Refinements of the Nash equilibrium concept Book

Well I have two pages of this text to go (since I have yet to read the concluding chapter, which I will do tomorrow), so I can go ahead and write my review early.

This is a good book. Despite what I may say here, it is at least a step in a great direction: I think it is a good book.

This book attempts to discover an equilibrium point for player's behavior in noncooperative and cooperative games (and all intermediate stages in between). It does so, in my opinion, rather poorly. BUT if you are not familiar with the Nash equilibrium, then this book is a great way to learn it. Quite frankly, seeing as Harsanyi won the Peace prize with Selten and Nash, Harsanyi really does know Nash's equilibrium quite well. It's the most simple, and accurate, part of the book, seeing as it isn't his own liberties within the field of mathematics to explain things.

This brings me to my main scruples with this book:
1. I wish he had focused more on the idea of the objective externalities vs. subjective interpretations of his 'social welfare function'. The 'social welfare function' is really anything but, as it comprises the individual himself. I can, through a series of mathematical and philosophically moral arguments, make this make sense. His function 'w' is fully elaborated in chapter 4 and makes a surprisingly intelligent and awe-inspiring comeback in chapter 13, as a measurement of the non-transferable utility the individual RECEIVES from the coalitions within society. I thought this was a fascinating connection: to connect the social welfare with his utility returns from them in the first place. I thought 'm' a rather abstract constant as well, supposed to denote the sensitivity of certain participants in the welfare function to certain moral values. All taken together if he decided to focus on THESE issues, this entire book would have been a lot better. That connection I just referred to is similar to Nietzschean morality or the Benthamite rendition of what it really means to 'give' to the community in a meaningful (ie utilitarian) way. That your contributions and receipts of utility are non-transferable I thought to be the crux of the argument: after all, morals are not traded in money, yet for some reason he does linearize utility throughout most of this book, so while not entirely monetary he stays true to Von Neumann in this respect, and maybe in this only.

Going back to the original point I had to make though: in any game, a strategy is effective and dominant if it is part of Alpha, the set of imputations that dominate all Beta strategies of himself and opponents. This is a complex set of imputations, but the imputations are discrete (not a compact, convex set, which I will get into in a second). These imputations are themselves sets of vectors which are dependent on externalities within the game, like expected behavior and relative preferences. It is important to know that these relative preferences though could easily be objective externalities. And to the extent that an individual's behavior is fundamentally deterministic through these is the extent that narrows the individual's strategy down to one choice, as indicated in the original Theory of Games book by Von Neumann. But these are all OBJECTIVE externalities. They have nothing to do with the psychological interpretations of others on this individual. A lot of the questions facing society could be tackled from the viewpoint of objective externalities, yet I did not see this quasi-mathematician do any of that, instead he whirled around saying everyone interprets everything differently, even society itself. Which, while I agree, is not entirely the case, obviously there are some objective externalities to discuss, which he basically tackles through the use of the constant 'm' in the fourth chapter and the different values of the social welfare function. But he never delves into this issue or defines it fully. It may be that I should write something about it, look into it more myself, but I can't see why he would focus on the subjectivities, which you cannot define, and not on the objectivies, which you can. I can guarantee you Von Neumann and Morgenstern would not have overlooked that, considering they have both (Harsanyi and Neumann) opted to linearize utility.
2. Harsanyi does not really understand Von Neumann's concepts of disjunct strategy imputations. The different vectors given for certain imputations within Von Neumann's Alpha set of strategies, implies certain behavioral differences that are discrete. Indeed, in Von Neumann's example of a 4 person game with preferences to one player, the player's strategies were scattered either in the center, or in the corner of the preferential player. These are disjunct areas. They are not compact nor convex. Apparently Harsanyi does not understand the meaning of having different strategies. If I find it EQUALLY (since we are assuming linear utility here) advantageous to post high bids 90% of the time and fold 10% of the time as I find it advantageous to fold 90% of the time and post high bids 10% of the time, does it make sense that I should then find the 'equiprobability mixture of all strongly dominating strategies' for the 'centroid best reply' theory? Under Harsanyi's absurd centroid best reply theory, he stipulates that if those strategies are strongly dominating all others, if I am facing a coalition of all the other players (which, by the way, would only happen theoretically if I had a strong preference like in Von Neumann's sample 4-person game) then I should take an equiprobability mixture of the strategies. That almost leaves me at 50/50 bet high and fold, which is most likely not even CLOSE to being as profitable as either of my most effective strategies. Absurd.

Imputations are frequently used in conjunction with external objective factors as well, and that is nowhere covered within Harsanyi's book either, implying that he might have misinterpreted that concept by Von Neumann as well.

3. Some of the mathematics is just... absurd... as are the terms. Like it or not, Harsanyi is a Nobel prize winner. That's why I read this. But the introduction of some of the terms (and their misuse, as I believe he misused Saddle Points and Imputations, as introduced by Von Neumann and Morgenstern) was absurd. By the end of the book I was wondering if anything was really defined well in this book at all. Sometimes concepts are brought up and introduced and layed out randomly, so I have to pay attention, because all the way up until the end of the book he is still defining new terms. Theoretically this wouldn't be a problem if the terminology even made sense. He is acting like he is defining a science, but in reality he is simply describing some of the original concepts as put forward by Von Neumann somewhat incorrectly. There are also times where I felt rigorous mathematics (like for the definition of constant M in chapter 4) COULD have been used (almost anywhere else in the book). And then there were times that I felt it SHOULD NOT have been used at all (for many of the concepts towards the end).

Overall though, if you are unfamiliar with the Nash equilibrium, Zeuthen's principle, or just bargaining in Game Theory, I would certainly give this a read as it would be beneficial to anyone in those fields. My main scruples with the book are HUGE terminological differences though, so if you're wondering if I am correct or not, and you have not yet read the original work by Von Neumann, do so immediately, preferably before beginning this work.