Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The …
Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen.
The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint.
In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems.
Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The …
Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen. The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint. In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems.
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent …
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of $p$ matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most $O(p)$, and this factor is also tight.
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent …
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a prophet who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most O(p), and this factor is also tight.
Beyond their interest as theorems about pure online algorithms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings.
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent …
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most O(p), and this factor is also tight. Beyond their interest as theorems about pure online algorithms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings.
Free order inequalities bound the ratio between the expected value obtained by two parties each selecting a value from a set of independent random variables: a prophet who knows the …
Free order inequalities bound the ratio between the expected value obtained by two parties each selecting a value from a set of independent random variables: a prophet who knows the value of each variable and may select the maximum one, and a who is free to choose the order in which to observe the values but must select one of them immediately after observing it, without knowing what values will be sampled for the unobserved variables. It is known that the gambler can always ensure an expected payoff at least $0.669\dots$ times as great as that of the prophet. In fact, there exists a threshold stopping rule which guarantees a gambler-to-prophet ratio of at least $1-\frac1e=0.632\dots$. In contrast, if the gambler must observe the values in a predetermined order, the tight bound for the gambler-to-prophet ratio is $1/2$.
In this work we investigate a model that interpolates between these two extremes. We assume there is a predefined set of permutations, and the gambler is free to choose the order of observation to be any one of these predefined permutations. Surprisingly, we show that even when only two orderings are allowed---namely, the forward and reverse orderings---the gambler-to-prophet ratio improves to $\varphi^{-1}=0.618\dots$, the inverse of the golden ratio. As the number of allowed permutations grows beyond 2, a striking double plateau phenomenon emerges: after increasing from $0.5$ to $\varphi^{-1}$, the gambler-to-prophet ratio achievable by threshold stopping rules does not exceed $\varphi^{-1}+o(1)$ until the number of allowed permutations grows to $O(\log n)$. The ratio reaches $1-\frac1e-\varepsilon$ for a suitably chosen set of $O(\text{poly}(\varepsilon^{-1})\cdot\log n)$ permutations and does not exceed $1-\frac1e$ even when the full set of $n!$ permutations is allowed.
Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting a value from a set of independent random variables: a "prophet" who knows …
Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting a value from a set of independent random variables: a "prophet" who knows the value of each variable and may select the maximum one, and a "gambler" who is free to choose the order in which to observe the values but must select one of them immediately after observing it, without knowing what values will be sampled for the unobserved variables. It is known that the gambler can always ensure an expected payoff at least $0.669\dots$ times as great as that of the prophet. In fact, there exists a threshold stopping rule which guarantees a gambler-to-prophet ratio of at least $1-\frac1e=0.632\dots$. In contrast, if the gambler must observe the values in a predetermined order, the tight bound for the gambler-to-prophet ratio is $1/2$. In this work we investigate a model that interpolates between these two extremes. We assume there is a predefined set of permutations, and the gambler is free to choose the order of observation to be any one of these predefined permutations. Surprisingly, we show that even when only two orderings are allowed---namely, the forward and reverse orderings---the gambler-to-prophet ratio improves to $\varphi^{-1}=0.618\dots$, the inverse of the golden ratio. As the number of allowed permutations grows beyond 2, a striking "double plateau" phenomenon emerges: after increasing from $0.5$ to $\varphi^{-1}$, the gambler-to-prophet ratio achievable by threshold stopping rules does not exceed $\varphi^{-1}+o(1)$ until the number of allowed permutations grows to $O(\log n)$. The ratio reaches $1-\frac1e-\varepsilon$ for a suitably chosen set of $O(\text{poly}(\varepsilon^{-1})\cdot\log n)$ permutations and does not exceed $1-\frac1e$ even when the full set of $n!$ permutations is allowed.
Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting one value from a set of independent random variables: a "prophet" who knows …
Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting one value from a set of independent random variables: a "prophet" who knows the value of each variable and may select the maximum one, and a "gambler" who is free to choose the order in which to observe the values but must select one of them immediately after observing it, without knowing what values will be sampled for the unobserved variables. It is known that the gambler can always ensure an expected payoff at least 0.669 … times as great as that of the prophet. In fact, even if the gambler uses a threshold stopping rule, meaning there is a fixed threshold value such that the gambler rejects every sample below the threshold and accepts every sample above it, the threshold can always be chosen so that the gambler-to-prophet ratio is at least . … In contrast, if the gambler must observe the values in a predetermined order, the tight bound for the gambler-to-prophet ratio is 1/2.In this work we investigate a model that interpolates between these two extremes. We assume there is a predefined set of permutations of the set indexing the random variables, and the gambler is free to choose the order of observation to be any one of these predefined permutations. Surprisingly, we show that even when only two orderings are allowed — namely, the forward and reverse orderings — the gambler-to-prophet ratio improves to …, the inverse of the golden ratio. As the number of allowed permutations grows beyond 2, a striking "double plateau" phenomenon emerges: after increasing from 0.5 to φ–1 when two permutations are allowed, the gambler-to-prophet ratio achievable by threshold stopping rules does not exceed φ–1 + o(1) until the number of allowed permutations grows to O(log n). The ratio reaches for a suitably chosen set of O(poly(∊–1) · log n) permutations and does not exceed even when the full set of n! permutations is allowed.
