Guest post by Bart de Keijzer.

This blog post is about a result in optimal stopping theory that relates to matroids, and an application of this result to mechanism design (a field of game theory).

The Classical Prophet Inequality

In the theory of optimal stopping, we deal with problems where a gambler observes a sequence of unknown and possibly random values, and has to decide when to take a particular action in order to optimize some objective value.  In a previous post, we have already seen the secretary problem, which is an example of a problem studied in optimal stopping theory. In that problem, we face a finite sequence of $n$ numbers, and each time a new number arrives we have to decide irrevocably whether we want to choose that number or not.

I will discuss in this post a slightly different problem from optimal stopping theory, where the permutation is not random, but the numbers are. We are given a sequence $X_1, \ldots, X_n$ of independent non-negative random variables, and the gambler observes this sequence. When he observes a number, he has to make an irrevocable decision about whether or not to stop. If he decides to stop, he receives a payoff equal to the number on which he chose to stop. If he decides not to stop, then the gambler simply moves on to the next number in the sequence. The classical prophet inequality of Krengel, Sucheston, and Garling [4] tells us that the gambler can get a payoff that is half of the expected maximum.

Theorem 1.  If $X_1, \ldots, X_n$ are independent non-negative distributions, then there exists a stopping rule $R$ such that $\mathbf{E}[R(X_1,\ldots,X_n)] \geq \mathbf{E}[\max\{X_i : 1 \leq i \leq n\}]/2$.

The stopping rule $R$ that achieves this particular guarantee is simple: stop at the first value that is at least half of the expected maximum. The name prophet inequality is because it shows that a gambler can achieve a payoff that is at least half of the value that would be picked by a prophet who is completely foresighted. Many variations and generalizations on this result have appeared in the literature. See [2] for an (old) survey.

Sequential Posted Price Mechanisms

We now move to a related scenario that is studied in mechanism design. Suppose that multiple identical products are being sold to a set of $n$ interested buyers. Each buyer is interested in obtaining at most one copy of the item, and each buyer has a valuation for the item that is for sale. The valuation of a buyer $i$ is expressed in terms of a real number which we denote by $v_i$, and is drawn from an independent probability distribution $D_i$. The valuations are privately held by the buyers and the auctioneer does not know them. However, the auctioneer does know the distributions $D_1, \ldots, D_n$.

The auctioneer wants to sell the item through a very simple process called a sequential posted price (SPP) mechanism. This means that the auctioneer has to make take-it-or-leave it offers to the buyers one by one. When a buyer $i$ receives an offer of $p_i$ he can choose to accept it or reject it. If he accepts, the buyer pays price $p_i$ and gets an item. The utility of buyer $i$ is then $v_i – p_i$. If the buyer rejects, he pays nothing, gets no item, and the auctioneer proceeds to the next buyer. The auctioneer makes at most one offer to each buyer and the buyers are assumed to optimize their utility, so that a buyer will accept an offer if and only if his valuation exceeds the price of the offer.

SPP mechanisms are studied because they are an abstraction of a sales mechanism that is frequently encountered in the real world. The popularity of posted price mechanisms is due to its simplicity, its transparency toward the participating buyers, and the efficiency with which it can be implemented.

The question that we are interested in is how well we can approximate the optimal social welfare by means of an SPP mechanism. The social welfare of an allocation is defined as the sum of valuations of all the buyers who get an item. In case there is only one unit for sale, this setting is very similar to the prophet inequality setting, and in fact we can use the prophet inequality to obtain a good approximation to the social welfare. The optimum social welfare is $\mathbf{E}[\max\{v_i : 1 \leq i \leq n\}]$, and the prophet inequality tells us that the auctioneer is able to achieve half of that by offering every buyer a price of $\mathbf{E}[\max\{v_i : 1 \leq i \leq n\}]/2$.

If there is an infinite supply or multiple copies of the item for sale, the original prophet inequality cannot be applied anymore.  Having multiple copies or infinite supply essentially means that we have a different allocation constraint on the set of buyers that we may allocate an item. In these cases the family of sets of buyers that we may allocate is a uniform matroid. Let us be even more ambitious, and assume that we have an arbitrary matroid constraint on the set of buyers that may get an item.

