arXiv subjects

Tuomas Sandholm

Publications and source records attributed to Tuomas Sandholm.

160 records · Page 9Linked to original sources

arXiv2014

Algorithms for Closed Under Rational Behavior (CURB) Sets

We provide a series of algorithms demonstrating that solutions according to the fundamental game-theoretic solution concept of closed under rational behavior (CURB) sets in two-player, normal-form games can be computed in polynomial time (we also discuss extensions to n-player games). First, we describe an algorithm that identifies all of a player's best responses conditioned on the belief that the other player will play from within a given subset of its strategy space. This algorithm serves as a subroutine in a series of polynomial-time algorithms for finding all minimal CURB sets, one minimal CURB set, and the smallest minimal CURB set in a game. We then show that the complexity of finding a Nash equilibrium can be exponential only in the size of a game's smallest CURB set. Related to this, we show that the smallest CURB set can be an arbitrarily small portion of the game, but it can also be arbitrarily larger than the supports of its only enclosed Nash equilibrium. We test our algorithms empirically and find that most commonly studied academic games tend to have either very large or very small minimal CURB sets.

cs.GT↗

arXiv2013

On the complexity of strong Nash equilibrium: Hard-to-solve instances and smoothed complexity

The computational characterization of game-theoretic solution concepts is a central topic in artificial intelligence, with the aim of developing computationally efficient tools for finding optimal ways to behave in strategic interactions. The central solution concept in game theory is Nash equilibrium (NE). However, it fails to capture the possibility that agents can form coalitions (even in the 2-agent case). Strong Nash equilibrium (SNE) refines NE to this setting. It is known that finding an SNE is NP-complete when the number of agents is constant. This hardness is solely due to the existence of mixed-strategy SNEs, given that the problem of enumerating all pure-strategy SNEs is trivially in P. Our central result is that, in order for a game to have at least one non-pure-strategy SNE, the agents' payoffs restricted to the agents' supports must, in the case of 2 agents, lie on the same line, and, in the case of n agents, lie on an (n - 1)-dimensional hyperplane. Leveraging this result, we provide two contributions. First, we develop worst-case instances for support-enumeration algorithms. These instances have only one SNE and the support size can be chosen to be of any size-in particular, arbitrarily large. Second, we prove that, unlike NE, finding an SNE is in smoothed polynomial time: generic game instances (i.e., all instances except knife-edge cases) have only pure-strategy SNEs.

cs.GT↗

arXiv2012

Combining local search techniques and path following for bimatrix games

Computing a Nash equilibrium (NE) is a central task in computer science. An NE is a particularly appropriate solution concept for two-agent settings because coalitional deviations are not an issue. However, even in this case, finding an NE is PPAD-complete. In this paper, we combine path following algorithms with local search techniques to design new algorithms for finding exact and approximate NEs. We show that our algorithms largely outperform the state of the art and that almost all the known benchmark game classes are easily solvable or approximable (except for the GAMUT CovariantGameRand class).

cs.GT↗

arXiv2012

Common Voting Rules as Maximum Likelihood Estimators

Voting is a very general method of preference aggregation. A voting rule takes as input every voter's vote (typically, a ranking of the alternatives), and produces as output either just the winning alternative or a ranking of the alternatives. One potential view of voting is the following. There exists a 'correct' outcome (winner/ranking), and each voter's vote corresponds to a noisy perception of this correct outcome. If we are given the noise model, then for any vector of votes, we can

cs.GT↗

arXiv2003

Complexity of Determining Nonemptiness of the Core

Coalition formation is a key problem in automated negotiation among self-interested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can do things more efficiently. However, motivating the agents to abide to a solution requires careful analysis: only some of the solutions are stable in the sense that no group of agents is motivated to break off and form a new coalition. This constraint has been studied extensively in cooperative game theory. However, the computational questions around this constraint have received less attention. When it comes to coalition formation among software agents (that represent real-world parties), these questions become increasingly explicit. In this paper we define a concise general representation for games in characteristic form that relies on superadditivity, and show that it allows for efficient checking of whether a given outcome is in the core. We then show that determining whether the core is nonempty is $\mathcal{NP}$-complete both with and without transferable utility. We demonstrate that what makes the problem hard in both cases is determining the collaborative possibilities (the set of outcomes possible for the grand coalition), by showing that if these are given, the problem becomes tractable in both cases. However, we then demonstrate that for a hybrid version of the problem, where utility transfer is possible only within the grand coalition, the problem remains $\mathcal{NP}$-complete even when the collaborative possibilities are given.

