A race across target states, or Bayes and Nakamoto together

My editorial

And so I continue prodding my idea of local, green energy systems, with different theories of probability. The three inside me – my curious ape, my austere monk, and my happy bulldog – are having a conversation with two wise men: reverend Thomas Bayes, and Satoshi Nakamoto. If you need to keep track of my last updates, you can refer to ‘Time puts order in the happening’ as well as to ‘Thomas Bayes, Satoshi Nakamoto et bigos’. And so I am at the lemmas formulated by Thomas Bayes, and at the basic analytical model proposed by Nakamoto. Lemma #1 by Thomas Bayes says: ‘The probability that the point o will fall between any two points in the line AB is the ratio of the distance between the two points to the whole line AB’. Although Thomas Bayes provides a very abundant geometric proof to this statement, I think it is one of those things you just grasp intuitively. My chances of ever being at the coast of the Pacific Ocean are greater than those of ever visiting one tiny, coastal village in the Hawaii, just because the total coastline of the Pacific is much bigger an expanse than one, tiny, Hawaiian village. The bigger is my target zone in relation to the whole universe of probability, the greater is my probability of hitting the target. Now, in lemma #2, we read pretty much the same, just with some details added: ‘The ball W having been thrown, and the line os drawn, the probability of the event M in a single trial is the ratio of Ao to AB’.

I think a little reminder is due in relation to those two Bayesian lemmas. As for the detailed Bayes’s logic, you can refer to Bayes, Price 1763[1], and I am just re-sketching the landscape, now. The whole universe of probability, in Thomas Bayes’s method, is a flat rectangle ABCD, with corners being named clockwise, starting from A at the bottom right, as if that whole universe started around 4 o’clock. AB is kind of width of anything that can happen. Although this universe is a rectangle, it is essentially unidimensional, and AB is that dimension. I throw two balls, W and O. I throw W as the first, at the point where it lands in the rectangle ABCD becomes a landmark. I draw a line through that point, perpendicular to AB, crossing AB at the point o, and CD and the point s. The line os becomes the Mississippi river of that rectangle: from now on, two sub-universes emerge. There is that sub-universe of M happening, or success, namely of the second ball, the O, landing between the lines os and AD (in the East). On the other hand, there are all those strange things that happen on the other side of the line os, and those things are generally non-M, and they are failures to happen. The probability of the second ball O hitting M, or landing between the lines os and AD, is equal to p, or p = P(M). The probability of the ball O landing west of Mississippi, between the lines os and BC, is equal to q, and this is the probability of a single failure.

On the grounds of those two lemmas, Thomas Bayes states one of the most fundamental propositions of his whole theory, namely proposition #8: ‘If upon BA you erect a figure BghikmA, whose property is this, that (the base BA being divided into any two parts, as Ab and Bb and at the point of division b a perpendicular being erected and terminated by the figure in m; and y, x, r representing respectively the ratio of bm, Ab, and Bb to AB, and E being the coefficient of the term in which occurs ap*bq when the binomial [a + b]p + q is expanded) y = E*xp*rq. I say that before the ball W is thrown, the probability the point o should fall between f and b, any two points named in the line AB, and that the event M should happen p times and fail q [times] in p + q = n trials, is the ratio of fghikmb, the part of the figure BghikmA intercepted between the perpendiculars fg, bm, raised upon the line AB, to CA the square upon AB’.

Right, I think that with all those lines, points, sections, and whatnot, you could do with some graphics. Just click on this link to the original image of the Bayesian rectangle and you will see it as I tried to recreate it from the original. I think I did it kind of rectangle-perfectly. Still, according to my teachers of art, at school, my butterflies could very well be my elephants, so be clement in your judgment. Anyway, this is the Bayesian world, ingeniously reducing the number of dimensions. How? Well, in a rectangular universe ABCD, anything that can happen is basically described by the powers ABBC or BCAB. Still, if I assume that things happen just kind of on one edge, the AB, and this happening is projected upon the opposite edge CD, and the remaining two edges, namely BC and DA, just standing aside and watching, I can reduce a square problem to a linear one. I think this is the whole power of geometry in mathematical thinking. Whilst it would be foolish to expect rectangular universes in our everyday life, it helps in dealing with dimensions.

