## Tuesday, August 18, 2015

### The Dungeons and Dragons approach to economics

 Dice roll with six-sided and twenty-sided dice.
Editor's note: This post has been languishing in the drafts folder for several months, so I gave it a quick finish and posted it.
In a footnote to the previous post I mentioned flipping a coin versus rolling a twenty sided die as a different way of thinking about the solution to paradox of value -- why diamonds cost more than water (we need water to survive, yet water is cheaper than diamonds).

The 19th century solution is marginal utility. And it works well.

If we look at the picture above, we need a total of about 33 bits to specify this particular result

$$11 \cdot \log_{2} 6 + 1 \cdot \log_{2} 20 \simeq 32.8 \text{ bits}$$

Each of the six-sided dice reveals about 2.6 bits, while the 20-sided die reveals about 4.3 bits.

In equilibrium, the supply effectively knows the demand's "roll". 'Quanta' of 4.3 bits are flowing back and forth in the diamond market and quanta of 2.6 bits are flowing back and forth in the water market. However if there is a change, a change in the 20-sided die in the demand's roll, it requires a flow of 4.3 bits to the supply side. A change in one of the six-sided dice requires a flow of 2.6 bits.

Therefore the information transfer index $k_{d}$ for diamonds is going to be larger than the index for water $k_{w}$. If there is something that functions as money (see here and here, Ed. note: these are later posts that more clearly demonstrate my point), then the information transfer index in terms of money will also be larger. Therefore, the price of diamonds will grow much faster than the price of water since for some $m$:

$$p_{d} \sim m^{k_{d} - 1} > p_{w} \sim m^{k_{w} - 1}$$

and therefore

$$\frac{d}{dm} p_{d} > \frac{d}{dm} p_{w}$$

if $k_{d} > k_{w}$.

The absolute price is not knowable in the information transfer model, but by simply being rarer (i.e. lower probability so that more information is revealed by specifying its allocation) the price will grow much faster. Thus eventually, regardless of the starting point, diamonds will be more valuable than water. The one caveat (assumption) is that both things must continue to have a market.

...

Update: fixed typos of 4.6 bits where it should read 4.3 bits (H/T Tom Brown in comments below).

1. Interesting. Twice in the text you mention "4.6 bits" but did you mean to write "4.3 bits?"

1. Yes, thanks. Fixed it.

2. Yes very interesting, makes something of intuitive sense, but as you state "...that both things must continue to have a market." Diamonds are rare, and expensive, because there is a market for them. One could imaging that certain types of natural minerals are as rare as diamonds, but not attractive when placed on fingers, and thus would not be very expensive, as the information from its rareness does not reach equilibrium with demand.

1. There are other factors to consider -- diamonds are crystals prized partially for their size and e.g. metals can be melted down. But even industrial diamonds are about as expensive, if not more expensive, than gold even without looking nice.

I should add that part of the reason non-industrial diamonds are more expensive than they 'should' be is that there is something of a cartel controlling supply.

But as there are markets now for some rare-earth metals that have uses in electronics, we might expect their price to outstrip diamonds eventually ...