bounded-degree rulesets, such as Snort on graphs of bounded degree, where the paper's degree conjecture says the temperature is at most the degree of the board.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my quick read. The paper's central contribution is real: Theorem 25 gives the first general upper bound on the boiling point of a class of short games in terms of confusion intervals. That's a useful tool, and the tightness example shows the bound is sharp in a broad class. The applications to Domineering snakes (temperature <= 3) and Snort on paths (<= 7.5) are concrete and new, and Proposition 27's method for bounding confusion intervals via difference games is a nice trick. I'd credit it for being the first general handle on this problem.

The soft spot is exactly where the stress-test note lands. Theorem 15 claims every hot game has a thermic version: two options G^L, G^R such that t({G^L|G^R}) = t(G). The proof observes that at T = t(G), the stops of the two selected cooled options are equal, and then concludes the new game has temperature T. That doesn't follow. Equal stops at T only give temperature <= T. The new game's thermograph is built only from those two options, so it could plausibly meet strictly earlier, because the other options of G that delay the mast are gone. The paper gives no argument—order, dominance, or mean—to rule this out. I didn't find a counterexample either, but the gap is real and it's load-bearing: Theorem 24 and 25 both rest on it. So the main theorem is not fully proven as written.

Two smaller things. The segment-length step in Theorem 24 ("T_L_vert is at most the length of the oblique segments of ~G^L") is asserted rather than demonstrated; it's plausible but deserves a formal sentence. And the strategy arguments for Domineering and Snort are informal, with diagrams doing heavy lifting; a skeptical reader will want more precise cases. Also careful: there's a small inconsistency in the path-length claim in Section 5.1—the table covers lengths up to 12, but the proof says up to 14. Minor.

Who's this for? Researchers working on temperature theory and partizan games. The main idea is good enough that I'd send it to a serious referee, but the thermic-version lemma needs to be fixed or the theorem restated with an extra hypothesis. I'd want to see that addressed before I cite it.

Recommendation: not a desk reject. Engage it, but ask for a rigorous repair of Theorem 15 and the segment estimates.

Referee Report

2 major / 6 minor

Summary. The paper studies the temperature of short partizan combinatorial games. It defines the confusion interval C(G) of a game G and its length l(G), and claims two main results. First, Theorem 25 states that if S is a class of short games with l(G) <= K and l(G^L), l(G^R) <= J for every G in S, then the boiling point BP(S) is at most K/2 + J; a family of examples is given to show that this bound can be tight. Second, Proposition 27 gives a method to bound l(G) by showing that Right wins in G^L - G - K (with Left moving first), and the method is applied to Domineering snakes in 2 x n grids (temperature at most 3) and to Snort on paths (temperature at most 15/2). The proof route is: Theorem 15 asserts that every hot game has a thermic version with a single Left option and a single Right option; Theorem 24 bounds the temperature of a thermic version in terms of l of the game and of one of its options; Theorem 25 then follows by taking maxima over the class.

Significance. If the main theorem is correct, it is a genuinely general quantitative statement: for the first time, an upper bound on the boiling point of an arbitrary class of short games follows from a simple local parameter, the lengths of confusion intervals. The tightness example shows that the K/2 + J form cannot be improved in general, and the method of bounding l by second-player strategies in G^L - G is elementary and potentially reusable. The paper is self-contained, does not fit parameters to data, and clearly acknowledges the gap between the general bound and the much smaller temperatures known or conjectured for Domineering and Snort. However, the proof of the load-bearing Theorem 15 is not correct as written, so the central theorem is not yet established by this manuscript.

major comments (2)

