Essays on Auction Theory

Abstract

This dissertation focuses on auction theory, especially multi-unit auction, since bidder's behaviour will differ a lot in multi-unit auctions than in single-unit auctions. In "Multi-Unit Auction In Discrete Type Space", we study a multi-unit pay-as-bid auction where there are two discrete types of bidders and each type of bidder demands two units. We find closed-form solutions for symmetric Bayesian Nash Equilibria for different proportions of types in the population and one main feature is identical bidding behaviour, where one particular type of bidder will use identical bidding prices for both of the units. This chapter also finds that distributions for mixed strategy equilibrium from different types will have overlapping support in bidding spaces. These two features will lead to inefficient allocations. The identical bidding behaviour is also reported in empirical literature studying treasury bill auctions. We also compare expected revenue between formats of multi-unit auctions and confirm that revenue equivalence does not hold in multi-unit settings with ambiguous ranking between revenue from pay-as-bid and Vickrey auctions, while both dominate uniform-price auction in expected revenue. We also show through examples that the identical bidding behaviour can also be extended to higher-unit settings.

In ``Package Bidding with Distinct Objects", we study a combinatorial auction where two discrete types of bidders are competing for two distinct objects. In this chapter we assume different bidders will have different favoured objects. In the setting of a combinatorial auction, we allow each bidder to propose an extra price for the bundle, besides prices for each single object. In this chapter, we focus on closed-form solutions for symmetric Bayesian Nash Equilibria with restriction to equilibria where bidders bid pure strategies on single objects and mixed strategies on the bundle. We continue to study their performance in terms of welfare.

In ``Smooth Ambiguity Averse Level-k Auction", we return to the context of single-unit auction and re-examine the elimination process of dominated bids in a first-price sealed bid auction. In particular, we study the first-price auction in a discretized unit interval and construct the upper and lower bounds of feasible bids in the process of elimination of implausible bids, with the help of smooth ambiguity averse model proposed by Klibanoff, Marinacci and Mukerji (2005) which allows bidders to aggregate smoothly over the support of subjective beliefs. We numerically construct upper and lower bounds of feasible bids in each round of elimination as well as the convergent stable bids for each type in discretized type and bidding spaces. We compare our result of stable bounds with results from Bayesian Nash Equilibrium for first price auctions. We also find that my approach is also similar to the level-k theory.