A Trémaux tree, named after the French mathematician Édouard Trémaux, is a structure used in graph theory, specifically in the context of exploring undirected graphs. It is used to represent the exploration of the graph and the paths taken during a traversal. Typically, a Trémaux tree is constructed during a depth-first search (DFS) or a breadth-first search (BFS) of a graph, where the edges represent the paths followed by the traversal.

Sarit Kraus is a prominent researcher and scholar in the field of artificial intelligence, specifically known for her work in areas such as multi-agent systems, game theory, and human-agent interaction. She has contributed significantly to the understanding of how autonomous agents can operate and collaborate in complex environments, including those involving strategic interaction and negotiation. Kraus has held academic positions at institutions such as Bar-Ilan University in Israel and has published numerous papers in journals and conferences related to AI.

Epsilon-equilibrium, often denoted as ε-equilibrium, is a concept used in game theory, particularly in the context of non-cooperative games. It extends the idea of Nash equilibrium by allowing for a tolerance level, ε, that accounts for the possibility of small deviations from optimal play by players in the game. In a standard Nash equilibrium, each player's strategy is a best response to the strategies chosen by the other players.

Perfect Bayesian Equilibrium (PBE) is a refinement of Bayesian Nash Equilibrium in the context of dynamic games with incomplete information. It incorporates the concepts of beliefs and sequential rationality to provide a detailed analysis of players' strategies and their updates based on observed actions. The key elements of Perfect Bayesian Equilibrium include: 1. **Beliefs**: Players have beliefs about the types of other players (i.e., their private information) based on prior probabilities.

Sequential equilibrium is a concept from game theory, particularly in the context of dynamic games, which are games where players make decisions at various points in time, and the decisions can depend on past actions. A sequential equilibrium is an extension of the Nash equilibrium that takes into account the order of moves and the information available to players at each decision point. It considers both the strategies of players and their beliefs about the game's state.

Poisson games are a type of strategic game theory model that incorporates the idea of players arriving randomly over time, in accordance with a Poisson process. This framework can be useful for analyzing situations where players independently choose actions from a set of strategies, and the timing of when players enter the game is stochastic. In a typical Poisson game, players have a common interest or goal, but their interaction is characterized by random arrivals.

The Complexity of Cooperation typically refers to the intricate dynamics and mechanisms involved in cooperative behavior among individuals, groups, or entities across various contexts, including social, economic, biological, and technological systems. This concept often intersects with multiple academic fields, such as sociology, psychology, evolutionary biology, economics, and computer science. In a social context, cooperation may involve the ways in which people or groups work together to achieve common goals, resolve conflicts, or share resources.

The Dictator Game is a widely studied economic experiment that explores concepts of altruism, fairness, and decision-making. It involves two players: one designated as the "dictator" and the other as the "recipient." The basic structure of the game is as follows: 1. **Endowment**: The dictator is given a certain amount of resources (commonly money, but it can be points or goods in different variations of the game).

The Grim Trigger is a concept in game theory, particularly in the study of repeated games. It refers to a specific strategy employed by a player in response to the actions of their opponent. The Grim Trigger strategy is characterized by its severe punishment mechanism for any deviation from cooperative behavior. Here’s how it works: 1. **Cooperation Phase**: Players start by cooperating with each other in the repeated game.

The lump of labour fallacy is an economic misconception that suggests there is a fixed amount of work available in an economy, implying that if one person gains employment, it must come at the expense of another person's job. This fallacy assumes that there is a limited amount of work to be done, leading to the belief that when jobs are created or taken away, the overall employment level remains unchanged.

The chess endgame is the final phase of a chess game that occurs after the middlegame and follows the reduction of material on the board. In this stage, each player's pieces have been reduced significantly, often to just a few pawns and pieces, such as kings, rooks, bishops, knights, or queens. The primary focus of the endgame is to promote pawns into queens or other pieces, checkmate the opponent's king, and leverage the material advantage effectively.

The Tarrasch Rule, named after the German chess player and theorist Dawid Tarrasch, is a guideline in pawn structure in chess. It states that in general, pawns on the fourth rank (for White, the rank is the 4th; for Black, it’s the 5th) are stronger than pawns that are advanced further.

The Two Knights Endgame is a situation in chess where only two knights are left on the board for one side, typically against a lone king or another minor piece (often a pawn). This endgame is distinct because it usually represents a challenging scenario for the player with the two knights, as they cannot checkmate a lone king without the assistance of a pawn or another piece. In its purest form, the most common scenario involves one player having two knights and the other player having just a king.

The term "wrong rook pawn" typically refers to a specific scenario in chess endgames, particularly in king and pawn endgames. It describes a situation where a pawn is on the corner file (a-file or h-file) of the board, and it is important because it can affect the ability to win or draw the game depending on the position of the opposing king.

Uzbek astronomers have made significant contributions to the field of astronomy throughout history, particularly during the Islamic Golden Age. One of the most notable figures is Ulugh Beg (1394–1449), an Uzbek ruler and astronomer who founded an important observatory in Samarkand. His work included the compilation of a star catalog and the development of astronomical tables that improved the accuracy of celestial measurements.

Symmetric fair cake-cutting refers to a method of dividing a "cake" (or any divisible resource) in such a way that all participants perceive the division to be fair and equitable, ensuring symmetry in their allocations. The concept stems from the fields of economics and game theory, where fairness in resource allocation is crucial. In symmetric fair cake-cutting: 1. **Symmetry** means that if two participants start with the same preferences and information, they should receive identical portions of the cake.

Map segmentation is a process used in geographic information systems (GIS), image processing, and various fields of computer vision to divide a map or an image into distinct regions or segments based on specific criteria. The goal of map segmentation is to facilitate analysis, interpretation, and understanding of spatial data by reducing complexity and enhancing relevant features.

Online fair division refers to the problem of allocating resources or dividing goods among agents in a dynamic environment where the agents arrive and make requests over time. In contrast to traditional fair division, where all agents and items are present from the beginning, online fair division must consider situations where agents show up sequentially, and decisions need to be made without the knowledge of future arrivals or requests.

Game balance refers to the process of ensuring that all elements of a game—such as characters, abilities, weapons, items, or mechanics—are designed and adjusted in a way that creates a fair, enjoyable, and challenging experience for players. Effective game balance aims to prevent any single aspect of the game from being overwhelmingly powerful or weak, which could lead to frustration or diminish the enjoyment of the game.