We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an $O(1)$-competitive algorithm. For …
We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an $O(1)$-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward-closed feasibility constraint, we give an $O(\log n \log^2 r)$-competitive algorithm (where $r$ is the cardinality of the largest feasible subset). Inspired by the proof of our subadditive prophet inequality, we also obtain an $O(\log n \cdot \log^2 r)$-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an $\Omega(\sqrt{n})$ lower bound by Bateni, Hajiaghayi, and Zadimoghaddam \cite{BHZ13-submodular-secretary_original} in a restricted query model. En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma for non-monotone submodular functions.
We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For …
We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward- closed feasibility constraint, we give an O(log n log2 r)- competitive algorithm (where r is the cardinality of the largest feasible subset).Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log2 r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model.En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for nonmonotone submodular functions.
In the classical prophet inequality settings, a gambler is given a sequence of $n$ random variables $X_1, \dots, X_n$, taken from known distributions, observes their values in this (potentially adversarial) …
In the classical prophet inequality settings, a gambler is given a sequence of $n$ random variables $X_1, \dots, X_n$, taken from known distributions, observes their values in this (potentially adversarial) order, and select one of them, immediately after it is being observed, so that its value is as high as possible. The classical \emph{prophet inequality} shows a strategy that guarantees a value at least half of that an omniscience prophet that picks the maximum, and this ratio is optimal. Here, we generalize the prophet inequality, allowing the gambler some additional information about the future that is otherwise privy only to the prophet. Specifically, at any point in the process, the gambler is allowed to query an oracle $\mathcal{O}$. The oracle responds with a single bit answer: YES if the current realization is greater than the remaining realizations, and NO otherwise. We show that the oracle model with $m$ oracle calls is equivalent to the \textsc{Top-$1$-of-$(m+1)$} model when the objective is maximizing the probability of selecting the maximum. This equivalence fails to hold when the objective is maximizing the competitive ratio, but we still show that any algorithm for the oracle model implies an equivalent competitive ratio for the \textsc{Top-$1$-of-$(m+1)$} model. We resolve the oracle model for any $m$, giving tight lower and upper bound on the best possible competitive ratio compared to an almighty adversary. As a consequence, we provide new results as well as improvements on known results for the \textsc{Top-$1$-of-$m$} model.
Prophet inequalities are fundamental optimal stopping problems, where a decision-maker observes sequentially items with values sampled independently from known distributions, and must decide at each new observation to either stop …
Prophet inequalities are fundamental optimal stopping problems, where a decision-maker observes sequentially items with values sampled independently from known distributions, and must decide at each new observation to either stop and gain the current value or reject it irrevocably and move to the next step. This model is often too pessimistic and does not adequately represent real-world online selection processes. Potentially, rejected items can be revisited and a fraction of their value can be recovered. To analyze this problem, we consider general decay functions $D_1,D_2,\ldots$, quantifying the value to be recovered from a rejected item, depending on how far it has been observed in the past. We analyze how lookback improves, or not, the competitive ratio in prophet inequalities in different order models. We show that, under mild monotonicity assumptions on the decay functions, the problem can be reduced to the case where all the decay functions are equal to the same function $x \mapsto \gamma x$, where $\gamma = \inf_{x>0} \inf_{j \geq 1} D_j(x)/x$. Consequently, we focus on this setting and refine the analyses of the competitive ratios, with upper and lower bounds expressed as increasing functions of $\gamma$.
The prophet inequalities problem has received significant study over the past decades and has several applications such as to online auctions. In this paper, we study two variants of the …
The prophet inequalities problem has received significant study over the past decades and has several applications such as to online auctions. In this paper, we study two variants of the i.i.d. prophet inequalities problem, namely the windowed prophet inequalities problem and the batched prophet inequalities problem. For the windowed prophet inequalities problem, we show that for window size $o(n)$, the optimal competitive ratio is $\alpha \approx 0.745$, the same as in the non-windowed case. In the case where the window size is $n/k$ for some constant $k$, we show that $\alpha_k < WIN_{n/k} \le \alpha_k + o_k(1)$ where $WIN_{n/k}$ is the optimal competitive ratio for the window size $n/k$ prophet inequalities problem and $\alpha_k$ is the optimal competitive ratio for the $k$ sample i.i.d. prophet inequalities problem. Finally, we prove an equivalence between the batched prophet inequalities problem and the i.i.d. prophet inequalities problem.
Numerous recent papers have studied the tension between thickening and clearing a market in (uncertain, online) long-time horizon Markovian settings. In particular, (Aouad and Saritaç EC'20, Collina et al. WINE'20, …
Numerous recent papers have studied the tension between thickening and clearing a market in (uncertain, online) long-time horizon Markovian settings. In particular, (Aouad and Saritaç EC'20, Collina et al. WINE'20, Kessel et al. EC'22) studied what the latter referred to as the Stationary Prophet Inequality Problem, due to its similarity to the classic finite-time horizon prophet inequality problem. These works all consider unit-demand buyers. Mirroring the long line of work on the classic prophet inequality problem subject to combinatorial constraints, we initiate the study of the stationary prophet inequality problem subject to combinatorially-constrained buyers.
