Consider the following game in multi-cryptocurrency mining with a large number of individual miners. Each miner has a limited amount of computing time, constrained by the hardware she possesses. Depending on the value of each cryptocurrency and the chance of successful mining in each cryptocurrency, she faces the strategic problem of how to optimality allocating her computing time to mining different cryptocurrencies, so as to maximize her own benefit. Such games have been studied under the name of Tullock contests
. In the Internet era, we can see analogous production economies
in crowdsourcing and P2P network applications.
In these large and distributed production economies, we desire stability. We ask: is there any distributed protocol the players can use to adjust their strategic choices, so that the economy eventually reaches a stable state, e.g. (approximate) Nash equilibrium?
The first choices of such protocols are game-theoretic motivated, e.g. Gradient Ascent and Best Response. However, we formally prove that Gradient Ascent can lead to unstable and even chaotic behavior, via the rigorous notion of Li-Yorke chaos
. The chaos can occur even in the very simple instance with only 2 homogeneous players and 1 cryptocurrency.
To remedy this, we observe that when such production economies becomes large
, i.e. there are a large number of miners and none of them has superior amount of computing time, we may model it as a Fisher market with quasilinear utilities
instead. Then by using the market-motivated protocol of Proportional Response
, which was shown to be remarkably stable in many markets via techniques in Numerical Optimization, the players will reach the market equilibrium of the Fisher market, while the largeness guarantees the market equilibrium is an approximate Nash equilibrium.