32 of 33 people found the following review helpful:
4.0 out of 5 stars
An Inimitable and Exhilarating Tour of Decision Theory, February 18, 2009
This review is from: Rational Decisions (The Gorman Lectures in Economics) (Hardcover)
Modern decision theory was founded by Frank Ramsey and Bruno di Finetti in the 1930's, but the definitive exposition is due to Leonard (Jimmie) Savage in the late 1950's. Bayesian decision theory has both ardent supporters and petulant detractors, but like Binmore, I believe it is about the best thing going, despite its limitations. Binmore sets himself the task of both expounding and defending theory, while stressing the limited domain over which it applies. His assessment is insightful and balanced, basically embracing Savage's own assessment of the theory.
Ken Binmore has always been a brilliant thinker, but he used to be an awful writer, combining the worst of repetitiveness and opaqueness. He has, however, become an excellent writer, as this slim volume attests. Binmore refuses to "stick to the facts," but rather offers his personal opinions at every turn. This can be extremely refreshing. Daniel Ellsberg, he tells us, "made a number of important contributions to decision theory before he heroically blew the whistle on Richard Nixon's cynical attitude to the loss of American lives in the Vietnam War by leaking what became known as the Pentagon Papers" (p. 89). Concerning the excesses of Bayesianism, he writes that "the ghost of the Reverend Thomas Bayes must be in a constant state of astonishment that our culture has embraced a philosophical doctrine called Bayesianism that treats the trivial manipulation of conditional probabilities that he discovered sometime before 1764 as the solution to the problem of scientific induction." (p. 126) Commenting on the Harsanyi Doctrine, Binmore writes: "John Harsanyi...advocated using a mind experiment to determine the one-and-only rational prior. If Pandora imagines that a veil of ignorance conceals all the information she has ever received, she will supposedly select the same prior as all other ideally rational folk in the same sate of sublime ignorance. But when I try this trick, no ideas about a suitable prior come to me at all." (p. 128) There are more expansive critiques of the Harsanyi Doctrine, but none so disarmingly accurate.
Binmore follows Savage in defending the "small world" application of the rational actor model. In "small world" decision theory, decision-makers know the payoffs and probabilities, and decisions come down to a simple choice among an array of clear alternatives. In the "large world" in which we live, Binmore asserts that Bayesian decision-making is virtually useless, or at least highly compromised. Indeed, human decision-makers share the capacity for rational choice with many other species, but only humans can make creative, insightful, and personally enhancing decisions under conditions of extreme uncertainty and partial ignorance. Similarly, new information in the small world context entails Bayesian updating using conditional probabilities, but in the large world, new information can lead to wholesale rejection of a mental framework in favor of an alternative. Binmore spends several short chapters developing a "muddling through" alternative that he believes might apply to the large world context. His exposition is creative and enlightening, although purely mathematical. There is a whole school of cognitive psychology currently working on this issue, including Alison Gopnik at UC Berkeley and Joshua Tenenbaum at MIT, who believe Bayesian insights apply to understanding the human brain, who might learn from Binmore's exposition (and conversely, Binmore might gain from the study of such cognitive research).
Instead of heaping more praise on this book, some alternative insights might serve the reader better. I can supply several. For one, Binmore follows the crowd in holding that the Nash equilibrium concept is the centerpiece of game theory. He writes: "There are two reasons why game theorists care about Nash equilibria. The first reason is that a game theory book can't authoritatively point to a pair of strategies (a,b) as the rational solution of a game unless it is a Nash equilibrium." (p. 26) In fact, in many games, we can authoritatively point to a range of non-Nash strategies that are routinely played by rational players, and have higher payoff that Nash strategies. For instance, in the finitely repeated Prisoner's Dilemma, the only Nash equilibrium is mutual defection on every round. However, real-life individuals often cooperate for many plays of the game, until the final period of play looms large on the horizon. Moreover, rational individuals rarely have an incentive to play a mixed strategy because all the underlying pure strategies have equal payoff.
Binmore gives as the second reason for stressing Nash equilibrium that all stable equilibria of evolutionary games are Nash equilibria of the underlying stage game. This is true, but only if errors take particular forms. Low levels of random mutation in such systems can maintain the evolutionary system far, far from a Nash equilibrium.
Binmore moreover strenuously supports the position that the only rational move in the Prisoner's Dilemma is to defect. Now, his argument is mainly directed against implausible philosophical arguments that are highly deserving of criticism. But in the real world, subjects often prefer to cooperate, provided their partner's cooperate, and there is certainly nothing irrational about such other-regarding or morally-influenced behavior. Binmore makes his point by arguing that if Alice and Bob prefer to cooperate in the Prisoner's Dilemma, then it really isn't a Prisoner's Dilemma at all, but rather a new "Reciprocator Game. This is just false. The Prisoner's Dilemma is defined by a set of available actions and a payoff matrix, independent of whether the players are self-regarding or other-regarding. I think is counterproductive to defend the principle that rationality implies defection in the Prisoner's Dilemma by redefining the game when rational players prefer not to defect.
One of the more controversial of Binmore's choices is to develop a measure-theoretic approach to probability theory. I am sure this is useful in some cases (e.g., when dealing with diffusion processes), but most of game theory and decision theory gets by very well by assuming finite probability measures. Moreover, the Axiom of Choice is a key assumption in justifying the Lebesgue measure/Kolmogorov probability approach, and the Axiom of Choice forces the analysis to be non-constructive. This feeds the economist's tendency to shy away from constructive approaches to model-building. My prejudice is to use non-constructive approaches only where absolutely necessary, and then to apologize that we cannont do better.