Numerous recent papers have studied the tension between thickening and clearing a market in (uncertain, online) long-time horizon Markovian settings. In particular, (Aouad and Sarita{\c{c}} EC'20, Collina et al. WINE'20, …
Numerous recent papers have studied the tension between thickening and clearing a market in (uncertain, online) long-time horizon Markovian settings. In particular, (Aouad and Sarita{\c{c}} EC'20, Collina et al. WINE'20, Kessel et al. EC'22) studied what the latter referred to as the Stationary Prophet Inequality Problem, due to its similarity to the classic finite-time horizon prophet inequality problem. These works all consider unit-demand buyers. Mirroring the long line of work on the classic prophet inequality problem subject to combinatorial constraints, we initiate the study of the stationary prophet inequality problem subject to combinatorially-constrained buyers. Our results can be summarized succinctly as unearthing an algorithmic connection between contention resolution schemes (CRS) and stationary prophet inequalities. While the classic prophet inequality problem has a tight connection to online CRS (Feldman et al. SODA'16, Lee and Singla ESA'18), we show that for the stationary prophet inequality problem, offline CRS play a similarly central role. We show that, up to small constant factors, the best (ex-ante) competitive ratio achievable for the combinatorial prophet inequality equals the best possible balancedness achievable by offline CRS for the same combinatorial constraints.
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Prophet Inequalities with Limited InformationPablo D. Azar, Robert Kleinberg, and S. Matthew WeinbergPablo D. …
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Prophet Inequalities with Limited InformationPablo D. Azar, Robert Kleinberg, and S. Matthew WeinbergPablo D. Azar, Robert Kleinberg, and S. Matthew Weinbergpp.1358 - 1377Chapter DOI:https://doi.org/10.1137/1.9781611973402.100PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V1, …, Vn and must decide, for each reward Vi, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value Vi. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose k items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards V1, …, Vn are drawn. The assumption that the gambler knows the distribution from which V1, …, Vn are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, even with full knowledge of the distribution. Specifically, we provide a novel single-sample algorithm when the gambler can choose any k elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related secretary problem to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. In addition, we apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders. Connections between prophet inequalities and posted-price mechanisms are already known, but applying the existing framework requires knowledge of the underlying distributions, as well as the so-called "virtual values" even when the underlying prophet inequalities do not. We therefore provide an extension of this framework that bypasses virtual values altogether, allowing our mechanisms to take full advantage of the limited information required by our new prophet inequalities. Previous chapter Next chapter RelatedDetails Published:2014ISBN:978-1-61197-338-9eISBN:978-1-61197-340-2 https://doi.org/10.1137/1.9781611973402Book Series Name:ProceedingsBook Code:PRDA14Book Pages:viii + 1885
In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or …
In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value $V_i$. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose $k$ items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards $V_1,...,V_n$ are drawn. The assumption that the gambler knows the distribution from which $V_1,...,V_n$ are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, \emph{even with full knowledge of the distribution}. Specifically, we provide a novel single-sample algorithm when the gambler can choose any $k$ elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related \emph{secretary problem} to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. We apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders.
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive …
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive policies are achievable using only a single sample from the distribution of each agent. A notable portion of their results relies on reducing the design of single-sample prophet inequalities (SSPIs) to that of order-oblivious secretary (OOS) policies. The above reduction comes at the cost of not fully utilizing the available samples. However, to date, this is essentially the only method for proving SSPIs for many combinatorial sets. Very recently, Rubinstein et al. [ITCS'20] give a surprisingly simple algorithm which achieves the optimal competitive ratio for the single-choice SSPI problem $-$ a result which is unobtainable going through the reduction to secretary problems. Motivated by this discrepancy, we study the competitiveness of simple SSPI policies directly, without appealing to results from OOS literature. In this direction, we first develop a framework for analyzing policies against a greedy-like prophet solution. Using this framework, we obtain the first SSPI for general (non-bipartite) matching environments, as well as improved competitive ratios for transversal and truncated partition matroids. Second, motivated by the observation that many OOS policies for matroids decompose the problem into independent rank-$1$ instances, we provide a meta-theorem which applies to any matroid satisfying this partition property. Leveraging the recent results by Rubinstein et al., we obtain improved competitive guarantees (most by a factor of $2$) for a number of matroids captured by the reduction of Azar et al. Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.
In this paper, we introduce an over-time variant of the well-known prophet inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, …
In this paper, we introduce an over-time variant of the well-known prophet inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, we decide for each step how long we select the value. Then we cannot select another value until this period is over. The goal is to maximize the expectation of the sum of selected values. We describe the structure of the optimal stopping rule and give upper and lower bounds on the prophet inequality. In online algorithms terminology, this corresponds to bounds on the competitive ratio of an online algorithm.
We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, …
We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, this algorithm is adaptive. Other approaches of [CHMS10] and [FSZ16] have used non-adaptive thresholds with a feasibility restriction; however, this translates to adaptively changing an item's threshold to infinity when it cannot be taken with respect to the additional feasibility constraint, hence the algorithm is not truly non-adaptive. A major application of prophet inequalities is in auction design, where non-adaptive prices possess a significant advantage: they convert to order-oblivious posted pricings, and are essential for translating a prophet inequality into a truthful mechanism for multi-dimensional buyers. The existing matroid prophet inequalities do not suffice for this application. We present the first non-adaptive constant-factor prophet inequality for graphic matroids.
