I'm David. I'm reading through the books in the MIRI research guide and will write a review for each as I finish them. By way of inspiration from how Nate did it.

Naive Set Theory

Halmos Naive Set Theory is a classic and dense little book on axiomatic set theory, from a "naive" perspective.

Which is to say, the book won't dig to the depths of formality or philosophy, it focuses on getting you productive with set theory. The point is to give someone who wants to dig into advanced mathematics a foundation in set theory, as set theory is a fundamental tool used in a lot of mathematics.

Summary

Is it a good book? Yes.

Would I recommend it as a starting point, if you would like to learn set theory? No. The book has a terse presentation which makes it tough to digest if you aren't already familiar with propositional logic, perhaps set theory to some extent already and a bit of advanced mathematics in general. There are plenty of other books that can get you started there.

If you do have a somewhat fitting background, I think this should be a very competent pick to deepen your understanding of set theory. The author shows you the nuts and bolts of set theory and doesn't waste any time doing it.

Perspective of this review

I will first refer you to Nate's review, which I found to be a lucid take on it. I don't want to be redundant and repeat the good points made there, so I want to focus this review on the perspective of someone with a bit weaker background in math, and try to give some help to prospective readers with parts I found tricky in the book.

What is my perspective? While I've always had a knack for math, I only read about 2 months of mathematics at introductory university level, and not including discrete mathematics. I do have a thorough background in software development.

Set theory has eluded me. I've only picked up fragments. It's seemed very fundamental but school never gave me a good opportunity to learn it. I've wanted to understand it, which made it a joy to add Naive Set Theory to the top of my reading list.

How I read Naive Set Theory

Starting on Naive Set Theory, I quickly realized I wanted more meat to the explanations. What is this concept used for? How does it fit in to the larger subject of mathematics? What the heck is the author expressing here?

I supplemented heavily with wikipedia, math.stackexchange and other websites. Sometimes, I read other sources even before reading the chapter in the book. At two points, I laid down the book in order to finish two other books. The first was Gödel's Proof, which handed me some friendly examples of propositional logic. I had started reading it on the side when I realized it was contextually useful. The second was Concepts of Modern Mathematics, which gave me much of the larger mathematical context that Naive Set Theory didn't.

Consequently, while reading Naive Set Theory, I spent at least as much time reading other sources!

A bit into the book, I started struggling with the exercises. It simply felt like I hadn't been given all the tools to attempt the task. So, I concluded I needed a better introduction to mathematical proofs, ordered some books on the subject, and postponed investing into the exercises in Naive Set Theory until I had gotten that introduction.

Chapters

In general, if the book doesn't offer you enough explanation on a subject, search the Internet. Wikipedia has numerous competent articles, math.stackexchange is overflowing with content and there's plenty additional sources available on the net. If you get stuck, do try playing around with examples of sets on paper or in a text file. That's universal advice for math.

I'll follow with some key points and some highlights of things that tripped me up while reading the book.

Axiom of extension

The axiom of extension tells us how to distinguish between sets: Sets are the same if they contain the same elements. Different if they do not.

Axiom of specification

The axiom of specification allows you to create subsets by using conditions. This is pretty much what is done every time set builder notation is employed.

Puzzled by the bit about Russell's paradox at the end of the chapter? http://math.stackexchange.com/questions/651637/russells-paradox-in-naive-set-theory-by-paul-halmos

Unordered pairs

The axiom of pairs allows one to create a new set that contains the two original sets.

Unions and intersections

The axiom of unions allows one to create a new set that contains all the members of the original sets.

Complements and powers

The axiom of powers allows one to, out of one set, create a set containing all the different possible subsets of the original set.

Getting tripped up about the "for some" and "for every" notation used by Halmos? Welcome to the club:
http://math.stackexchange.com/questions/887363/axiom-of-unions-and-its-use-of-the-existential-quantifier
http://math.stackexchange.com/questions/1368073/order-of-evaluation-in-conditions-in-set-theory

Using natural language rather than logical notation is commmon practice in mathematical textbooks. You'd better get used to it:
http://math.stackexchange.com/questions/1368531/why-there-is-no-sign-of-logic-symbols-in-mathematical-texts

The existential quantifiers tripped me up a bit before I absorbed it. In math, you can freely express something like "Out of all possible x ever, give me the set of x that fulfill this condition". In programming languages, you tend to have to be much more... specific, in your statements.

