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## Competitive Auctions

This week I will present the Theory Discussion Group about Competitive Auctions. It is mainly a serie of results in papers from Jason Hartline, Andrew Goldberg, Anna Karlin, Amos Fiat, … The first paper is Competitive Auctions and Digital Goods and the second is Competitive Generalized Auctions. My objective is to begin with a short introduction about Mechanism Design, the concept of truthfulness and the characterization of Truthful Mechanisms for Single Parameter Agents. Then we describe the Random Sampling Auction for Digital Goods and in the end we discuss open questions. I thought writting a blog post was a good way of organizing my ideas to the talk.

1. Mechanism Design and Truthfulness

A mechanism is an algorithm augmented with economic incentives. They are usually applied in the following context: there is an algorithmic problem and the input is distributed among several agents that have some interest in the final outcome and therefore they may try manipulate the algorithm. Today we restrict our attention to a specific class of mechanisms called single parameter agents. In that setting, there is a set ${N}$ consisting of ${n}$ agents and a service. Each agent ${i \in N}$ has a value ${v_i}$ for receiving the service and ${0}$ otherwise. We can think of ${v_i}$ as the maximum player ${i}$ is willing to pay for that service. We call an environment ${\mathcal{X} \subseteq 2^N}$ the subsets of the bidders that can be simultaneously served. For example:

1. Single item auction: ${\mathcal{X} =\{S; \vert S \vert \leq 1\}}$
2. Multi item auction: ${\mathcal{X} =\{S; \vert S \vert \leq k\}}$
3. Digital goods auction: ${\mathcal{X} =2^N}$
4. Matroid auctions: ${\mathcal{X}}$ is a matroid on ${N}$
5. Path auctions: ${N}$ is the set of edges in a graph and ${\mathcal{X}}$ is the set of ${s-t}$-paths in the graph
6. Knapsack auctions: there is a size ${s_i}$ for each ${i \in N}$ and ${S \in \mathcal{X}}$ iff ${\sum_{i \in S} s_i \leq C}$ for a fixed ${C}$

Most mechanism design problems focus in maximizing (or approximating) the social welfare, i.e., finding ${S \in \mathcal{X}}$ maximizing ${\sum_{i \in S} v_i}$. Our focus here will be maximizing the revenue of the auctioneer. Before we start searching for such a mechanism, we should first see which properties it is supposed to have, and maybe even first that that, define what we mean by a mechanism. In the first moment, the agents report their valuations ${v_i}$ (which can be their true valuations or lies), then the mechanism decides on an allocation ${S \subseteq N}$ (in a possibly randomized way) and charges a payment ${P_i}$ for each allocated agents. The profit of the auctioneer is ${\sum_{i \in S} P_i}$ and the utility of a bidder is:

\displaystyle u_i = \left\{ \begin{aligned} v_i - P_i &, i \in S \\ 0 &, i \notin S \end{aligned} \right.

The agents will report valuations so to maximize their final utility. We could either consider a general mechanism e calculate the profit/social welfare in the game induced by this mechanism or we could design an algorithm that gives incentives for the bidders to report their true valuation. The revelation principle says there is no loss of generality to consider only mechanisms of the second type. The intuition is: the mechanisms of the first type can be simulated by mechanisms of the second type. So, we restrict our attention to mechanisms of the second type, which we call truthful mechanisms. This definnition is clear for deterministic mechanisms but not so clear for randomized mechanisms. There are two such definitions:

• Universal Truthful mechanisms: distribution over deterministic truthful mechanisms, i.e., some coins are tossed and based on those coins, we choose a deterministic mechanism and run it. Even if the players knew the random coins, the mechanism would still be truthful.
• Truthful in Expectation mechanisms: Let ${u_i(b_i)}$ be the utility of agent ${i}$ if he bids ${b_i}$. Since it is a randomized mechanism, then it is random variable. Truthful in expectation means that ${\mathop{\mathbb E}[u_i(v_i)] \geq \mathop{\mathbb E}[u_i(b_i)], \forall b_i}$.

Clearly all Universal Truthful mechanisms are Truthful in Expectation but the converse is not true. Now, before we proceed, we will redefine a mechanism in a more formal way so that it will be easier to reason about:

Definition 1 A mechanism ${\mathcal{M}}$ is a function that associated for each ${v \in {\mathbb R}^N}$ a distribution over elements of ${\mathcal{X}}$.

