# Roughing For a Formal Synthesis of Nash, Satoshi, Szabo’s Implementation of (asymptotically) Ideal Money

Bitgold As a Formal Implementation of John Nash’s Ideal Money

From Nick Szabo’s BitGold:

….however, since bit gold is timestamped, the time created as well as the mathematical difficulty of the work can be automatically proven. From this, it can usually be inferred what the cost of producing during that time period was.

The total supply of bitgold:

Supply = sum of the periods of (problem difficulty / total architecture output)

Thus for each period:

Diffp = Sp (Wp)

With bitgold, anyone could mined as much as they want, but timestamps allowed for the distinctions of periods in which bitgold was mined. Szabo noticed we can equate pieces of bitgold by considering their value as ratios of their supply in a given period which values them at a ratio of their total POW.

Wp = Diffp/Sp

If 100 production units are mined in a time period and 200 production units in the next time period, the first bundle of 100 production units might be called 1 bitgold unit but the 200 production units mined in the next time period would also be 1 additional bitgold unit.

Bitgold is a Eureka moment. In proto-cryptoexchangelandea all of a sudden there is digital gold everywhere.

Interestingly Nash calls for a money that has a supply governed in relation to the underlying production cost. Pieces of bitgold assayed with respect to the cost/production ratio are units of Nashian Ideal Money (whereas the production units are comparatively not Ideal).

The differences in production units per period and the effect on cost can be thought of as ore sent to be refined. We can consider 100 production units in a period versus 200 production units in a period each equalling 1 unit of Bitgold as if the former went through a doubly efficient refinement process.

Supply = sum of the periods of (problem difficulty / total architecture output)

Consider our formula where produced units are constant such that refinement is effectively at its optimum or negated as a factor. This is not unlike a scenario where miners are extracting ore and refining it at their optimums.

In real life this is a reasonable consideration also because it removes the need for an assaying mechanism/market.

The difference between the latter scenario as the optimal schedule and the former as the measured is the adjustment needed to recalibrate the ‘production unit supply’ aka stabilize refinement.

Consider Nick Szabo’s essay on dead reckoning which requires tuples involving direction, speed, time:

A dead reckoning itinerary can be specified as a sequence of tuples { direction, speed, time }. It can be drawn as a diagram of vectors laid down head-to-tail. However, as mentioned above, this diagram by itself, for nontrivial sea and ocean voyages, contains insufficient information to map the arrows accurately onto a Ptolemaic map (i.e. maps as we commonly understand them, based on celestial latitudes and longitudes), yet sufficient at least in theory to guide a pilot following such directions to their destination.

Here the direction gets us to our goal which is supply/period and thus to be over or under the rate in a given period is to be missing our direction (perhaps by left or right or west versus east etc). Every period the captain wakes up, reprimands his crews for veering off course, straightens the course and goes back to sleep.

With bitgold for any next period we can ‘assay’ the sum of the previous periods and adjust the difficulty based on the assumption of the previous period’s architecture (the assumption being in an efficient market the previous period is the best guess of the POW of the next period).

## This is comparable to governing by moving averages.

When, for each new period, we measure and re-target for a constant supply of production units the adjustment is the deviation of the average cost to produce/unit from Nash’s Ideal Money, For this the process is such that it asymptotically fluctuates around what is Ideal by Nash’s accord.

Notice we haven’t limited the supply but only governed quality of units from the Nashian and Szabonian sense (of unforgeable costliness for Szabo).