In this paper we show that, using only mild assumptions, dynamic multidimensional blockchain fee markets have strong performance guarantees, even against worst-case adversaries. In particular, we show that the average welfare gap between the following two scenarios is at most O(1/√T), where T is the length of the time horizon considered. In the first scenario, the designer knows all future actions by users and is allowed to fix the optimal prices of resources ahead of time, based on the designer’s oracular knowledge of those actions. In the second, the prices are updated by a very simple algorithm that does not have this oracular knowledge, special cases of which are EIP-4844 and EIP-1559, both fee mechanisms used by the Ethereum blockchain. Roughly speaking, this means that, on average, over a reasonable timescale, there is no difference in welfare between "correctly" fixing the prices, with oracular knowledge of the future, when compared to the proposed algorithm. We show a matching lower bound of Ω(1/√T) for any implementable algorithm and also separately consider the case where the adversary is known to be stochastic.

Constant function market makers (CFMMs) are the most popular type of decentralized trading venue for cryptocurrency tokens. In this paper, we give a very general geometric framework (or 'axioms') which encompass and generalize many of the known results for CFMMs in the literature, without requiring strong conditions such as differentiability or homogeneity. One particular consequence of this framework is that every CFMM has a (unique) canonical trading function that is concave, homogeneous, and nondecreasing, showing that many results known only for homogeneous trading functions are actually fully general. We also show that CFMMs satisfy a number of intuitive and geometric composition rules, and give a new proof, via conic duality, of the equivalence of the portfolio value function and the trading function. Many results are extended to the general setting where the CFMM is not assumed to be path-independent, but only one trade is allowed. Finally, we show that all 'path-independent' CFMMs have a simple geometric description that does not depend on any notion of a 'trading history'. Many of the results here depend on a type of duality for concave, homogeneous, and nondecreasing functions which may be of independent interest.

Public blockchains implement a fee mechanism to allocate scarce computational resources across competing transactions. Most existing fee market designs utilize a joint, fungible unit of account (e.g., gas in Ethereum) to price otherwise non-fungible resources such as bandwidth, computation, and storage, by hardcoding their relative prices. Fixing the relative price of each resource in this way inhibits granular price discovery, limiting scalability and opening up the possibility of denial-of-service attacks. As a result, many prominent networks such as Ethereum and Solana have proposed multi-dimensional fee markets. In this paper, we provide a principled way to design fee markets that efficiently price multiple non-fungible resources. Starting from a loss function specified by the network designer, we show how to compute dynamic prices that align the network's incentives (to minimize the loss) with those of the users and miners (to maximize their welfare), even as demand for these resources changes. Our pricing mechanism follows from a natural decomposition of the network designer's problem into two parts that are related to each other via the resource prices. These results can be used to efficiently set fees in order to improve network performance.

We consider the problem of optimally executing an order involving multiple crypto-assets, sometimes called tokens, on a network of multiple constant function market makers (CFMMs). When we ignore the fixed cost associated with executing an order on a CFMM, this optimal routing problem can be cast as a convex optimization problem, which is computationally tractable. When we include the fixed costs, the optimal routing problem is a mixed-integer convex problem, which can be solved using (sometimes slow) global optimization methods, or approximately solved using various heuristics based on convex optimization. The optimal routing problem includes as a special case the problem of identifying an arbitrage present in a network of CFMMs, or certifying that none exists.

Automated market makers, first popularized by Hanson's logarithmic market scoring rule (or LMSR) for prediction markets, have become important building blocks, called 'primitives,' for decentralized finance. A particularly useful primitive is the ability to measure the price of an asset, a problem often known as the pricing oracle problem. In this paper, we focus on the analysis of a very large class of automated market makers, called constant function market makers (or CFMMs) which includes existing popular market makers such as Uniswap, Balancer, and Curve, whose yearly transaction volume totals to billions of dollars. We give sufficient conditions such that, under fairly general assumptions, agents who interact with these constant function market makers are incentivized to correctly report the price of an asset and that they can do so in a computationally efficient way. We also derive several other useful properties that were previously not known. These include lower bounds on the total value of assets held by CFMMs and lower bounds guaranteeing that no agent can, by any set of trades, drain the reserves of assets held by a given CFMM.