Efficient and truthful mechanisms to price resources on servers/machines have been the subject of much work in recent years due to the importance of the cloud market. This paper considers …
Efficient and truthful mechanisms to price resources on servers/machines have been the subject of much work in recent years due to the importance of the cloud market. This paper considers revenue maximization in the online stochastic setting with non-preemptive jobs and a unit capacity server. One agent/job arrives at every time step, with parameters drawn from the underlying distribution. We design a posted-price mechanism which can be efficiently computed and is revenue-optimal in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic, depending only on the length of the allotted time interval and on the earliest time the server is available. We also prove that the proposed pricing strategy is robust to imprecise knowledge of the job distribution and that a distribution learned from polynomially many samples is sufficient to obtain a near-optimal truthful pricing strategy.
We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both …
We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both problems. This is close to the Omega(n/log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log(n) log(r))-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(loglog(n) log(r)) factor) an open question posed to us by Bobby Kleinberg.
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is …
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. To study these settings, we introduce a novel unified framework of batched prophet inequalities that captures online settings where elements arrive in batches; in particular it captures matching under the two aforementioned arrival models. Our algorithms rely on the construction of suitable online contention resolution schemes (OCRS). We first extend the framework of OCRS to batched-OCRS, we then establish a reduction from batched prophet inequality to batched OCRS, and finally we construct batched OCRSs with selectable ratios of 0.337 and 0.5 for edge and vertex arrival models, respectively. Both results improve the state of the art for the corresponding settings. For vertex arrival, our result is tight. Interestingly, pricing-based prophet inequalities with comparable competitive ratios are unknown.
We consider the problem of maximizing the expected revenue from selling $k$ homogeneous goods to $n$ unit-demand buyers who arrive sequentially with independent and identically distributed valuations. In this setting …
We consider the problem of maximizing the expected revenue from selling $k$ homogeneous goods to $n$ unit-demand buyers who arrive sequentially with independent and identically distributed valuations. In this setting the optimal posted prices are dynamic in the sense that they depend on the remaining numbers of goods and buyers. We investigate how much revenue is lost when a single static price is used instead for all buyers and goods, and prove upper bounds on the ratio between the maximum revenue from dynamic prices and that from static prices. These bounds are tight for all values of $k$ and $n$ and vary depending on a regularity property of the underlying distribution. For general distributions we obtain a ratio of $2-k/n$, for regular distributions a ratio that increases in $n$ and is bounded from above by $1/(1-k^k/(e^{k}k!))$, which is roughly $1/(1-1/(\sqrt{2πk}))$. The lower bounds hold for the revenue gap between dynamic and static prices. The upper bounds are obtained via an ex-ante relaxation of the revenue maximization problem, as a consequence the tight bounds of $2-k/n$ in the general case and of $1/(1-1/(\sqrt{2πk}))$ in the regular case apply also to the potentially larger revenue gap between the optimal incentive-compatible mechanism and the optimal static price. Our results imply that for regular distributions the benefit of dynamic prices vanishes while for non-regular distributions dynamic prices may achieve up to twice the revenue of static prices.
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case …
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here n agents with subadditive valuation functions compete for the assignment of m items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of O(log m). We make major progress on this question by providing an O(log log m) prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an O(log log m) approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
We consider prophet inequalities in a setting where agents correspond to both elements in a matroid and vertices in a graph. A set of agents is feasible if they form …
We consider prophet inequalities in a setting where agents correspond to both elements in a matroid and vertices in a graph. A set of agents is feasible if they form both an independent set in the matroid and an independent set in the graph. Our main result is an ex-ante 1/(2d+2)-prophet inequality, where d is a graph parameter upper-bounded by the maximum size of an independent set in the neighborhood of any vertex. We establish this result through a framework that sets both dynamic prices for elements in the matroid (using the method of balanced thresholds), and static but discriminatory prices for vertices in the graph (motivated by recent developments in approximate dynamic programming). The threshold for accepting an agent is then the sum of these two prices. We show that for graphs induced by a certain family of interval-scheduling constraints, the value of d is 1. Our framework thus provides the first constant-factor prophet inequality when there are both matroid-independence constraints and interval-scheduling constraints. It also unifies and improves several results from the literature, leading to a 1/2-prophet inequality when agents have XOS valuation functions over a set of items and use them for a finite interval duration, and more generally, a 1/(d+1)-prophet inequality when these items each require a bundle of d resources to procure.
Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 - 1/e on the approximation factor of their …
Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 - 1/e on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is 1/1+1/e ≃ 0.731. This conjecture remained open prior to this paper for over 30 years. In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of 1/1+1/e, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-uniform distributions and discuss its applications in mechanism design.
We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of …
We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of items, and the aim of a mechanism is to improve the social welfare by arranging purchases and sales of the items. A mechanism is given prior distributions on the agents’ valuations of the items, but not the actual valuations; thus, the aim is to maximise the expected social welfare over these distributions. As in previous work, we are interested in the worst-case ratio between the social welfare achieved by a truthful mechanism and the best social welfare possible. Our main result is an incentive compatible and budget balanced constant-factor approximation mechanism in a setting where buyers have XOS valuations and sellers’ valuations are additive. This is the first such approximation mechanism for a two-sided market setting where the agents have combinatorial valuation functions. To achieve this result, we introduce a more general kind of demand query that seems to be needed in this situation. In the simpler case that sellers have unit supply (each having just one item to sell), we give a new mechanism whose welfare guarantee improves on a recent one in the literature. We also introduce a more demanding version of the strong budget balance (SBB) criterion, aimed at ruling out certain “unnatural” transactions satisfied by SBB. We show that the stronger version is satisfied by our mechanisms.
