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In combinatorial game theory, a branch of mathematics, a hot game is one in which each player can improve their position by making the next move.

By contrast, a cold game is one where each player can only worsen their position by making the next move. The class of cold games are equivalent to the class of surreal numbers and so can be ordered by value, while hot games can have other values.[1]

There are also tepid games, which are games with a temperature of exactly zero. Tepid games are formed by the class of strictly numerish games: that is, games that are equivalent to a number plus an infinitesimal.

Hackenbush can only represent cold and tepid games (by its decomposition[2] into a purple mountain and a green jungle).

Example

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For example, consider a game in which players alternately remove tokens of their own color from a table, the Blue player removing only blue tokens and the Red player removing only red tokens, with the winner being the last player to remove a token. Obviously, victory will go to the player who starts off with more tokens, or to the second player if the number of red and blue tokens are equal. Removing a token of one's own color leaves the position slightly worse for the player who made the move, since that player now has fewer tokens on the table. Thus each token represents a "cold" component of the game.

Now consider a special purple token bearing the number "100", which may be removed by either player, who then replaces the purple token with 100 tokens of their own color. (In the notation of Conway, the purple token is the game {100|−100}.) The purple token is a "hot" component, because it is highly advantageous to be the player who removes the purple token. Indeed, if there are any purple tokens on the table, players will prefer to remove them first, leaving the red or blue tokens for last. In general, a player will always prefer to move in a hot game rather than a cold game, because moving in a hot game improves their position, while moving in a cold game injures their position.

Temperature

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The temperature of a game is a measure of its value to the two players. A purple "100" token has a temperature of 100 because its value to each player is 100 moves. In general, players will prefer to move in the hottest component available. For example, suppose there is a purple "100" token and also a purple "1,000" token which allows the player who takes it to dump 1,000 tokens of their own color on the table. Each player will prefer to remove the "1,000" token, with temperature 1,000 before the "100" token, with temperature 100.

To take a slightly more complicated example, consider the game {10|2} + {5|−5}. {5|−5} is a token which either player may replace with 5 tokens of their own color, and {10|2} is a token which the Blue player may replace with 10 blue tokens or the Red player may replace with 2 blue tokens.

The temperature of the {10|2} component is ½(10 − 2) = 4, while the temperature of the {5|−5} component is 5. This suggests that each player should prefer to play in the {5|−5} component. Indeed, the best first move for the Red player is to replace {5|−5} with −5, whereupon the Blue player replaces {10|2} with 10, leaving a total of 5; had the Red player moved in the cooler {10|2} component instead, the final position would have been 2 + 5 = 7, which is worse for Red. Similarly, the best first move for the Blue player is also in the hotter component, from {5|−5} to 5, even though moving in the {10|2} component produces more blue tokens in the short term.

Snort

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In the game of Snort, Red and Blue players take turns coloring the vertices of a graph, with the constraint that two vertices that are connected by an edge may not be colored differently. As usual, the last player to make a legal move is the winner. Since a player's moves improve their position by effectively reserving the adjacent vertices for them alone, positions in Snort are typically hot. In contrast, in the closely related game Col, where adjacent vertices may not have the same color, positions are usually cold.

Applications

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The theory of hot games has found some application in the analysis of endgame strategy in Go.[3][4]

See also

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References

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Hot game

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from Grokipedia
In combinatorial game theory, a hot game is a game position in which the player to move has a significant incentive to do so, as there exist options for both the Left player (typically the maximizing player) and the Right player (the minimizing player) that are strictly better for them than the current position; formally, it is defined as a game where some Left option is strictly greater than some Right option. This contrasts with "cold" games, such as numbers, where moving generally disadvantages the player by leaving a worse position for themselves. Hot games are central to advanced analyses in the field, particularly through temperature theory, which quantifies the "heat" or urgency of a position by assigning a temperature value—often calculated as (x−y)/2(x - y)/2 for a simple switch {x∣y}\{x \mid y\} where x>yx > y—to guide optimal play in sums of games. The concept originates from the foundational work of mathematicians like , , and Richard Guy, who developed it in the 1970s and 1980s to handle non-numeric game values in impartial and partizan games. In practice, hot games appear in analyses of board games like Go or chess variants, where positions encourage aggressive moves due to their dynamic imbalance. Temperature theory further allows "cooling" hot games by introducing move costs, revealing their mean value (the expected outcome under optimal play) and thermostatic strategies, where players prioritize moves in the hottest components of a game sum to maximize advantage. For instance, the game {1∣−1}\{1 \mid -1\} is a classic hot game, as both players benefit from moving to a favorable number, leading to a temperature of 1 and a mean value of 0. Hot games have broader implications in , enabling the evaluation of complex positions that cannot be reduced to simple numbers, and they underpin algorithms for finding optimal stops (best moves) and values in heated scenarios. Their study continues to influence applications, such as AI game-solving and algorithmic , emphasizing the strategic depth beyond zero-sum equilibria.

