Thursday, December 22, 2016

local optimization

Among the methodological similarities between physics and economics is the frequent use of optimization techniques; in fact, both disciplines often involve optimizing something they call a "Lagrangian", though the meaning of that word is rather different in the two subjects!  In both cases, though, there's some sense in which what is frequently sought is a partial optimum, rather than a full global optimum.

Suppose you have a marble in a bowl on a table, and you want to figure out where it goes.  Roughly speaking, you expect it to seek to lower its potential energy.  Usually, though, it will go toward the middle of the bowl, even though it would get a lower potential energy by jumping out of the bowl onto the floor.  Quantum systems tend to "do better" at finding a global minimum than classical systems; liquid helium, in its superfluid state, will actually climb out of bowls to find lower potential energy states.  Even quantum systems, though, often end up more or less in states where the first-order conditions are satisfied, rather than the actual global maximization problem.  This is perhaps most elegantly achieved with path-integrals; you associate a quantum mechanical amplitude with each point in your state space, make it undulate as a function of the optimand, and integrate it, and where the optimand isn't constant it cancels itself out, leaving only the effect of its piling up where the optimand satisfies the first-order conditions.

In economics and game theory, "equilibrium" will typically maximize agents' utility functions, each subject to variation only in the corresponding agent's choice variables; externalities are, somewhat famously, left out.  I'm tempted to try to apply a path-integral technique, but in game theory in particular the optimum is often at a "corner solution" where a constraint binds, and where the optimand doesn't therefore satisfy usual first-order conditions.  Something complicated with lagrange multipliers might be a possibility, but I suspect the use of (misleadingly named) "smoothed utility functions" will effectively do the same thing, but more easily.  I might then try to integrate "near" an equilibrium, but only in the dimensions corresponding to one particular agent's choice variables.

I wonder whether I can make something useful of that.

Wednesday, December 14, 2016

dividing systems

This will be a bit different, and may well not be terribly original, but I want to think about some epistemological issues (perhaps with some practical values) associated with dividing up complex systems into parts.

In particular, suppose a complex system is parameterized by a (large) set of coordinates, and the equations of motion are such as to minimize an expression in the coordinates and their first time derivatives as is typical of Lagrangians in physics; I'll simply refer to it as the Lagrangian going forward, though in some contexts it might be a thermodynamic potential or a utility function or the like.  A good division amounts to separating the coordinates into three subsets, say A, B, and C, where A and C at least are nonempty and the (disjoint) union of the three is the full set of coordinates of the system.  Given values of the coordinates (and their derivatives) in A and B, we can calculate the optimal value of the Lagrangian by optimizing over values that C can take, and similarly given B and C I can optimize over A, and I get effective Lagrangians for (A and B) and (B and C) respectively.  Where this works best, though, is where the optimizing coordinates in C (in the first case) or A (in the second) depend only on the coordinates in B; conditional on B, A and C are independent of each other.  This works even better if B is fairly small compared to both A and C, and might in that case even be quite useful if the conditional independence is only approximate.

In general there will be many ways to write the Lagrangian as LA+LB+LC+LAB+LBC+LAC+LABC, with each term depending only on coordinates in that set, but it will be particularly useful to write a Lagrangian this way if the last two terms are small or zero. If we are "optimizing out" the C coordinates, the effective Lagrangian for A and B is LA+LB+LAB plus what we get from optimizing LC+LBC+LAC+LABC; this will depend only on B if the terms with both A and C are absent.  Thus on some level a good decomposition of a system is one in which the Lagrangian can be written as LAB+LB+LBC, where I've absorbed previous LA and LC terms into the first and last terms; for given evolutions of B variables, the A variables will optimize LAB and the C variables will optimize LBC and these two optimizations can be done separately.

Wednesday, September 28, 2016

short sales

Matt Levine this morning writes (ctrl-F "blockchain") about what short sales would look like on a blockchain, and it's pretty straightforwardly correct; you lift the process we have now with all the sales taking place the way they do on a blockchain and get some of the additional transparency that comes with it. Fungibility on the blockchain is a bit less than it is without that transparency; one of the things being addressed specifically in his passage is that right now, if people "own" 110% of the outstanding shares of an issue, nobody knows whether their shares are among the 10% that in some sense don't count.