The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items …
The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item.
Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.
The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset …
The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives.
We survey the main results from [Dütting, Kesselheim, and Lucier 2020]: 1 a simple posted-price mechanism for subadditive combinatorial auctions with m items that achieves an O (log log m …
We survey the main results from [Dütting, Kesselheim, and Lucier 2020]: 1 a simple posted-price mechanism for subadditive combinatorial auctions with m items that achieves an O (log log m ) approximation to the optimal welfare, plus a variant with entry fees that approximates revenue. These are based on a novel subadditive prophet inequality.
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is …
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. To study these settings, we introduce a novel unified framework of batched prophet inequalities that captures online settings where elements arrive in batches; in particular it captures matching under the two aforementioned arrival models. Our algorithms rely on the construction of suitable online contention resolution scheme (OCRS). We first extend the framework of OCRS to batched-OCRS, we then establish a reduction from batched prophet inequality to batched OCRS, and finally we construct batched OCRSs with selectable ratios of 0.337 and 0.5 for edge and vertex arrival models, respectively. Both results improve the state of the art for the corresponding settings. For the vertex arrival, our result is tight. Interestingly, a pricing-based prophet inequality with comparable competitive ratios is unknown.
A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and …
A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and selects one of them is at least an α fraction of the maximum value in the sequence. We obtain three distinct improvements for a setting that was first studied by Correa et al. (EC, 2019) and is particularly relevant to modern applications in algorithmic pricing. In this setting, the random variables are i.i.d. from an unknown distribution and the gambler has access to an additional βn samples for some β ≥ 0. We first give improved lower bounds on α for a wide range of values of β; specifically, α ≥ (1 + β)/e when β ≤ 1/(e − 1), which is tight, and α ≥ 0.648 when β = 1, which improves on a bound of around 0.635 due to Correa et al. (SODA, 2020). Adding to their practical appeal, specifically in the context of algorithmic pricing, we then show that the new bounds can be obtained even in a streaming model of computation and thus in situations where the use of relevant data is complicated by the sheer amount of data available. We finally establish that the upper bound of 1/e for the case without samples is robust to additional information about the distribution, and applies also to sequences of i.i.d. random variables whose distribution is itself drawn, according to a known distribution, from a finite set of known candidate distributions. This implies a tight prophet inequality for exchangeable sequences of random variables, answering a question of Hill and Kertz (Contemporary Mathematics, 1992), but leaves open the possibility of better guarantees when the number of candidate distributions is small, a setting we believe is of strong interest to applications.
In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: …
In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: see text] is probability-competitive if every element from the optimum appears with probability [Formula: see text] in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of [Formula: see text] for laminar matroids. Additionally, we modify Kleinberg’s [Formula: see text] utility-competitive algorithm for uniform matroids of rank [Formula: see text] in order to obtain a [Formula: see text] probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.
We show that the matroid secretary problem is equivalent to correlated contention resolution in the online random-order model. Specifically, the matroid secretary conjecture is true if and only if every …
We show that the matroid secretary problem is equivalent to correlated contention resolution in the online random-order model. Specifically, the matroid secretary conjecture is true if and only if every matroid admits an online random-order contention resolution scheme which, given an arbitrary (possibly correlated) prior distribution over subsets of the ground set, matches the balance ratio of the best offline scheme for that distribution up to a constant. We refer to such a scheme as universal. Our result indicates that the core challenge of the matroid secretary problem lies in resolving contention for positively correlated inputs, in particular when the positive correlation is benign in as much as offline contention resolution is concerned.
Our result builds on our previous work which establishes one direction of this equivalence, namely that the secretary conjecture implies universal random-order contention resolution, as well as a weak converse, which derives a matroid secretary algorithm from a random-order contention resolution scheme with only partial knowledge of the distribution. It is this weak converse that we strengthen in this paper: We show that universal random-order contention resolution for matroids, in the usual setting of a fully known prior distribution, suffices to resolve the matroid secretary conjecture in the affirmative.
Our proof is the composition of three reductions. First, we use duality arguments to reduce the matroid secretary problem to the matroid prophet secretary problem with arbitrarily correlated distributions. Second, we introduce a generalization of contention resolution we term labeled contention resolution, to which we reduce the correlated matroid prophet secretary problem. Finally, we combine duplication of elements with limiting arguments to reduce labeled contention resolution to classical contention resolution.
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip …
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca ReiffenhäuserConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca Reiffenhäuserpp.1298 - 1325Chapter DOI:https://doi.org/10.1137/1.9781611977073.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold …
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
We design novel mechanisms for welfare maximization in two-sided markets. That is, there are buyers willing to purchase items and sellers holding items initially, both acting rationally and strategically in …
We design novel mechanisms for welfare maximization in two-sided markets. That is, there are buyers willing to purchase items and sellers holding items initially, both acting rationally and strategically in order to maximize utility. Our mechanisms are designed based on a powerful correspondence between two-sided markets and prophet inequalities. They satisfy individual rationality, dominant-strategy incentive compatibility, and budget balance constraints and give constant factor approximations to the optimal social welfare. We improve previous results in several settings. Our main focus is on matroid double auctions. Here, sellers hold identical items, and the set of buyers that obtain an item needs to be independent in a matroid. We construct two mechanisms, the first being a 1/3 approximation of the optimal social welfare-satisfying strong budget balance and requiring the agents to trade in a customized order and the second being a 1/2 approximation weakly budget balanced and able to deal with online arrival determined by an adversary. In addition, we construct constant factor approximations in two-sided markets with identical items when buyers need to fulfill a knapsack constraint. Also, in combinatorial double auctions with heterogeneous items, where buyers have valuation functions over item bundles instead of being interested in only one item, using similar techniques, we design a mechanism that is a 1/2 approximation of the optimal social welfare, is strongly budget balanced, and can deal with the online arrival of agents in an adversarial order. Funding: A. Braun was funded by the Deutsche Forschungsgemeinschaft [Grant 437739576].
