Asymmetric Budget Constraints in a First Price Auction

Bobkova, Nina (2017): Asymmetric Budget Constraints in a First Price Auction.

Abstract

I solve a first-price auction for two bidders with asymmetric budget distributions and known valuations for one object. I show that in any equilibrium, the expected utilities and bid distributions of both bidders are unique. If budgets are sufficiently low, the bidders will bid their entire budget in any equilibrium. For sufficiently high budgets, mass points in the equilibrium strategies arise. A less restrictive budget distribution could make both bidders strictly worse off. If the budget distribution of a bidder is dominated by the budget distribution of his opponent in the reverse-hazard rate order, the weaker bidder will bid more aggressively than his stronger opponent. In contrast to existing results for symmetric budget distributions, with asymmetric budget distributions, a second-price auction can yield a strictly higher revenue than a first-price auction. Under an additional assumption, I derive the unique equilibrium utilities and bid distributions of both bidders in an all-pay auction.

• Item Type: MPRA Paper
• Language: English
• Keywords: Budget Constraints; First Price Auctions; Asymmetric Bidders
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games; D - Microeconomics > D4 - Market Structure, Pricing, and Design > D44 - Auctions; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D82 - Asymmetric and Private Information ; Mechanism Design

References

Amann, E. and W. Leininger (1996): Asymmetric all-pay auctions with incomplete information: the two-player case. Games and Economic Behavior, 14:1-18.

Bagnoli, M. and T. Bergstrom (2005): Log-concave probability and its applications. Economic Theory, 26:445-469.

Baye, M. R.; D. Kovenock; and C. G. de Vries (1996): The all-pay auction with complete information. Economic Theory, 8:291-305.

Benoît, J.-P. and V. Krishna (2001): Multiple-object auctions with budget constrained bidders. Review of Economic Studies, 68:155-179.

Bergemann, D. and M. Pesendorfer (2007): Optimal information disclosure in auctions. Journal of Economic Theory, 137:580-609.

Blume, A. (2003): Bertrand with fudge. Economic Letters, 78:167-168.

Cantillon, E. (2008): The effect of bidders' asymmetries on expected revenue in auctions. Games and Economic Behavior, 62:1-25.

Che, Y.-K. and I. Gale (1996): Expected revenue of all-pay auctions and first-price sealed-bid auctions with budget constraints. Economic Letters, 50:373-379.

Che, Y.-K. and I. Gale (1998): Standard auctions with financially constrained bidders. Review of Economic Studies, 65:1-21.

Che, Y.-K. and I. Gale (2000): Optimal mechanism for selling to a budget constrained buyer. Journal of Economic Theory, 92:198-233.

Che, Y.-K. and I. Gale (2006): Revenue comparisons for auctions when bidders have arbitrary types. Theoretical Economics, 1:95-118.

Cheng, H. (2006): Ranking sealed high-bid and open asymmetric auctions. Journal of Mathematical Economics, 42:471-498.

Dobzinski, S.; R. Lavi; and N. Nisan (2012): Multi-unit auctions with budget limits. Games and Economic Behavior, 74:486-503.

Fibich, G. and A. Gavious (2003): Asymmetric first price auctions - a perturbation approach. Mathematics of Operations Research, 28:836-852.

Fibich, G.; A. Gavious; and A. Sela (2004): Revenue equivalence in asymmetric auctions. Journal of Economic Theory, 115:309-321.

Gavious, A. and Y. Minchuk (2014): Ranking asymmetric auctions. International Journal of Game Theory, 43:369-393.

Kamenica, E. and M. Gentzkow (2011): Bayesian persuasion. American Economic Review, 101:2590-2615.

Kaplan, T.R. and S. Zamir (2012): Asymmetric first-price auctions with uniform distributions: analytic solutions to the general case. Economic Theory, 50:269-302.

Kotowski, M. (May 2018): First price auctions with budget constraints. Working Paper.

Kotowski, M. and F. Li (2014): On the continuous equilibria of affiliated-value, all-pay auctions with private budget constraints. Games and Economic Behavior, 85:84-108.

Lebrun, B. (2009): Auctions with almost homogenous bidders. Journal of Economic Theory, 144:1341-1351.

Malakhov, A. and R. Vohra (2008): Optimal auctions for asymmetrically budget constrained bidders. Review of Economic Design, 12:245-257.

Maskin, E. and J. Riley (2000): Asymmetric auctions. The Review of Economic Studies, 67:413-438.

Plum, M. (1992): Characterization and computation of Nash-equilibria for auctions with incomplete information. International Journal of Game Theory, 20:393-418.

Salant, D. (1997): Up in the air: GTE's experience in the MTA Auction for personal communications Sevices Licences. Journal of Economics and Management Strategy, 6:549-572.

Zheng, C. Z. (2001): High bids and broke winners. Journal of Economic Theory, 100:129- 171.