Fundamentals

Definition

In combinatorial game theory, a hot game GG is formally defined as a position where both Left and Right have options leading to positions that provide a higher value for the player about to move, specifically such that the value of GG exceeds its Left options GLG^L but is less than its Right options GRG^R in the partial ordering of games. This condition arises when some Left option is strictly greater than some Right option, indicating an "active" position where the sets of options overlap in value, preventing GG from simplifying to a single numeric outcome. Hot games are distinct from in that they are non-numeric positions; unlike cold games, which equate to surreal numbers with temperature zero, hot games feature a value that lies between the bounds of their component options due to this overlap, creating strategic volatility. The value, often computed as the average outcome over repeated plays of the game, underscores this excess, positioning hot games outside the ordered surreal number line. A central feature of hot games is the urgency to move: the first player to act from such a position gains a significant advantage, leaving the second player at a disadvantage, as the active nature compels immediate play to capture the higher-valued options. The concept of hot games emerged in the context of during the and 1990s, pioneered by , John H. Conway, and as an extension of Conway's surreal numbers framework, with key advancements by Bill Spight applying it to loopy games like Go endgames.

Properties

Hot games in are denoted using the standard form G={L∣R}G = \{ L \mid R \}, where LL represents the set of positions available to the Left player and RR the set to the Right player, with the value of GG exceeding the values of its options such that value(G)>value(L)\mathrm{value}(G) > \mathrm{value}(L) for all L∈LL \in L and value(G)<value(R)\mathrm{value}(G) < \mathrm{value}(R) for all R∈RR \in R. This notation underscores the recursive structure of games, where options lead to subpositions of earlier birthday, ensuring the inequality holds by construction in the surreal number framework extended to partizan games. Unlike cold games, which equate to surreal numbers, hot games possess non-numeric values characterized as "switches" or more intricate structures that do not simplify to a single numerical equivalent. A basic switch takes the form {a∣b}\{ a \mid b \} with numbers a>ba > b, where the mean value lies midway but the game favors the player who moves first due to the straddling options. These structures arise because hot games are non-numbers, meaning not all Left options are less than all Right options, leading to strategic complexity beyond numerical evaluation. In disjunctive sums involving and games, the hot component dominates strategic considerations owing to its elevated urgency, as the overall of the sum approximates that of the hottest , compelling players to prioritize moves within it. Specifically, adding a game (a number) to a game preserves the temperature of the hot game, ensuring its influence overrides the numerical component in determining optimal play. A game GG is deemed hotter than another HH if its t(G)t(G) exceeds t(H)t(H), a metric that dictates move priority in sums by indicating the relative urgency of responding in each . Positive t(G)>0t(G) > 0 defines hot games, distinguishing them from tepid or ones and guiding the thermostatic where players target the with the highest .

Examples

Token-Based Example

To illustrate the of a hot game, consider a simple token-based model that builds for positions where both players benefit from making the next move. Imagine separate stacks representing positions favorable to each player: blue tokens for Left (positive value) and red tokens for Right (negative value from Left's perspective). Introduce a special "hot" purple token positioned between the stacks, valued at 100 units, which either player can remove on their turn and replace with 100 tokens of their own color—blue for Left or red for Right. This token creates a hot position because the player to move gains a significant advantage by acting immediately. For the mover, removing the purple token and adding 100 of their color shifts the overall position by +100 (if Left moves) or -100 (if Right moves), improving their relative standing compared to leaving the purple token in place. In formal terms, this corresponds to the game position denoted as {100 \mid -100}, which has a mean value of 0—the simplest number between its Left and Right options—but is hot because both available moves enhance the mover's prospects relative to the current neutral stance. In step-by-step play, assume Left starts facing the purple token. Left removes it and adds 100 blue tokens, resulting in a position valued at +100, which favors Left strongly. Right now confronts this +100 stack, a position where any move (removing some blue tokens) only worsens Right's situation further, as it reduces the value toward 0 or below without gaining an equivalent benefit. This demonstrates how the initial hot position compels immediate action, transferring the advantage decisively to the first mover.

