Sorry if this is more suited to r/learnmath, but I thought there might be a quicker response here. I’m aware that there are great notes which help when reading through baby rudin for example, so I was wondering if there was a collection of these for more textbooks?
So decision theory is basically the theory of what decisions rational agents make, for different definitions of rational (think game theory, except you usually talk about 1 player or completely cooperative games). One proposal was that rational agents maximize the expected value of a utility function. This means that there was some function from outcomes to real numbers representing how much they “liked” that outcome. When making a decision, the agent figures out the probability that each outcome for each option they could choose, and then they choose the option that maximizes the expected value of the utility of the outcome. This theory was able to explain why people avoided risky financial situations even if on average they would make a net positive of money. Their utility function “cared” more about not losing money than gaining it. This intuition behind why someone would have a utility function like that is the fact that quality of life scales sublinearly with wealth (if you gain $1,000, you’ll probably be rather happy. If you lose $1,000, you might end up homeless).
Anyways John von Neumann and Oskar Morgenstern proved a theorem about this hypothesis called the Von Neumann–Morgenstern utility theorem. They introduced a new concept called VNM-rational. A VNM-rational agent satisfies 4 axioms, stated in the article. The theorem then proved that if an agent is VNM-rational, then there exists some utility function (commonly called the VNM utility function) such that the agents decisions coincide with the decisions that maximize that utility function, even if they are not aware of it. This was surprising, since the axioms for VNM-rationality seemed much weaker than the expected utility hypothesis.
Anyways, enough background. Why I am writing this post? Well, the cool thing is that the VNM theorem is constructive. In fact, for a small number of outcomes, you can efficiently find the VNM utility function of an agent by answering a series of “would you rather” type questions. The activity below tells you how to calculate it.
Note that this activity will only give a completely accurate result if you are VNM rational. (This is not a given, since some people think that being VNM rational is not required to be rational in the regular sense of the word.) Also, this activity is meant for fun; although the results may be interesting, they will not turn you into a hyperrational robot person or anything like that. Feel free to give silly answers to the questions.
List some possible outcomes your life could have. Ideally you’d list all of them, but because that could make the list too long, only list a couple. For example, “creating a robot dinosaur army, taking over southern France, and then dying peaceful” could be an outcome.
Determine which outcome you like the best, and which you like the least. If there is a tie, choose one arbitrarily. However, if you are indifferent between all the outcomes, assign each outcome a utility of 0, and jump to step 4. Otherwise, call the best outcome A, and the worst outcome B. Assign A a utility of 100, and B a utility of 0.
Now, for every other scenario C, determine C’s utility using these questions. (Note that there is a shortcut at the end. It is quicker, but more complicated conceptually.)
(OPTIONAL) Pick any positive real number q and any real number r (it does not matter which ones you chose). For each outcome, replace its utility with qx + r, where x is the old utility.
And you are done! Now, for any decision, figure out what probability you will have for each outcome given each option. The one that maximizes the utility function is the VNM-rational choice, assuming you did the above steps correctly.
Now, some more observations. You probably noticed that during option 4, you could had many different options for q and r. This is because the Von Neumann utility function is not unique. Any q and r will give you a valid one. However, you can get from any Von Neumann utility function to another using step 4. In other words it is unique up to addition of constants and multiplication of positive constants. This seemingly small problem actually has a lot of implications. Utilitarians, for example, propose that we should take actions that maximize the average or total utility across all individuals (or some other aggregate of utilities). The fact that VNM utility is not unique means that you can not simply calculate everyone’s VNM utilities and use that in your calculation. If you want to use VNM utility functions, you have to specify which one to use for each person. (If you are using average utility, you actually only need to be unique up to an additive constant, since the additive constants will not change the ordering of decisions under the average. Multiplicative constants will still change the result though.)