Ordered pairs

Cartesian products are used to represent plenty of mathematical concepts, notably coordinate systems.

Relations

Equivalence relations and equivalence classes are important concepts in mathematics.

Functions

Halmos is using some dated terminology and is in my eyes a bit inconsistent here. In modern usage, we have: injective, surjective, bijective and functions that are none of these. Bijective is the combination of being both injective and surjective. Replace Halmos' "onto" with surjective, "one-to-one" with injective, and "one-to-one correspondence" with bijective.

He also confused me with his explanation of "characteristic function" - you might want to check another source there.

Families

This chapter tripped me up heavily because Halmos mixed in three things at the same time on page 36: 1. A confusing way of talking about sets. 2. Convoluted proof. 3. n-ary cartesian product.

Families are an alternative way of talking about sets. An indexed family is a set, with an index and a function in the background. A family of sets means a collection of sets, with an index and a function in the background. For Halmos build-up to n-ary cartesian products, the deal seems to be that he teases out order without explicitly using ordered pairs. Golf clap. Try this one for the math.se treatment: http://math.stackexchange.com/questions/312098/cartesian-products-and-families

Inverses and composites

The inverses Halmos defines here are more general than the inverse functions described on wikipedia. Halmos' inverses work even when the functions are not bijective.

Numbers

The axiom of infinity states that there is a set of the natural numbers.

The Peano axioms

The peano axioms can be modeled on the the set-theoretic axioms. The recursion theorem guarantees that recursive functions exist.

Arithmetic

The principle of mathematical induction is put to heavy use in order to define arithmetic.

Order

Partial orders, total orders, well orders -- are powerful mathematical concepts and are used extensively.

Some help on the way:
http://math.stackexchange.com/questions/1047409/sole-minimal-element-why-not-also-the-minimum
http://math.stackexchange.com/questions/367583/example-of-partial-order-thats-not-a-total-order-and-why
http://math.stackexchange.com/questions/225808/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct
http://math.stackexchange.com/questions/160451/difference-between-supremum-and-maximum

Also, keep in mind that infinite sets like subsets of w can muck up expectations about order. For example, a totally ordered set can have multiple elements without a predecessor.

Axiom of choice

The axiom of choice lets you, from any collection of non-empty sets, select an element from every set in the collection. The axiom is necessary to do these kind of "choices" with infinite sets. In finite cases, one can construct functions for the job using the other axioms. Though, the axiom of choice often makes the job easier in finite cases so it is used where it isn't necessary.

Zorn's lemma

Zorn's lemma is used in similar ways to the axiom of choice - making infinite many choices at once - which perhaps is not very strange considering ZL and AC have been proven to be equivalent.

robot-dreams offers some help in following the massive proof in the book.

Well ordering

A well-ordered set is a totally ordered set with the extra condition that every non-empty subset of it has a smallest element. This extra condition is useful when working with infinite sets.

The principle of transfinite induction means that if the presence of all strict predecessors of an element always implies the presence of the element itself, then the set must contain everything. Why does this matter? It means you can make conclusions about infinite sets beyond w, where mathematical induction isn't sufficient.

Transfinite recursion

Transfinite recursion is an analogue to the ordinary recursion theorem, in a similar way that transfinite induction is an analogue to mathematical induction - recursive functions for infinite sets beyond w.

In modern lingo, what Halmos calls a "similarity" is an "order isomorphism".

Ordinal numbers

The axiom of substitution is called the axiom (schema) of replacement in modern use. It's used for extending counting beyond w.

Sets of ordinal numbers

The counting theorem states that each well ordered set is order isomorphic to a unique ordinal number.

Ordinal arithmetic

The misbehavior of commutativity in arithmetic with ordinals tells us a natural fact about ordinals: if you tack on an element in the beginning, the result will be order isomorphic to what it is without that element. If you tack on an element at the end, the set now has a last element and is thus not order isomorphic to what you started with.

The Schröder-Bernstein theorem

The Schröder-Bernstein theorem states that if X dominates Y, and Y dominates X, then X ~ Y (X and Y are equivalent).

Countable sets

Cantor's theorem states that every set always has a smaller cardinal number than the cardinal number of its power set.

Cardinal arithmetic

Read this chapter after Cardinal numbers.