Algebraic Example

A canonical algebraic example of a hot game is the simple switch G={1∣−1}G = \{ 1 \mid -1 \}. In this position, Left's sole option is to move to the number 11, a positive value that favors Left since the opponent (Right) would then face a position where Left has a winning regardless of who starts. Similarly, Right's option is to move to −1-1, a negative value favoring Right, as the opponent (Left) would then be to move in a position where Right wins. This configuration renders GG hot because both players prefer to move from it: each option exceeds the current position in value for the respective player, creating a strong for the player to act rather than pass. The game value is ±1\pm 1, a switch with mean value 0 and temperature 1, reflecting its volatility and first-player advantage (fuzzy with 0). The algebraic notation here abstracts the intuitive token-based setup from prior examples, where a neutral component allows replacement with a favorable one. When summed with the zero game, G+0={1∣−1}G + 0 = \{ 1 \mid -1 \}, the result remains hot and equivalent to GG alone, as the first player can win by moving in the hot component while ignoring the cold 0 (no moves). This illustrates how hot positions dominate cold ones like numeric values, where the sum's outcome is determined by the hotter element. By contrast, the cold position 0={∣}0 = \{ \mid \} offers no moves, so the player to move loses immediately under normal play; there is no advantage to acting, and the second player wins with optimal play. Unlike the hot switch, 00 lacks any incentive for the current player, exemplifying a stable, non-volatile position.

Thermography

Temperature Concept

In , the temperature t(G)t(G) of a hot game GG serves as a quantitative measure of the urgency for the player to move, defined as the smallest non-negative t≥0t \geq 0 such that the cooled version GtG_t of the game is infinitesimally close to . This captures the volatility of the position, approximating the rate at which the game's value fluctuates under optimal play as moves are made. Hot games, where both players prefer to move rather than pass, exhibit positive , distinguishing them from colder positions where the incentive to act diminishes. Intuitively, a higher indicates a "hotter" position with greater strategic heat, creating a strong incentive for the current player to act immediately to capitalize on the volatility before the opponent does. This can be likened to a cooling process in , where successive optimal moves by players gradually reduce the game's , stabilizing the value toward its mean as the position simplifies and the urgency fades. In practice, guides strategic decisions in sums of games by prioritizing moves in the hottest components, where the potential swing in advantage is largest. The relates closely to the game's mean value and its boundary values, known as the left stop LS(G)LS(G) and right stop RS(G)RS(G), which represent the outcomes if one player were to play indefinitely without opposition. For hot games, t(G)>0t(G) > 0 and LS(G)>RS(G)LS(G) > RS(G). The width of the confusion interval is ℓ(G)=LS(G)−RS(G)\ell(G) = LS(G) - RS(G), and after cooling by amounts greater than or equal to t(G), the position becomes infinitesimally close to its mean value, behaving like a stable number. This difference between the stops and the mean highlights the dynamic nature of hot games, where the mean value alone underestimates the immediate tactical importance. The temperature concept originated in the work of , building on John Horton Conway's foundational framework in On Numbers and Games (1976), and was formalized in analyses of complex games like . Berlekamp's contributions, including in The Economist's View of Combinatorial Games (1994), emphasized temperature's role in economic interpretations of game values, while subsequent extensions by researchers like Bill Spight applied it to thermographic methods for games with loops, such as Go endgames.