Binmore's treatment of epistemic game theory is very exciting and insightful, but probably too breezy for all but the most knowledgeable readers. Gödel's contributions are taken for granted, as well as Turing's and Church's. Binmore makes an argument inspired by the Turing halting problem that "seeks to discredit the use of Bayesian epistemology in worlds in which...self -reference cannot be avoided." (p. 146) The argument is quite sophisticated, but it is not complete (it assumes a certain Turing machine L "sometimes answers NO when asked suitable coded questions" without suggesting why, or when, this might be the case). In fact, as Binmore notes, there is an argument due to Kaplan and Montague (Mind 1960) that makes such an argument based on Gödel's constructions. The whole self-referencing literature is important and critical for game theory. It is wonderful to see even a brief treatment in this book.
I have found in several of Binmore's writings what seems to me to be a confusion of rationality and common knowledge of rationality (CKR). This occurs clearly in Rational Decisions. Referring to Robert Aumann's famous proof that CKR implies backward induction in extensive form games, Binmore says "But rational players stay on the equilibrium path because of what would happen if they were to deviate. In the counterfactual world that would be created by such a deviation, the players would have to live with the fact that their knowledge that nobody will play irrational has proved fallible." (p. 149) In fact, it is not rational players, but players who accept CKR, who stay on the equilibrium path, and counterfactuals have nothing whatever to do with it. Moreover, it is not a violation of rationality to move off the equilibrium path, but rather a violation of CKR. Indeed, as both Aumann and Binmore understand, and as I explain at length in my book, The Bounds of Reason (Princeton, 2009), CKR is precisely the sort of self-referencing condition that should be avoided. Indeed, I argue that CKR is not a "higher form" of rationality, but rather a technical condition that has no epistemic justification.
Just to check my memory, I went back to an article Binmore wrote in 1996, "A Note on Backward Induction," Games and Economic Behavior 17, 135-137. In this paper, Binmore objects to Aumann's proof (1995) that CKR necessarily implies backward induction. Here are three examples of Binmore equating rationality with CKR.
First, Binmore writes "According to Aumann (1995), common knowledge of rationality in the Centipede makes it irrational for player I to choose across at his opening move." In fact, of course, it does not make choosing across irrational; rather choosing across violates CKR. Second, Binmore writes "If down is the only Bayesian-rational action at the opening, then p < ½." In fact, down is not the only Bayesian-rational action; rather it is the only action compatible with CKR. Finally, a rather long but important argument: "if nothing can be said about what would happen off the backward-induction path, then it seems obvious that nothing can be said about the rationality of remaining on the backward-induction path. How else do we...
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4 of 14 people found the following review helpful:
4.0 out of 5 stars
Binmore is completely ignorant of Keynes's lower-upper interval approach to probability, February 26, 2009
This review is from: Rational Decisions (The Gorman Lectures in Economics) (Hardcover)
Binmore realizes that the current ,Bayesian decision theory approach,which is a generalization of Jeremy Bentham's approach,based on the claim that all probabilities are subjective, unique,precise,exact,additive single number estimates ,as are all outcomes,has broken down.It can only be applied in Savage's "small " world, as opposed to the real world.On pp. 163-169,he proposes to use a variant of the lower-upper interval approach to probability that he credits to Good as a way of making Bayesianism more relevant and applicable.Binmore argues ,correctly,that Bayesianism can't deal with uncertainty (p.35), as opposed to risk ,or ignorance (p.154).He presents a decent discussion of lower-upper probabilities on pp.88-93,although he has no idea at all that Keynes had already provided a much more broader discussion of such interval estimates in the A T
reatise on Probability.However,we are then treated to the following summary,provided after Binmore categorically dismisses and rejects Keynes's logical approach to probability ,which Keynes firmly anchored to his own modified version of Boole's (The Laws of Thought,1854)original lower -upper probability approach in chapters 15,17,20,22,and 26 of the A Treatise on Probability in 1921 [the reader should note that this material is also in Keynes's 1907 (unsuccessful) and 1908 (successful) Fellowship dissertations done at Cambridge]: "The prevailing orthodoxy in economics is Bayesianism,which I take to be the philosophical position that Bayesian decision theory always applies to all decision problems.In particular,it proceeds as though the subjective probabilities of Savage's theory can be reinterpreted as logical probabilities without any hassle "(Binmore,p.96).Binmore is correct only in the very special case where Keynes's weight of the evidence,w, specified on the unit interval from 0 to 1,is equal to 1.This is the case of linear risk.It means that there is no uncertainty or ignorance facing the decision maker,only risk.Binmore attempted to deal with uncertainty earlier in his book when he went over the Ellsberg Paradox,where Ellsberg's ambiguity would be substituted for Keynes's uncertainty from the GT or Keynes's weight of the evidence,w, from the TP.
In summary,Binmore has successfully reinvented the wheel.His suggested solution is inferior to that first presented by J M Keynes over a 100 years ago.Binmore,of course,is correct.What is amazing is his complete and total ignorance of the work in this area of J M Keynes . Most likely this is due to his accepting at face value two totally preposterous book reviews of Keynes's TP written by the original Bayesian subjectivist,Frank Ramsey.These two book reviews,writtten in 1922 and 1926,respectively,are the source of the prevalent, incorrect claims that Keynes used " non-numerical " probabiities and that these non-numerical probabilities involved no numbers at all except in the few, very special cases where the principle of indifference could be applied.This makes no sense at all today or in 1922 or 1926.
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