We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of …
We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of items, and the aim of a mechanism is to improve the social welfare by arranging purchases and sales of the items. A mechanism is given prior distributions on the agents' valuations of the items, but not the actual valuations; thus the aim is to maximise the expected social welfare over these distributions. As in previous work, we are interested in the worst-case ratio between the social welfare achieved by a truthful mechanism, and the best social welfare possible.
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold …
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper …
Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper considers revenue maximization in the online stochastic setting with non-preemptive jobs and a unit capacity server. One agent/job arrives at every time step, with parameters drawn from an underlying unknown distribution.
We design a posted-price mechanism which can be efficiently computed, and is revenue-optimal in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic, depending only on the length of the allotted time interval and on the earliest time the server is available. If the distribution of agent's type is only learned from observing the jobs that are executed, we prove that a polynomial number of samples is sufficient to obtain a near-optimal truthful pricing strategy.
In a single-parameter mechanism design problem, a provider is looking to sell some service to a group of potential buyers. Each buyer i has a private value vi for receiving …
In a single-parameter mechanism design problem, a provider is looking to sell some service to a group of potential buyers. Each buyer i has a private value vi for receiving this service, and some feasibility constraint restricts which subsets of buyers can be served simultaneously. Recent work in economics introduced (deferred-acceptance) clock auctions as a superior class of auctions for this problem, due to their transparency, simplicity, and very strong incentive guarantees. Subsequent work in computer science focused on evaluating these auctions with respect to their social welfare approximation guarantees, leading to strong impossibility results: in the absence of prior information regarding the buyers' values, no deterministic clock auction can achieve a bounded approximation, even for simple feasibility constraints with only two maximal feasible sets.
Efficient and truthful mechanisms to price time on remote servers/machines have been the subject of much work in recent years due to the importance of the cloud market. This paper …
Efficient and truthful mechanisms to price time on remote servers/machines have been the subject of much work in recent years due to the importance of the cloud market. This paper considers online revenue maximization for a unit capacity server, when jobs are non preemptive, in the Bayesian setting: at each time step, one job arrives, with parameters drawn from an underlying distribution. We design an efficiently computable truthful posted price mechanism, which maximizes revenue in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic. We also show the pricing mechanism is robust to learning the job distribution from samples, where polynomially many samples suffice to obtain near optimal prices.
We study generalizations of the Prophet Inequality and Secretary Problem, where the algorithm is restricted to an arbitrary downward-closed set system. For {0,1}-values, we give O(log n)-competitive algorithms for both …
We study generalizations of the Prophet Inequality and Secretary Problem, where the algorithm is restricted to an arbitrary downward-closed set system. For {0,1}-values, we give O(log n)-competitive algorithms for both problems. This is close to the \Omega(log n / loglog n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log n log r)-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(log r loglog n) factors) an open question posed to us by Bobby Kleinberg.
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case …
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the arrival of each applicant, their cost per time step is revealed, and …
We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the arrival of each applicant, their cost per time step is revealed, and we have to fix the duration of employment, starting immediately. This decision is irrevocable; that is, we can neither extend a contract nor dismiss a candidate once hired. In every time step, at least one candidate needs to be under contract, and our goal is to minimize the total hiring cost, which is the sum of the applicants’ costs multiplied with their respective employment durations. We provide a competitive online algorithm for the case that the applicants’ costs are drawn independently from a known distribution. Specifically, the algorithm achieves a competitive ratio of 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants’ costs is unknown; (iii) the total number n of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than [Formula: see text] if concurrent employment is forbidden.
The prophet and secretary problems demonstrate online scenarios involving the optimal stopping theory. In a typical prophet or secretary problem, selection decisions are assumed to be immediate and irrevocable. However, …
The prophet and secretary problems demonstrate online scenarios involving the optimal stopping theory. In a typical prophet or secretary problem, selection decisions are assumed to be immediate and irrevocable. However, many online settings accommodate some degree of revocability. To study such scenarios, we introduce the $\ell-out-of-k$ setting, where the decision maker can select up to $k$ elements immediately and irrevocably, but her performance is measured by the top $\ell$ elements in the selected set. Equivalently, the decision makes can hold up to $\ell$ elements at any given point in time, but can make up to $k-\ell$ returns as new elements arrive.
We give upper and lower bounds on the competitive ratio of $\ell$-out-of-$k$ prophet and secretary scenarios. These include a single-sample prophet algorithm that gives a competitive ratio of $1-\ell\cdot e^{-\Theta\left(\frac{\left(k-\ell\right)^2}{k}\right)}$, which is asymptotically tight for $k-\ell=\Theta(\ell)$. For secretary settings, we devise an algorithm that obtains a competitive ratio of $1-\ell e^{-\frac{k-8\ell}{2+2\ln \ell}} - e^{-k/6}$, and show that no secretary algorithm obtains a better ratio than $1-e^{-k}$ (up to negligible terms). In passing, our results lead to an improvement of the results of Assaf et al. [2000] for $1-out-of-k$ prophet scenarios.