Theorem 2 Let ${x_i(v) = \sum_{i \in S \in \mathcal{X}} Pr_{\mathcal{M}(v)}[S]}$ be the probability that ${i}$ is allocated by the mechanism given ${v}$ is reported. The mechanism is truthful iff ${x_i(v)}$ is monotone and each allocated bidder is charged payment:

$\displaystyle P_i = v_i - \frac{1}{x_i(v_i, v_{-i})} \int_0^{v_i} x_i(w, v_{-i}) dw$

This is a classical theorem by Myerson about the characterization of truthful auctions. It is not hard to see that the auction define above is truthful. We just need to check that ${x_i(v_i, v_{-i}) (v_i - P(v_i, v_{-i})) \geq x_i(v'_i, v_{-i}) (v_i - P(v'_i, v_{-i}))}$ for all ${v'_i}$. The opposite is trickier but is also not hard to see.

Note that this characterization implies the following characterization of deterministic truthful auctions, i.e., auctions that map each ${v \in {\mathbb R}^N}$ to a set ${S \in \mathcal{X}}$, i.e., the probability distribution is concentrated in one set.

Theorem 3 A mechanism is a truthful deterministic auction iff there is a functions ${f_i(b_{-i})}$ such that for each we allocate to bidder ${i}$ iff ${b_i \geq f_i(b_{-i})}$ and in case it is allocated, we charge payment ${f_i(b_{-i})}$.

It is actually easy to generate this function. Given a mechanism, ${b_i \mapsto x_i(b_i, b_{-i})}$ is a monotone and is a ${\{0,1\}}$-function. Let ${f_i(b_{-i})}$ the point where it transitions from ${0}$ to ${1}$. Now, we can give a similar characterization for Universal Truthful Mechanism:

Theorem 4 A mechanism is a universal truthful randomized auction if there are functions ${f_i(r,b_{-i})}$ such that for each we allocate to bidder ${i}$ iff ${b_i \geq f_i(r,b_{-i})}$ and in case it is allocated, we charge payment ${f_i(r,b_{-i})}$, where ${r}$ are random bits.

2. Profit benchmarks

Let’s consider a Digital Goods auction, where ${\mathcal{X} = 2^N}$. Two natural goals for profit extraction would be ${\mathcal{T}(v) = \sum_i v_i}$ and ${\mathcal{F}(v) = \max_i i v_i}$ where we can think of ${v_1 \geq v_2 \geq \hdots \geq v_n}$, the first is the best profit you can extract charging different prices and the second is the best profit you can hope to extract by charging a fixed price. Unfortunately it is impossible to design a mechanism that even ${O(1)}$-approximates both benchmarks on every input. The intuition is that ${v_1}$ can be much larger then the rest, so there is no way of setting ${f_1(b_{-1})}$ in a proper way. Under the assumption that the first value is not much larger than the second, we can do a good profit approximation, though. This motivates us to find an universal truthful mechanism that approximates the following profit benchmark:

$\displaystyle \mathcal{F}^{(2)}(v) = \max_{i \geq 2} i v_i$

which is the highest single-price profit we can get selling to at least ${2}$ agents. We will show a truthful mechanism that ${4}$-approximates this benchmark.

3. Profit Extractors

Profit extractor are building blocks of many mechanisms. The goal of a profit extractor is, given a constant target profit ${C}$, extract that profit from a set of agents if that is possible. In this first moment, let’s see ${C}$ as an exogenous constant. Consider the following mechanism called CostShare${_C (v)}$: find the largest ${k}$ s.t. ${k \cdot v_k \geq C}$. Then allocate to

Lemma 5 CostShare${_C}$ is a truthful profit-extractor that can extract profit ${C}$ whenever ${\mathcal{F}(v) = \max_i i v_i \geq C}$.

Proof: It is clear that it can extract profit at most ${C}$ if ${\mathcal{F}(v) \geq C}$. We just need to prove it is a truthful mechanism and this can be done by checking the characterization of truthful mechanisms. Suppose that under CostShare${_C (v)}$ exacly ${k}$ bidders are getting the item, then let’s look at a bidder ${i}$. If bidder ${i}$ is not getting the item, then his value is smaller than ${C/k}$, otherwise we could incluse all bidders up to ${i}$ and sell for a price ${C/k_1}$ for some ${k_1 > k}$. It is easy to see that bidder ${i}$ will get the item just if he changes his value ${v_i}$ to some value greater or equal than ${C/k}$.