Now, you can see the essence of the original Bayesian approach: imagine a universe of occurrences, give it some depth by adding dimensions, then give it some simplicity by taking some dimensions away from it, and map your occurrences in thus created an expanse of things that can happen. Now, I jump to Satoshi Nakamoto and his universe. I will quote, to give an accurate account of the original logic: ‘The success event is the honest chain being extended by one block, increasing its lead by +1, and the failure event is the attacker’s chain being extended by one block, reducing the gap by -1. The probability of an attacker catching up from a given deficit is analogous to a Gambler’s Ruin problem. Suppose a gambler with unlimited credit starts at a deficit and plays potentially an infinite number of trials to try to reach breakeven. We can calculate the probability he ever reaches breakeven, or that an attacker ever catches up with the honest chain, as follows:

p = probability an honest node finds the next block

q = probability the attacker finds the next block

qz = probability the attacker will ever catch up from z blocks behind

Now, I rephrase slightly the original Nakamoto’s writing, as the online utilities I am using on my mutually mirroring blogs – https://discoversocialsciences.com and https://researchsocialsci.blogspot.com – are not really at home with displaying equations. And so, if p ≤ q, then qz = 1. If, on the other hand, p > q, my qz = (q/p)z. As I mentioned it in one of my previous posts, I use the original Satoshi Nakamoto’s thinking in the a contrario way, where my idea of local green energy systems is the Nakamoto’s attacker, and tries to catch up, on the actual socio-economic reality from z blocks behind. For the moment, and basically fault of a better idea, I assume that my blocks can be carved in time or in capital. I explain: catching from z blocks behind might mean catching in time, like from a temporal lag, or catching up across the expanse of the capital market. I take a local community, like a town, and I imagine its timeline over the 10 years to come. Each unit of time (day, week, month, year) is one block in the chain. Me, with my new idea, I am the attacker, and I am competing with other possible ideas for the development and/or conservation of that local community. Each idea, mine and the others, tries to catch over those blocks of time. The Nakamoto’s logic allows me to guess the right time frame, in the first place, and my relative chances in competition. Is there any period of time, over which I can reasonably expect my idea to take over the whole community, sort of qz = 1 ? This value z can also be my time advantage over other projects. If yes, this will be my maximal planning horizon. If not, I just simulate my qz with different extensions of time (different values of z), and I try to figure out how does my odds change as z changes.

If, instead of moving through time, I am moving across the capital market, my initial question changes: is there any amount of capital, like any amount z of capital chunks, which makes my qz = 1 ? If yes, what is it? If no, what schedule of fundraising should I adopt?

Mind you, this is a race: the greater my z, the lower my qz. The more time I have to cover in order to have my project launched, the lower my chances to ever catch on. This is a notable difference between the Bayesian framework and that by Satoshi Nakamoto. The former says: your chances to succeed grow as the size of your target zone grows in relation to everything that can possibly happen. The more flexible you are, the greater are your chances of success. On the other hand, in the Nakamoto’s framework, the word of wisdom is different: the greater your handicap over other projects, ideas, people and whatnot, in terms of time or resources to grab, the lower your chances of succeeding. The total wisdom coming from that is: if I want to design a business plan for those local, green energy systems, I have to imagine something flexible (a large zone of target states), and, in the same time, something endowed with pretty comfortable a pole position over my rivals. I guess that, at this point, you will say: good, you could have come to that right at the beginning. ‘Be flexible and gain some initial advantage’ is not really science. This is real life. Yes, but what I am trying to demonstrate is precisely the junction between the theory of probability and real life.

[1] Mr. Bayes, and Mr Price. “An essay towards solving a problem in the doctrine of chances. by the late rev. mr. bayes, frs communicated by mr. price, in a letter to john canton, amfrs.” Philosophical Transactions (1683-1775) (1763): 370-418

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