Cores of Convex Games

Cores of Convex Games by Lloyd Shapley studies the mathematical theory of cooperative games, examining how groups of players distribute benefits and costs. The work focuses on the concept of the "core" - a way to allocate payoffs that gives each potential coalition at least as much as they could achieve on their own. The book introduces key concepts by using both formal mathematical proofs and explanatory examples drawn from economics and social scenarios. Through rigorous analysis, Shapley establishes fundamental theorems about when game cores exist and how they behave under different conditions. The text builds up from basic definitions to advanced results about infinite games and non-atomic markets. Multiple chapters explore applications to economics, particularly market mechanisms and resource allocation problems. This foundational work connects abstract game theory to concrete questions about fairness, stability, and efficiency in human cooperation. The implications extend beyond mathematics into economics, political science, and the social sciences broadly.

There are not enough internet reviews to create a summary of this book. Instead, here is a summary of reviews of Lloyd Shapley's overall work: Note: Lloyd Shapley published academic papers and mathematical works rather than books for general readers, so traditional reader reviews are limited. His work is primarily discussed in academic contexts. Academic readers value Shapley's clear mathematical proofs and elegant solutions to complex allocation problems. His papers on cooperative game theory receive citations for their precise formulations and practical applications. Several researchers note how his Shapley value concept provides intuitive solutions to fair division problems. PhD students and researchers sometimes find his papers challenging due to dense mathematical notation and assumptions of advanced knowledge. Some note that additional explanatory material would help accessibility. On Google Scholar, his most-cited works include: - "College Admissions and Stability of Marriage" (23,000+ citations) - "On Cores and Indivisibility" (3,000+ citations) - "Values of Large Games" (2,800+ citations) Traditional review sites like Goodreads and Amazon do not have ratings for Shapley's academic publications. His work appears primarily in economics journals and mathematical proceedings.

Theory of Games and Economic Behavior by John von Neumann, Oskar Morgenstern. This foundational text establishes the mathematical framework for game theory and cooperative solutions that Shapley's work builds upon.

The Theory of Value by Gerard Debreu. The book presents mathematical approaches to economic equilibrium through convex analysis and set theory, complementing Shapley's treatment of convex games.

Convex Analysis by R. Tyrrell Rockafellar. This text provides the mathematical tools and theorems for understanding convex functions and sets that underpin the analysis of convex games.

Game Theory and Applications by Tatsuro Ichiishi. The book expands on cooperative game theory concepts with applications to economics and social sciences, following Shapley's theoretical framework.

Algorithmic Game Theory by Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani. The text connects classical game theory concepts to computational methods, including implementations of Shapley value calculations.

🎲 Lloyd Shapley received the Nobel Prize in Economics in 2012 for his work on cooperative game theory, including the concepts explored in this book about game cores. 🏆 The core of a game, which is central to this book's focus, represents a set of payoff allocations where no coalition of players can achieve better results by breaking away from the grand coalition. 📚 Shapley developed what became known as the "Shapley value" - a solution concept for fairly distributing both gains and costs among players in cooperative games. 🔄 The book's exploration of convex games helped establish fundamental principles that are now applied in economics, particularly in matching markets and resource allocation problems. 🎓 While at RAND Corporation, where much of this work was developed, Shapley collaborated with David Gale to create the Gale-Shapley algorithm, which is used today in matching medical residents to hospitals and students to schools.