Option values in sequential auctions with time-varying valuations
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Option values in sequential auctions with time-varying valuations Amir Ban1
· Ron Lavi2
Accepted: 29 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We investigate second-price sequential auctions of unit-demand bidders with timevariable valuations under complete information. We describe how a bidder figures willingness to pay by calculating option values, and show that when bidders bid their option value, and a condition of consistency is fulfilled, a subgame-perfect equilibrium is the result. With no constraints on valuations, equilibria are not necessarily efficient, but we show that when bidder valuations satisfy a certain constraint, an efficient equilibrium always exists. This result may be extended to a model with arrivals of bidders. We show how the equilibrium allocation, bids, and bidder utilities are calculated in the general case. We prove constructively that a pure subgame-perfect equilibrium always exists, and show how all pure equilibria can be found by the method of option values Keywords Sequential auctions · Option value · Bidding equilibrium · Complete information
1 Introduction This paper studies a setting where several identical items are sold sequentially via a second-price auction to a set of bidders. In each round, a bidder may have a different value for the item, and this value may change in arbitrary ways. This can capture, e.g., situations where the bidder is absent in some periods (indicated by having zero value), situations with discount factors, etc. Bidders may win at most a single item, and have complete information about the setting. This scenario is common in many
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Amir Ban [email protected] Ron Lavi [email protected]
1
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
2
Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology, Haifa, Israel
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A. Ban, R. Lavi
computational settings in which resources are allocated periodically. For example, in Bitcoin, every 10 min a new block is mined and the transaction slots of this block are allocated to the users. In cloud computing, computing resources are sold periodically. And so on. In such settings, a bidder’s value for the item is often time-dependent, e.g., different bidders have different arrival times, different departure times, different urgencies, and so on. Bidder values may decrease to reflect a preference to get the item sooner rather than later, or it may increase, e.g., when bidders value the flexibility to change their mind about needing the item. We investigate strategic behavior in these settings. Even in a complete-information setting, when valuations change between rounds, it is unclear how a bidder should approach the bidding decision and how much should be his maximal willingness to pay in the current round. A natural approach that was suggested by Bernhardt and Scoones (1994) is the option-value approach: a player recursively determines his resulting utility in future rounds as
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