Coordinate Systems for Games
Simplifying the "me" and "we" Interactions
- 2019
- Buch
- Verfasst von
- Daniel T. Jessie
- Prof. Donald G. Saari
- Verlag
- Springer International Publishing
Über dieses Buch
This monograph develops a method of creating convenient coordinate systems for game theory that will allow readers to more easily understand, analyze, and create games at various levels of complexity. By identifying the unique characterization of games that separates the individual’s strategic interests from the group’s collective behavior, the authors construct a single analytical methodology that readers will be able to apply to a wide variety of games. With its emphasis on practicality and approachability, readers will find this book an invaluable tool, and a viable alternative to the ad hoc analytical approach that has become customary for researchers utilizing game theory.
The introductory chapters serve two important purposes: they review several games of fundamental importance, and also introduce a dynamic that is inherent in games, but has gone unexplored until now. After this has been established, readers will advance from simple 2 x 2 games to games with more player strategies and dynamics. For interested readers, a rigorous treatment of the underlying mathematics is conveniently gathered at the end of the book. Additional topics of interest, such as extensive form and coalitional games, are presented to help readers visualize more complex settings that will be vital in aiding the understanding of advanced topics, such as coalition-free Nash points, multi-player repeated games, and more.
Coordinate Systems for Games is ideal for a wide variety of researchers interested in game theory, including social scientists, economists, mathematicians, computer scientists, and more. The authors' approachable style also makes this accessible to an audience at any scale of experience, from beginning non-specialists to more practiced researchers.
Inhaltsverzeichnis
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Frontmatter
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Chapter 1. Introduction
Daniel T. Jessie, Donald G. SaariAbstractMost readers probably know something about game theory, so selective fundamentals are described as a quick primer for those who are not familiar with the topic and as a hasty review for all others. What differs from a standard introduction is that certain basic structures of games are outlined by borrowing concepts from dynamical systems, and it offers an intuitive commentary of the book’s peculiar “me and we” subtitle, which refers to personal opportunity versus possible mutual benefits. -
Chapter 2. Two-Player, Two-Strategy Games
Daniel T. Jessie, Donald G. SaariAbstractWhile the analysis in this chapter is straightforward, a considerable amount of new information is developed. Thus, a quick preview will help to pull it together. -
Chapter 3. Consequences
Daniel T. Jessie, Donald G. SaariAbstractAs indicated in the Preface, a measure of whether a coordinate system is appropriately designed is if it simplifies discovering and proving new results. A reason this feature holds with our decomposition is that the associated coordinates explicitly separate and display the inherent tension between individual preferences and group advantages or disadvantages. Indeed, as shown in Chap. 2, the \(\mathcal G^B\) component can launch unexpected complexity into a game. But what about other choices of coordinates? In several places of this chapter, modified coordinates are introduced to reflect new needs. -
Chapter 4. Two-Person Multi-strategy Games
Daniel T. Jessie, Donald G. SaariAbstractMuch of what has been developed in previous chapters holds for games with more players and/or pure strategies. Rather than carrying out fairly obvious extensions, we identify attributes that differ from \(2\times 2\) games. This includes explaining why certain \(2\times 2\) assertions collapse with more strategies, differences in \(\mathcal G^B\) features, a mystery associated with the Ultimatum game, and the Nash structure of \(k_1\times k_2\) games (including possible hiding places for NME). -
Chapter 5. Extensive Form Games
Daniel T. Jessie, Donald G. SaariAbstractThere are other ways to represent games, and so we show how to adapt the coordinate system. (Readers more interested in normal form games with more players could jump ahead to the next chapter.) In a single-shot normal form game, it is standard to assume that each player makes their strategic choices simultaneously, or, at least, without knowledge of any other player’s strategic choices. -
Chapter 6. Multiplayer Games
Daniel T. Jessie, Donald G. SaariAbstractEchoing the “Two is company, but three is a crowd” expression, adding players to a game can create interesting, unexpected, and perhaps unwanted differences. On the other hand, modeling truly multiplayer settings with two-player games can be counterproductive. -
Chapter 7. The Underlying Mathematics
Daniel T. Jessie, Donald G. SaariAbstractThis chapter, which essentially is the concluding one (it is followed by a brief summary), develops the mathematical foundation for our decomposition. It is written in a manner to encourage readers to apply these notions to other concerns. -
Chapter 8. Summary
Daniel T. Jessie, Donald G. SaariAbstractThis skinny chapter is positioned off by itself because not all readers will wade through the previous chapter to appreciate homomorphisms, modules, character tables, and the contributions of Maschke and Schur. Yet, expect many to return because fields of rewards involving other social, behavioral, and managerial issues are waiting to be harvested. Think of it this way; if much of what is being done in an area involves, as true with voting and game theories, symmetries with linear processes (equations, algebra, comparisons, etc.), then it is worth learning the Chap. 7 material. And if anyone utters the phrase, “Here are some symmetries that are satisfied, but they are useless, so forget them,” then definitely review Chap. 7! -
Backmatter
- Titel
- Coordinate Systems for Games
- Verfasst von
-
Daniel T. Jessie
Prof. Donald G. Saari
- Copyright-Jahr
- 2019
- Electronic ISBN
- 978-3-030-35847-1
- Print ISBN
- 978-3-030-35846-4
- DOI
- https://doi.org/10.1007/978-3-030-35847-1
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