• Chawla, Hartline, Malec and Sivan [1] prove that in this case there still exists an SPP mechanism $M$ that achieves a $1/2$-approximation to the optimal social welfare. The mechanism they propose assumes that the order in which to make offers to the buyers can be controlled. The way in which $M$ works is as follows.
• For every bidder $i$ let $r_i$ be the probability that buyer $i$ gets an item in the social-welfare-optimizing allocation. (Note that this allocation is random, because the valuations are random: for every vector of valuations of the buyers there is a (possibly distinct) optimal allocation.) For buyer $i$, set $p_i$ to be the price such that $\mathbf{Pr}[v_i \geq p_i] = r_i$.
• Iteratively offer the next buyer $i$ a take-it-or-leave it price $p_i$. Make the offers in order of non-increasing $p_i$, and only make $i$ an offer in case allocating them an item will yield an independent set in the matroid.

The following can then be proved.

Theorem 2.  The expected social welfare of $M$ is at least half of the expected optimum social felware.

The above can be seen as a prophet-inequality-type result for a setting where the gambler/auctioneer can select an independent set in a given matroid, instead of only a single item. However, the result requires that the gambler can choose the order of the random variables in advance.

Matroid Prophet Inequalities

Kleinberg and Weinberg [3] strengthen the above result by proving that it is not needed to have control over the order in which to offer the prices to the buyers.
This yields a true generalization of the original prophet inequality and strengthens the above result on SPP mechanisms.

Theorem 3.  Suppose that $X_1, \ldots, X_n$ is a sequence of nonnegative real values, arriving one by one, drawn from independent and non-negative distributions. If there is a matroid constraint $\mathcal{M}$ over the index set of values that may be selected, then there exists a selection rule such that
$\mathbf{E}[\text{ Total value of the selected elements }] \geq \mathbf{E}[\max\{\sum_{i \in I} X_i : I \in \mathcal{M}\}]/2$.

The selection rule that achieves this sets a specific threshold value for each number that arrives, and selects the arriving element if and only if its value exceeds the threshold and the element can be added without violating the matroid constraint. The threshold at iteration $i$ may depend on the numbers that have arrived in previous iterations. The authors prove the existence of the desired threshold values in three steps.

• They first define the notion of $\alpha$-balancedness, which is a property that a threshold-based selection rule may posess. Let $T_1, \ldots, T_n$ be a set of thresholds, where $T_i$ is the threshold value that is used in the $i$th iteration. Let $A$ be the set of elements that would be selected when using these thresholds, and let $B$ be the random set that a prophet selects (i.e., a basis of the matroid that maximizes the total value after observing the values). Let $R(A)$ be the (also random) subset of $B \subset A$ of maximum total value such that $A \cup R(A)$ is a basis. The threshold rule is $\alpha$-balanced iff: (i.) For every realization of $T_1, \ldots, T_n$, the total threshold corresponding to the values in $A$ is at least $1/\alpha$ times the expected total value of the elements in $B \setminus R(A)$. Informally this means that the value of the set of elements that a foresighted prophet selects instead of $A$ is within a $1/\alpha$ factor of the set selected by the threshold selection rule.
  (ii.) For every realization of $T_1, \ldots, T_n$, let $V$ be any set disjoint from $A$ such that $A \cup V$ is a matroid basis. Then the total threshold corresponding to the values in $V$ must be at most $(1 – 1/\alpha)$ times the expected total value of $R(A)$. Informally, this means that the total weight of the elements rejected by the threshold rule is small (not more than a $(1 – 1/\alpha)$ times the value of the set $R(A)$ that a prophet would add in expectation).
• They then prove that using $\alpha$-balanced thresholds yields a selection rule that achieves an expected value within $1/\alpha$ times the expected optimum value.
• Finally they prove that there exists $2$-balanced thresholds. The threshold for the $i$th value $T_i$ is set to $\frac{1}{2}\mathbf{E}[w(R(A_{i-1})) – w(R(A_{i-1} \cup \{i\}))]$. Note that these thresholds are actually adaptive: the way $T_i$ is set depends on the set that was selected among the first $i-1$ elements.

References

[1] Shuchi Chawla, Jason D. Hartline, David L. Malec, and Balasubramanian Sivan. Multi-parameter mechanism design and sequential posted pricing. In Proceedings of the Forty-second ACM Symposium on Theory of Computing (STOC), pages 311–320. ACM, 2010.

[2] Theodore P. Hill and Robert P. Kertz. A survey of prophet inequalities in optimal stopping theory. Contemporary Mathematics, 125, 1992.

[3] Robert Kleinberg and Seth Matthew Weinberg. Matroid prophet inequalities. In Proceedings of the Forty-fourth ACM Symposium on Theory of Computing (STOC), pages 123–136. ACM.

[4] Ulrich Krengel and Louis Sucheston. On semiamarts, amarts, and processes with finite value. Advances in Probability Related Topics.