Rational Decisions

by Ken Binmore

It is widely held that Bayesian decision theory is the final word on how a rational person should make decisions. However, Leonard Savage--the inventor of Bayesian decision theory--argued that it would be ridiculous to use his theory outside the kind of small world in which it is always possible to "look before you leap." If taken seriously, this view makes Bayesian decision theory inappropriate for the large worlds of scientific discovery and macroeconomic enterprise. When is it correct to use Bayesian decision theory--and when does it need to be modified? Using a minimum of mathematics, Rational Decisions clearly explains the foundations of Bayesian decision theory and shows why Savage restricted the theory's application to small worlds.


The book is a wide-ranging exploration of standard theories of choice and belief under risk and uncertainty. Ken Binmore discusses the various philosophical attitudes related to the nature of probability and offers resolutions to paradoxes believed to hinder further progress. In arguing that the Bayesian approach to knowledge is inadequate in a large world, Binmore proposes an extension to Bayesian decision theory--allowing the idea of a mixed strategy in game theory to be expanded to a larger set of what Binmore refers to as "muddled" strategies.


Written by one of the world's leading game theorists, Rational Decisions is the touchstone for anyone needing a concise, accessible, and expert view on Bayesian decision making.

Genres: Economics, Philosophy, Psychology, Science, Nonfiction, Mathematics, Textbooks

224 pages, Hardcover

First published January 1, 2008

About the author

Professor of Economics at UCL, after holding corresponding positions at LSE and the University of Pennsylvania and Michigan. Onetime Professor of Mathematics at LSE.

Author of 77 published papers and 11 books. Research in evolutionary game theory, bargaining theory, experimental economics, political philosophy, mathematics and statistics.

Grants from National Science Foundation (3), ESRC (1), STICERD (2) and others. Chairman of LSE Economics Theory Workshop (10 years), Director of Michigan Economic Laboratory (5 years). Fellow of the Econometric Society and British Academy. Extensive collaboration with 25 co-authors.

Awarded the CBE in the New Years Honours List 2001 largely for his role in designing the UK 3G Spectrum Auction.

Reviews

[Manny · Rating 3 out of 5]

[Original review, Jun 21 2021]
I don't know enough game theory to be able to give a competent opinion on this book, but like everything from Binmore it's well-written, full of provocative thoughts, and based on a phenomenal amount of reading. The central idea is characteristically imaginative: maybe there are types of game strategies that no one has even considered?

To give some background, Binmore spends quite a lot of time here discussing a toy game he calls micro-poker. This is played between two people we'll refer to as Q and JK, using a deck of three cards which contains a Jack, a Queen and a King. At the start of the game, Q gets the Queen face-up, and JK gets either the Jack or the King face-down, with a 50% chance of receiving either one. JK can look at his card but Q doesn't know what it is.

Now JK has to decide whether to check or to raise. If he checks, he loses one dollar. If he raises, Q has to decide whether to fold or to see. If he folds, he loses one dollar. If he sees, the player with the higher card wins two dollars and the other one loses two dollars. What are the best strategies for each player?

Evidently JK will raise if he has the King, but what does he do if he's dealt the Jack? If he always raises, then after a while Q will notice this and always call. Then JK will win when he has the King and lose when he has the Jack, so he'll break even. Similarly, if he never raises with the Jack, then Q will again notice after a while that JK's only raising half the time. He'll stop seeing altogether, and again he'll break even.

As any fule kno, both players should choose mixed strategies. If JK has the Jack, he should raise with some probability x, and if he raises then Q should see with some probability y, using some kind of method (a "randomising box") to choose the number in a way that can't be predicted. The question is what the optimal values are for the probabilities. It's fun to use a little algebra to work it out! JK's payoff is:

0.5(2y + (1-y)) + 0.5(-(1-x) + x(-2y + (1-y))) =

0.5(2y + (1-y) + -(1-x) + x(-2y + (1-y))) =

0.5(y + x + x(-3y + 1)) =

0.5(y + 2x - 3xy) [P]

If JK chooses x = 1/3, the formula for the payoff P reduces to

0.5(y + 2*1/3 - 3*1/3*y) = 1/3

which doesn't depend on y. So whatever strategy Q chooses, JK will come out ahead in the long run, gaining an average of 33 cents per game. Similarly, if Q chooses y = 2/3, he can make sure that X doesn't do any better than this, and the two choices form a Nash equilibrium.

