Abstract
This chapter applies the analysis of equilibrium bidding behavior in first-price auctions from Chap. 2 to answer different questions. In Exercise 3.1, we measure the seller’s expected revenue in this auction format when bidders draw their valuations from a generic distribution function and then evaluate this expected revenue in the case that bidders’ valuations are uniformly distributed.
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Notes
- 1.
Recall the integration by parts formula: \(\int uv^{\prime }=uv-\int u^{\prime }v\). Let u = x and v ′ = G α(x)′, then u ′ = 1 and \(v=G_{\alpha }(x)=x^{\frac {N-1}{\alpha }}\).
- 2.
Recall that we apply backward induction to find the subgame perfect equilibrium (or equilibria) of a sequential-move game. When applying backward induction, we start with the last stage of the game, finding optimal actions for the player(s) called to move in this stage. Then, we move to the second-to-last mover who, anticipating equilibrium behavior in the last stage, chooses his optimal action. We can repeat this process by moving one step closer to the first stage of the game, successively finding equilibrium behavior of each stage. For more details on this solution concept, see the Game Theory Appendix at the end of the book, and for more examples, see Munoz-Garcia and Toro-Gonzalez (2020, Chapter 4).
- 3.
The above expression considers that bidders submitting bids \(b_{i}\left ( v_{i}\right ) \geq r\) participate in the auction, while those with valuations leading to low bids \(b_{i}\left ( v_{i}\right ) <r\) do not. In addition, it considers that the participating bidder with the lowest valuation submits a bid \(b_{i}\left ( r\right ) =r\) where his valuation is v i = r. To see this point, note that he must be indifferent between participating and not participating in the auction, so his expected utility must satisfy F(v i)N−1(v i − b i) = 0. Substituting b i = r, we obtain F(v i)N−1(v i − r) = 0, which holds only if v i = r, i.e., if his valuation coincides with the reservation price. Therefore, the participation condition using bids, \(b_{i}\left ( v_{i}\right ) \geq r\), implies an analogous condition using valuations, which we can write as v i ≥ r, explaining that the lower bound of integration is v i = r in the above expressions.
- 4.
We need this condition to ensure a one-to-one mapping between the entry fee and the critical bidder’s valuation. The monotonically increasing bidding function b i(v i) ensures that bidders with valuations above v e obtain a positive utility from participating in the auction (after paying the entry fee E) and thus submit a positive bid for the object.
References
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Choi, PS., Munoz-Garcia, F. (2021). First-Price Auctions: Extensions. In: Auction Theory. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-69575-0_3
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