Optimal Bid Sequences for Multiple-Object Auctions with Unequal Budgets

TL;DR

This paper develops an optimal bidding algorithm for two bidders with unequal budgets in position-randomized multiple-object auctions, challenging the assumption that winning probabilities are proportional to budgets.

Contribution

It introduces a new optimal bidding algorithm for two bidders with unequal budgets, extending previous models and providing an efficient O(n) solution.

Findings

Optimal algorithm for two bidders with unequal budgets

Winning probabilities are not proportional to budgets

Algorithm runs in optimal O(n) time

Abstract

In a multiple-object auction, every bidder tries to win as many objects as possible with a bidding algorithm. This paper studies position-randomized auctions, which form a special class of multiple-object auctions where a bidding algorithm consists of an initial bid sequence and an algorithm for randomly permuting the sequence. We are especially concerned with situations where some bidders know the bidding algorithms of others. For the case of only two bidders, we give an optimal bidding algorithm for the disadvantaged bidder. Our result generalizes previous work by allowing the bidders to have unequal budgets. One might naturally anticipate that the optimal expected numbers of objects won by the bidders would be proportional to their budgets. Surprisingly, this is not true. Our new algorithm runs in optimal O(n) time in a straightforward manner. The case with more than two bidders is open.