Cardinal arithmetic is an arithmetic where just about all the standard operators do nothing (beyond the finite cases).

Cardinal numbers

Read this chapter before Cardinal arithmetic.

The continuum hypothesis asserts that there is no cardinal number between that of the natural numbers and that of the reals. The generalized continuum hypothesis asserts that, for all cardinal numbers including aleph-0 and beyond aleph-0, the next cardinal number in the sequence is the power set of the previous one.

Concluding reflections

I am at the same time humbled by the subject and empowered by what I've learned in this episode. Mathematics is a truly vast and deep field. To build a solid foundation in proofs, I will now go through one or two books about mathematical proofs. I may return to Naive Set Theory after that. If anyone is interested, I could post my impressions of other mathematical books I read.

I think Naive Set Theory wasn't the optimal book for me at the stage I was. And I think Naive Set Theory probably should be replaced by another introductory book on set theory in the MIRI research guide. But that's a small complaint on an excellent document.

If you seek to get into a new field, know the prerequisites. Build your knowledge in solid steps. Which I guess, sometimes requires that you do test your limits to find out where you really are.

The next book I start on from the research guide is bound to be Computability and Logic.

Following in the path of So8res and others, I’ve decided to work my way through the textbooks on the MIRI Research Guide. I’ve been working my way through the guide since last October, but this is my first review. I plan on following up this review with reviews of Enderton’s A Mathematical Introduction to Logic and Sipser’s Introduction to the Theory of Computation. Hopefully these reviews will be of some use to you.

Discrete Mathematics and Its Applications

Discrete Mathematics and Its Applications is wonderful, gentle introduction to the math needed to understand most of the other books on the MIRI course list. It successfully pulls off a colloquial tone of voice. It spends a lot of time motivating concepts; it also contains a lot of interesting trivia and short biographies of famous mathematicians and computer scientists (which the textbook calls “links”). Additionally, the book provides a lot of examples for each of its theorems and topics. It also fleshes out the key subjects (counting, proofs, graphs, etc.) while also providing a high level overview of their applications. These combine to make it an excellent first textbook for learning discrete mathematics.

However, for much the same reasons, I would not recommend it nearly as much if you’ve taken a discrete math course. People who’ve participated in math competitions at the high school level probably won’t get much out of the textbooks either. Even though I went in with only the discrete math I did in high school, I still got quite frustrated at times because of how long the book would take to get to the point. Discrete Mathematics is intended to be quite introductory and it succeeds in this goal, but it probably won’t be very suitable as anything other than review for readers beyond the introductory level. The sole exception is the last chapter (on models of computation), but I recommend picking up a more comprehensive overview from Sipser’s Theory of Computation instead.

I still highly recommend it for those not familiar with the topics covered in the book. I’ve summarized the contents of the textbook below:

Contents:

1.     The Foundations: Logic and Proofs

2.   Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

3.     Algorithms

4.     Number Theory and Cryptography

5.     Induction and Recursion

6.     Counting

7.     Discrete Probability

8.     Advanced Counting Techniques

9.     Relations

10.  Graphs

11.  Trees

12.  Boolean Algebra

13.  Modeling Computation

The Foundations: Logic and Proofs

This chapter introduces propositional (sentential) logic, predicate logic, and proof theory at a very introductory level. It starts by introducing the propositions of propositional logic (!), then goes on to introduce applications of propositional logic, such as logic puzzles and logic circuits. It then goes on to introduce the idea of logical equivalence between sentences of propositional logic, before introducing quantifiers and predicate logic and its rules of inference. It then ends by talking about the different kinds of proofs one is likely to encounter – direct proofs via repeated modus ponens, proofs by contradiction, proof by cases, and constructive and non-constructive existence proofs.

This chapter illustrates exactly why this book is excellent as an introductory text. It doesn’t just introduce the terms, theorems, and definitions; it motivates them by giving applications. For example, it explains the need for predicate logic by pointing out that there are inferences that can’t be drawn using only propositional logic. Additionally, it also explains the common pitfalls for the different proof methods that it introduces.

Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

This chapter introduces the different objects one is likely to encounter in discrete mathematics. Most of it seemed pretty standard, with the following exceptions: functions are introduced without reference to relations; the “cardinality of sets” section provides a high level overview of a lot of set theory; and the matrices section introduces zero-one matrices, which are used in the chapters on relations and graphs.

