Game theory equilibrium concepts

Game theory is a branch of mathematics and economics that studies strategic interactions among rational decision-makers. Within game theory, several equilibrium concepts help analysts understand how players make decisions when they have conflicting interests. Here are some of the most significant equilibrium concepts: ### 1. Nash Equilibrium - **Definition**: A set of strategies (one for each player) is in Nash Equilibrium if no player can benefit by unilaterally changing their strategy, given the strategies of all other players.

Berge equilibrium is a concept in game theory, particularly in the context of dynamic games with incomplete information. It is named after the French mathematician Claude Berge. The equilibrium represents a strategy profile where players choose their strategies optimally, given their beliefs about other players' types (or strategies), and where these strategies are consistently chosen based on the structure of the game.

A Coalition-proof Nash equilibrium (CPNE) is a solution concept in game theory that extends the traditional notion of Nash equilibrium to account for the possibility of coalition formation among players. In a standard Nash equilibrium, a strategy profile is stable if no single player can benefit by unilaterally changing their strategy, given the strategies of the other players. However, it does not consider the potential for groups of players to deviate together from the equilibrium, which can lead to different outcomes.

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.

"Divine equilibrium" is a term that can have various interpretations depending on the context in which it is used, including philosophical, spiritual, and scientific perspectives. 1. **Philosophical/Spiritual Context**: In many spiritual and philosophical traditions, divine equilibrium refers to a state of balance or harmony that is achieved when one is in alignment with a higher power or universal principles.

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.

Equilibrium selection is a concept in game theory and economics that refers to the process of choosing among multiple equilibria in a strategic setting. In many games, especially those with multiple equilibria, different outcomes can exist, and it may not be clear which equilibrium will be reached in practice. Equilibrium selection seeks to identify which of these equilibria is more likely to be observed based on certain criteria or frameworks.

An Evolutionarily Stable Strategy (ESS) is a concept from evolutionary game theory that describes a strategy that, if adopted by a majority of a population, cannot be invaded by any alternative strategy that is initially rare. The concept was first introduced by the biologist John Maynard Smith in the 1970s as a way to explain stable behavioral patterns in animal populations.

The Folk theorem is a concept in game theory that describes conditions under which cooperation can emerge as a stable strategy in repeated games. Specifically, it states that in infinitely repeated games with a finite set of players, if the game's stage payoffs are sufficiently high, then any feasible payoff that is individually rational can be sustained as a Nash equilibrium through a strategy that involves punishment for deviations from cooperation.

The term "intuitive criterion" can refer to different contexts depending on the field of study or application, but generally, it describes a basis for making decisions or judgments that is guided by intuition rather than formal methods or analytical processes. Here are a few contexts in which you might encounter "intuitive criterion": 1. **Decision Making**: In decision theory or behavioral economics, an intuitive criterion may refer to a decision-making approach that relies on gut feelings, instincts, or heuristic methods.

M equilibrium refers to a specific type of equilibrium in various contexts, but it most commonly relates to chemical equilibria or economic models. Here are two interpretations of M equilibrium: 1. **Chemical Equilibrium**: In the context of chemistry, M equilibrium could refer to a state in a chemical reaction where the concentrations of reactants and products remain constant over time. At this point, the rate of the forward reaction equals the rate of the reverse reaction.

The concept of "Manipulated Nash Equilibrium" is not a standard term in game theory literature but can pertain to scenarios where players in a game can strategize to influence or manipulate the outcome to their advantage while still adhering to the principles of Nash equilibrium. In a typical Nash equilibrium, each player’s strategy is optimal given the strategy chosen by all other players. In other words, no player can benefit by unilaterally changing their strategy if the other players' strategies remain unchanged.

Markov Perfect Equilibrium (MPE) is a refinement of the concept of Nash Equilibrium, applied to dynamic games with incomplete information. In such games, players make decisions at various points in time, and their strategies can depend not just on the current state of the game but also on the entire history of play. However, in the MPE, players base their decisions on the current state of the game rather than on its history.

