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.

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 L_A+L_B+L_C+L_AB+L_BC+L_AC+L_ABC, 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 L_A+L_B+L_AB plus what we get from optimizing L_C+L_BC+L_AC+L_ABC; 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 L_AB+L_B+L_BC, where I've absorbed previous L_A and L_C terms into the first and last terms; for given evolutions of B variables, the A variables will optimize L_AB and the C variables will optimize L_BC and these two optimizations can be done separately.