On the Efficiency of Hybrid Objective Multi-Unit Auctions
Twan Kroll

Abstract:
In this thesis, we analyze the efficiency of equilibria in multi-unit auctions (discriminatory and uniform price) in the presence of budgets where each player has their own objective that is intermediate between value and utility maximizing. We prove in both multi-unit auctions the existence of pure Nash equilibria and in the case of discriminatory multi-unit auction give a complete description of their efficiency in the presence of budgets, but also in the budgetless case. The main result of this thesis is the method of analyzing the price of anarchy in the presence of budgets. The notion of price of anarchy we use was first coined by Caragiannis & Voudouris (SAGT 2014) and recently received more attention in the autobidding survey by Aggarwal et al. (ACM 2024). By transforming a multi-unit auction with budgets into a welfare objective auction we manage to bound the price of anarchy (POA) of the former by bounding the POA in the latter through smoothness. The smoothness framework, introduced by Syrgkanis & Tardos (STOC 2013), utilizes the extension to different types introduced by Colini-Baldeschi et al. (arXiv, 2025). The upper bounds obtained through this method on the POA are valid for coarse correlated equilibria and for some classes even tight. The method can be extended to simultaneous first-price multi-unit auctions where players have submodular valuations. Through this, we manage to prove that the coarse correlated POA of utility maximizing first-price simultaneous auctions where bidders have additive valuations is approximately 2.18, exactly the same value Deng et al. (NeurIPS 2024) found for the price of anarchy of mixed Nash equilibria of budgetless simultaneous auctions in mixed settings.