The Ore extension, named after the mathematician Ole Johan Dahl Ore, is a concept in algebra that pertains to the extension of rings and modules. In particular, it is used to construct new rings from a given ring by adding new elements and defining new operations. The most common application of Ore extensions occurs in the context of noncommutative algebra, where it is used to form the Ore localization of a polynomial ring. This involves extending a ring by introducing new elements that satisfy specific relations concerning multiplication.

In the context of ring theory, a branch of abstract algebra, a **perfect ring** is a specific type of ring that has certain characteristics relating to its structure, particularly concerning ideals and their relations to other elements in the ring.

An **orthodox semigroup** is a specific type of algebraic structure that arises in the study of semigroups. A semigroup is a set equipped with an associative binary operation. The concept of an orthodox semigroup relates to the structure of its idempotent elements, which significantly influence the semigroup's properties.

A subdirectly irreducible algebra is a concept from universal algebra, a branch of mathematics that studies algebraic structures. Specifically, an algebraic structure (such as a group, ring, or lattice) is called subdirectly irreducible if it cannot be represented as a non-trivial subdirect product of other algebras. ### Definition An algebra \( A \) is said to be subdirectly irreducible if: 1. It is non-trivial, i.

Post's theorem, named after Emil Post, is a result in the field of mathematical logic and computability theory. It specifically deals with the properties of recursively enumerable sets, particularly in the context of formal languages and decision problems. The theorem states that: **"For any countable set of recursive (or computable) functions, there exists a recursively enumerable set that captures all the functions from the set.

Bid shading is a strategy often used in auction markets, particularly in the context of online advertising and real-time bidding (RTB) environments. It refers to the practice where bidders intentionally lower their bids from the maximum price they are willing to pay in order to increase their expected return on investment (ROI) or efficiency of their ad spend.

Tarski's undefinability theorem is a result in mathematical logic that deals with the concept of truth within formal languages. Named after the logician Alfred Tarski, the theorem asserts that the notion of truth cannot be defined within a sufficiently expressive formal language that can express arithmetic truths about itself.

Bidding is the process of making an offer to purchase or secure a product, service, or asset, often in a competitive environment. It is commonly used in various contexts, including auctions, procurement, real estate, and online platforms. Here are a few key aspects of bidding: 1. **Types of Bidding**: - **Open bidding**: Participants can see others' bids, and it often continues until no higher bids are placed.

Shashi and Ravi Ruia are prominent Indian businessmen and industrialists known for their significant contributions to the Indian business landscape, particularly through their involvement in the Essar Group. The Essar Group is a multinational conglomerate with interests in sectors such as steel, power, oil and gas, telecommunications, and infrastructure. The brothers co-founded the Essar Group in the 1960s, and under their leadership, the group grew into one of India's largest and most influential corporate entities.

Ariel Rubinstein is an Israeli economist and game theorist known for his significant contributions to the fields of economics and game theory. He has worked on various topics including rationality, bargaining, and strategic interactions. Rubinstein is also recognized for his role in the development of concepts such as extensive-form games and perfect equilibrium. In addition to his academic work, he has published extensively and has been involved in discussions about the foundations of economic theory and the implications of game theory in real-world scenarios.

Robin Farquharson does not appear to be a widely recognized public figure or a prominent entity based on available information up to October 2023. It is possible that he could be a private individual or a lesser-known person in a specific field.

Thomas Schelling (1921-2021) was an American economist and Nobel laureate known for his work in game theory, bargaining, and strategic decision-making. He was awarded the Nobel Prize in Economic Sciences in 2005 for his analysis of conflict and cooperation through game-theoretic concepts. Schelling's most notable contributions include his work on the dynamics of segregation in societies, as outlined in his influential book "Micromotives and Macrobehavior," published in 1978.

The Ultimatum Game is a classic experiment in economics and psychology that demonstrates the concepts of fairness, negotiation, and decision-making in social situations. It involves two players: 1. **The Proposer**: This player is given a specific amount of money (or another resource) and makes an offer on how to split that amount with the second player.

A bimatrix game is a type of game in game theory that involves two players, each of whom has a finite set of strategies. The outcomes of their choices are represented in the form of a matrix (or two matrices) that captures the payoffs for each combination of strategies chosen by the players. In a bimatrix game, the strategies of the two players can be represented as follows: - Player 1's strategies are represented in one matrix (let's call it Matrix A).

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.

"Co-opetition" is a business strategy concept that combines elements of cooperation and competition, and it is also the title of a book written by Adam M. Brandenburger and Barry J. Nalebuff, published in 1996. The authors introduce the idea that companies can benefit from cooperating with each other while still maintaining competitive edges. The book outlines how businesses can create value in markets by understanding and managing the interplay between cooperative and competitive strategies.

An N-player game is a type of strategic game in which multiple players (N) participate, each making decisions that can affect not only their own outcomes but also the outcomes of other players. These games can involve various strategies, rules, and objectives, and they can be either cooperative or competitive in nature. Key characteristics of N-player games include: 1. **Multiple Players**: The game involves more than two players, with "N" representing the number of players involved.