Papers by Peter Bro Miltersen
arXiv (Cornell University), Feb 17, 2012
Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stoc... more Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games. When the number of positions of the game is constant, our algorithms run in polynomial time.
In this paper we consider two party communication complexity when the input sizes of the two play... more In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the "asymmetric" case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds, and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our "round elimination" lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the "round reduction" techniques used in many previous lower bound proofs.
arXiv (Cornell University), Dec 22, 2011
Gimbert and Horn gave an algorithm for solving simple stochastic games with running time O(r!n) w... more Gimbert and Horn gave an algorithm for solving simple stochastic games with running time O(r!n) where n is the number of positions of the simple stochastic game and r is the number of its coin toss positions. Chatterjee et al. pointed out that a variant of strategy iteration can be implemented to solve this problem in time 4 r r O(1) n O(1). In this paper, we show that an algorithm combining value iteration with retrograde analysis achieves a time bound of O(r2 r (r log r + n)), thus improving both time bounds. While the algorithm is simple, the analysis leading to this time bound is involved, using techniques of extremal combinatorics to identify worst case instances for the algorithm.
BRICS Report Series, Dec 11, 2003
See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained... more See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting:
arXiv (Cornell University), Jul 6, 2013
We consider the fundamental mechanism design problem of approximate social welfare maximization u... more We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare maximization in this setting. With m being the number of alternatives, we exhibit a randomized truthful-in-expectation ordinal mechanism implementing an outcome whose expected social welfare is at least an Ω(m −3/4) fraction of the social welfare of the socially optimal alternative. On the other hand, we show that for sufficiently many agents and any truthful-in-expectation ordinal mechanism, there is a valuation profile where the mechanism achieves at most an O(m −2/3) fraction of the optimal social welfare in expectation. Furthermore, we prove that no truthful-in-expectation (not necessarily ordinal) mechanism can achieve 0.94-fraction of the optimal social welfare. We get tighter bounds for the natural special case of m = 3, and in that case furthermore obtain separation results concerning the approximation ratios achievable by natural restricted classes of truthful-in-expectation mechanisms. In particular, we show that for m = 3 and a sufficiently large number of agents, the best mechanism that is ordinal as well as mixed-unilateral has an approximation ratio between 0.610 and 0.611, the best ordinal mechanism has an approximation ratio between 0.616 and 0.641, while the best mixed-unilateral mechanism has an approximation ratio bigger than 0.660. In particular, the best mixed-unilateral non-ordinal (i.e., cardinal) mechanism strictly outperforms all ordinal ones, even the non-mixed-unilateral ordinal ones.
Springer eBooks, 2008
We consider approximating the minmax value of a multiplayer game in strategic form. Tightening re... more We consider approximating the minmax value of a multiplayer game in strategic form. Tightening recent bounds by Borgs et al., we observe that approximating the value with a precision of ǫ log n digits (for any constant ǫ > 0) is NP-hard, where n is the size of the game. On the other hand, approximating the value with a precision of c log log n digits (for any constant c ≥ 1) can be done in quasi-polynomial time. We consider the parameterized complexity of the problem, with the parameter being the number of pure strategies k of the player for which the minmax value is computed. We show that if there are three players, k = 2 and there are only two possible rational payoffs, the minmax value is a rational number and can be computed exactly in linear time. In the general case, we show that the value can be approximated with any polynomial number of digits of accuracy in time n O(k). On the other hand, we show that minmax value approximation is W [1]-hard and hence not likely to be fixed parameter tractable. Concretely, we show that if k-CLIQUE requires time n Ω(k) then so does minmax value computation. ⋆ Work supported by Center for Algorithmic Game Theory, funded by the Carlsberg Foundation. ⋆⋆ Supported by a postdoc fellowship from the Carlsberg Foundation.
arXiv (Cornell University), Feb 28, 2016
Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, whi... more Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, while several buyers bring monetary endowments. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as Walrasian envy-free pricing do. In this paper we design an optimal envy-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources and, in particular, is incompatible with (Pareto) efficiency.
Springer eBooks, 2003
See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained... more See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting:
Lecture Notes in Computer Science, 2003
ABSTRACT
arXiv (Cornell University), Dec 2, 2008
It is NP-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in... more It is NP-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in strategic form with integer payoffs is trembling hand perfect.
BRICS Report Series, Jan 30, 1996
Addressing a problem of Fredman and Willard, we implement fusion trees in deterministic linear sp... more Addressing a problem of Fredman and Willard, we implement fusion trees in deterministic linear space using AC 0 instructions only.
We consider existence and computation of symmetric Pure Strategy Nash Equilibrium (PSNE) in singl... more We consider existence and computation of symmetric Pure Strategy Nash Equilibrium (PSNE) in single-item, sealedbid, first-price auctions with integral valuations and bids. For the most general case, we show that existence of PSNE is NP-hard. Then, we present algorithmic results for the case of independent valuations and two ways of breaking ties: Vickrey tie-breaking and random tie-breaking.
