Evolutionary games

My editorial for today

My mind is wandering a bit, this morning. I am experiencing that peculiar state of creative lack in my focus, as if my brain was hesitating which compound parcel of information to treat. I am having a quick glance at ‘Théorie de la spéculation’ by Louis Bachelier (Bachelier 1900[1]), as I am almost always interested in practical applications of uncertainty. That, in turn, suggests me to add some data about stock markets to my database anchored in Penn Tables 9.0 (Feenstra et al. 2015[2]). On the other hand, I would like to continue on the path that I have been developing for a few days, about innovation and technological change understood as a game of adaptation in a complex social structure. As I am trying to connect those two dots, I end up reading papers like that by Frederic Scherer (Scherer 1996[3]) on the probability of big, fat profits from an invention . Finally, I have that idea of reconciling the game-theoretic approach with the evolutionary one, in modelling processes of adaptation under uncertainty.

Maybe if I start from the end and advance towards the beginning, as I love doing in my research, I will come to some interesting results? Good, let’s waltz. So I am studying evolution. Evolution of anything is about breeding, i.e. about having fun, first, and being a responsible parent in subsequent periods. At a given time ‘t’ , in a place (market) MR, we have a set N of n inventions, N = {N₁, N₂, …, N_n}, and each of those inventions can give rise to some fun on the spot, and to responsible parenthood next, or, in other words, it can be implemented as a technology, and maintained in exploitation for a given period of time. Right, so I need a set of established technologies, as well. I call this set TC = {TC₁, TC₂, …, TC_m}, and this long name with curly brackets in it means that my set of established technologies contains ‘m’ of them.

Let’s suppose that inventions are male and established technologies are children. We need females, if we want to be serious about evolution. A female organism is what gets fecundated by some seed coming from the male. Logically, in this specific process, investment capital is female. Thus, I imagine a set CH = {CH₁, CH₂, …, CH_k} of ‘k’ capital holders, who are logically female but who, in fact, can be male by physicality, or even be kind of transgender if they are corporations or governments. Anyway, the set N of inventions is having fun with the set CH of capital holders, which subsequently gives rise to responsible parenthood regarding the set TC. Some inventions stay out of the evolutionary game fault of proper mating, just as some capital holders remain infecund fault of catching the right opportunity. I know, it sounds cruel, but this is not the first time I learn how far from conventional decency are the actual ways of the world.

What is interesting, is the process of selection. As we are in the world of technological change, lipstick, short skirts, six-pack abs and Ferraris are being replaced by rules of selection, which give systematic preference to some matches between i-th invention and j-th capital holder, to the detriment of others. As I read some recent evolutionary literature (see for example Young 2001[4]), there is some inclination, in the theory, to considering the female mechanisms of selection, i.e. those applied by females regarding males, as dominant in importance. In the logic I have just developed, it generally holds: I can safely assume that capital holders select inventions to finance, rather than inventions picking up their investors. Yes, it is a simplification, and so is a cafe Americano, but it works (when you order an espresso, a good barista should serve it with a glass of cold, still water in accompaniment; still, when the barista can suspect you are not really a gourmet in coffee, they basically heat up that water, mix it with a slightly thinned espresso, and you get cafe Americano).

Anyway, we have j-th capital holder CH_j picking up the i-th invention N_i, to give birth to o-th established technology TC_o. Evolutionary theory assumes, in general, that this process is far from being random. It has an inherent function of selection, and this function is basically the stuff that makes hierarchy in the social structure of the corresponding species. The function of selection defines the requirements that a successful match should have. In this case, it means that a hierarchy of inventions is being formed, with the top inventions being the most likely to be selected, and subsequent levels of hierarchy being populated with inventions displaying decreasing f****ability. As usually in such cases, spell: ‘f-asterisk-asterisk-asterisk-asterisk-ability’.

As I am a scientist, I am keen on understanding the mating mechanism, and so I take my compound database made of Penn Tables 9.0, glued together with data from the World Bank about patent applications. I hypothesise something very basic, namely that the number of resident patent applications per year, in a given country, significantly depends on the amount of capital being invested in fixed assets. Patent applications stand for inventions, and capital stands for itself. Now, I can do two types of tests for this hypothesis. I can go linear, or I can follow Milton Friedman. Being linear means that I posit my selection function as something in the lines of:

n = a*k + residual

Of course, we keep in mind that ‘n’ is the number of inventions (patent applications in this case), and ‘k’ stands for the number of capital holders, which I approximate with the aggregate amount of capital (variable ‘ck’ in Penn Tables 9.0). Now, I squeeze it down to natural logarithms, in order to provide for non-stationarity in empirical data, and I test: ln(Patent Applications) = a1*ln(ck) + residual ln.  For the mildly initiated, non-stationarity means that data usually jumps up and down, sometimes even follows local trends. It generates some noise, and natural logarithms help to quiet it all down. Anyway, as I test this logarithmic equation in my dataset, I get a sample of 2 623 valid observations, and they yield a coefficient of determination R² = 0,478. It means, once again for the mildly initiated, that my equation explains some 47,8% of variance observable in ln(Patent Applications), and more specifically, the whole things spells: ln(Patent Applications) = 0,825*ln(ck) + residual ln -4,204.

