# Monotonicity of second price auctions with bidder-specific reserve prices

I am having difficulty understanding the problem below:

Given CDFs $$F_1, ..., F_n$$, prove that the allocation rule of a second-price auction with bidder-specific reserve prices is monotone. The item is rewarded to the bidder with the highest bid that meets his reserve price. The reserve price of bidder $$i$$ is $$r_i = 𝜙_i^{-1}(t)$$ where $$𝜙_i$$ is the virtual valuation function

$$\phi_i(v_i) = v_i - \frac{1 - F_i(v_i)}{f_i(v_i)}$$

and $$t$$ is chosen such that

$$Pr\left[\max_i 𝜙_i(v_i) \ge t\right] = 1/2.$$ Since the virtual valuations are determined before the auction (if I understand it correctly), how does the reserve prices affect the monotonicity of the allocation rule? The behavior is still the same: if you keep on increasing your bid, you'll have a higher chance of getting the price. How should I go about this?

• What exactly is your problem? As you said from the perspective of any buyer this is just a SPA with a reserve price. Hence, everyone bids their value and the allocation probability weakly increases in the own valuation. – Bayesian Jun 15 '19 at 18:45
• @Bayesian My problem is proving the monotonicity of the allocation rule and the proof must involve the use of the virtual valuation function for the reserve prices. Does the reserve price change if the bid changes? – Pet Bottle Jun 16 '19 at 8:55
• Would it be possible to solve for $t$ or $r_i$? – Pet Bottle Jun 16 '19 at 13:01
• In the definition of $\phi$, are the cdf's indexed by $i$? – Regio Jun 16 '19 at 17:34
• @Regio Yes. They're not necessarily identical distributions, but we can assume that they are nondecreasing for all $i$ – Pet Bottle Jun 17 '19 at 10:20

Some thoughts. If this was a SPA without a reserve price, the monotonicity of the allocation rule is trivial. When the reserve price is uniform across bidders then the allocation rule is clearly weakly monotonic now; since for low bids, increasing them might not change the probability of allocation. The only thing that makes this problem different is that the reserve prices are bidder-specific. So one possibility is that you only need to show that the logic for the previous monotonicity is still valid in this setup. Probably while trying to show that the logic still goes through you will find out if there is some extra argument you will have to make in order to still have the monotonicity.

For example, I might worry that by increasing my bid I might increase my own reserve price, or decrease the reserve price of some other bidder and thus reduce my probability of being allocated the good.

• That's my main dilemma here, not sure how the bids affect the reserve prices. My understanding is that the virtual valuations are computed by the auctioneer before the auction, so the bids should not affect the reserve prices. I'm sure there's more to that that I'm not picking up – Pet Bottle Jun 17 '19 at 10:24
• That's right, seems that you only need to argue that knowing the distribution of valuations, for bidders is enough to compute $t$, and knowing $t$ and the $F_i$'s is sufficient to compute the $r_i$'s. Therefore, the reserve prices are independent of bidders' bids. Once reserve prices are constant with respect to a player's bids, the logic of previous cases should follow. – Regio Jun 17 '19 at 16:24
• The probability operator suggests that $t$ is determined before the auction. The way $r_i$ is formulated suggests that $\phi_i$ is invertible (most likely because $\phi_i$ is increasing, regularity). Hence, $r_i$ is determined before the auction, independent of the bids. There does not seem to be anything special to this task. Why exactly are you interested in this auction? – Bayesian Jun 17 '19 at 20:21