arXiv subjects

Paul W. Goldberg

Publications and source records attributed to Paul W. Goldberg.

At least 37 records · Page 2Linked to original sources

arXiv2023

Solving Strong-Substitutes Product-Mix Auctions

This paper develops algorithms to solve strong-substitutes product-mix auctions. That is, it finds competitive equilibrium prices and quantities for agents who use this auction's bidding language to truthfully express their strong-substitutes preferences over an arbitrary number of goods, each of which is available in multiple discrete units. (Strong substitutes preferences are also known, in other literatures, as $M^\natural$-concave, matroidal and well-layered maps, and valuated matroids). Our use of the bidding language, and the information it provides, contrasts with existing algorithms that rely on access to a valuation or demand oracle to find equilibrium. We compute market-clearing prices using algorithms that apply existing submodular minimisation methods. Allocating the supply among the bidders at these prices then requires solving a novel constrained matching problem. Our algorithm iteratively simplifies the allocation problem, perturbing bids and prices in a way that resolves tie-breaking choices created by bids that can be accepted on more than one good. We provide practical running time bounds on both price-finding and allocation, and illustrate experimentally that our allocation mechanism is practical.

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arXiv2023

The Computational Complexity of Single-Player Imperfect-Recall Games

We study single-player extensive-form games with imperfect recall, such as the Sleeping Beauty problem or the Absentminded Driver game. For such games, two natural equilibrium concepts have been proposed as alternative solution concepts to ex-ante optimality. One equilibrium concept uses generalized double halving (GDH) as a belief system and evidential decision theory (EDT), and another one uses generalized thirding (GT) as a belief system and causal decision theory (CDT). Our findings relate those three solution concepts of a game to solution concepts of a polynomial maximization problem: global optima, optimal points with respect to subsets of variables and Karush-Kuhn-Tucker (KKT) points. Based on these correspondences, we are able to settle various complexity-theoretic questions on the computation of such strategies. For ex-ante optimality and (EDT,GDH)-equilibria, we obtain NP-hardness and inapproximability, and for (CDT,GT)-equilibria we obtain CLS-completeness results.

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arXiv2023

Best-Response Dynamics in Lottery Contests

We study the convergence of best-response dynamics in lottery contests. We show that best-response dynamics rapidly converges to the (unique) equilibrium for homogeneous agents but may not converge for non-homogeneous agents, even for two non-homogeneous agents. For $2$ homogeneous agents, we show convergence to an $ε$-approximate equilibrium in $Θ(\log\log(1/ε))$ steps. For $n \ge 3$ agents, the dynamics is not unique because at each step $n-1 \ge 2$ agents can make non-trivial moves. We consider a model where the agent making the move is randomly selected at each time step. We show convergence to an $ε$-approximate equilibrium in $O(β\log(n/(εδ)))$ steps with probability $1-δ$, where $β$ is a parameter of the agent selection process, e.g., $β= n$ if agents are selected uniformly at random at each time step. Our simulations indicate that this bound is tight.

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arXiv2023

The Complexity of Gradient Descent: CLS = PPAD $\cap$ PLS

We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain $[0,1]^2$ is PPAD $\cap$ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) - which was defined by Daskalakis and Papadimitriou as a more "natural" counterpart to PPAD $\cap$ PLS and contains many interesting problems - is itself equal to PPAD $\cap$ PLS.

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arXiv2022

Consensus Division in an Arbitrary Ratio

We consider the problem of partitioning a line segment into two subsets, so that $n$ finite measures all have the same ratio of values for the subsets. Letting $α\in[0,1]$ denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which $α=\frac{1}{2}$. Stromquist and Woodall showed that for any $α$, there exists a solution using $2n$ cuts of the segment. They also showed that if $α$ is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values $α$. For $α= \frac{\ell}{k}$, we show a lower bound of $\frac{k-1}{k} \cdot 2n - O(1)$ cuts; we also obtain almost matching upper bounds for a large subset of rational $α$. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1. when using the minimal number of cuts for each instance is required, the problem is NP-hard for any $α$; 2. for a large subset of rational $α= \frac{\ell}{k}$, when $\frac{k-1}{k} \cdot 2n$ cuts are available, the problem is in PPA-$k$ under Turing reduction; 3. when $2n$ cuts are allowed, the problem belongs to PPA for any $α$; more generally, the problem belong to PPA-$p$ for any prime $p$ if $2(p-1)\cdot \frac{\lceil p/2 \rceil}{\lfloor p/2 \rfloor} \cdot n$ cuts are available.

