Bounding game temperature using confusion intervals

Abstract

We consider bounds for the temperatures of combinatorial games. Our first result gives an upper bound on the temperatures of the positions of a ruleset in terms of the lengths of the confusion intervals of these positions. We give an example to show that this bound is tight. Our second main result is a method to find a bound for the lengths of the confusion intervals. This pair of results constitutes the first general technique to bound temperatures. As examples of the bound and the method, we consider the temperature of subsets of positions in Domineering and Snort.

Introduction

You are playing a game, you found what you believe to be a good move. But is it good, great, or good enough? How can you tell? Exhaustive search would take too long and be too boring. Using a computer probably would be illegal and still might not provide an answer. We provide a method that partially answers the questions about great and good enough moves. Deciding whether a move is good is an aesthetic question and will be left to the player.

One method of estimating the worth of a position is to ask how much of a penalty would a player be willing to pay to move in it. That is, for a game G, find a number (penalty) x such that both Left and Right are indifferent between (i) playing in G and paying the penalty, and (ii) not playing at all. On a naive, but useful level, it appears that $G^L-x=G^R+x$. Thus, if the penalty is less than x, then a player gains by playing. If the penalty is x, the result is the same regardless of who plays. If $G^L-x=G^R+x=a$, for some number a, then $G=a+\{x|-x\}=a\pm x$. The mean of G is a and the temperature is x. The temperature represents the gain over the mean that a player obtains by playing.

For the disjunctive sum of switches

$$G=a_1\pm x_1+a_2\pm x_2+\dots a_k\pm x_k=a_1+a_2+\dots +a_k\pm x_1\pm x_2\pm \dots \pm x_k,$$

the best strategy is to play the switches in order from largest to smallest, i.e. play the component of highest temperature. If every position could be approximated by a switch, then, in a disjunctive sum, each could be replaced by the approximation. A great (best possible!) move would be playing in a switch of the highest temperature! A good-enough move is one in a switch not of the greatest temperature, but that gives the player sufficient advantage to win going second in the rest of the switches.

The problem is that not every position can be approximated by a switch. The game $\{\{10|2\}|-2\}$ could be replaced by ±2 since Left neither gains nor loses anything in $\{\{10|2\}|-2\}-2$ and neither does Right in $\{\{10|2\}|-2\}+2$. However, Left's move to $\{10|2\}$ is to a game with temperature 4, which is greater than 2, the temperature of ±2. Consequently, $\{\{10|2\}|-2\}+G_2+\dots +G_k$ is not the same as $\{2|-2\}+G_2+\dots +G_k$ since after a Left move in the first component of each, Right may have to respond in $\{10|2\}$ and there is no corresponding move in $2+G_2+\dots +G_k$.

We give the precise definition of temperature later but for the rest of the exposition replacing ‘temperature’ by ‘temperature of an approximating switch’ is enough to understand the gist of our results. A ‘great’ move will still be a move in a position of maximum temperature, and ‘good-enough’ moves are those of sufficiently high temperature that, whilst not best, still allow the player to force a win. The approach of this paper is to find the maximum temperature, or at least an upper bound, for all the positions in a set. For example, in game played on a grid, restricting the positions to those on a $2\times n$ grid. This maximum temperature, we call the boiling point of the restricted game. If G is a position in the set whose boiling point is known, and if the temperature of some $G^L$ is close to the boiling point, then Left's move to $G^L$ is probably ‘pretty good’.

Moves that increase the temperature are called threats and occur in many games. For example, ko threats in Go are an integral part of the game. Our approach is to find a parameter which is easier to calculate than the temperature but that limits the maximum temperature of all positions in the game. Threats are still allowed in the games but their impact is limited by the parameter.

The confusion interval of a game G is the set of numbers with which G is confused and, unsurprisingly given the name, the set is an interval. The parameter of interest to us is the length of the confusion interval. For example, the confusion interval of $G=\{\{10|2\}|-2\}$ is $[-2,2)$, which has length 4. The lengths of the confusion intervals can be arbitrary large. For example, Snort played on the complete bipartite graph $K_{1,m}$ has value $\{m|-m\}$ and the length of the confusion interval is 2m.

The first main result of this paper is that, given a game, if the lengths of the confusion intervals of all positions of the game are bounded by a constant K, then the maximum possible temperature of any position in the game is bounded by $3K/2$. However, this can be improved upon if the lengths of the confusion intervals of the options are less than K. This is given in Theorem 21.

For Theorem 21 to be useful the lengths of the confusion intervals must be known. In general, this is a difficult task since, similar to temperature, it involves considering all positions in the game tree. However, for games where there is a bound, Proposition 25 gives a method for determining it. Proposition 25 states that if there is a number K and an infinitesimal ϵ and Right wins $G^L-G-K+ϵ$, then the confusion interval is bounded by K.

The results and techniques from Section 3 are the first to give temperature upper bounds for general games. They cover games in which the temperature and length of the confusion interval can be greater for an option than that of the original position. Berlekamp's orthodox theory (see [13]) considers games where a move cannot increase the temperature. There have been many interesting results for such games, but as yet there is no general upper bound for their temperatures.

In the last sections, we apply our results to finding upper bounds for Domineering and Snort when played on a restricted set of graphs.

In general, Domineering is played on (a subset of) a checkerboard, with Left placing dominoes vertically and Right placing horizontally. Possibly the longest-standing conjecture in combinatorial game theory belongs to Berlekamp. In the 1970s, he conjectured that the boiling point of Domineering is 2 [3] (see also Problem B4 of [7]). The values and temperatures of Domineering positions have been the main point of interest in many papers (see for example [1], [6], [8], [9], [12], [14], [15]), all of which support Berlekamp's conjecture. It is known that if the conjecture is correct, then the bound is tight as the position in Fig. 1, found by Drummond-Cole in 2004 [5], has temperature 2. However, there is no known, non-trivial, upper bound for the temperatures of Domineering. A survey of the temperature results specific to the boards known as ‘snakes’ is given at the beginning of Section 4.

For Snort on a graph G, players place tokens on the vertices of G but they are not allowed to place a token next to one of the opponent's tokens. Despite being introduced in the first edition of Winning Ways, little is known about the game. If G is $K_{1,n}$, then the value is $\{n|-n\}$, where the best move is to play the central vertex. Since the temperature of $\{n|-n\}$ is n, the boiling point of Snort is infinite. This suggests that the maximum degree of a graph, $\mathrm{\Delta }(G)$, seems to play an important role in its temperature. An early conjecture of ours was that the temperature of G is bounded by $\mathrm{\Delta }(G)$. However, at the Capital Games 2020 workshop, examples were found where the temperature is of the order of $\mathrm{\Delta }{(G)}^2$. No non-trivial bounds, upper or lower, on the temperature are known.

For examples of finite boiling points see [10] for some splitting games, and [11] for a restricted version of Partizan Geography. The recent survey by Berlekamp [2] gives information on temperatures of many other games.