Abstract
Motivated by auctions for carbon emission permits, we examine the social welfare of multi-buyer sequential auctions for identical items. Assuming the buyers have weakly-decreasing incremental (concave) valuation functions, we study subgame perfect equilibria of a repeated second-price auction with n buyers and T time periods. We show that these auctions admit envy-free subgame-perfect equilibria that \((1-1/e)\)-approximate the optimal welfare. The equilibria we construct have a natural interpretation: each bidder guarantees for herself the best outcome she could obtain if all other bidders were non-strategic. Without the envy-freeness condition, the price of anarchy can be as bad as \(\varTheta (1/T)\) even when restricting to equilibria that satisfy a no-overbidding condition. We also consider the restricted class of envy-free subgame perfect equilibria that survive iterated deletion of weakly undominated strategies. For this class of equilibria we prove constant bounds on the price of anarchy for three settings with differing levels of market competitiveness, based on the number of buyers with oligopsony power (market power).
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Notes
- 1.
The price of stability is the worst case ratio over all instances between the welfare of the best equilibrium (for a specified class of equilibria) and the optimal welfare, and the price of anarchy is the worst case ratio over all instances between the welfare of the worst equilibrium (for a specified class of equilibria) and the optimal welfare.
- 2.
In alternate settings, such as when reserve prices are incorporated, it may be the case that \(\sum _{i \in [n]} \pi _i(h|b) < 1\).
- 3.
For example, take a single-item second-price auction with two buyers. Suppose \(v_1(1)=1\) and \(v_2(1)=\epsilon \). The bids \(b_1(0)=0\) and \(b_2(0)=2\) form an equilibrium with social welfare \(\epsilon \), but the optimal welfare is 1.
- 4.
We note that this is an equilibrium refinement rather than a restriction of the action space. Bidders are still able to consider deviations in which they overbid.
- 5.
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Acknowledgements
The authors are very grateful for advice from and discussions with Craig Golding, Tom Johnson and Alex Wood of the former Ontario Ministry of the Environment and Climate Change, and from Christopher Regan, Dale Beugin and Jason Dion of Canada’s Ecofiscal Commission.
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Ahunbay, M.Ş., Lucier, B., Vetta, A. (2021). The Price of Stability of Envy-Free Equilibria in Multi-buyer Sequential Auctions. In: Caragiannis, I., Hansen, K.A. (eds) Algorithmic Game Theory. SAGT 2021. Lecture Notes in Computer Science(), vol 12885. Springer, Cham. https://doi.org/10.1007/978-3-030-85947-3_2
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