Adaptive Agents and Multi-Agents Systems, May 6, 2013
ArXiv, 2016
We study the envy-free pricing problem in multi-unit markets with budgets, where there is a selle... more We study the envy-free pricing problem in multi-unit markets with budgets, where there is a seller who brings multiple units of an item, while several buyers bring monetary endowments (budgets). Our goal is to compute an envy-free (item) price and allocation---i.e. an outcome where all the demands of the buyers are met given their budget constraints---which additionally achieves a desirable objective. We analyze markets with linear valuations, where the buyers are price takers and price makers, respectively. For the price taking scenario, we provide a polynomial time algorithm for computing the welfare maximizing envy-free pricing, followed by an FPTAS and exact algorithm---that is polynomial for a constant number of types of buyers---for computing a revenue maximizing envy-free pricing. In the price taking model, where the buyers can strategize, we show a general impossibility of designing strategyproof and efficient mechanisms even with public budgets. On the positive side, we pro...
Proceedings of the AAAI Conference on Artificial Intelligence, 2019
In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set ... more In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set of buyers that have monetary endowments. While a Walrasian equilibrium does not always exist in this model, natural relaxations of the concept that retain its desirable fairness properties do exist. We study the dynamics of (Walrasian) envy-free pricing mechanisms in this environment, showing that for any such pricing mechanism, the best response dynamic starting from truth-telling converges to a pure Nash equilibrium with small loss in revenue and welfare. Moreover, we generalize these bounds to capture all the (reasonable) Nash equilibria for a large class of (monotone) pricing mechanisms. We also identify a natural mechanism, which selects the minimum Walrasian envy-free price, in which for n=2 buyers the best response dynamic converges from any starting profile. We conjecture convergence of the mechanism for any number of buyers and provide simulation results to support our conjecture.
We consider existence and computation of symmetric Pure Strategy Nash Equilibrium (PSNE) in singl... more We consider existence and computation of symmetric Pure Strategy Nash Equilibrium (PSNE) in single-item, sealedbid, first-price auctions with integral valuations and bids. For the most general case, we show that existence of PSNE is NP-hard. Then, we present algorithmic results for the case of independent valuations and two ways of breaking ties: Vickrey tie-breaking and random tie-breaking.
Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)
In this paper we show several results about monotone planar circuits. We show that monotone plana... more In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform AC 0. This provides a striking contrast to the non-planar case, where exactly NC 1 is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the outer face only, can be solved in LOGDCFL SC, improving a LOGCFL upper bound due to Dymond and Cook. We show that for monotone planar circuits, with inputs on the outer face only, excessive depth compared to width is useless; any function computed by a monotone planar circuit of width w with inputs on the outer face can be computed by a monotone planar circuit of width O(w) and depth w O(1). Finally, we show that monotone planar read-once circuits, with inputs on the outer face only, can be efficiently learned using membership queries.
Proceedings 15th Annual IEEE Conference on Computational Complexity, 2000
The CD complexity of a string x is the length of the short• est polynomial time program which acc... more The CD complexity of a string x is the length of the short• est polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CDA::;n complexity of all strings x in some set A.. The best known upper bound for this problem is 2log(l!ASn!I) + O(log(n)), due to Buhrman and Fortnow. We show that the constant factor 2 in this bound is optimal. We also give new bounds for a certain kind of random sets R ~ {O, 1 r. for which we show an upper bound oflog(!!RS"!I) + O(log(n)). 1 Introduction Kolmogorov complexity is a notion that measures the amount of regularity in a finite string. It has turned out to be a very useful tool in theoretical computer science. A simple counting argument showing that for each length there exist random strings, i.e. strings with no regularity, has had many applications (see [LV97J). Early in the history of computational complexity resource bounded notions of Kolmogorov complexity were studied [Har83, Lon90, Lon86]. In particular Sipser [Sip83] introduced a new version of resource bounded Kolmogorov complexity, CD complexity, where one considers the size of the smallest program that accepts the given string and no others. Sipser showed that one can approximate the size of sets using CD complexity with random advice and then used this to show that BPP ~ PH.
Proceedings of 37th Conference on Foundations of Computer Science
In this paper we consider solutions to the static dictionary problem on ACo RAMs, i.e. random acc... more In this paper we consider solutions to the static dictionary problem on ACo RAMs, i.e. random access machines where the only restriction on theJinite instruction set is that all computationcil instructions are in ACO. Our main result is a tight upper and lower bound of 0 (Jlog n / log log n) on the time for (answering membership queries in a set of size n when reasonable space is used for the data structure storing the set; the upper bound can be obtained using O(n) space, and the lower bound holds even if we allow Several varitztions of this result are also obtained. Among others, we show a tradeoff between time and circuit depth under the unit-cost assumption: any RAM instruction set which purrtits a linear space, constant query time solution to the static dictionary problem must have an instruction of depth R(1og w / log log w), where w is the word size of the machine (and log the size ofthe universe). This matches the depth of multiplication and integer division, used in the perfect hashing scheme by Fredman, Komlds and Szemerkdi.
Lecture Notes in Computer Science, 1995
We study dynamic membership problems for the Dyck languages, the class of strings of properly bal... more We study dynamic membership problems for the Dyck languages, the class of strings of properly balanced parentheses. We also study the Dynamic Word problem for the free group. We present deterministic algorithms and data structures which maintain a string under replacements of symbols, insertions, and deletions of symbols, and language membership queries. Updates and queries are handled in polylogarithmic time. We also give both Las Vegas-and Monte Carlo-type randomised algorithms to achieve better running times, and present lower bounds on the complexity for variants of the problems.
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Papers by Peter Bro Miltersen