Metric distortion in social choice is a framework for evaluating how well voting rules minimize social cost when both voters and candidates exist in a shared metric space, with a voter's cost defined by their distance to a candidate. Voters submit rankings, and the rule aggregates these rankings to determine a winner. We extend this framework to incorporate probabilistic voting, recognizing that real-world voters exhibit randomness in how they vote. Our extension includes various probability functions, notably the widely studied Plackett-Luce (PL) model.

Batch auctions are a classical market microstructure, acclaimed for their fairness properties, and have received renewed interest in the context of blockchain-based financial systems. Constant function market makers (CFMMs) are another market design innovation praised for their computational simplicity. Liquidity provision in batch exchanges is an important problem, and CFMMs have recently shown promise in being useful within batch exchanges. Different real-world implementations have used fundamentally different approaches towards integrating CFMMs in batch exchanges, and there is a lack of formal understanding of the trade-offs of different design choices.

We study low sample complexity mechanisms in participatory budgeting (PB), where each voter votes for a preferred allocation of funds to various projects, subject to project costs and total spending constraints. We analyse the distortion that PB mechanisms introduce relative to the minimum-social-cost outcome in expectation. The Random Dictator mechanism for this problem obtains a distortion of 2. In a special case where every voter votes for exactly one project, [Fain et al., 2017] obtain a distortion of 4/3. We show that when PB outcomes are determined as any convex combination of the votes of two voters, the distortion is 2. When three uniformly randomly sampled votes are used, we give a PB mechanism that obtains a distortion of at most 1.66, thus breaking the barrier of 2 with the smallest possible sample complexity. We give a randomized Nash bargaining scheme where two uniformly randomly chosen voters bargain with the disagreement point as the vote of a voter chosen uniformly at random. This mechanism has a distortion of at most 1.66. We provide a lower bound of 1.38 for the distortion of this scheme. Further, we show that PB mechanisms that output a median of the votes of three voters chosen uniformly at random, have a distortion of at most 1.80.

Constant Function Market Makers (CFMMs) are a tool for creating exchange markets, have been deployed effectively in prediction markets, and are now especially prominent in the Decentralized Finance ecosystem. We show that for any set of beliefs about future asset prices, an optimal CFMM trading function exists that maximizes the fraction of trades that a CFMM can settle. We formulate a convex program to compute this optimal trading function. This program, therefore, gives a tractable framework for market-makers to compile their belief function on the future prices of the underlying assets into the trading function of a maximally capital-efficient CFMM. Our convex optimization framework further extends to capture the tradeoffs between fee revenue, arbitrage loss, and opportunity costs of liquidity providers. Analyzing the program shows how the consideration of profit and loss leads to a qualitatively different optimal trading function. Our model additionally explains the diversity of CFMM designs that appear in practice. We show that careful analysis of our convex program enables inference of a market-maker's beliefs about future asset prices, and show that these beliefs mirror the folklore intuition for several widely used CFMMs. Developing the program requires a new notion of the liquidity of a CFMM, and the core technical challenge is in the analysis of the KKT conditions of an optimization over an infinite-dimensional Banach space.

Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in a uniformly random order and the edge-weights are arbitrary positive numbers. For the capacitated online bipartite matching problem in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the online general (non-bipartite) matching problem, where all vertices arrive online, we give a 3.34-competitive algorithm. We also study the online roommate matching problem, in which each room (offline vertex) holds 2 persons (online vertices). Persons derive non-negative additive utilities from their room as well as roommate. In this model, with the goal of maximizing the sum of utilities of all persons, we give a 7.96-competitive algorithm. This is an improvement over the 24.72 approximation factor in prior work.