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arXiv2022

The Hairy Ball Problem is PPAD-Complete

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of (a) computing an approximate zero is PPAD-complete, and (b) computing an exact zero is FIXP-hard. We also consider the Hairy Ball Theorem on toroidal instead of spherical domains and show that the approximate problem remains PPAD-complete. On a conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely on the investigation of multiple-source variants of END-OF-LINE, the canonical PPAD-complete problem. Our results on these new END-OF-LINE variants are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the IMBALANCE problem defined by Beame et al. in 1998.

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arXiv2022

PPAD-Complete Pure Approximate Nash Equilibria in Lipschitz Games

Lipschitz games, in which there is a limit $λ$ (the Lipschitz value of the game) on how much a player's payoffs may change when some other player deviates, were introduced about 10 years ago by Azrieli and Shmaya. They showed via the probabilistic method that $n$-player Lipschitz games with $m$ strategies per player have pure $ε$-approximate Nash equilibria, for $ε\geqλ\sqrt{8n\log(2mn)}$. Here we provide the first hardness result for the corresponding computational problem, showing that even for a simple class of Lipschitz games (Lipschitz polymatrix games), finding pure $ε$-approximate equilibria is PPAD-complete, for suitable pairs of values $(ε(n), λ(n))$. Novel features of this result include both the proof of PPAD hardness (in which we apply a population game reduction from unrestricted polymatrix games) and the proof of containment in PPAD (by derandomizing the selection of a pure equilibrium from a mixed one). In fact, our approach implies containment in PPAD for any class of Lipschitz games where payoffs from mixed-strategy profiles can be deterministically computed.

cs.GT↗

arXiv2022

Simultaneous Contests with Equal Sharing Allocation of Prizes: Computational Complexity and Price of Anarchy

We study a general scenario of simultaneous contests that allocate prizes based on equal sharing: each contest awards its prize to all players who satisfy some contest-specific criterion, and the value of this prize to a winner decreases as the number of winners increases. The players produce outputs for a set of activities, and the winning criteria of the contests are based on these outputs. We consider two variations of the model: (i) players have costs for producing outputs; (ii) players do not have costs but have generalized budget constraints. We observe that these games are exact potential games and hence always have a pure-strategy Nash equilibrium. The price of anarchy is $2$ for the budget model, but can be unbounded for the cost model. Our main results are for the computational complexity of these games. We prove that for general versions of the model exactly or approximately computing a best response is NP-hard. For natural restricted versions where best response is easy to compute, we show that finding a pure-strategy Nash equilibrium is PLS-complete, and finding a mixed-strategy Nash equilibrium is (PPAD$\cap$PLS)-complete. On the other hand, an approximate pure-strategy Nash equilibrium can be found in pseudo-polynomial time. These games are a strict but natural subclass of explicit congestion games, but they still have the same equilibrium hardness results.

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arXiv2021

Equivalence of Models of Cake-Cutting Protocols

The cake-cutting problem involves dividing a heterogeneous, divisible resource fairly between $n$ agents. Brânzei et al. [6] introduced {\em generalised cut and choose} (GCC) protocols, a formal model for representing cake-cutting protocols as trees with "cut" and "choose" nodes corresponding to the agents' actions, and if-else statements. In this paper, we identify an alternative and simpler extensive-form game model for cake-cutting protocols, that we call {\em branch choice} (BC) protocols. We show that the class of protocols we can represent using this model is invariant under certain modifications to its definition. We further prove that any such protocol can be converted to a restricted form in which the agents first cut the cake and then get to choose between various branches leading to different allocations. Finally, we show that this model has the same expressive power as GCC protocols, i.e. they represent the same class of protocols up to a notion of equivalence involving the bounds on envy that each agent can guarantee for themselves. For this purpose, we introduce a new notion of envy-equivalence of protocols.

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arXiv2021

Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games

Nearly a decade ago, Azrieli and Shmaya introduced the class of $λ$-Lipschitz games in which every player's payoff function is $λ$-Lipschitz with respect to the actions of the other players. They showed that such games admit $ε$-approximate pure Nash equilibria for certain settings of $ε$ and $λ$. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding $ε$-approximate pure Nash equilibria of $n$-player, binary-action, $λ$-Lipschitz games that is exponential in $\frac{nλ}ε$. In addition, we introduce ``Multi-Lipschitz games,'' a generalization involving player-specific Lipschitz values, and provide a reduction from finding equilibria of these games to finding equilibria of Lipschitz games, showing that the value of interest is the sum of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding $ε$-approximate correlated equilibria of $n$-player, $m$-action, $λ$-Lipschitz games for strong values of $ε$, motivating the consideration of explicitly randomized algorithms in the above results. Our proof is arguably simpler than those previously used to show similar results.