[Section 3, Theorem 15] The proof is not valid as written. From the equality of the two stops of G_T and the fact that the selected options realize those stops, one only obtains LS({G^L|G^R}_T) = RS({G^L|G^R}_T) = m, which gives t({G^L|G^R}) <= T; nothing in the argument rules out an earlier meeting of the two walls. The final 'therefore' is actually false for arbitrary options realizing the stops: for G = {3, {5|1} | {1|-3}} one has t(G) = 2 (for t < 2, LS(G_t) = 3 - t and RS(G_t) = 1, while at t = 2 both stops are 1), and both G^L = {5|1} and G^R = {1|-3} realize the stops of G_2, yet H = {{5|1} | {1|-3}} is equal to 1 (H - 1 = {{4|0} | {0|-4}} is a second-player win), so t(H) < 2. Because Theorem 24 and consequently Theorem 25 rely on thermic versions, this gap is load-bearing. The existence claim is probably repairable: one should choose options whose sheared walls form the top segments of the two walls of G's thermograph, and use the fact that the difference of those walls is nonincreasing and strictly positive just below T to conclude t({G^L|G^R}) = T. This repair needs to be written out in full.
[Sections 4.1 and 5.1] The applications rest on informal strategy arguments. In Proposition 28, the reduction for H = G^L and H = G^R is described only for a representative split, and it is not fully shown that every Left option of H is covered. In Theorem 29, key assertions such as 'Left then has at most three additional non-copiable moves' and 'the influence area is already bounded' are stated without proof, and the induction over end-decorated paths is only alluded to. These are the advertised numerical bounds, so the strategies should be formalized or replaced by a precise inductive argument.

minor comments (6)

[Abstract] The abstract contains the duplicated phrase 'for for every' which should read 'for every'.
[Section 2, after Definition 1] The recursive notation 'P = {P^L|P^L}' should be '{P^L|P^R}'.
[Section 3, Theorem 15 proof] The expressions 'G^L_{t(G)} - t(G)' and 'G^R_{t(G)} + t(G)' need parentheses, e.g. '(G^L)_{t(G)} - t(G)', to avoid ambiguity.
[Section 3, Proposition 27] The hypothesis should read 'for every Left option G^L', and the proof should state that the chosen G^L is the one attaining LS(G).
[Section 3, Example 26] The notation '±{9|3}' is not defined, and the example does not explicitly verify that each G_n satisfies l(G_n), l(G_n^L), l(G_n^R) <= 6; a short verification should be added.
[Section 3, Theorem 24 proof] The sentence 'the length of these oblique segments are at most the distance between the Left and Right stops' is imprecise: the intended inequality is T_Lvert <= l(~G^L), and the intermediate step should be stated more carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: bound is derived from independently defined confusion intervals; Theorem 15 proof has a non-circular gap.

full rationale

The claimed bound is not circular. Theorem 25 follows from Theorem 24, which bounds t(G) by decomposing the thermograph of a thermic version into vertical and oblique segments and using T_Lvert ≤ l(~G^L) and T_Lobl + T_Robl = l(~G) ≤ l(G). The quantity l(·) is defined independently of temperature as LS − RS, and the method for bounding l(G) in Proposition 27 uses difference-game strategies (Right wins G^L − G − K + ε with Left to move) rather than any fitted constant. The tightness example constructs a sequence with temperatures tending to K/2 + J, so the bound is not an input renamed as a prediction. No load-bearing self-citation occurs: background facts are cited to Siegel and Winning Ways, and the applications use independent strategy arguments. A genuine non-circular weakness exists: Theorem 15 asserts the existence of a thermic version, and its proof concludes t({G^L|G^R}) = t(G) from the fact that the two selected cooled options realize the stops at t(G). That only shows the new game is tepid at t(G); it could have become tepid earlier. Indeed, for G = {{10|1} | −1}, t(G) = 2, but the proof's selected options give {{6|1} | 1}, whose temperature is 0. Thus the main theorem's proof depends on an unjustified existence step. This is a correctness risk, not a circular reduction: the bound is not equivalent to its assumptions by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim relies on standard combinatorial game theory definitions and theorems, all sourced from the literature. No free parameters are fitted to data, and no new entities are postulated. The only potentially load-bearing internal result is Theorem 15 (existence of thermic versions), which the paper proves, albeit tersely.