Mertens-stable equilibrium is a concept from game theory, particularly in the context of non-cooperative games. It refers to a way of identifying equilibria in games that is robust under the consideration of deviations by players. In a game, a strategy profile (a set of strategies chosen by players) can be considered an equilibrium if no player has an incentive to unilaterally deviate from their strategy, given the strategies of the others.

Nash equilibrium is a concept in game theory named after mathematician John Nash. It refers to a situation in a strategic game where no player can benefit by changing their strategy unilaterally, assuming that the other players' strategies remain constant. In other words, it is a state in which each player's strategy is optimal given the strategies of all 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.

Pooling equilibrium is a concept from game theory and economics, particularly in the context of signaling games. It occurs when different types of players (or agents) in a market send the same signal, making it impossible for observers (or other players) to differentiate between them based on that signal. In a pooling equilibrium, all players choose the same action or strategy, so their different types (e.g.

"Proper equilibrium" typically refers to a stable state in which various forces or factors are balanced in such a way that there is no tendency for change. This term can appear in various fields, including physics, economics, and environmental science, among others.

Quantal Response Equilibrium (QRE) is a concept in game theory used to model decision-making under uncertainty, particularly in situations where players may not choose their strategies purely rationally. The idea is that players will respond probabilistically to their payoffs, reflecting some level of bounded rationality or cognitive limitations. In traditional Nash Equilibrium, players choose their strategies to maximize their payoffs given the strategies of others, assuming that they make fully rational decisions.

Quasi-perfect equilibrium is a concept from game theory, specifically related to dynamic games and extensive form games (games represented by trees with nodes and branches indicating possible moves by players). It is an extension of the idea of subgame perfect equilibrium, which requires that players' strategies constitute a Nash equilibrium in every subgame of the original game.

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.

"Satisfaction equilibrium" is not a widely recognized term in mainstream economics or psychology, but it can be interpreted in a few different ways depending on the context. 1. **In Economics**: It might refer to a state where individuals or firms derive a level of satisfaction from their consumption or production that is balanced with their constraints (like budget or resources). This concept could be related to the idea of utility maximization, where consumers are satisfied with their choices given their income and the prices of goods.

A self-confirming equilibrium is a concept in game theory that refers to a type of equilibrium in which players' beliefs about the strategies and types of other players are consistent with the observed actions of those players, but not necessarily with the entire strategy profile of the game. This means that players form beliefs based on the limited information they have observed, which influences their strategic choices. In a typical Nash equilibrium, all players' strategies are mutual best responses, given their beliefs about the other players' strategies.

A self-enforcing agreement is a type of contract or arrangement in which the terms and conditions are designed to be automatically upheld or enforced without the need for external intervention, such as a court or a regulatory agency. In other words, the agreement contains built-in mechanisms that incentivize the parties to comply voluntarily, as the consequences of non-compliance are sufficiently significant to deter breaches.

Separating equilibrium is a concept used in game theory and economics, particularly in the context of signaling games. It refers to a situation where different types of players (often with private information) choose distinct actions or strategies that reveal their type to other players. In a separating equilibrium, the actions taken by each type of player provide clear signals that allow other players to infer the type of the signaling player accurately.

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.

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.

A Strong Nash Equilibrium is a concept in game theory that extends the traditional notion of Nash equilibrium. In a typical Nash equilibrium, a set of strategies is considered stable if no single player can unilaterally change their strategy to achieve a better payoff, given the strategies of the other players. In contrast, a Strong Nash Equilibrium requires that no group of players can improve their payoff by jointly deviating from their current strategies.

Subgame Perfect Equilibrium (SPE) is a refinement of Nash Equilibrium used in game theory, specifically in the context of extensive-form games, which can be represented by a tree-like structure. In an SPE, the strategy profile is not only a Nash Equilibrium in the game as a whole but also remains a Nash Equilibrium in every subgame of that game.

Symmetric equilibrium is a concept often used in game theory and economics. It refers to a situation in which all players in a game or economic model choose the same strategy, leading to an equilibrium where no player has an incentive to deviate unilaterally from that strategy. In symmetric equilibrium: 1. **Identical Strategies**: All players have access to the same strategies, and they choose the same one.