Another thing you probably noticed is that this is not a super practical way of making decisions. First off, your life usually has an extremely large number of outcomes, and second off, calculating how small decisions (like which shoe to tie first) effect their probabilities is basically impossible. Although this activity is mostly for entertainment value, there are some cases ways to make it more practical decision making tool, if you are so inclined. One thing you can do is only use it for one, big, decision at a time, and list the outcomes of the scenario it effects, instead of looking all the way to the end of your life. For example, you could try using these to decide what major to choose based on how it will affect your income (calculating utils of income is much faster than a general set of outcomes, since utility is usually monotonic (but not linear) with respect to wealth). For more complicated decisions, you can use what is known as an Influence diagram, which breaks a situation done into chunks, and then depicts how they are related. These are apparently actually used by large businesses, although I am having trouble finding any specific examples (which would happen either way, since businesses are usually pretty secretive about decision making). They have been widely studied by decision theorists, though.
So, what do you all think? Anyone want to share their results?
By now I imagine most everyone has seen the Numberphile’s video for the Trapped Knight with Neil Sloane of OEIS.org. Link below.
Well I’m curious if folks would like to work together towards implementing a new ruleset. Perhaps the Rook and Queen might be interesting to study also. Here goes the suggested rule change.
So on the infinite spiral, we start at 1 like usual and hop the knight’s pattern to the lowest available tile. But in this variant, we’d combine for the sum of the tile we’re on and the tile we travel to. So the first jump is to 10 from 1, 1+10=11. Here we would cross 11 off wherever it is to be found on the board, and remain to jump from 10. So 1 and 10 can both be returned to still, in fact 1 will always be returnable to because no 2 numbers will sum to it, unless perhaps something at the end game occurs and it travels back to 1 and gets trapped there, then it maybe it might be worth considering that 1 has finished. What does everyone think? Would this be difficult to implement into a program displaying the board with an option to select the size of the board displayed?
EDIT: The only additional rule is no returning to the previous square you just traveled from, obviously.
Also, let’s say to travel to the lowest available you need to still have that the sum available to be crossed off, which comes up once moving from 5, where you’d like to move to 8, except 13 has already been removed, and so therefore you’re stuck hopping to 14, and crossing out 19.
There are actually a few end game variants that are springing into mind, but one step at a time.
Thanks for reading!
TLDR: I feel like I understand things, but I can’t solve problems. How do I put my issues into questions I actually ask my professor or TA, without seeming like I just glanced over the problem.
I’m a grad student in physics, and studying a lot of pure math right now (working my way through Lang – amazing book, but rock hard). I noticed that I have a very particular problem that makes it very difficult for me to seek out help.
When I read the chapters, I usually don’t have much of a problem reading the proofs and theorems, occasionally I fill out the details of a proof or prove some theorem we didn’t prove in the lecture. I feel really confident at that stage, and although I appreciate the difficulty of the subject, I don’t feel like I am struggling to understand it.
But then there’s the problems. Often here, the story is entirely different. I often find myself just staring at a problem for hours without an idea on how to even start approaching it. Then I go back to the theorems and everything seems clear again.
This makes it really hard for me to ask specific questions. I feel like I understand the theorems (although I’m starting to suspect that this feeling is misleading) – but I just can’t solve the problems. I don’t want to bug my professor with specific problems, but I would much rather have a step I just don’t understand, or something I can describe rather than just this feeling of powerlessness in front of a problem.
I went on and proved every single problem we had so far on my own. This improved things, but not to the extent I expected it to. To be fair though, I am about 4 problem sheets behind on that, but even the oldest problems I still have trouble solving.
So what do you think? Am I approaching the whole thing from the wrong end? Is there some tiny step I’m missing out on? I think my studying strategy definitely needs some improvement, since I am performing at the very bottom of my class, which is very unusual for me, I usually find myself within the upper quarter (not bragging, just noticing a sharp discontinuity in my performance) .
Thanks for reading. Any help is greatly appreciated!
Recently, I have been carefully reading through the homotopy type theory book, as well as some related research papers. I figured it would make good conversation to share my opinion on the topic.
Before I criticize the theory, I should point out what makes it amazing.