Algorithms

This chapter presents … surprise, surprise… algorithms! It starts by introducing the notion of algorithms, and gives a few examples of simple algorithms. It then spends a page introducing the halting problem and showing its undecidability. (!) Afterwards, it introduces big-o, big-omega, and big-theta notation and then gives a (very informal) treatment of a portion of computation complexity theory. It's quite unusual to see algorithms being dealt with so early into a discrete math course, but it's quite important because the author starts providing examples of algorithms in almost every chapter after this one.

Number Theory and Cryptography

This section goes from simple modular arithmetic (3 divides 12!) to RSA, which I found extremely impressive. (Admittedly, I’ve only ever read one other discrete math textbook.) After introducing the notion of divisibility, the textbook takes the reader on a rapid tour through base-n notation, the fundamental theorem of arithmetic, the infinitude of primes, the Euclidean GCD algorithm, Bezout’s theorem, the Chinese remainder theorem, Fermat’s little theorem, and other key results of number theory. It then gives several applications of number theory: hash functions, pseudorandom numbers, check digits, and cryptography. The last of these gets its own section, and the book spends a large amount of it introducing RSA and its applications.

Induction and Recursion

This chapter introduces mathematical induction and recursion, two extremely important concepts in computer science. Proofs by mathematical induction, basically, are proofs that show that a property is true of the first natural number (positive integer in this book), and if it is true of an integer k it is true of k+1. With these two results, we can conclude that the property is true of all natural numbers (positive integers). The book then goes on to introduce strong induction and recursively defined functions and sets. From this, the book then goes on to introduce the concept of structural induction, which is a generalization of induction to work on recursively-defined sets. Then, the book introduces recursive algorithms, most notably the merge sort, before giving a high level overview of program verification techniques.

Counting

The book now changes subjects to talk about basic counting techniques, such as the product rule and the sum rule, before (interestingly) moving on to the pigeonhole principle. It then moves on to permutations and combinations, while introducing the notion of combinatorial proof, which is when we show that two sides of the identity count the same things but in different ways, or that there exists a bijection between the sets being counted on either side. The textbook then introduces binomial coefficients, Pascal’s triangle, and permutations/combinations with repetition. Finally, it gives algorithms that generate all the permutations and combinations of a set of n objects.

Compared to other sections, I feel that a higher proportion of readers would be familiar with the results of this chapter and the one on discrete probability that follows it. Other than the last section, which I found quite interesting but not particularly useful, I felt like I barely got anything from the chapter.

Discrete Probability

In this section the book covers probability, a topic that most of LessWrong should be quite familiar with. Like most introductory textbooks, it begins by introducing the notion of sample spaces and events as sets, before defining probability of an event E as the ratio of the cardinality of E to the cardinality of S. We are then introduced to other key concepts in probability theory: conditional probabilities, independence, and random variables, for example. The textbook takes care to flesh out this section with a discussion about the Birthday Problem and Monte Carlo algorithms. Afterwards, we are treated to a section on Bayes theorem, with the canonical example of disease testing for rare diseases and a less-canonical-but-still-used-quite-a-lot example of Naïve Bayes spam filters. The chapter concludes by introducing the expected value and variances of random variables, as well as a lot of key results (linearity of expectations and Chebyshev’s Inequality, to list two). Again, aside from the applications, most of this stuff is quite basic.

Advanced Counting Techniques

This chapter, though titled “advanced counting techniques”, is really just about recurrences and the principle of inclusion-exclusion. As you can tell by the length of this section, I found this chapter quite helpful nevertheless.  

We begin by giving three applications of recurrences: Fibonacci’s “rabbit problem”, the Tower of Hanoi, and dynamic programming. We’re then shown how to solve linear homogenous relations, which are relations of the form

a_n = c₁ a_n-1 + c₂ a_n-2 + … + c_k a_n-k+ F(n)

Where c1, c2, …, ck are constants, ck =/= 0, and F(n) is a function of n. The solutions are quite beautiful, and if you’re not familiar with them I recommend looking them up. Afterwards, we’re introduced to divide-and-conquer algorithms, which are recursive algorithms that solve smaller and smaller instances of the problem, as well as the master method for solving the recurrences associated with them, which tend to be of the form

f(n) = a f(n/b) + cn^d

After these algorithms, we’re introduced to generating functions, which are yet another way of solving recurrences.