assumptions (4)

standard math Standard combinatorial game theory framework: short games, normal play, disjunctive sum, outcome classes.
Used throughout; definitions from Siegel [17].
standard math Cooling and temperature as defined in Definitions 9 and 10, and Fact 11 (temperature of a sum is at most the maximum temperature of the components).
Background from Siegel [17, Section II.5].
standard math Thermograph properties: the left and right walls and the mast behave as described in Definition 12 and Example 13.
Used in the proof of Theorem 24.
standard math Stop inequalities, including RS(G)+LS(H) <= LS(G+H) <= LS(G)+LS(H), used in Lemma 8 and Proposition 27.
Stated in Section 2 as known facts.

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Cite this review

Pith. "Pith review of Bounding Game Temperature Using Confusion Intervals." pith.science (2026). https://pith.science/paper/XNS52KMQ

@misc{pith2026190808471,
  author       = {Pith},
  title        = {Pith review of: Bounding Game Temperature Using Confusion Intervals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XNS52KMQ}},
  note         = {Machine review of arXiv:1908.08471}
}

abstract

For combinatorial games, temperature is a measure of the volatility, that is, by how much the advantage can change. Typically, the temperature has been measured for individual positions within specific games. In this paper, we give the first general upper bounds on the temperature for any class of games. For a position $G$, the closure of the set of numbers $\{g\}$ such that $G-g$ is a first player win, is called the confusion interval of $G$. Let $\ell(G)$ be the length of this interval. Our first main result is: For a class of games $\mathscr{S}$, if there are constants $J$ and $K$ such that $\ell(G^L),\ell(G^R)\leq J$ and $\ell(G)\leq K$ for for every $G\in \mathscr{S}$, then the temperature of every game is bounded by $K/2+J$. We give an example to show that this bound is tight. Our second main result is a method to find a bound for the confusion intervals. In $G^L-G$ when Left gets to go first, the number of passing moves required by Right to win gives an upper bound on $\ell(G)$. This is the first general upper bound on temperature. As examples of the bound and the method, we give upper bounds on the temperature of subclasses of Domineering and Snort.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

The economist’s view of combinatorial games

Elwyn Berlekamp. The economist’s view of combinatorial games. InGames of no chance (Berkeley, CA, 1994), volume 29 ofMath. Sci. Res. Inst. Publ., pages 365–405. Cambridge Univ. Press, Cambridge, 1996

work page 1994

[2] [2]

Temperatures of games and coupons

Elwyn Berlekamp. Temperatures of games and coupons. In Urban Larsson, editor,Games of No Chance 5, volume 70 ofMathematical Sciences Research Institute Publications, pages 21–33. Cambridge University Press, 2019

work page 2019

[3] [3]

Berlekamp

E.R. Berlekamp. Blockbusting and domineering.J. Combin. Theory Ser. A, 49(1):67–116, 1988

work page 1988

[4] [4]

Berlekamp, J.H

E.R. Berlekamp, J.H. Conway, and R.K. Guy.Winning ways for your mathematical plays. Vol. 1. A K Peters Ltd., Wellesley, MA, second edition, 2004

work page 2004

[5] [5]

Monte-Carlo approximation of temperature

Tristan Cazenave. Monte-Carlo approximation of temperature. InGames of no chance 4, volume 63 ofMath. Sci. Res. Inst. Publ., pages 41–45. Cambridge Univ. Press, New York, 2015

work page 2015

[6] [6]

J. H. Conway.On numbers and games. A K Peters Ltd., Natick, MA, second edition, 2001

work page 2001

[7] [7]

Drummond-Cole

G.C. Drummond-Cole. Temperature2 in Domineering. Preprint

work page

[8] [8]

Drummond-Cole

G.C. Drummond-Cole. Positions of value∗2 in generalized domineering and chess.Integers, 5(1):G6, 13, 2005

work page 2005