One of the things that's highlighted here, though, is that the short-sale concept is perhaps not what you would create if you were designing the market system top-down from whole cloth:
Just transfer 10 shares from B to A, in exchange for a smart contract to return them, and then sell those shares from A to C over the blockchain. Easy as blockchain. C now owns the shares on the blockchain's ledger, while A also "owns" them in the sense that she has a recorded claim to get them back from B.
This is how short-selling works; if A wants to sell short, A borrows the stock from someone (B) and then sells it to someone else (C).  If you introduce brokers, the way our current system works, the actual beneficial owner B won't even know that the shares have been lent out; both B and C think they own the shares.  The big change the blockchain makes is that, at least in principle, B can see that the "shares" B owns are actually an agreement by A to deliver them in the future.

There's some sense in which the borrow and sale are superfluous, though; the promise to (re-)deliver in the future is what you're trying to create by doing a short sale.  What you would think, from first principles in the absence of market structure concerns, would be the way to get there is let C buy the shares from B while A enters a forward contract with B, or, if C is just as happy to be on the receiving end of a forward contract, leave B out of it altogether and have a forward contract from A to deliver shares to C.  There are exchanges for stocks, and a less centralized market for lending securities, and these grew up (one and then the other) to facilitate short sales; in our current world, then, it's hard (especially for retail customers) to enter bilateral forward contracts, and the institutions for effecting the same result are set up to facilitate it in a somewhat baroque manner.  If you're moving to blockchain for settlement, and need to change the structure of the market to accommodate that, then
A blockchain would need to do something similar: let some people create new securities on the blockchain, but carefully control who gets that access.
doesn't seem to me like my first choice approach; what would make more sense to me would be a market in which buyers see offers to enter into forward contracts as well, and where the borrow gets left out altogether.

Tuesday, August 9, 2016

simplified heuristics and Bellman equations

An idea I've probably mentioned is that certain behavioral biases are perhaps simplifications that, on average, at least in the sort of environment in which the human species largely evolved, work very well.  We can write down our von Neumann / Morgenstern / Friedman / Savage axioms and argue that a decision-maker that is not maximizing expected utility (for some utility and some probability measure) is, by its own standards, making mistakes, but the actual optimization, in whatever sense it's theoretically possible with the agent's information, may be very complicated, and simple heuristics may be much more practical, even if they occasionally create some apparent inconsistencies.

Consider a standard dynamic programming (Bellman) style set-up: there's a state space, and the system moves around within the state space, with a transition function specifying how the change in state is affected by the agent's actions; the agent gets a utility that is a function of the state and the agent's action, and a rational agent attempts to choose actions to optimize not just the current utility, but the long-run utility.  Solving the problem typically involves (at least in principle) finding the value function, viz. the long-run utility that is associated with each state; where one action leads to a higher (immediate) utility than the other but favors states that have lower long-run utility, the magnitudes of the effects can be compared.  The value function comprises all the long-run considerations you need to make, and the decision-making process at that point is superficially an entirely myopic one, trying in the short-run to optimize the value function (plus, weighted appropriately, the short-run utility) rather than the utility alone.

A problem that I investigated a couple of years ago, at least in a somewhat simple setting, was whether the reverse problem could be solved: given a value function and a transition rule, can I back out the utility function?  It turns out that, at least subject to certain regularity conditions, the answer is yes, and that it's generally mathematically easier than going in the usual direction.  So here's a project that occurs to me: consider such a problem with a somewhat complex transition rule, and suppose I can work out (at least approximately) the value function, and then I take that value function with a much simpler transition function and try to work out a utility function that gives the same value function with the simpler transition function.  I have a feeling I would tend to reach a contradiction; the demonstration that I can get back the utility function supposed that it was in fact there, and if there is no such utility function I might find that the math raises some objection.  If there is such a utility function that exactly solves the problem, of course, I ought to find it, but there seems to me at least some hope that, even if there isn't, the math along the way will hint how to find a utility function, preferably a simple one, that gives approximately the same value function.  This, then, would suggest that a seemingly goal-directed agent pursuing a comparatively simple goal would behave the same way as the agent pursuing the more complicated goal.

cf. Swinkels and Samuelson (2006): "Information, evolution and utility," Theoretical Economics, 1(1): 119--142, which pursues the idea that a cost in complication in animal design would make it evolutionarily favorable for the animal to be programmed directly to seek caloric food, for example, rather than assess at each occasion whether that's the best way to optimize long-run fecundity.