I though all games were in principle similar, just more complicated. But Binmore says, suppose you could make choices that don't have an associated probability? He goes to some lengths to specify exactly what this means (there is quite a lot of nontrivial mathematics), but at the end of the day he has enough machinery to be able to define what he calls a "totally muddling box". This takes the concept of a randomising box a step further: a totally muddling box produces numbers randomly, and you can't assign a probability to any given thing happening. He shows you a simple game where he says a totally muddling box can give you a better strategy than any ordinary randomising box.

And then, just when I thought things had got really interesting, the book stops! Damn. Why?! Did I miss something? Couldn't he think of a suitable way to continue? Had he made his point and felt no need to do any more? Those mathematicians. Sometimes they're too smart for their own good.
______________________
[Update, Jun 23 2021]

I showed the micro-poker problem to Not and another strong bridge-player we know, and they both gave the same answer after only a few seconds: if they were playing JK and were dealt the Jack, they'd raise all the time.

I pointed out that this is suboptimal according to the game-theoretical analysis above, but Not, a practical card-player with vast experience, was unimpressed. If we set x = 1 (i.e. JK raises all the time on the Jack), the value for the payoff formula P reduces to y - 1. If Q sees at their game-theoretical optimal rate of 2/3, JK will do as well raising all the time as they would with the game-theoretical optimal rate of raising one-third of the time. For JK to do worse by raising all the time, Q has to realise that they should be seeing more often than two-thirds of the time. It could take them a while to figure out that they need to adopt this seemingly risky strategy. As Binmore points out several times in the book, most people are instinctively risk-averse. The bottom line is that raising all the time can't do worse than break even, is likely to do as well as the theoretically optimal strategy, and gives the opponent plenty of chances to go wrong.

I'm reminded of a famous quote from Karpov, one of the world's great pragmatic chessplayers. A journalist asked Karpov what he'd do given a choice between a brilliant but complicated sacrificial combination which might lead to a forced win, and an endgame with a tiny advantage.

"The endgame, without even thinking," said Karpov.

[Hélio Steven · Rating 4 out of 5]

Binmore is a well known authority on game theory, especially its classical version. In this book he takes a step back to tackle issues in decision theory -- more specifically, he's interested in developing an approach to how agents can make rational decisions in large worlds, which are worlds with a whole lot more of complexity and noise, and thus riddled with risk and uncertainty. To this end, Binmore goes back to the foundations of decision theory and starts off by clearing the conceptual ground.

Binmore is well aware that "rationality" has multiple meanings, and so seeks to avoid confusion by providing a working definition of what it is to be a rational agent. He resorts to Paul Samuelson's revealed preference theory (RPT), according to which an agent is taken to be rational if she observes consistency, stability and transitivity in her choices. Binmore is also careful to make clear that "preference" here is not to be understood as a psychological variable, but rather as a summary of behavioral patterns: an agent that consistently chooses A over B thereby reveals a preference relation A > B. Binmore warns us that reading what was just said as "the choice of A over B was caused by the agent's experiencing a greater utility choosing A instead of choosing B" is to fall for what he calls the causal utility fallacy. Under RPT, it's the other way around: it's because an agent chooses A over B that we say she prefers A to B, and then we construct an utility function for such agent where A has higher utility than B.

Armed with this thin notion of rationality, Binmore powers through a range of topics related to decision theory. Along the way he makes impotant distinctions that tend to get bundled on discussions about rational decisions. And so we are reminded that risk, uncertainty and ambiguity should be regarded as different things, and also that there are at least three different notions of probability (objective, subjective, and logical) that can't be conflated. And then we arrive at Bayesian decision theory, which models rational decisions as those decisions that result from the application of the famous Bayes' rule. Binmore tells us that this procedure was first deployed by Savage to model decision making in small worlds, and that Savage himself also warned not to extrapolate this modeling approach to decision problems in worlds larger than the ones he was considering. However, it turns out that a lot of people did exactly that, originating what Binmore calls "Bayesianism", an approach according to which Bayesian updating is the way to go in every decision making and learning context. Binmore rejects Bayesianism while at the same time setting himself the goal to extend Bayesian decision theory to larger worlds. In this process it's especially interesting how Binmore deals with the problem of establishing priors. Here it becomes much clearer why Savage was wary to apply Bayesian decision theory to larger worlds. In small worlds, it's possible for an agent to establish their priors by what Binmore calls the "massaging" of their posterior probabilities of future events given some amount of present information. Since we're in small worlds, it's easier for an agent to gather enough information and get into this massaging process and subsequently coming up with a prior before making a decision. As worlds get larger and larger, though, uncertainty scales up much more, to the point where the usefulness of this massaging process is unclear for constructing an agent's priors. This was especially illuminating for me because for some time now I have seen the application of Bayesian reasoning as very important and useful, while at the same time having reservations about how to properly select priors. My impression is that this is not an insurmountable problem for the Bayesian approach, but Binmore's discussion certainly has made it clear to me that dealig with this problem needs more careful thinking than I had imagined.