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arXiv2021

Learning Strong Substitutes Demand via Queries

This paper addresses the computational challenges of learning strong substitutes demand when given access to a demand (or valuation) oracle. Strong substitutes demand generalises the well-studied gross substitutes demand to a multi-unit setting. Recent work by Baldwin and Klemperer shows that any such demand can be expressed in a natural way as a finite list of weighted bid vectors. A simplified version of this bidding language has been used by the Bank of England. Assuming access to a demand oracle, we provide an algorithm that computes the unique list of weighted bid vectors corresponding to a bidder's demand preferences. In the special case where their demand can be expressed using positive bids only, we have an efficient algorithm that learns this list in linear time. We also show super-polynomial lower bounds on the query complexity of computing the list of bids in the general case where bids may be positive and negative. Our algorithms constitute the first systematic approach for bidders to construct a bid list corresponding to non-trivial demand, allowing them to participate in `product-mix' auctions.

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arXiv2020

Consensus Halving for Sets of Items

Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally. Prior work has shown that when the resource is represented by an interval, a consensus halving with at most $n$ cuts always exists, but is hard to compute even for agents with simple valuation functions. In this paper, we study consensus halving in a natural setting where the resource consists of a set of items without a linear ordering. When agents have additive utilities, we present a polynomial-time algorithm that computes a consensus halving with at most $n$ cuts, and show that $n$ cuts are almost surely necessary when the agents' utilities are drawn from probabilistic distributions. On the other hand, we show that for a simple class of monotonic utilities, the problem already becomes PPAD-hard. Furthermore, we compare and contrast consensus halving with the more general problem of consensus $k$-splitting, where we wish to divide the resource into $k$ parts in possibly unequal ratios, and provide some consequences of our results on the problem of computing small agreeable sets.

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arXiv2020

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.

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arXiv2019

Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $ε$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}ε \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}ε \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.

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arXiv2018

Multi-unit Bilateral Trade

We characterise the set of dominant strategy incentive compatible (DSIC), strongly budget balanced (SBB), and ex-post individually rational (IR) mechanisms for the multi-unit bilateral trade setting. In such a setting there is a single buyer and a single seller who holds a finite number k of identical items. The mechanism has to decide how many units of the item are transferred from the seller to the buyer and how much money is transferred from the buyer to the seller. We consider two classes of valuation functions for the buyer and seller: Valuations that are increasing in the number of units in possession, and the more specific class of valuations that are increasing and submodular. Furthermore, we present some approximation results about the performance of certain such mechanisms, in terms of social welfare: For increasing submodular valuation functions, we show the existence of a deterministic 2-approximation mechanism and a randomised e/(1-e) approximation mechanism, matching the best known bounds for the single-item setting.

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arXiv2018

The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches

We resolve the computational complexity of two problems known as NECKLACE-SPLITTING and DISCRETE HAM SANDWICH, showing that they are PPA-complete. For NECKLACE SPLITTING, this result is specific to the important special case in which two thieves share the necklace. We do this via a PPA-completeness result for an approximate version of the CONSENSUS-HALVING problem, strengthening our recent result that the problem is PPA-complete for inverse-exponential precision. At the heart of our construction is a smooth embedding of the high-dimensional Möbius strip in the CONSENSUS-HALVING problem. These results settle the status of PPA as a class that captures the complexity of "natural" problems whose definitions do not incorporate a circuit.

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arXiv2018

Hardness Results for Consensus-Halving

We study the consensus-halving problem of dividing an object into two portions, such that each of $n$ agents has equal valuation for the two portions. The $ε$-approximate consensus-halving problem allows each agent to have an $ε$ discrepancy on the values of the portions. We prove that computing $ε$-approximate consensus-halving solution using $n$ cuts is in PPA, and is PPAD-hard, where $ε$ is some positive constant; the problem remains PPAD-hard when we allow a constant number of additional cuts. It is NP-hard to decide whether a solution with $n-1$ cuts exists for the problem. As a corollary of our results, we obtain that the approximate computational version of the Continuous Necklace Splitting Problem is PPAD-hard when the number of portions $t$ is two.

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arXiv2017

Consensus Halving is PPA-Complete

We show that the computational problem CONSENSUS-HALVING is PPA-complete, the first PPA-completeness result for a problem whose definition does not involve an explicit circuit. We also show that an approximate version of this problem is polynomial-time equivalent to NECKLACE SPLITTING, which establishes PPAD-hardness for NECKLACE SPLITTING, and suggests that it is also PPA-complete.

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