Correlated equilibrium is a concept in game theory that extends the idea of Nash equilibrium. It was introduced by Robert Aumann in the 1970s. The key idea behind correlated equilibrium is that players can achieve better outcomes by coordinating their strategies based on some form of common signal or correlation, rather than choosing their strategies independently. ### Definition: In a correlated equilibrium, a mediator (or correlation device) sends signals to the players, suggesting which strategies they should play.

Risk dominance is a concept from game theory that helps determine which of several potential strategies or equilibria in a game is more likely to be chosen by players when they are unsure of the actions of others. It is particularly useful in coordination games, where players have to make decisions without knowing what others will choose.

Stochastically stable equilibrium is a concept used in the field of evolutionary game theory and dynamic systems to describe a state of a system that remains stable over time under stochastic (random) influences. It represents an equilibrium point that is not only stable in a deterministic sense but also resilient to small random fluctuations or perturbations that may occur in the system.

An extensive-form game is a type of game used in game theory to model situations where players make decisions sequentially over time. It represents the structure of a game in a tree-like format, illustrating the various possible moves, strategies, and outcomes that can arise in the game. Each node of the tree represents a decision point for a player, and branches represent the possible actions that can be taken from that point.

A hedonic game is a concept in cooperative game theory, where the preferences of players are based on their individual utilities derived from the coalition they form with others. In a hedonic game, players derive satisfaction or "hedonic value" from being in groups with certain other players, while their utility is typically not affected by the group's size or the overall wealth of the coalition.

An **adherent point** is a concept in topology, a branch of mathematics. In the context of a topological space, an adherent point (or limit point) of a set refers to a point that is either in the set itself or is a limit point of that set.

In topology, the concept of a boundary is associated with the idea of the closure and interior of a set in a topological space.

Either-or topology, also known as the "discrete topology," is a simple kind of topology that can be defined on a set. In this topology, every subset of the set is considered an open set. The discrete topology is characterized by the following properties: 1. **Open Sets**: Every subset of the set is in the topology. This includes the empty set and the entire set itself.

A **sequential game** is a type of game in game theory where players make decisions one after another, rather than simultaneously. This structure allows players to observe previous actions before making their own decisions, which can influence their strategies and outcomes. ### Key Characteristics of Sequential Games: 1. **Order of Moves**: Players take turns making decisions. The order in which players move can affect the strategies they choose.

A **Cauchy space** is a concept from the field of topology and analysis, named after the mathematician Augustin-Louis Cauchy. It generalizes certain properties of sequences and convergence in metric spaces, allowing for a more abstract setting in which to study convergence and completeness. In more formal terms, a **Cauchy space** is defined in the following way: 1. **Set and Filter**: Start with a set \( X \).

A signaling game is a concept from game theory that involves two players: a sender and a receiver. The sender has some information that the receiver does not have, and the sender can choose to send a signal to convey that information. The receiver, in turn, interprets the signal to make a decision or form a belief about the sender's type or the state of the world.

A symmetric game is a type of game in game theory where the payoffs for each player depend only on the strategies chosen and not on the specific identity of the players. In other words, the game remains unchanged when the players' roles are switched. This means that if player 1's strategy results in a certain outcome, the same strategy used by player 2 will lead to the same outcome, provided their roles are reversed.

Cofiniteness is a concept often discussed in the context of model theory and formal languages, particularly related to the properties of certain mathematical structures. In general, a property or structure is said to exhibit cofiniteness when the complement set (or the set of elements that do not belong to it) is finite.

Thinning in the context of mathematical morphology is a morphological operation used primarily in image processing and computer vision. It is a technique that reduces the thickness of objects in a binary image while preserving their connectivity and shape. The goal of thinning is to simplify the representation of features in an image, often used for tasks like shape analysis, object recognition, or preprocessing for further analysis.

The term "neighbourhood system" can have different meanings depending on the context. Here are a few interpretations: 1. **Urban Planning and Geography**: In urban planning, a neighbourhood system refers to the arrangement and organization of communities within a larger city or metropolitan area. It encompasses residential areas, commercial zones, parks, and public spaces, and focuses on the interactions and relationships between these components.

In topology, a connected space is a fundamental concept that refers to a topological space that cannot be divided into two disjoint, non-empty open sets. More formally, a topological space \( X \) is called connected if there do not exist two open sets \( U \) and \( V \) such that: 1. \( U \cap V = \emptyset \) 2. \( U \cup V = X \) 3.

The Eberlein compactum is a specific topological space that is an example of a compact space which is not metrizable. It is constructed using the properties of certain compact sets in the space of continuous functions. More formally, an Eberlein compactum can be described as a subspace of the space of all bounded sequences of real numbers, specifically the closed bounded interval [0,1] or some analogous bounded topological space. The compactum is named after the mathematician P.

Interlocking interval topology is a concept in the field of topology, specifically dealing with spaces constructed using intervals that have a particular relationship with one another. Here's a basic overview of the concept: ### Definitions: 1. **Intervals:** In a typical setting (especially in \(\mathbb{R}\)), intervals can be open, closed, or half-open.

In topology and mathematical analysis, an **isolated point** (or isolated point of a set) is a point that is a member of a set but does not have other points of the set arbitrarily close to it.

The Cataclysmic Pole Shift Hypothesis is a theory that suggests significant and rapid changes in the Earth's geographic poles could lead to catastrophic effects on the planet's environment, climate, and life. This idea encompasses several concepts, including the possibility that the Earth's crust could shift relative to its molten core, leading to a sudden reorientation of the planet's surface.