Thermographic Analysis

The cooling process in thermographic involves iteratively applying a penalty, or tax, to the options of a hot game, effectively reducing its while preserving the value, until the position becomes tepid with 0. This is achieved by cooling the hottest components first, replacing each hot subposition with its value in a step-by-step manner, which simplifies the game for evaluation without altering the overall strategic equilibrium. For simple hot games, such as switches of the form {a \mid b} where a > 0 > b, the t(G) is approximated by (value of Left option - value of Right option) / 2, though more complex positions require the full thermographic method. In general, is computed via the thermogram, which provides a precise measure beyond this approximation. A thermogram is a graphical representation plotting the cooled values of the game against iterations of the cooling process, displaying the mean value as the height of the "mast" and the range of possible values as boundaries that converge to a tepid position. The left and right walls of the thermogram indicate the stops for Left and Right players under varying s, facilitating analysis of how the game's value evolves as it cools. For example, in the position {a \mid b} with a > 0 > b and symmetric magnitudes (e.g., a = -b = x > 0), the simplifies to t = (a - b)/2 = x, as cooling by x units yields a tepid star position {0 \mid 0}. In sums of hot games, thermographic analysis prioritizes the components with the highest , cooling them sequentially to determine the overall mean and , ensuring optimal play focuses on the most volatile subgames first.

Snort

Snort is a partizan played on the vertices of a graph, where two players, often called Left and Right, alternate coloring uncolored vertices with their respective colors under the restriction that a player may not color a vertex adjacent to one already colored by the opponent. The proceeds until no legal moves remain, with the last player to move declared the winner under normal play convention. This setup creates dynamic interactions, as players seek to expand their colored regions while blocking the opponent, often leading to clustered same-color components separated by uncolorable boundaries. The hot nature of Snort arises from the incentive structure of its moves, where playing in certain positions often leaves the opponent facing cooler (less advantageous) options, heightening the urgency to seize high-temperature spots before the adversary does. In combinatorial game theory terms, hot games like Snort feature positions with positive , a measure quantifying how much the player to move benefits from acting first, typically exceeding zero in active regions and driving aggressive play toward "heating up" central or connected areas. Game values in Snort manifest as switches—bipartite forms like { positive\ options \mid negative\ options }, such as ↑ = {0 \mid *}, rather than simple numerical values, reflecting the partizan and strategic volatility. In contrast to the cold game , where the restriction prohibits coloring adjacent to one's own color (enforcing separation of same-color vertices), Snort inverts this by forbidding adjacency to the opponent's color, allowing expansion near one's own territories but creating defensive hotspots. This duality positions Snort as the hot counterpart to Col's cooler dynamics, with Snort promoting contiguous same-color growth and Col emphasizing dispersed placements, as detailed in foundational analyses. Unlike purely impartial games analyzed via Grundy numbers (mex of option Grundies), Snort's partizan structure yields non- values through its disjunctive sum decomposition, where the minimum excludant (mex) over partizan options produces switch-like forms that capture the imbalance, such as temperatures rising with graph connectivity. A representative example occurs on a P_5 (five vertices in a line), where an early central move by the first player heats up adjacent vertices, pressuring the opponent to respond nearby to avoid losing control of the heated segment, ultimately favoring the first player under optimal play due to the odd length yielding a positive value.

Col and Other Map-Coloring Games

Col is a map-coloring game played on a graph, where players alternate turns coloring uncolored vertices with their designated color—typically black for the first player and white for the second—such that no two adjacent vertices share the same color, adhering to the proper condition. This partizan rule set ensures that each player faces distinct move options based on their color, rendering Col positions "cold" in terms, where game values often simplify to numbers or numbers plus infinitesimal components like stars, indicating low urgency to move and predictable second-player advantages on certain boards. In contrast to hot games like its counterpart Snort on the same family of graphs but with rules forbidding adjacency to the opponent's color (allowing same-color adjacencies), Col exhibits a hot-cold duality wherein its positions equate to numerical values, while Snort's feature hot switches with positive temperatures that incentivize immediate play. This duality highlights fundamental differences in strategic volatility: Col's cold nature requires partizan tools to evaluate positions, whereas Snort's hot dynamics require thermographic tools to evaluate move urgency. Both games belong to the broader class of bicoloring graph games, analyzed extensively on rectangular boards where Col often favors the second player on even-dimensional grids. Other map-coloring and tiling games further illustrate hot-cold distinctions, such as Cram, an impartial variant of the partizan cold game Domineering played on grid graphs. In Domineering, one player places only vertical dominoes and the other horizontal, leading to cold positions amenable to numerical evaluation; Cram, however, allows both players unrestricted domino placements, introducing hot elements where first-player wins dominate on even-by-odd boards due to asymmetric territorial control. This shift underscores how relaxing partisan restrictions can heat up a game, transforming stable subgames into volatile ones. Strategically, hot variants like Snort or Cram enable the first player to often force the opponent into isolated cold subgames, such as numerical Col-like components, thereby dictating the overall temperature and securing advantages through selective overheating. These dynamics were pioneered in thermographic analysis by in the , who developed methods to compute positional temperatures in Winning Ways, revealing how hot games decompose into cooler, analyzable parts for optimal play.

