Information is not fair. Information is asymmetric. - xh3b4sd

Information as a concept is one fundamental part of our universe. It describes how some thing is configured or looks like. The way you appear in the mirror is described by the information content of your DNA and its subsequent biological expressions. The way you look is defined by information, encoded by many biological functions. The way we speak as humans is described by information too. The shape of the sounds and words and sentences that we produce are all configurations, specified by the information content encoded in our language.

Nothing can be described without information. And the signals allowing us to perceive any information do often only provide incomplete representations, because signals are lossy channels of information content. Signals are a way to consume information, more or less completely. Information content is a concept describing that parts of a signal are useful, and that information can be gained by consuming it. We are human, and we are naturally inclined to maximize some utility function. In the context of information theory, that means that we are usually trying to maximize the information content or information gain of a system.

Once we understand the concept of information gain, we can see it in many aspects of our lives. The foundation of any game is the information content that allows its players to either win or lose. Any information theorists out there may want to weigh in on the following take. Regardless, I would say that all games are games of imperfect information, exactly because future behaviour of other players or the environment are uncertain. There are many interesting dynamics to discover amidst various game designs, because games can be complete, incomplete, or imperfect in information. All the fascinating game designs aside, what we want to look at a little bit closer this week are real world applications of maximizing the information gain of a system.

Information theory has a concept of increasing the information gain by some clever mechanism. Several thought experiments exist in the form of logic puzzles. One of them is called the "12 ball problem", which poses the following question. Given 12 equal balls, of which only 1 has a slightly different weight, and given a balance scale, what is the minimum amount of weighings required in order to find the odd ball?

The least efficient way to go about this problem may also result in the highest possible resource consumption. We could simply weigh 11 of the 12 balls against any random ball and find the odd one of the lot eventually. This approach comes with a lot of variation in its performance profile. Sometimes it takes 7 weighings, rarely 1, and in worst case 11. The naive sequential approach does not provide any execution guarantees. So what do Senpai?

I wanted to discuss this problem because the solution to the problem requires the kind of thinking that is usually only found outside of the colloquial box. Our objective is to maximize the information gain, because that automatically implies to maximize the execution performance and subsequently minimize resource consumption. Weighing every ball separately results in the smallest possible information gain per weighing. The outside of the box thinking approach would then maybe be to weigh all the balls at once. And when we do that, then we effectively create batches of resource constrained processes, which reduces the amount of operations necessary in order to complete a task. And if we can do that, then we are also increasing the efficiency of the system.

The way in which we define our batches must be strategic to some extend. For instance, we would want to minimize the cases in which we are weighing the same ball multiple times. But that by itself does not tell us how large our batches should be. An indicator might be the amount of distinct information gains achieved per operation, because that amount can be the basis of our batching formula. A balance scale has two sides. We have to way one thing against another. The distinct amounts of gained information per operation is therefore 2, because we learn about the order of two items in comparison relative to one another. 2 is then already greater than 1, which is a pretty big deal if we were to execute some operation infinitely often for eternity.

What is so fascinating about the 12 ball problem is not only the information gained by the actions that we are doing, but also the information gained by the actions that we are not doing. Weighing one batch against another tells us how two batches compare. And the question for us is whether we should just weigh a random selection of 6 balls against the other 6, because it turns out that this approach would result in a slightly reduced information gain per operation compared to the optimal strategy.

Let's break down how 6 vs 6 would fare. With a total of 12 balls, the only possible outcome of the first 6/6 weighing is tilt in either direction, because the odd ball can now only be on either side of the scale. The second weighing would then split the tilted balls into a 3/3 weighing. The third weighing would then weigh two random balls of the tilted 3, where the possible results are two times tilt, plus one times balance, each of which refers to only a single ball. The information gain of the first weighing in this case is 50%, because we ruled out one half of the population. The second weighing rules out another 25%, leaving the third weighing to solve for the last quarter, where something remarkable may happen. If the third weighing produces a balance, then that process itself hasn't provided any direct information gain, because both balls are of equal weight after all. But the information inferred by the lack of information gain tells us that the last ball that we did not weigh on the third try must be the odd one, simply because we have ruled out the rest of the population at this point already. We utilized knowledge of an action that we have not done on the last try. And it is pretty cool that something like this is even possible.

The optimal solution though is slightly more efficient, because it utilizes knowledge of an action that we have not done on the first try, instead of the last one. The secret lies in the way in which we can craft our formula to derive the optimal batch size to start with. Remember, a balance scale gives us two clues at once, and if we can, then we should also choose to gain information without doing anything for it. That means 2 clues of an action we are doing, plus 1 clue of an action we are not doing, gives us an optimal 3 clues per weighing. Every clue is associated with a batch, and having 3 batches across a population of 12 provides us with a batch size of 4 if my math is strong today.

Let's break it down once more. With a total of 12 balls, the outcomes of the first 4/4 weighing is one of two tilts and one balance. The balance case gives away the clue of the action that we did not do. From here the second weighing is simply a 2/2 in any event, and the third weighing is a straight forward 1/1. The 4/4 approach allows us to rule out 66.66% of the population at first trial. Utilizing all characteristics of the environment provides us with over 16% more efficiency at the start despite having smaller batches to begin with, which in itself may save a lot of money on day.

Batching for efficient information gain is a widely adopted primitive across all walks of life. Medical diagnostics run tests against batches of samples in order to rule out large amounts of clients at once. Epidemiologically most samples are usually negative anyway, and so most batches are too. The few batches that result in a positive test can then be dissected with more effort individually, but only if there is a real reason to expend more resources on individual samples. Binary search algorithms split sorted lists in ever smaller halves for fast lookups that do not require the verification of every single item. And last, but not least, fraud proofs bundle blocks of many transactions so that only the minority of fraudulent transactions require in-depth verification.

Any form of hash tree allows any root hash to be verified efficiently within distributed systems. And as long as everyone computed the same root hash, no more work has to be done in order to provide certain verification guarantees of the underling system. Only a conflict in root hash verification triggers more workloads to be executed. In a similar fashion, zero knowledge proofs generate a single cryptographic proof that guarantees the correctness of an entire batch of computations mathematically. That way validating every single step of a large set of transactions is not necessary anymore. Verifiers can then easily check the relevant zero knowledge proof in constant time, regardless of the underlying batch size.

And this, ladies and gentlement, is then also exactly how we scale Ethereum. The largest, economically useful smart contract blockchain network, home to over 130 billion USD in stablecoin TVL. Every other day I am flubbergasted by the sheer amount of intellectual property constantly generated on top of Ethereum, most of which is economically relevant. These days the Eigenlayer ecosystem is firing on all cylinders, leaning heavingly into trust minimized autonomous agents. AI that is for you, dear reader. Narrative traders and proponents of overvalued infrastructure projects can go eat dirt, because agentic onchain banking is about to unfold on the public blockchain with the strongest security guarantees.

All of the above brings us finally around to the number of the week, because BlackRock's BUIDL fund is now sitting at over 1 billion USD in TVL, of which roughly 83% are managed on, you guessed it, Ethereum. There is no second best, Michael Saylor. Bring it on.