The Mathematics of 2048: Optimal Play with Markov Decision Processes (2018)

In real world practice, i.e., where we don't get to drive the stochastic part of the problem with our own random number generator, the Markov assumption (past and future conditionally independent given the present) can be, commonly is, difficult to justify.

But, then in just the theory, every stochastic process is Markov -- just let the state be the history -- I intend this as a joke, but actually, technically in the math, it is true.

If we DO make the state the history and have a problem in continuous time, then, at least in theory, we are conditioning on uncountably infinitely many random variables, and just defining how to do that is a bit interesting.

Then for our decisions. policy, we can encounter the problem of measurable selection, also a bit interesting and surprising.

I.e., dynamic programming in continuous time gets us into measure theory, the Radon-Nikodym theorem, etc.

Stochastic dynamic programming, when find a good application and can build a good solution, i.e., keep the computational demands within the super computer range, can seem really smart, of course is not prescient but first cut, intuitively, can seem to be. Your solution likely also seems prescient.

More applications would be nice.

Here is a potential, potentially large, collection of new applications: Develop a spreadsheet, for an example, for, say, budgeting for the next 5 years. To have a simple description, have one column for each month, for a initial column and then one more for the end of each of the 60 months, and have one row for each variable. Some of the spreadsheet cells might have random quantities (have them all independent, to keep the Markov assumption easy to justify), and some of the cells might be empty and available for decisions.

Now, presto, bingo, whether we knew or intended it or not, we now have formulated a problem in discrete time stochastic dynamic programming, and since in practice problem formulation is a common obstacle, we have maybe made some progress.

In principle at this point the software for a solution is well defined. And since long ago people wrote general purpose stochastic dynamic programming software, the software we would need is in a sense actually simple (if we don't care about execution time).

Now just feed this problem into a suitably large computer and get the resulting decisions and the expected earnings, we wanted to maximize, at the end of the 5 years.

If make the software more complicated, then can put in some ideas to make the computations faster by some factors of 10.

Nemhauser and Dreyfus and Law are intended as intros and are quite well written. Bellman, who likely should be named the father of dynamic programming, is the oldest book on my list and is not intended as an intro. But if can get Bellman for less than $10, maybe for the price of a Big Mac, then eat a lighter lunch and get the book.

Bellman's book likely covers the Hamilton–Jacobi–Bellman equation, not so easy to understand via the other intros.

Not really a biggie but worth knowing, a big part of this subject in computation is the value function, and at least Bertsekas used neural nets as a means to have an efficient approximation. So more recent treatments may include this idea.

Dynamic programing is one of the leading examples of "the curse of dimensionality".

The field has not been still; e.g., multivariate splines have been an idea for handling the value function in computation.

Of course can have software go through the motions of stochastic dynamic programming when the Markov assumption is not satisfied; the software won't object. But when the assumption does old, and the rest of the effort has good quality, can argue that the work provides an optimal solution, the best possible, of course, in expected value, for any means of handling the data. In short, we have what is called "the principal of optimality".

The continuous time case, with all the math details filled in, is more difficult; so for an intro, stay with discrete time.

There has been a lot of attention to this subject in the ORFE (Operations Research and Financial Engineering) department at Princeton. Long the chair was E. Çinlar. The best course I took in school was from a Çinlar student.

Of course, if expect artificial intelligence to be really smart, then as a special case it should be able to be as good as stochastic optimal control. And apparently now some of reinforcement learning is using the core idea of stochastic dynamic programming. But without the Markov assumption, might count that application as a heuristic with no more than weak claims about optimality.

I spent a lot of time in the field. Eventually I concluded that generally in the US economy a lot of knowledge of stochastic optimal control and a dime wouldn't cover a 10 cent cup of coffee. Would have a lot better chance buying a house and supporting a family getting paid to develop Web sites, e.g., to sell, say, used math books, including Bellman's. Maybe there have been and are some niche applications (maybe somewhere in US national security), but that niche is likely much sharper than any razor.

There is something of an organizational problem for a math guy getting hired to apply stochastic optimal control: The person hiring you, your hiring manager, will likely know much less about the math than the math guy, likely know nothing about the math. So this manager will be very reluctant to allocate much of his (her) budget and risk his job to support work he doesn't understand and, that, indeed, might justify the C-level suits promoting the math guy over his manager.

Due to such issues, the math guy with an application potentially valuable in the economy could be better off doing a corresponding startup. Else, to avoid scaring hiring managers, he might be better off omitting such math from his resume. So, for getting hired as an employee, a lot of math background on a resume can be from useless down to, say, a felony conviction.

A recipe for rabbit stew starts out "First catch a rabbit.". A recipe for applied math might start, "First find an application." Can't argue that some topic in math will be useless forever outside of math and in the economy, but forever is a long time. As it is, it can appear that some math papers and books, including what looks like applied math, are written before seeing any rabbits.

Yet, the best of pure/applied math is in some senses super terrific stuff, way up there with the best of Bach and Beethoven, etc., and at times there are good applications. E.g., there is some pure and applied math at the core of my startup; the pure math provides some special support, a version of optimality, for the whole effort. Just what that pure math says is terrific, gives a solid guarantee where intuitively we have only confusion, is so good it's tough to believe, but still it's true. The applied math is original with me, powerful for my startup but no biggie as research.

Good luck.

https://simogo.com/work/spl-t/

I think it would be interesting to try some of the methods for approximately solving MDPs on this game to see how complicated the machine learning model that approximates the value function (or state-action function) has to be to get reasonable accuracy. Maybe some simple rules would emerge...