Beyond the contribution to online algorithms and optimal stopping theory, our results have implications to mechanism design. In particular, we use our prophet algorithms to derive {\em overbooking} mechanisms with good welfare and revenue guarantees; these are mechanisms that sell more items than the seller's capacity, then allocate to the agents with the highest values among the selected agents.
Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper …
Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper considers revenue maximization in the online stochastic setting with non-preemptive jobs and a unit capacity server. One agent/job arrives at every time step, with parameters drawn from an underlying unknown distribution. We design a posted-price mechanism which can be efficiently computed, and is revenue-optimal in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic, depending only on the length of the allotted time interval and on the earliest time the server is available. If the distribution of agent's type is only learned from observing the jobs that are executed, we prove that a polynomial number of samples is sufficient to obtain a near-optimal truthful pricing strategy.
In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, …
In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is $\alpha$ probability-competitive if every element from the optimum appears with probability $1/\alpha$ in the output. We present a technique to design algorithms with strong probability-competitive ratios, improving the guarantees for almost every matroid class considered in the literature: e.g., we get ratios of 4 for graphic matroids (improving on $2e$ by Korula and P\'al [ICALP 2009]) and of 5.19 for laminar matroids (improving on 9.6 by Ma et al. [THEOR COMPUT SYST 2016]). We also obtain new results for superclasses of $k$ column sparse matroids, for hypergraphic matroids, certain gammoids and graph packing matroids, and a $1+O(\sqrt{\log \rho/\rho})$ probability-competitive algorithm for uniform matroids of rank $\rho$ based on Kleinberg's $1+O(\sqrt{1/\rho})$ utility-competitive algorithm [SODA 2005] for that class. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids of rank $\rho$. We devise an $O(\log \rho)$ probability-competitive algorithm and an $O(\log\log \rho)$ ordinal-competitive algorithm, a weaker notion of competitiveness but stronger than the utility variant. These are based on the $O(\log\log \rho)$ utility-competitive algorithm by Feldman et al.~[SODA 2015].
We consider a stochastic online problem where $n$ applicants arrive over time, one per time step. Upon arrival of each applicant their cost per time step is revealed, and we …
We consider a stochastic online problem where $n$ applicants arrive over time, one per time step. Upon arrival of each applicant their cost per time step is revealed, and we have to fix the duration of employment, starting immediately. This decision is irrevocable, i.e., we can neither extend a contract nor dismiss a candidate once hired. In every time step, at least one candidate needs to be under contract, and our goal is to minimize the total hiring cost, which is the sum of the applicants' costs multiplied with their respective employment durations. We provide a competitive online algorithm for the case that the applicants' costs are drawn independently from a known distribution. Specifically, the algorithm achieves a competitive ratio of 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants' costs is unknown; (iii) the total number $n$ of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than $\Omega(\sqrt{n} / \log n)$ if concurrent employment is forbidden.
We consider the Bayesian online selection problem of a matching in bipartite graphs, that is, the weighted online matching problem where the edges arrive online and edge weights are generated …
We consider the Bayesian online selection problem of a matching in bipartite graphs, that is, the weighted online matching problem where the edges arrive online and edge weights are generated from a known distribution. This setting corresponds to the intersection of two matroids in the work of Kleinberg and Weinberg [ 40 ] and Feldman et al. [ 27 ]. We study a simple class of nonadaptive policies that we call vertex-additive policies. A vertex-additive policy assigns static prices to every vertex in the graph and accepts only those edges whose weight exceeds the sum of the prices on the edge endpoints. We show that there exists a vertex-additive policy with the expected payoff of at least one-third of the prophet’s payoff and present a gradient descent algorithm that quickly converges to the desired vector of vertex prices. Our results improve on the adaptive online policies of Kleinberg and Weinberg and Feldman et al. for the intersection of two matroids in two ways: our policy is nonadaptive and has a better approximation guarantee of 3 instead of the previous guarantees of 5.82 in Kleinberg and Weinberg and 5.43 in Feldman et al. We give a complementary lower bound of 2.25 for any online algorithm in the bipartite matching setting. Funding: This work was supported by Fundamental Research Funds of the Central Universities in China, a Science and Technology Innovation 2030 major project [“New Generation of Artificial Intelligence,” Project 2018AAA0100903], the Shanghai Municipal Education Commission Innovation Program, and the Program for Innovative Research Team of the Shanghai University of Finance and Economics (IRTSHUFE).
