Online Random Sampling for Budgeted Settings

Abstract

We study online multi-unit auctions in which each agent’s private type consists of the agent’s arrival and departure times, valuation function and budget. Similarly to secretary settings, the different attributes of the agents’ types are determined by an adversary, but the arrival process is random. We establish a general framework for devising truthful random sampling mechanisms for online multi-unit settings with budgeted agents. We demonstrate the applicability of our framework by applying it to different objective functions (revenue and liquid welfare), and a range of assumptions about the agents’ valuations (additive or general) when selling identical divisible items. Our main result is the design of mechanisms for additive bidders with budget constraints that extract a constant fraction of the optimal revenue (under a standard large market assumption). We also show a mechanism that extracts a constant fraction of the optimal liquid welfare for general valuations.

Appendices

Appendix A: Proofs of the Theorems from Borgs et al.

Recall that P_k(S) is the optimal price for selling k items to a set S of agents and that $\overline {\epsilon } = \epsilon (S,k)$. Let $r^{\infty }_{S}(p)$ denote the revenue achieved by selling an unlimited number of items (digital goods) to set S of agents at price per item p. In order to show the proofs of the lemmas in Section 3.3, we need the following lemma from [8]:

Lemma 20

[8] For anyδ > 0 andfor anyp ≥ P_k(S),the probability that$\left |{r^{\infty }_{S_1}(p)-r^{\infty }_{S_2}(p)} \right | \leq \delta \cdot OPT(S,k)$isat least$1-2e^{-\delta ^2/4\overline {\epsilon }}$.

We now turn to prove the lemmas stated in Section 3.3.

Proof

of Lemma3 Since the revenue obtained from selling items at a given price per item is either bounded by the agents’ budgets or by the number of items for sale, we get that $OPT(S,k)=\min \left \{P_k(S)\cdot k,r^{\infty }_{S}(P_k(S))\right \}$. Therefore, we get

$$ \begin{array}{@{}rcl@{}} OPT(S,k)\leq P_{k}(S)\cdot k, \end{array} $$

(7)

and

$$ \begin{array}{@{}rcl@{}} OPT(S,k)\leq r^{\infty}_{S}(P_{k}(S)). \end{array} $$

(8)

When unrestricted by the number of items, we have that

$$ \begin{array}{@{}rcl@{}} r^{\infty}_{S_{1}}(P_{k}(S))+r^{\infty}_{S_{2}}(P_{k}(S))=r^{\infty}_{S}(P_{k}(S))\geq OPT(S,k). \end{array} $$

(9)

Whenever $\left |{r^{\infty }_{S_1}(P_k(S))-r^{\infty }_{S_2}(P_k(S))}\right |\leq \delta \cdot OPT(S,k)$, then

$$ \begin{array}{@{}rcl@{}} r^{\infty}_{S_{2}}(P_{k}(S)) \leq r^{\infty}_{S_{1}}(P_{k}(S))+\delta\cdot OPT(S,k), \end{array} $$

(10)

and

$$ r^{\infty}_{S_{1}}(P_{k}(S)) \leq r^{\infty}_{S_{2}}(P_{k}(S))+\delta\cdot OPT(S,k). $$

(11)

Combining (10) with (9) yields $r^{\infty }_{S_1}(P_k(S))\geq \frac {1-\delta }{2}OPT(S,k)$. Similarly, combining (11) with (9) yields $r^{\infty }_{S_2}(P_k(S))\geq \frac {1-\delta }{2}OPT(S,k)$. By (7) we clearly have $P_k(S)\cdot \frac {k}{2}\geq OPT(S,k)/2$. Since both $r_{S_1}=\min \left \{r^{\infty }_{S_1}(P_k(S)),P_k(S)\cdot \frac {k}{2}\right \}$ and $r_{S_2}=\min \left \{r^{\infty }_{S_2}(P_k(S)),P_k(S)\cdot \frac {k}{2}\right \}$ we get that whenever $\left |{r^{\infty }_{S_1}(P_k(S))-r^{\infty }_{S_2}(P_k(S))}\right |\leq \delta \cdot OPT(S,k)$ it holds that $r_{S_1} \geq \frac {1-\delta }{2}OPT(S,k)$ and $r_{S_2} \geq \frac {1-\delta }{2}OPT(S,k)$. The proof is now concluded by Lemma 20. □

In order to prove Lemma 4, we also need the following lemma from [8].⁸

Lemma 21

[8] For allδ,p > 0 both$r^{\infty }_{S_1}(p)\geq \min \left \{r^{\infty }_{S_2}(p)-\delta \cdot OPT(S,k),\frac {1-\delta }{2}\cdot OPT(S,k)\right \}$and$r^{\infty }_{S_2}(p)\geq \min \left \{r^{\infty }_{S_1}(p)-\delta \cdot OPT(S,k),\frac {1-\delta }{2}\cdot OPT(S,k)\right \}$withprobability at least$1-2e^{-\delta ^2/4\overline {\epsilon }}$.