On the other hand, it ${i}$ is currently getting the item under ${v}$, then increasing his value won’t make it change. It is also clear that for any value ${v_i \geq C/k}$, he will still get the item. For ${v_i < C/k}$ he doesn’t get it. Suppose it got, then at least ${k+1}$ people get the item, because the price they sell it to ${i}$ must be less than ${v_i < C/k}$. Thefore, increasing ${v_i}$ back to its original value, we could still sell it to ${k+1}$ players, what is a contradiction, since we assumed we were selling to ${k}$ players.

We checked monotonicity and we also need to check the payments, but it is straightforward to check they satisfy the second condition, since ${x_i(v_i) = 1}$ for ${v_i \geq C/k}$ and zero instead. $\Box$

4. Random Sampling Auctions

Now, using that profit extractor as a building block, the main idea is to estimate ${C}$ smaller than ${\mathcal{F}(v)}$ for one subset of the agents and extract that profit from them using a profit extractor. First we partition ${N}$ is two sets ${N'}$ and ${N''}$ tossing a coin for each agent to decide in which set we will place it, then we calculate ${\mathcal{F}' = \mathcal{F}(v_{N'})}$ and ${\mathcal{F}'' = \mathcal{F}(v_{N''})}$. Now, we run CostShare${_{\mathcal{F}'} (v'')}$ and CostShare${_{\mathcal{F}''} (v')}$. This is called Random Cost Sharing Auction.

Theorem 6 The Random Cost Sharing Auction is a truthful auction whose revenue ${4}$-approximates the benchmark ${\mathcal{F}^{(2)}(v)}$.

Proof: Let ${\mathcal{R}}$ be a random variable associated with the revenue of the Sampling Auction mechanism. It is clear that ${\mathcal{R} = \min \{ \mathcal{F}', \mathcal{F}'' \}}$. Let’s write ${\mathcal{F}^{(2)}(v) = kp}$ meaning that we sell ${k}$ items at price ${p}$. Let ${k = k' + k''}$ where ${k'}$ and ${k''}$ are the items among those ${k}$ items that went to ${N'}$ and ${N''}$ respectively. Then, clearly ${\mathcal{F}' \geq p k'}$ and ${\mathcal{F}'' \geq p k''}$, what gives us:

$\displaystyle \frac{\mathcal{R}}{\mathcal{F}^{(2)}} = \frac{\min\{\mathcal{F}', \mathcal{F}''\}}{\mathcal{F}^{(2)}} \geq \frac{\min\{k'p, k''p\}}{kp} = \frac{\min\{k', k''\}}{k}$

and from there, it is a straighforward probability exercise:

$\displaystyle \frac{\mathop{\mathbb E}[{\mathcal{R}}]}{\mathcal{F}^{(2)}} = \mathop{\mathbb E}\left[{ \frac{\min\{k', k''\}}{k} }\right] = \frac{1}{k} \sum_{i=1}^{k-1} \min\{ i, k-i \} \begin{pmatrix} k \\ i \end{pmatrix} 2^{-k}$

since:

$\displaystyle \frac{k}{2} = \sum_{i = 0}^k i \begin{pmatrix} k \\ i \end{pmatrix} 2^{-k} \leq \frac{k}{4} + 2 \sum_{i = 0}^{k/2} i \begin{pmatrix} k \\ i \end{pmatrix} 2^{-k}$

and therefore:

$\displaystyle \frac{\mathop{\mathbb E}[{\mathcal{R}}]}{\mathcal{F}^{(2)}} \geq \frac{1}{4}$ $\Box$

This similar approximations can be extended to more general environments with very little change. For example, for multi-unit auctions, where ${\mathcal{X} = \{ S; \vert S \vert \leq k \}}$ we use the benchmark ${\mathcal{F}^{(2,k)} = \max_{2 \leq i \leq k} i v_i}$ and we can be ${O(1)}$-competitive against it, by random-sampling, evaluating ${\mathcal{F}^{(1,k)} = \max_{\leq i \leq k} i v_i}$ on both sets and running a profit extractor on both. The profit extractor is a simple generalization of the previous one.

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