All in all, this was a stimulating book to read. However, if someone doesn't have a fresh and decent background in some mathematics, especially algebra, some basic geometry and probability, they'll likely struggle quite a bit in many parts of the book. I'm not proud to say that I'm saying this from experience, as my algebra and my geometry are still in terrible shape. As a result I most certainly didn't get as much from the book as other people that have their maths straight can get. Still, I felt it was worth the time and effort. Other than the main project, along the book Binmore also gives us some interesting personal takes on side discussions, such as how one can conceptualize knowledge in scientific enquiry as commitment to a model.

Coincidentally, a week ago I came across a more recent paper Binmore wrote that basically summarizes the main argument of the book in a math-free way, so people who want to have a better idea of what exactly is discussed in the book and are reluctant to start it due to math anxiety (a feeling I know too well) should read this paper for more encouragement. The paper's title is "On the Foundations of Decision Theory" (you can read it for free here: https://link.springer.com/article/10....).

[Rif A. Saurous · Rating 4 out of 5]

I have become super interested in Leonard Jimmie Savage's small vs. big distinction, particularly the idea that Bayesian inference only makes sense in small worlds, where we have a complete probabilistic model of the relevant situation. This book is often recommended (by AI) as the modern version of that story. I was slightly underwhelmed in that regard; the book discusses this only in passing, agreeing with my-view-of-Savage's-view (I haven't gone back and read Savage's 1950's work) without expanding on it. Instead, the book discusses the basics of decision theory and the foundations of probability, and concludes with a final chapter on how to extend Bayesian inferences to *slightly* larger worlds using upper and lower probabilities, which was interesting but not what I was looking for.

Generally well-written and readable, although I could've used a little more review rather than assuming I'd remembered the complicated arguments from previous chapters. Droll in places.

[Dio Mavroyannis · Rating 5 out of 5]

This is a great little book, lots of paradoxes, a great way of explaining things. Goes through small/large world distinctions, Bayesian decision theory etc. I consider most of the content here to be approachable by a non-expert in decision theory(though to be realistic, you probably need some math background to be comfortable from the first to the last chapter). The technical appendix is also brilliantly written with some absolutely fantastic measure-theoretic examples. No book like it!

[Jalaluddin Abdullah · Rating 5 out of 5]

This is a must-read as it presents a comprehensive set of the principles and axioms behind neo-classical economics. Binmore is a mathematician, hence everything is mapped correctly, clearly, and thoroughly.
I spent several days in a seminar with Binmore and was surprised to discover, from his arguments, that much of the criticism against the foundations of decision theory is a strawman. The idea doesn't say what people think it says. It may have some problems (such as knowledge of probability and understanding of future payoffs) but not the problems discussed in the behavioral and heterodox literature that appear to be violated by people in their experiments. Binmore writes the following gem: "Nor does the theory [Revealed Preferences] insist that people are selfish, as its critics mischievously maintain. It has no difficulty modeling the kind of saintly folk who would sell the shirt off their back rather than see a baby cry". Binmore doesn't say it explicitly, but hints that even the highly influential critiques of Amartya Sen in "Rational Fools" and elsewhere appear to be a bit straw mannish. The book is short and dense enough to be a reference.

[Adrien Mogenet · Rating 3 out of 5]

Lots of insights on the different paradigms ruling the decision theories. I don't necessarily agree with all of Ken's views on rationality, but his theories on the limitations of bayesianism are definitely worth a read. It's well written, and it's easy to spot the sections you want to dig and the ones you can skip without losing the essence of the theories. A great recipe!

[Maximus · Rating 3 out of 5]

Worth reading if you are interested in decision theory, but lacks any (?) real world applicability. Granted while I didn’t skim trough it, I could have read it more thoroughly but it was just to boring for that.

As for real world applicability you are much better served reading Nassim Taleb or even the behavioral economists.

[Denis Vasilev · Rating 4 out of 5]

Сложновато без серьезной подготовки в Теории игр и Теории вероятностей

[Marius Heje Mæhle]

Binmore prøver å utvide og gjøre game theory mer applicable i ett bredere perspektiv. Min matematiske kunnskap strekker ikke til på dette nivået

[Badal]

- https://x.com/RichardMCNgo/status/211...

[Dennis · Rating 4 out of 5]

http://pro-libertate.net/20091026/100...

[Ai Herrera huerta]

Creo que será un buen libro para entender las decisiones