A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: Given a sequence of random variables $X_1,\dots,X_n$ drawn independently from a distribution …
A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: Given a sequence of random variables $X_1,\dots,X_n$ drawn independently from a distribution $F$, the goal is to choose a stopping time $\tau$ so as to maximize $\alpha$ such that for all distributions $F$ we have $\mathbb{E}[X_\tau] \geq \alpha \cdot \mathbb{E}[\max_tX_t]$. What makes this problem challenging is that the decision whether $\tau=t$ may only depend on the values of the random variables $X_1,\dots,X_t$ and on the distribution $F$. For quite some time the best known bound for the problem was $\alpha\geq1-1/e\approx0.632$ [Hill and Kertz, 1982]. Only recently this bound was improved by Abolhassani et al. [2017], and a tight bound of $\alpha\approx0.745$ was obtained by Correa et al. [2017]. The case where $F$ is unknown, such that the decision whether $\tau=t$ may depend only on the values of the first $t$ random variables but not on $F$, is equally well motivated (e.g., [Azar et al., 2014]) but has received much less attention. A straightforward guarantee for this case of $\alpha\geq1/e\approx0.368$ can be derived from the solution to the secretary problem. Our main result is that this bound is tight. Motivated by this impossibility result we investigate the case where the stopping time may additionally depend on a limited number of samples from~$F$. An extension of our main result shows that even with $o(n)$ samples $\alpha\leq 1/e$, so that the interesting case is the one with $\Omega(n)$ samples. Here we show that $n$ samples allow for a significant improvement over the secretary problem, while $O(n^2)$ samples are equivalent to knowledge of the distribution: specifically, with $n$ samples $\alpha\geq1-1/e\approx0.632$ and $\alpha\leq\ln(2)\approx0.693$, and with $O(n^2)$ samples $\alpha\geq0.745-\epsilon$ for any $\epsilon>0$.
We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the …
We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the case where the online algorithm has only limited access to the value distributions through samples. First, we give a $16$-approximate prophet inequality for matching in general graphs under edge arrivals that uses only a single sample from each value distribution as prior information. Then, for bipartite matching and (one-sided) vertex arrivals, we show an improved bound of $8$ that also uses just a single sample from each distribution. Finally, we show how to turn our $16$-approximate single-sample prophet inequality into a truthful single-sample mechanism for online bipartite matching with vertex arrivals.
A central issue in applying auction theory in practice is the problem of dealing with budget-constrained agents. A desirable goal in practice is to design incentive compatible, individually rational, and …
A central issue in applying auction theory in practice is the problem of dealing with budget-constrained agents. A desirable goal in practice is to design incentive compatible, individually rational, and Pareto optimal auctions while respecting the budget constraints. Achieving this goal is particularly challenging in the presence of nontrivial combinatorial constraints over the set of feasible allocations. Toward this goal and motivated by AdWords auctions, we present an auction for polymatroidal environments satisfying the above properties. Our auction employs a novel clinching technique with a clean geometric description and only needs an oracle access to the submodular function defining the polymatroid. As a result, this auction not only simplifies and generalizes all previous results, it applies to several new applications including AdWords Auctions, bandwidth markets, and video on demand. In particular, our characterization of the AdWords auction as polymatroidal constraints might be of independent interest. This allows us to design the first mechanism for Ad Auctions taking into account simultaneously budgets, multiple keywords and multiple slots.
Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined …
Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined by $t(c) = \text{smallest} i$ for which $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent …
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of $p$ matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most $O(p)$, and this factor is also tight.
For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to the mechanism design problem for each individual buyer. Our frame- …
For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to the mechanism design problem for each individual buyer. Our frame- work can be applied to any setting which roughly satisfies the following assumptions: (i) the buyer's types must be distributed independently (not necessarily identically), (ii) the objective function must be linearly separable over the set of buyers, and (iii) the supply constraints must be the only constraints involving more than one buyer. Our framework is general in the sense that it makes no explicit assumption about any of the following: (i) the buyer's valuations (e.g., submodular, additive, etc), (ii) The distribution of types for each buyer, and (iii) the other constraints involving individual buyers (e.g., budget constraints, etc). We present two generic ra-buyer mechanisms that use 1- buyer mechanisms as black boxes. Assuming that we have an α-approximate 1-buyer mechanism for each buyer and assuming that no buyer ever needs more than 1/k of all copies of each item for some integer k ≥ 1, then our generic n- buyer mechanisms are γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> · α-approximation of the optimal n-buyer mechanism, in which γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> is a constant which is at least 1 - 1/√(k+3). Observe that γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> is at least1/2 (for k = 1) and approaches 1 as k increases. As a byproduct of our construction, we improve a generalization of prophet inequalities. Furthermore, as applications of our main theorem, we improve several results from the literature.
We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in …
We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC).We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity.
For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to single buyer sub-problems. Our framework can be applied to any …
For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to single buyer sub-problems. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) buyers' types must be distributed independently (not necessarily identically), (ii) objective function must be linearly separable over the buyers, and (iii) except for the supply constraints, there should be no other inter-buyer constraints. Our framework is general in the sense that it makes no explicit assumption about buyers' valuations, type distributions, and single buyer constraints (e.g., budget, incentive compatibility, etc).
We present two generic multi buyer mechanisms which use single buyer mechanisms as black boxes; if an $\alpha$-approximate single buyer mechanism can be constructed for each buyer, and if no buyer requires more than $\frac{1}{k}$ of all units of each item, then our generic multi buyer mechanisms are $\gamma_k\alpha$-approximation of the optimal multi buyer mechanism, where $\gamma_k$ is a constant which is at least $1-\frac{1}{\sqrt{k+3}}$. Observe that $\gamma_k$ is at least 1/2 (for $k=1$) and approaches 1 as $k \to \infty$. As a byproduct of our construction, we present a generalization of prophet inequalities. Furthermore, as applications of our framework, we present multi buyer mechanisms with improved approximation factor for several settings from the literature.
Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The …
Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen.
The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint.
In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems.