Belief-Based Stability in Coalition Formation with Uncertainty…

“In this book we present several novel concepts in cooperative game theory, but from a computer scientist’s point of view. Especially, we will look at a type of games called non-transferable utility games. […] In this book, we extend the classic stability concept of the non-transferable utility core by proposing new belief-based stability criteria under uncertainty, and illustrate how the new concept can be used to analyse the stability of a new type of belief-based coalition formation game. Mechanisms for reaching solutions of the new stable criteria are proposed and some real life application examples are studied. […] In Chapter 1, we first provide an introduction of topics in game theory that are relevant to the concepts discussed in this book. In Chapter 2, we review some relevant works from the literature, especially in cooperative game theory and multi-agent coalition formation problems. In Chapter 3, we discuss the effect of uncertainty in the agent’s beliefs on the stability of the games. A rule-based approach is adopted and the concepts of strong core and weak core are introduced. We also discuss the effect of precision of the beliefs on the stability of the coalitions. In Chapter 4, we introduce private beliefs in non-transferable utility (NTU) games, so that the preferences of the agents are no longer common knowledge. The impact of belief accuracy on stability is also examined. In Chapter 5, we study an application of the proposed belief-based stability concept, namely the buyer coalition problem, and we see how the proposed concept can be used in the evaluation of this multi-agent coalition formation problem. In Chapter 6, we combine the works of earlier chapters and produce a complete picture of the introduced concepts: non-transferable utility games with private beliefs and uncertainty. We conclude this book in Chapter 7.”

The above quote is from the preface of the book, which I finished yesterday. It deals with some issues I was slightly annoyed about not being covered in a previous micro course; my main problem being that it seemed to me back then that the question of belief accuracy and the role of this variable was not properly addressed in the models we looked at (‘people can have mistaken beliefs, and it seems obvious that the ways in which they’re wrong can affect which solutions are eventually reached’). The book makes the point that if you look at coalition formation in a context where it is not reasonable to assume that information is shared among coalition partners (because it is in the interest of the participants to keep their information/preferences/willingness to pay private), then the beliefs of the potential coalition partners may play a major role in determining which coalitions are feasible and which are ruled out. A key point is that in the model context explored by the authors, inaccurate beliefs of agents will expand the number of potential coalitions which are available, although coalition options ruled out by accurate beliefs are less stable than ones which are not. They do not discuss the fact that this feature is unquestionably a result of implicit assumptions made along the way which may not be true, and that inaccurate beliefs may also in some contexts conceivably lead to lower solution support in general (e.g. through variables such as disagreement, or, to think more in terms of concepts specifically included in their model framework, higher general instability of solutions which can feasibly be reached, making agents less likely to explore the option of participating in coalitions in the first place due to the lower payoffs associated with the available coalitions likely to be reached – dynamics such as these are not included in the coverage). I decided early on to not blog the stuff in this book in major detail because it’s not the kind of book where this makes sense to do (in my opinion), but if you’re curious about how they proceed, they talk quite a bit about the (classical) Core and discuss why this is not an appropriate solution concept to apply in the contexts they explore, and they then proceed to come up with new and better solution criteria, developed with the aid of some new variables and definitions along the way, in order to end up with some better solution concepts, their so-called ‘belief-based cores’, which are perhaps best thought of as extensions of the classical core concept. I should perhaps point out, as this may not be completely clear, that the beliefs they talk about deal both with the ‘state of nature’ (which in part of the coverage is assumed to be basically unobservable) and the preferences of agents involved.

If you want a sort of bigger picture idea of what this book is about, I should point out that in general you have two major sub-fields of game theory, dealing with cooperative and non-cooperative games respectively. Within the sub-field of cooperative games, a distinction is made between games and settings where utilities are transferable, and games/settings where they are not. This book belongs in the latter category; it deals with cooperative games in which utilities are non-transferable. The authors in the beginning make a big deal out of the distinction between whether or not utilities are transferable, and claim that the assumption that they’re not is the more plausible one; whereas they do have a point, I however also actually think the non-transferability assumption might in some of the specific examples included in the book be a borderline questionable assumption. To give an example, the non-transferability assumption seems in one context to imply that all potential coalition partners have the same amount of bargaining power. This assumption is plausible in some contexts, but wildly implausible in others (and I’m not sure the authors would agree with me about which contexts would belong to which category).

The professor teaching the most recent course in micro I took had a background in computer science, rather than economics – he was also Asian, but this perhaps goes without saying. This book is supposedly a computer science book, and they argue in the introduction that: “instead of looking at human beings, we study the problem from an intelligent software agent’s perspective.” However I don’t think a single one of the examples included in the book would be an example you could not also have found in a classic micro text, and it’s really hard to tell in many parts of the coverage that the authors aren’t economists with a background in micro – there seems to be quite a bit of field overlap here (this field overlap incidentally extends to areas of economics besides micro, is my impression; one econometrics TA I had, teaching the programming part of the course, was also a CS major). In the book they talk a bit about coalition formation mechanisms and approaches, such as propose-and-evaluate mechanisms and auction approaches, and they also touch briefly upon stuff like mechanism design. They state in the description that: “The book is intended for graduate students, engineers, and researchers in the field of artificial intelligence and computer science.” I think it’s really weird that they don’t include (micro-)economists as well, because this stuff is obviously quite close to/potentially relevant to the kind of work some of these people are working on.

There are a lot of definitions, theorems, and proofs in this book, and as usual when doing work on game theory you need to think very carefully about the stuff they cover to be able to follow it, but I actually found it reasonably accessible – the book is not terribly difficult to read. Though I would probably advise you against reading the book if you have not at least read an intro text on game theory. Although as already mentioned the book deals with an analytical context in which utilities are non-transferable, it should be pointed out that this assumption is sort of implicit in the coverage, in the sense that the authors don’t really deal with utility functions at all; the book only deals with preference relations, not utility functions, so it probably helps to be familiar with this type of analysis (e.g. by having studied (solved some problems) dealing with the kind of stuff included in the coverage in chapter 1 of Mas-Colell).

Part of the reason why I gave the book only two stars is that the authors are Chinese and their English is terrible. Another reason is that as is usually the case in game theory, these guys spend a lot of time and effort being very careful to define their terms and make correct inferences from the assumptions they make – but they don’t really end up saying very much.

February 28, 2015 - Posted by | books, Computer science, economics

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