In repeated games, strategies are often evaluated by their ability to guarantee the performance of the single best action that is selected in hindsight (a property referred to as Hannan consistency, or no-regret). However, the effectiveness of the single best action as a yardstick to evaluate strategies is limited, as any static action may perform poorly in common dynamic settings. We propose the notion of dynamic benchmark consistency, which requires a strategy to asymptotically guarantee the performance of the best dynamic sequence of actions selected in hindsight subject to a constraint on the number of action changes the corresponding dynamic benchmark admits. We show that dynamic benchmark consistent strategies exist if and only if the number of changes in the benchmark scales sublinearly with the horizon length. Further, our main result establishes that the set of empirical joint distributions of play that may emerge, when all players deploy such strategies, asymptotically coincides with the set of Hannan equilibria (also referred to as coarse correlated equilibria) of the stage game. This general characterization allows one to leverage analyses developed for frameworks that consider static benchmarks, which we demonstrate by bounding the social efficiency of the possible outcomes in our setting. Together, our results imply that dynamic benchmark consistent strategies introduce the following Pareto-type improvement over no-regret strategies: They enable stronger individual guarantees against arbitrary strategies of the other players, while maintaining the same worst-case guarantees on the social welfare, when all players adopt these strategies.

Game theoretic solution concepts, such as Nash equilibrium strategies that are optimal against worst case opponents, provide guidance in finding desirable autonomous agent behaviour. In particular, we wish to approximate solutions to complex, dynamic tasks, such as negotiation or bidding in auctions. Computational game theory investigates effective methods for computing such strategies. Solving human-scale games, however, is currently an intractable problem. Counterfactual Regret Minimization (CFR), is a regret-minimizing, online learning algorithm that dominates the Annual Computer Poker Competition (ACPC) and lends itself readily to various sampling and abstraction techniques. Abstract games are created to mirror the strategic elements of an original game in a more compact representation. The abstract game can be solved and the abstract game solution can be translated back into the full game. But crafting an abstract game requires domain-specific knowledge, and an abstraction can interact with the game solving process in unintuitive and harmful ways. For example, abstracting a game can create pathologies where solutions to more granular abstractions can be more exploitable against a worst-case opponent in the full game than those derived from simpler abstractions. An abstraction that could be dynamically changed and informed by the solution process could produce better solutions more consistently. We suggest that such abstractions can be largely subsumed by a regressor on game features that estimates regret during CFR. Replacing abstraction with a regressor allows the memory required to approximate a solution to a game to be proportional to the complexity of the regressor rather than the size of the game itself. Furthermore, the regressor essentially becomes a tunable, compact, and dynamic abstraction of the game that is informed by and adapts to the particular solution being computed. These properties will allow this technique to scale to previously intractable domains. We call this new algorithm Regression CFR (RCFR). In addition to showing that this approach is theoretically and practically sound, we improve RCFR by combining it with regret-matching+. Experiments involving two small poker games show that RCFR and its extension, RCFR+, show that it can approximately solve games with regressors that are drastically less complex than the game itself. In comparisons with traditional static abstractions of similar complexity, RCFR variants tend to produce less exploitable strategies.

Computer science and economics have engaged in a lively interaction over the past fifteen years, resulting in the new field of algorithmic game theory. Many problems that are central to modern computer science, ranging from resource allocation in large networks to online advertising, involve interactions between multiple self-interested parties. Economics and game theory offer a host of useful models and definitions to reason about such problems. The flow of ideas also travels in the other direction, and concepts from computer science are increasingly important in economics. This book grew out of the author's Stanford University course on algorithmic game theory, and aims to give students and other newcomers a quick and accessible introduction to many of the most important concepts in the field. The book also includes case studies on online advertising, wireless spectrum auctions, kidney exchange, and network management.

In this monograph, the focus is on two extreme cases in which the analysis of regret is particularly simple and elegant: independent and identically distributed payoffs and adversarial payoffs. Besides the basic setting of finitely many actions, it analyzes some of the most important variants and extensions, such as the contextual bandit model.

A comprehensive and rigorous introduction for graduate students and researchers, with applications in sequential decision-making problems.

The first textbook to explain the principles of epistemic game theory.

How do we begin to cope with loss that cannot be resolved? The COVID-19 pandemic has left many of us haunted by feelings of anxiety, despair, and even anger. In this book, pioneering therapist Pauline Boss identifies these vague feelings of distress as caused by ambiguous loss, losses that remain unclear and hard to pin down, and thus have no closure. Collectively the world is grieving as the pandemic continues to change our everyday lives. With a loss of trust in the world as a safe place, a loss of certainty about health care, education, employment, lingering anxieties plague many of us, even as parts of the world are opening back up again. Yet after so much loss, our search must be for a sense of meaning, and not something as elusive and impossible as "closure." This book provides many strategies for coping: encouraging us to increase our tolerance of ambiguity and acknowledging our resilience as we express a normal grief, and still look to the future with hope and possibility.