Finally, after a long trip through various recurrence-solving methods, the textbook introduces the principle of inclusion-exclusion, which lets us figure out how many elements are in the union of a finite number of finite sets.

Relations

Finally, 7 chapters after the textbook talks about functions, it finally gets to relations. Relations are defined as sets of n-tuples, but the book also gives alternative ways of representing relations, such as matrices and directed graphs for binary relations. We’re then introduced to transitive closures and Warshall’s algorithm for computing the transitive closure of a relation. We conclude with two special types of relations: equivalence relations, which are reflexive, symmetric, and transitive; and partial orderings, which are reflexive, anti-symmetric, and transitive.

Graphs

After being first introduced to directed graphs as a way of representing relations in the previous chapter, we’re given a much more fleshed out treatment in this chapter. A graph is defined as a set of vertices and a set of edges connecting them. Edges can be directed or undirected, and graphs can be simple graphs (with no two edges connecting the same pair of vertices) or multigraphs, which contain multiple edges connecting the same pair of vertices. We’re then given a ton of terminology related to graphs, and a lot of theorems related to these terms. The treatment of graphs is quite advanced for an introductory textbook – it covers Dijkstra’s algorithm for shortest paths, for example, and ends with four coloring. I found this chapter to be a useful review of a lot of graph theory.

Trees

After dealing with graphs, we move on to trees, or connected graphs that don’t have cycles. The textbook gives a lot of examples of applications of trees, such as binary search trees, decision trees, and Huffman coding. We’re then presented with the three ways of traversing a tree – in-order, pre-order, and post-order. Afterwards, we get to the topic of spanning trees of graphs, which are trees that contain every vertex in the graph. Two algorithms are presented for finding spanning trees – depth first search and breadth first search. The chapter ends with a section on minimum spanning trees, which are spanning trees with the least weight. Once again we’re presented with two algorithms for finding minimum spanning trees: Prim’s Algorithm and Kruskal’s algorithm. Having never seen either of these algorithms before, I found this section to be quite interesting, though they are given a more comprehensive treatment in most introductory algorithms textbooks.  

Boolean Algebra

This section introduces Boolean algebra, which is basically a set of rules for manipulating elements of the set {0,1}. Why is this useful? Because, as it turns out, Boolean algebra is directly related to circuit design! The textbook first introduces the terminology and rules of Boolean algebra, and then moves on to circuits of logic gates and their relationship with Boolean functions. We conclude with two ways to minimize the complexity of Boolean functions (and thus circuits) – Karnaugh Maps and the Quine-McCluskey Method, which are both quite interesting. 

Modeling Computation

Who should read this?

If you’re not familiar with discrete mathematics, this is a great book that will get you up to speed on the key concepts, at least to the level where you’ll be able to understand the other textbooks on MIRI’s course list. Of the three textbooks I’m familiar with that cover discrete mathematics, I think that Rosen is hands down the best. I also think it’s quite a “fun” textbook to skim through, even if you’re familiar with some of the topics already.

However, I think that people familiar with the topics probably should look for other books, especially if they are looking for textbooks that are more concise. It might also not be suitable if you’re already really motivated to learn the subject, and just want to jump right in. There are a few topics not normally covered in other discrete math textbooks, but I feel that it’s better to pick up those topics in other textbooks.

What should I read?

In general, the rule for the textbook is: read the sections you’re not familiar with, and skim the sections you are familiar with, just to keep an eye out for cool examples or theorems.

In terms of chapter-by-chapter, chapters 1 and 2 seem like they’ll help if you’re new to mathematics or proofs, but probably can be skipped otherwise. Chapter 3 is pretty good to know in general, though I suspect most people here would find it too easy. Chapters 4 through 12 are what most courses on discrete mathematics seem to cover, and form the bulk of the book – I would recommend skimming them once just to make sure you know them, as they’re also quite important for understanding any serious CS textbook. Chapter 13, on the other hand, seems kind of tacked on, and probably should be picked up in other textbooks.

Final Notes

Of all the books on the MIRI research guide, this is probably the most accessible, but it is by no means a bad book. I’d highly recommend it to anyone who hasn’t had any exposure to discrete mathematics, and I think it’s an important prerequisite for the rest of the books on the MIRI research guide.