Proof

Assuming $\left |{r^{\infty }_{S_1}(P_k(S))-r^{\infty }_{S_2}(P_k(S))}\right |\leq \delta \cdot OPT(S,k)$ we get that both $r^{\infty }_{S_2}(p)\geq r^{\infty }_{S_1}(p)-\delta \cdot OPT(S,k)$ and $r^{\infty }_{S_1}(p)\geq r^{\infty }_{S_2}(p)-\delta \cdot OPT(S,k)$ with probability greater than $1-2e^{-\delta ^2/4\overline {\epsilon }}$ for p ≥ P_k(S).

Recall that in the proof of Lemma 3, we showed that both $r^{\infty }_{S_1}(P_k(S))\geq \frac {1-\delta }{2}OPT(S,k)$ and $r^{\infty }_{S_2}(P_k(S))\geq \frac {1-\delta }{2}OPT(S,k)$ whenever $\left |{r^{\infty }_{S_1}(P_k(S))-r^{\infty }_{S_2}(P_k(S))}\right |\leq \delta \cdot OPT(S,k)$. Since with an unlimited supply of items, $r^{\infty }_{S}(p_1)\geq r^{\infty }_{S}(p_2)$ whenever p₁ < p₂ (only more agents exhaust their budget), we get that for every p ≤ P_k(S) both $r^{\infty }_{S_1}(p)\geq \frac {1-\delta }{2}OPT(S,k)$ and $r^{\infty }_{S_2}(p)\geq \frac {1-\delta }{2}OPT(S,k)$.

To conclude, we recall that according to Lemma 20, $\left |{r^{\infty }_{S_1}(P_k(S))-r^{\infty }_{S_2}(P_k(S))}\right |\leq \delta \cdot OPT(S,k)$ occurs with probability at least $1-2e^{-\delta ^2/4\overline {\epsilon }}$. □

We now proceed to show the proof of Lemma 4.

Proof

of Lemma4 We first assume that $\left |{r^{\infty }_{S_1}(P_k(S))-r^{\infty }_{S_2}(P_k(S))}\right |\leq \delta \cdot OPT(S,k)$. By definition, $OPT(S_1,k/2)\geq r_{S_1}$. In this case, as shown in the proof of Lemma 3, $OPT(S_1,k/2)\geq \frac {1-\delta }{2}OPT(S,k)$. Therefore, it is clear that both

$$ \begin{array}{@{}rcl@{}} P_{k/2}(S_{1})\cdot\frac{k}{2}\geq \frac{1-\delta}{2}OPT(S,k), \end{array} $$

(12)

and

$$ \begin{array}{@{}rcl@{}} r^{\infty}_{S_{1}}(P_{k/2}(S_{1}))\geq \frac{1-\delta}{2}OPT(S,k). \end{array} $$

(13)

Combining the last inequality with Lemma 21, we get that

$$ \begin{array}{@{}rcl@{}} r^{\infty}_{S_{2}}(P_{k/2}(S_{1}))\geq \frac{1-3\delta}{2}OPT(S,k). \end{array} $$

(14)

Since in Offline-Rev-Maximization either the agents of S₂ exhaust their budget, or k/2 items are sold, we get that the revenue of mechanism Offline-Rev-Maximization from agents in set S₂ is not smaller than: $\min \left \{r^{\infty }_{S_2}(P_{k/2}(S_1)),P_{k/2}(S_1)\cdot \frac {k}{2}\right \}\geq \frac {1-3\delta }{2}OPT(S,k)$. To conclude, we note that according to Lemma 20, $\left |{r^{\infty }_{S_1}(P_k(S))-r^{\infty }_{S_2}(P_k(S))}\right |\leq \delta \cdot OPT(S,k)$ occurs with probability at least $1-2e^{-\delta ^2/4\overline {\epsilon }}$. □

Appendix B: Tie Breaking

In this section, we show how to perform tie breaking when an agents is allowed to report the same arrival time as another agent, but cannot report an earlier arrival time than her real arrival time. The tie breaking rule we present is as follows. Whenever an agent i arrives at time a_i, the mechanism chooses a uniformly at random value $\tilde {a}_i\sim [0,1]$ and sets agent i’s arrival time to be $\langle a_i,\tilde {a}_i\rangle$. The mechanism orders the agents according to the following rule: Agent i precedes agent j if a_i < a_j or if a_i = a_j and $\tilde {a}_i< \tilde {a}_j$.

We change mechanisms Online-RS and SP-One-Dominant according to the above rule. We claim that these mechanisms are truthful. The only problematic case is when an agent reports an earlier arrival time such that she is placed in set A instead of being placed in set B (the proof of all the other cases is similar to the proof of Theorem 1). Since an agent cannot affect the random value assigned to her, and since we specifically prevent an agent from reporting an earlier arrival time, this is impossible.