Hi there @Sandro,

This is a question that I had as well a while ago, it is not trivial to make a PoS blockchain that abides the things we love about PoW. Though I do not have a paper to reference at the moment, I can inform you the best as I can (it exists but I can’t find it right now, Ill send it later). This is also a good moment for me to communicate that what I learned Two weeks ago I attended an online conference about the game theory behind Ouroborus and general PoX (where X can be anything). The results that where shown baffled me and are in my opinion profound (I took some screenshots of graphs to show). A disclaimer: the following dialogue needs to be taken with a grain of salt. The arguments made here are of mathematical nature and though game theory is a nice field it is based on assumptions (the main one being that players are rational). Game theory can exclude certain end states of a game but can not predict with certainty the outcome (it is simply not the real world).

I do not know your background but some preknowledge is needed for understanding the power of Ouroborus and the general game theory of PoX. In this post I will be less precise in my definitions to get the concepts clear. Most of the concepts that you need are Nash equilibrium and a bit of measure theory. The former is a game theory definition that says something like this, a Nash equilibrium is **a state** in a game in which all parties have no incentive to deviate from their position. That is if they deviate they will lose out on some notion of value. The latter concept is more mathematical of nature, it describes a field in mathematics that tries to “measure” certain objects and **relates** sizes between objects. A measure is a function that has certain properties and in the blockchain space it used to value the impact sizes of block producers in PoX protocols with a reward function. That is, how many blocks can a player produce given the resources X that the player has and what rewards should the player get.

Now what Prof. Aggelos Kiayias (chief scientist of IOG) et al did in one of their papers, They made mathematically abstracted the concepts of ANY PoX protocol and made precise what the game theory results of certain protocols where. With this knowledge/insight Ouroboros was designed to have certain properties.

The setting: In any PoX protocol validator play a game to produce blocks. They try to optimize their resources X to produce the maximal amount of block that they can and collect rewards for this (there is a reward function just like the measure function). In a general PoX protocol X is some value that can not be create out of nothing, if could be produce out of thin air centralization would be imminent. In PoW hash power is this X and it scales **linear** with the amount of block you produce (though a good network for block propagation is also something to account for). Note that this X is something valuable outside the blockchain, the insight of newly emerging PoX protocols is to chose X something that is directly related to something that live on the blockchain (but still is valuable so that it is not easily gathered). In PoS the valuable resource is the “in-game” token, in this way not value can be extracted outside the network and it completely disconnects the networks health to outside influence (like the price of hash rate that greatly influences the BTC price).

Now what makes a good “measure” and “reward” function such that centralization is not a Nash equilibrium? Below is a screenshot of some axions (things that are true by assumtion) for a these functions.

Some definition, the reward and resource measure functions above both take in some abstract value P that represents the resources available to a player. Again it could be any resource! I will go through them for clarification.

- The first one is continuity, this says that the function should value the resources in a uninterrupted way. As I stated above, bitcoin scales linear in its hash rate and as we all know a line is continous (it hase no holes in it).
- The second is fungibility and it states that the rewards for resources should be the same it playes have the same amount of it (here Q and P are different but equivalent resources).
- The third is agains sybil attacts, it says something like this; the rewards for combined resources should pay out more or the same than the rewards for splitting up these resource. (something like the triangle inequality for the more mathematical inclined ones with us).
- The forth is egalitarianism and it states something like;

With this out of the way, the paper goes in one how to make such a reward scheme and then make a statement on the Nash equilibria. The result is some theorem that states that certain valuations of resources lead provable to centralization. (I will follow up on this topic). This insight gave rise to Ouroborus that has a Nash equilibrium with k validators (currently k=500 of Cardano). The simulations gave the following plots. The line in the graph at the end of iterations is when the equilibrium is reached.

Hope this helps a bit