We've already had a lengthy (and still active) thread attempting to address the question "What are the best textbooks, and why are they better than their rivals?". That's excellent, but no one is going to post there unless they're prepared to claim: Textbook X is the best on its subject. But surely many of us have read many texts for which we couldn't say that but could say "I've read X and Y, and here's how they differ". A good supply of such comparisons would be extremely useful.

I propose this thread for that purpose. Rules:

• Each top-level reply should concern two or more texts on a single subject, and provide enough information about how they compare to one another that an interested would-be reader should be able to tell which is likely to be better for his or her purposes.
• Replies to these offering or soliciting further comparisons in the same domain are encouraged.
• At least one book in each comparison should either
  • be a very good one, or at least
    
  • look like a very good one even though it isn't.

If this gets enough responses that simply looking through them becomes tiresome, I'll update the article with (something like) a list of textbooks, arranged by subject and then by author, with links for the comments in which they're compared to other books and a brief summary of what's said about them. (I might include links to comments in Luke's thread too, since anything that deserves its place there would also be acceptable here.)

See also: magfrump's request for recommendations of basic science books; "Recommended Rationalist Reading" (narrower subject focus, and without the element of comparison).

Neuroeconomics is the application of advances in neuroscience to the fundamentals of economics: choice and valuation. Foundations of Neuroeconomic Analyis by Paul Glimcher, an active researcher in this area, presents a summary of this relatively new field to psychologists and economists. Although written as a serious work, the presentation is made across disciplines, so it should be accessible to anyone interested without much background knowledge in either area. Although the writing is so-so, the book covers multiple Less Wrong-relevant themes, from reductionism to neuroscience to decision theory. If nothing else, the results discussed provide a wonderful example of how no one knows what science doesn't know. I doubt many economists are aware researchers can point to something very similar to utility on a brain scanner and would scoff at the very notion.

Because of the book's wide target audience, there is not enough detail for specialists, but possibly a little too much for non-specialists. If you are interested in this topic, the best reason to pick up the book would be to track down further references. I hope the following summary does the book justice for everyone else.

Are book summaries of this sort useful? The recent review/summary of Predictably Irrational appears to have gone over well. Any suggestions to improve possible future reviews?

---

Introduction

Many economists think economics is fundamentally separate from psychology and neuroscience; since they take choices as primitives, little if any knowledge would be gained from understanding the mechanisms underlying choice. However, science steadily brings reduction and linkages between previously unrelated disciplines. A striking amount has already been discovered about the exact processes in the brain governing choice and valuation. On the other side, neuroscientists and psychologist underestimate the ability of economists to say whether claims about the brain are logically coherent or not.

Section I: The Challenge of Neuroeconomics

Consider a man and woman who have an affair with each other at a professional conference, which they later consider a mistake. An economist looking at this situation would treat their choice to sleep together as revealing a preference, regardless of their verbal claims. A psychologist would consider how mental states mediated this decision, and would be more willing to consider whether the decision was a mistake or not. Biologists would be more likely to point to ancestral benefits of extra-pair copulations, not considering the reflective judgements as directly relevant. These explanations largely speak past each other, hinting that a unified theory could do much better in predicting behavior.

The key to this is establishing linkages between the logical primitives of each discipline. Behavior could be explained on the level of physics, biology, psychology, or economics, but whether low-level explanations are practical is a different matter. Realistically, linking disciplines will strengthen both fields by mutually constraining the theories available to them.

With the neoclassical revolution, economics developed concepts of utility as reflecting ordinal relationships over revealed preferences. Choices that satisfied certain consistency conditions could be treated as if generated by a utility function. Additional axioms allowed consistent choice under uncertainty to be added to the theory. There are notable problems with this approach, but the core ideas of utility and maximization have surprisingly close neural analogues. Rather than operating "as if" individuals act on the basis of utility, a hard theory of "because" is being developed.

A look at visual perception reveals our subjective experience of light intensity varies subtantially depending on the wavelength of the light. Brightness is concept that resides in the mind, and furthermore sensitivity to different wavelengths corresponds precisely to the absorption spectra of the chemical rhodopsin in our retinas. All perceptions are represented in the mind along a power scale with some variance. Because the distributions of perceptions overlap, subjects can report accurately that a dimmer light is perceptually brighter. This suggests random utility models developed for statisical purposes might be directly explain what happens in the brain. One interesting consequence about the power scaling law is that risk aversion would be embedded at the level of perception.