Wednesday, July 27, 2016

policing police

This is a bit outside the normal bailiwick of this blog, but is the sort of off-the-wall, half-baked idea that seems to fit here at least in that way.

Police work, at least as done in modern America, requires special authority, sometimes including the authority to use force in ways that wouldn't be allowed to a private citizen; sometimes the police make mistakes, and it is important to create systems that reduce the likelihood of that, but allowances also need to be made that they are human beings put in situations where they are likely to believe they lawfully have certain authority; if a police officer arrests an innocent man, the officer will face no legal repercussions, while a private citizen would, even if the private citizen had "reasonable cause" to suspect the victim.  It is appropriate that this leeway be made, at least as for legal repercussions; if a particular police officer shows a pattern of making serious mistakes, even if they are clearly well-intended, it is just common sense[1] that that officer should be directed to more suitable employment, but being an officer trying to carry out the job in good faith should be a legal defense to criminal charges.

That extra authority, though, comes — morally if not legally — with a special duty not to intentionally abuse it.  This is the case not least because the task of police work is much more feasible where the citizens largely trust that an order appearing to come from a police officer is lawful than where they don't.  A police officer in Alabama was reported, not long ago, to have sexually assaulted someone he had detained, and in a situation like that the initial crime is additional to the societal cost of eroding trust people have that the officer is at least trying to be on the side of law.  This erosion of trust is also the primary reason that impersonating a police officer is a serious crime.[2]  I propose, then, upon the showing of mens rea in the commission of a serious crime by a police officer while using that office to facilitate the crime, that the officer be fired retroactively --- and brought up additionally on the impersonation charges.[3]




[1] I mean, it should be.  My impression is that it is too difficult to remove bad cops, but that's not an especially well-informed impression.

[2] Pressed to give secondary reasons, they would also line up pretty well between impersonating an officer and abusing the office.

[3] This policy would have an interesting relationship to the "no true Scotsman fallacy"; no true police officer would intentionally commit a heinous crime, and we'll redefine who was an officer when if we have to to make it true.

Tuesday, July 26, 2016

liquidity and efficiency of goods and services

Years ago, I went to a barber and got a haircut that took no more than five minutes.  I go with simple haircuts, and he had basically run some clippers over my head and used scissors to blend what was left.  At first, I was a bit taken aback, and thought that perhaps I should tip less than usual (and indeed wondered whether I should be charged less than usual altogether), but very quickly realized that this was perverse; the haircut I had received was not, in the context of my preferences, inferior in any way to other haircuts I have received, and I'm better off having the other (say) 15 minutes of my time to (say) squander writing blog posts on the internet.  Ceteris paribus, we both benefit from his having finished more quickly; I left my usual tip, leaving the pecuniary terms of trade unchanged from those in which we both lose more time.

Liquidity, like speed, is a benefit to both the buyer and the seller; both are a bit hard to analyze with supply and demand for this reason.  (My go-to deep neoclassical model, from Arrow-Debreu, treats a quick haircut as a different service from a slow haircut, and as such treats them as different markets, but they are such close substitutes that it's obviously useful to treat them as in some sense "almost" the same market.)  There may well be other ways in which different instances of a good or service differ in ways such that the quality that is better for the buyer is naturally better for the seller as well.  My interest especially is in market liquidity, and I wonder whether distilling out this aspect provides useful models for some of the important phenomenology around that.