When a loved one dies we mourn our loss. We take comfort in the rituals that mark the passing, and we turn to those around us for support. But what happens when there is no closure, when a family member or a friend who may be still alive is lost to us nonetheless? How, for example, does the mother whose soldier son is missing in action, or the family of an Alzheimer's patient who is suffering from severe dementia, deal with the uncertainty surrounding this kind of loss? In this sensitive and lucid account, Pauline Boss explains that, all too often, those confronted with such ambiguous loss fluctuate between hope and hopelessness. Suffered too long, these emotions can deaden feeling and make it impossible for people to move on with their lives. Yet the central message of this book is that they can move on. Drawing on her research and clinical experience, Boss suggests strategies that can cushion the pain and help families come to terms with their grief. Her work features the heartening narratives of those who cope with ambiguous loss and manage to leave their sadness behind, including those who have lost family members to divorce, immigration, adoption, chronic mental illness, and brain injury. With its message of hope, this eloquent book offers guidance and understanding to those struggling to regain their lives. Table of Contents: 1. Frozen Grief 2. Leaving without Goodbye 3. Goodbye without Leaving 4. Mixed Emotions 5. Ups and Downs 6. The Family Gamble 7. The Turning Point 8. Making Sense out of Ambiguity 9. The Benefit of a Doubt Notes Acknowledgments Reviews of this book: You will find yourself thinking about the issues discussed in this book long after you put it down and perhaps wishing you had extra copies for friends and family members who might benefit from knowing that their sorrows are not unique...This book's value lies in its giving a name to a force many of us will confront--sadly, more than once--and providing personal stories based on 20 years of interviews and research. --Pamela Gerhardt, Washington Post Reviews of this book: A compassionate exploration of the effects of ambiguous loss and how those experiencing it handle this most devastating of losses ... Boss's approach is to encourage families to talk together, to reach a consensus about how to mourn that which has been lost and how to celebrate that which remains. Her simple stories of families doing just that contain lessons for all. Insightful, practical, and refreshingly free of psychobabble. --Kirkus Review Reviews of this book: Engagingly written and richly rewarding, this title presents what Boss has learned from many years of treating individuals and families suffering from uncertain or incomplete loss...The obvious depth of the author's understanding of sufferers of ambiguous loss and the facility with which she communicates that understanding make this a book to be recommended. --R. R. Cornellius, Choice Reviews of this book: Written for a wide readership, the concepts of ambiguous loss take immediate form through the many provocative examples and stories Boss includes, All readers will find stories with which they will relate...Sensitive, grounded and practical, this book should, in my estimation, be required reading for family practitioners. --Ted Bowman, Family Forum Reviews of this book: Dr. Boss describes [the] all-too-common phenomenon [of unresolved grief] as resulting from either of two circumstances: when the lost person is still physically present but emotionally absent or when the lost person is physically absent but still emotionally present. In addition to senility, physical presence but psychological absence may result, for example, when a person is suffering from a serious mental disorder like schizophrenia or depression or debilitating neurological damage from an accident or severe stroke, when a person abuses drugs or alcohol, when a child is autistic or when a spouse is a workaholic who is not really 'there' even when he or she is at home...Cases of physical absence with continuing psychological presence typically occur when a soldier is missing in action, when a child disappears and is not found, when a former lover or spouse is still very much missed, when a child 'loses' a parent to divorce or when people are separated from their loved ones by immigration...Professionals familiar with Dr. Boss's work emphasised that people suffering from ambiguous loss were not mentally ill, but were just stuck and needed help getting past the barrier or unresolved grief so that they could get on with their lives. --Asian Age Combining her talents as a compassionate family therapist and a creative researcher, Pauline Boss eloquently shows the many and complex ways that people can cope with the inevitable losses in contemporary family life. A wise book, and certain to become a classic. --Constance R. Ahrons, author of The Good Divorce A powerful and healing book. Families experiencing ambiguous loss will find strategies for seeing what aspects of their loved ones remain, and for understanding and grieving what they have lost. Pauline Boss offers us both insight and clarity. --Kathy Weingarten, Ph.D, The Family Institute of Cambridge, Harvard Medical School

These authors draw on nearly 50 years of combined teaching and consulting experience to give readers a straightforward yet systematic approach for making estimates about the likelihood and consequences of future events -- and then using those assessments to arrive at sound decisions. The book's real-world cases, supplemented with expository text and spreadsheets, help readers master such techniques as decision trees and simulation, such concepts as probability, the value of information, and strategic gaming; and such applications as inventory stocking problems, bidding situations, and negotiating.