Section II: The Choice Mechanism

Due to its relative simplicity, eye movement serves as a model for motor control and perhaps decisions broadly. The superior colliculus represents possible eye movements topographically with "hills" of activity. Eventually, the tissue transitions to a bursting state where the most active hill becomes much more active and the rest are inhibited via a "winner-take-all" or "argmax" mechanism. All inputs have to eye motion have to pass through the superior colliculus, so this represents a common final pathway of processed sensory signals. By giving monkeys varying awards for eye-motion tasks, activity in the lateral intraparietal area (LIP) correlates strongly with the probability and size of reward in an area known to trigger action before the action is taken. In other words, this appears to be a direct neural representation of subjective expected valuation. If monkey subjects play a game with mixed strategies in equilibrium, neuron firing rates are all roughly equal, matching the conclusion that expected utilities of actions are equalized when an opponent is mixing.

Cortial neurons fire almost like independent Poisson processes, resulting in neurons down the line being able to easily extract the mean firing rate of the inputs. Interneuronal correlation can vary according to the task at hand, resulting in greater or lesser variation of the final decision, so descriptive decision theories must incorporate randomness in choice. This also provides support for mixed strategies being represented directly in the brain.

Subjective valuations are normalized, and are only considered relative to the other options at hand. This normalization maximizes the joint information of neurons, increasing the efficiency of value representation. One consequence is that as the choice set increases, valuations start overlapping, and choice becomes essentially random. Activity also varies according to the delay of rewards, matching previous findings of hyperbolic discounting. While these findings are largely based on eye-movements in monkeys, this provides a clear path of how choice can be reduced to neural mechanisms.

Section III: Valuation

Back to visual perception, our judgements are made relative to other elements in the environment. Color looks roughly the same indoors and outdoors, even though there can be six orders of magnitude more illumination outside. Drifting reference points make absolute values unrecoverable. Local irrationalies due to reliance on a reference point arise because evolution is trading off between accurate sensory encoding and the costs of these irrationalities.

One promising way to specify the reference point is as the discounted sum of our future wealth. Learning depends on the difference between actual and expected rewards, so valuation compared to a reference point arises from the learning process. In the brain, reward prediction errors are encoded through dopamine. Dopamine firing rates are well-described by an exponentially weighted sum of previous awards subtracted from the most recent award. Hebb's law, which says "cells that fire together, wire together", describes how long-term predictions work.

Valuation appears to be orginally constructed in the striatum and medial prefrontal cortex. The reference level encoded there can be directly observed with brain scanners. Various other regions provide inputs to construct value. For instance, the orbitofrontal cortex (OFC) provides an assessment of risk. Subjects with lesions in this area exhibit almost perfect risk neutrality. Values might also be stored in the OFC, again in a compressed and encoded way. Longer-term valuations might be stored in the amygdala.

Because valuations are encoded relatively and don't work well over large choice sets, humans might edit out options by sequentially considering particular attributes until the choice set become manageable. Sorting by attributes can lead to irrational choices, unsurprisingly.

Probabilistic valuations depend on whether the expectation was learned experientially or symbolically. Symbolically communicated probabilities, where the person is told a number, are overweighed near zero and underweighted near one. Experientially communicated probabilities, where the person samples the lotteries directly, exhibit the opposite pattern. This suggests at least two mechanisms at work, especially with the ability to deal with symbolic probabilities arising relatively late in our evolutionary history. Also, while experiential expected values incorporate probabilities implicitly, this information can't be extracted. When probabilities change, the only means to change valuations is to relearn them from scratch.

Section IV: Summary and Conclusions

Here the author presents formalized models of the descriptive theory. The normative uses of this theory are still unclear. Even if we can identify subjective valuations in the brain, does this have any relation to welfare?

The four critical observations of neuroeconomics are reference-dependence, the lack of an absolute measure of anything in the brain, stochasticity in choice, and the influence of learning on choice. Along with the question of the welfare implications of these findings, six primary questions are currently unanswered:

1. Where is subjective value stored and how does it get to choice?
2. What part of the brain governs when it is "time to choose"?
3. What neural mechanism guides complementarity between goods?
4. How does symbolic probability work?
5. How does the state of the world and utility interact?
6. How does the brain represent money?