- It is far easier to write computer checkable proofs in the language of HoTT than in the language of set theory.
- HoTT gives a beautiful foundation for synthetic homotopy theory. It has already taught us fascinating new things about the fundamental nature of homotopy. For example, we have learned that Whitehead’s theorem is fundamentally analytic in nature because it is not provable in “vanilla” HoTT. What I mean by “analytic” is that any proof of Whitehead’s theorem necessarily uses the “CW” part of CW-complexes, or some equivalent principle in the model of homotopy types you use. However, other parts of homotopy theory, like homotopy groups of spheres, have been shown to have purely homotopical proofs in homotopy type theory, which is actually amazing to me.
Now for my criticisms. I think that as a foundation for all mathematics, homotopy type theory is not the best possible choice. There is a difference between what makes a good foundation for computer assisted proofs, and what makes a good foundation for human thought. Here are some of my complaints.
1A) Sets are more intuitive than homotopy types. You can explain what sets are to a 5 year old, and their intuition for the concept will immediately be pretty close to the mathematical formalism. However, it can take years of study to build accurate intuition for homotopy types. This makes informal reasoning through homotopy type theory something that is much harder for humans to do, compared to informal reasoning through set theory.
1B) You cannot formally learn homotopy type theory without already knowing some homotopy theory. It is impossible. I express my deepest condolences to any graduate students who attempt to do this.
Combining 1A and 1B, we see that HoTT is harder to learn than set theory for humans at all levels of intellectual and mathematical development. Even if it is easier for computers, it is harder for humans. I don’t think we should be adding barriers to learning the foundations of mathematics. Foundations should be as easy to understand as possible.
2) It is harder to prove theorems in homotopy theory synthetically. If we accept homotopy type theory as a foundation for mathematics, then we are forced to accept the types of HoTT as the fundamental form of homotopy types. Then, any theorems of homotopy theory must be proven synthetically, and many of our “theorems” will no longer even be provable. As of yet, there is no transfer principle that lets you take ZFC proofs of theorems about some nice model of HoTT, and turn them into HoTT proofs about types. Furthermore, when proving something in homotopy theory, it is often way easier to just work with CW complexes or manifolds, allowing yourself to use analytic and geometric techniques if needed. It takes 3 lines to sketch a proof of pi_4(S^3 ) = Z/2Z classically: Take the inverse image of a regular value, look at the holonomy of the framing to get an isomorphism to pi_1(SO(3)), get an isomorphism to Z/2Z using the universal cover. (I admit there are details involved, but this captures the important ideas of the proof.) However, a 200 page phd thesis (that of Guillaume Brunerie) was devoted to proving this fact in homotopy type theory. What we can learn from this is that proving simple facts in homotopy theory purely homotopically is *hard*. Do we really want to limit ourselves like this at the foundational level?
3) You can’t really do point-set topology nicely in HoTT without modifying the axioms. This is what cohesive HoTT is for. I think cohesive HoTT is probably a better foundation that HoTT, but it is a messier theory. For how long are we going to keep adding axioms? This ties into…
4) HoTT is axiomatically complicated. There is not yet a nice short list of axioms/axiom schema like ZFC has, and there probably never will be. Seriously, I challenge you to make a list of all the axioms of HoTT which is fully formal and fits on one page. Even ETCS is better in this regard.
5) The way we are supposed to do analysis and combinatorics in HoTT is to define “sets” to be 0-types, and then just do set theory as usual. So from the perspective of analysts and combinatorialists, there should be absolutely no reason to use HoTT. (Unless you want to write computer checkable proofs.)
I have other minor complaints, but I think I have summed up my issues with thinking of HoTT as a foundation for human mathematics. I think it is better to think of it as a model for computer assisted proofs and synthetic homotopy theory. Despite my complaints, HoTT is undeniably a beautiful theory which will profoundly influence mathematics in the future. You absolutely should learn about it if you have the time.
All dissenting opinions are appreciated. 🙂