Tuesday, July 12, 2016

risk and uncertainty

A century ago, an economist named Frank Knight wrote a book on "Risk and Uncertainty", where by "risk" he meant what economists generally alternate between calling "risk" and "uncertainty" today and by "uncertainty" he meant something economists haven't given as much attention in the past seventy years, but have tended to call "ambiguity" when they do.[1]  The distinction is how well the relevant ignorance can be quantified; a coin toss is "risky", rather than "ambiguous", because we have pretty high confidence that the "right" probability is 50%, while the possibility of a civil war in a developed nation in the next ten years is perhaps better described as "ambiguous".  Here is a link to the wikipedia page on the Ellsberg paradox.  Weather in the next few days would have been "ambiguous" when Knight wrote, but was becoming risky, and is well quantifiable these days.

Perhaps one of the reasons the study of ambiguity fell out of favor, and has largely stayed there for more than half a century since then,[2] is that a strong normative case for the assignment of probabilities to events was developed around World War II; in short, there is a set of appealing assumptions about how a person would behave that imply that they would act so as to maximize "expected utility", where "utility" is a real-valued function of the outcome of the individual's actions and "expected" means some kind of weighted average over possible outcomes.  In perhaps simpler terms, if a reasonably intelligent person who understands the theorem were presented with actions that person had taken that were not all consistent with expected utility maximization, that person would probably say, "Yeah, I must have made a mistake in one of those decisions," though it would probably still be a matter of taste as to which of the decisions was wrong.

To be a bit more concrete, suppose an entrepreneur is deciding whether or not to build a factory.  The factory is likely to be profitable under some scenarios and unprofitable under others, and the entrepreneur will not know for sure which will obtain; if certain risks are likelier than some threshold, though, building the factory will have been a bad idea, and if they're less likely, than it will have been a good idea.  Whether or not the factory is built, then, implies at least a range of probabilities that the entrepreneur must impute to the risks; an entrepreneur making other decisions that are bad for any of those probabilities is making a mistake somewhere, such that changing multiple decisions guarantees a better outcome, though which decision(s) should be changed may still be up for debate (or reasoned assessment).  The rejoinder, then, to the assertion that a probability can't be put on a particular event, is that often probabilities are, at least implicitly, being put on unquantifiable events; it is certainly not necessarily the case that the best way to make those decisions is to start by trying to put probabilities on the risks, but it probably is worth trying to make sure that there is some probabilistic outlook that is consistent with the entire schedule of decisions, and, if there isn't, to consider which decisions are likely to be in error.[3]

There is a class of situations, though, in which something that resembles "ambiguity aversion" makes a lot of sense, and that is being asked to (in some sense) quote a price for a good in the face of adverse selection.  If, half an hour after a horse race, you remark to someone "the favored horse probably won," and she says, "You want to bet?", then, no, you don't.  In general, I should suppose that other people have some information that I don't, and if I expect that they have a lot of information that I don't, then my assessment of the value of an item or the probability of an event may be very different if I condition on some of their information than if I don't; if I set a price at which I'm willing to sell, and can figure out from the fact that someone is willing to buy at that price that I shouldn't have sold at that price, I'm setting the price too low, even if it's higher than I initially think is "correct".

In a lot of contexts in which people seem to be avoiding "ambiguity", this may well fit a model of a certain willingness to accept other's probability assessments; e.g. I'm not willing to bet at any price on a given proposition, because, conditional on others' assessments, my assessment is very close to theirs.


[1] There's a nonzero chance that I have his terms backward, but that nonzero chance is hard to quantify; in any case, the concepts here are what they are, and I'll try to keep my own terminology mostly consistent with itself.

[2] I'm pretty sure Pesendorfer and/or Gul, both of whom I believe are or were at Princeton, have produced some models since the turn of the millennium attempting to model "ambiguity aversion", and I should probably read Stauber (2014): "A framework for robustness to ambiguity of higher-order beliefs," International Journal of Game Theory, 43(3): 525--550.  This isn't quite my field.

[3] In certain institutional settings, certain seemingly unquantifiable events may be very narrowly pinned down; I mostly have in mind events that are tied to options markets.  If a company has access to options markets on a particular event, it is likely that there is a probability above which not buying (more) options is a mistake, and another below which not selling (more) options is a mistake, and those probabilities may well be very close to each other.  If you think you should build a factory, and the options-implied probability suggests you shouldn't, buying the options instead might strictly dominate building the factory; if you think you shouldn't and the market thinks you should, your best plan might be to build the factory and sell the options.