I am going to return to those linear results a bit later. Now, just to grasp that nice contrast between methodologies, I start following the intuitions of Milton Friedman. I mean the intuitions expressed in his equation of quantitative monetary equilibrium. Why do I follow this path? Well, I try to advance at the frontier of economics and evolutionary theory. In the latter, I have that concept of selection function. In the former, one of the central theoretical concepts is that of equilibrium. I mention Milton Friedman, because he used to maintain quite firmly that equilibriums in economic systems, if they are true equilibriums, are expressed by some kind of predictable proportion. In his equation of monetary equilibrium, it was expressed by the velocity of money, or the speed of circulation of money between different units of real output.

Here, I translate it as the velocity of capital regarding patent applications, or the speed of circulation in capital between patentable inventions. In short, it is the ratio of fixed capital ‘ck’ divided by the number of resident patent applications. Yes, I know that I labelled capital as female, and inventions as male, and I propose to drop the metaphor at this very point. Velocity of circulation in females between males is not something that decent scientists should discuss. Well, some scientists could, like biologists. Anyway, I hypothesise that if my selection function is really robust, it should be constant over space and time. In probabilistic terms, it means that the mean ratio of fixed capital per one patent application, in millions of 2011 US$ at current Purchasing Power Parities, should be fairly constant between countries and over time, as well as it should display relatively low and recurrent variability (standard deviation divided by mean) inside places and years. The more disparity will I be able to notice, like the more arithmetical distance between local means or the more local variability around them, the more confidently can I assume there are many selection functions in that evolutionary game.

So I computed the mean value of this ratio and its variance across countries (get it from this link), as well as over years (this is downloadable, too). On the whole, the thing is quite unstable. There is a lot of disparity between mean values, as well as around them. Still, geographical disparity seems to be stronger, in terms of magnitude, than changes observable over time. It allows me to formulate two, quite tentative, and still empirically verifiable hypotheses. Firstly, at a given time, there is a lot of different, local selection functions between capital holders and patentable inventions. Secondly, the geographical disparity in these selection functions is relatively recurrent over time. We have an internally differentiated structure, made of local evolutionary mechanisms, and this structure seems to reproduce itself in time. Returning to my linear model, I could go into nailing down linear functions of ln(Patent Applications) = a1*ln(ck) + residual ln for each country separately. Still, I have one more interesting path: I can connect that evolutionary game to the issue of food deficit, which I explored a bit in my previous posts, namely in ‘Cases of moderate deprivation’  and in ‘‘Un modèle mal nourri’ . I made a pivot out of my database and I calculated mean capital per patent application, as well as its variance across different depths of food deficit . Interesting results turn out, as usually with this cruel and unpleasant variable. Between different classes of food deficit, regarding the ratio of capital per one patent application, disparities are significant, and, what seems even more interesting, disparity around the local means, inside particular classes of food deficit, is strongly idiosyncratic. Different depths of food deficit are accompanied by completely different selection functions in my evolutionary game.

So far, I have come to more interesting results by applying the logic of economic equilibrium in my evolutionary game, than by using plain linear regression. This is the thing about correlations and regressions: if you want them to be really meaningful, you need to narrow down your hypotheses really tight, just as Milton Friedman used to say. By the way, you can see here, at work, the basic methodology of statistical analysis, as presented, for example, by a classic like Blalock[5]: first, you do your basic maths with descriptive statistics, and only in your next step you jump to correlations and whatnot.

[1] Bachelier, Louis. Théorie de la spéculation. Gauthier-Villars, 1900.

[2] Feenstra, Robert C., Robert Inklaar and Marcel P. Timmer (2015), “The Next Generation of the Penn World Table” American Economic Review, 105(10), 3150-3182, available for download at http://www.ggdc.net/pwt

[3] Scherer, Frederic M. “The size distribution of profits from innovation.” The Economics and Econometrics of Innovation. Springer US, 2000. 473-494.

[4] Young, H. Peyton. Individual strategy and social structure: An evolutionary theory of institutions. Princeton University Press, 2001.

[5] Blalock, Hubert Morse. Social Statistics: 2d Ed. McGraw-Hill, 1972.