Applications

Go Endgame Strategy

In the endgame phase of Go, assigns temperatures to positions based on the urgency of moves, where higher temperatures indicate greater point value swings that demand immediate attention to prevent opponent gains. High-temperature moves, such as capturing a ko , must be prioritized because they represent opportunities worth more than one point in the current ambient temperature, while lower-temperature moves can be delayed without significant loss. This concept stems from , which decomposes endgame positions into components reflecting move priorities, ensuring players cool the board efficiently by resolving urgent threats first. A representative example is a simple , modeled as a hot game in the form {gain | loss}, where the equals the point swing—typically 1 for a standard one-point ko, as the capture yields a direct territorial advantage equivalent to that value under orthodox play. In such scenarios, the ko acts as a because both players compete for the next move, with the dictating the ko-master's ability to enforce threats larger than half the ko's value to secure it. This analysis highlights how even basic endgame elements exhibit hot game dynamics, where failing to address the leads to suboptimal scoring. Bill Spight's work in the 1990s advanced Go thermography by addressing the challenges of loopy positions like , decomposing endgames into hot and cold subgames for precise valuation. In his 1999 paper, Spight extended thermography to handle multiple by defining thermographs based on enriched environments that account for ko threats and cycles, allowing computation of mean values and temperatures even in complex configurations. His 2001 analysis of the Jiang-Rui endgame further demonstrated this by breaking down a professional game into thermographic components, revealing how hot interact with cold dame to guide optimal play. These contributions enabled systematic evaluation of endgame urgency beyond traditional heuristics. The core strategic rule derived from this framework is to iteratively play the hottest move available, which reduces the overall board step by step until reaching a , state where all positions are gote or settled. This approach ensures maximal point efficiency by aligning move order with thermographic priorities, such as resolving high-temperature before filling low-value spaces. Recent developments post-2020 have explored integrating AI tools like with thermographic principles for automated estimation in Go endgames, comparing outputs against exact combinatorial solutions to refine move valuations.

Broader Combinatorial Uses

In , has been applied to analyze impartial games by computing s that facilitate the evaluation of complex sums, particularly in games like Hackenbush where hot stalks—linear configurations of edges that exhibit positive —represent urgent positions for both players. For instance, in Blue-Red Hackenbush, reveals how sums of hot stalks can be simplified through cooling processes, allowing players to determine optimal moves by approximating the game's mean value and volatility. This approach extends to broader impartial settings, such as Dawson's Chess variants, where hot positions emerge from symmetric structures but require adjustments to resolve sums. Algorithmic advancements since the have enabled efficient computation of temperatures in large game trees, often using recursive cooling methods that iteratively reduce a hot game's volatility to a cooler form amenable to analysis. Early implementations focused on polynomial-time algorithms for linear structures like Hackenbush stalks, while more recent techniques, such as Temperature Discovery Search (TDS), approximate temperatures through forward search in complex trees, achieving practical scalability for games with hundreds of positions. These tools, implemented in software like CGSuite, support recursive evaluation by propagating thermographs upward from terminal positions. Connections to economics leverage hot games to model scenarios with asymmetric urgency, such as auctions where players bid to control moves, akin to Richman games that transform standard CGT positions into bidding contests. In this framework, the temperature quantifies bidding intensity, as seen in hot auctions where high volatility encourages aggressive offers to secure advantageous turns. Similarly, bargaining models draw on hot game dynamics to represent negotiations with time-sensitive concessions, where surreal number values capture unequal move advantages. Despite these applications, analysis remains incomplete for partizan hot games, where differing Left and Right options complicate uniform assignment beyond impartial cases. Ongoing research as of 2025 extends frameworks to better accommodate such partizan structures, addressing gaps in infinite game representations. For example, hot positions in infinite poset games—ordered sets where moves remove elements and subordinates—exhibit temperatures that challenge traditional recursive cooling, highlighting needs for generalized thermographic tools. These efforts build on practical cases like Go endgames but emphasize theoretical unification across combinatorial domains.
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