cs.GT↗

arXiv2003

Definition and Complexity of Some Basic Metareasoning Problems

In most real-world settings, due to limited time or other resources, an agent cannot perform all potentially useful deliberation and information gathering actions. This leads to the metareasoning problem of selecting such actions. Decision-theoretic methods for metareasoning have been studied in AI, but there are few theoretical results on the complexity of metareasoning. We derive hardness results for three settings which most real metareasoning systems would have to encompass as special cases. In the first, the agent has to decide how to allocate its deliberation time across anytime algorithms running on different problem instances. We show this to be $\mathcal{NP}$-complete. In the second, the agent has to (dynamically) allocate its deliberation or information gathering resources across multiple actions that it has to choose among. We show this to be $\mathcal{NP}$-hard even when evaluating each individual action is extremely simple. In the third, the agent has to (dynamically) choose a limited number of deliberation or information gathering actions to disambiguate the state of the world. We show that this is $\mathcal{NP}$-hard under a natural restriction, and $\mathcal{PSPACE}$-hard in general.

cs.AI↗

arXiv2003

Universal Voting Protocol Tweaks to Make Manipulation Hard

Voting is a general method for preference aggregation in multiagent settings, but seminal results have shown that all (nondictatorial) voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. A number of recent papers study the complexity of manipulating existing protocols. This paper is the first work to take the next step of designing new protocols that are especially hard to manipulate. Rather than designing these new protocols from scratch, we instead show how to tweak existing protocols to make manipulation hard, while leaving much of the original nature of the protocol intact. The tweak studied consists of adding one elimination preround to the election. Surprisingly, this extremely simple and universal tweak makes typical protocols hard to manipulate! The protocols become NP-hard, #P-hard, or PSPACE-hard to manipulate, depending on whether the schedule of the preround is determined before the votes are collected, after the votes are collected, or the scheduling and the vote collecting are interleaved, respectively. We prove general sufficient conditions on the protocols for this tweak to introduce the hardness, and show that the most common voting protocols satisfy those conditions. These are the first results in voting settings where manipulation is in a higher complexity class than NP (presuming PSPACE $\neq$ NP).

cs.GT↗

arXiv2003

BL-WoLF: A Framework For Loss-Bounded Learnability In Zero-Sum Games

We present BL-WoLF, a framework for learnability in repeated zero-sum games where the cost of learning is measured by the losses the learning agent accrues (rather than the number of rounds). The game is adversarially chosen from some family that the learner knows. The opponent knows the game and the learner's learning strategy. The learner tries to either not accrue losses, or to quickly learn about the game so as to avoid future losses (this is consistent with the Win or Learn Fast (WoLF) principle; BL stands for ``bounded loss''). Our framework allows for both probabilistic and approximate learning. The resultant notion of {\em BL-WoLF}-learnability can be applied to any class of games, and allows us to measure the inherent disadvantage to a player that does not know which game in the class it is in. We present {\em guaranteed BL-WoLF-learnability} results for families of games with deterministic payoffs and families of games with stochastic payoffs. We demonstrate that these families are {\em guaranteed approximately BL-WoLF-learnable} with lower cost. We then demonstrate families of games (both stochastic and deterministic) that are not guaranteed BL-WoLF-learnable. We show that those families, nevertheless, are {\em BL-WoLF-learnable}. To prove these results, we use a key lemma which we derive.

cs.GT↗

arXiv2003

How many candidates are needed to make elections hard to manipulate?

In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. The complexity of manipulating realistic elections where the number of candidates is a small constant was recently studied (Conitzer 2002), but the emphasis was on the question of whether or not a protocol becomes hard to manipulate for some constant number of candidates. That work, in many cases, left open the question: How many candidates are needed to make elections hard to manipulate? This is a crucial question when comparing the relative manipulability of different voting protocols. In this paper we answer that question for the voting protocols of the earlier study: plurality, Borda, STV, Copeland, maximin, regular cup, and randomized cup. We also answer that question for two voting protocols for which no results on the complexity of manipulation have been derived before: veto and plurality with runoff. It turns out that the voting protocols under study become hard to manipulate at 3 candidates, 4 candidates, 7 candidates, or never.

cs.GT↗

arXiv2003

AWESOME: A General Multiagent Learning Algorithm that Converges in Self-Play and Learns a Best Response Against Stationary Opponents

A satisfactory multiagent learning algorithm should, {\em at a minimum}, learn to play optimally against stationary opponents and converge to a Nash equilibrium in self-play. The algorithm that has come closest, WoLF-IGA, has been proven to have these two properties in 2-player 2-action repeated games--assuming that the opponent's (mixed) strategy is observable. In this paper we present AWESOME, the first algorithm that is guaranteed to have these two properties in {\em all} repeated (finite) games. It requires only that the other players' actual actions (not their strategies) can be observed at each step. It also learns to play optimally against opponents that {\em eventually become} stationary. The basic idea behind AWESOME ({\em Adapt When Everybody is Stationary, Otherwise Move to Equilibrium}) is to try to adapt to the others' strategies when they appear stationary, but otherwise to retreat to a precomputed equilibrium strategy. The techniques used to prove the properties of AWESOME are fundamentally different from those used for previous algorithms, and may help in analyzing other multiagent learning algorithms also.

cs.GT↗

arXiv2002

Vote Elicitation: Complexity and Strategy-Proofness

Preference elicitation is a central problem in AI, and has received significant attention in single-agent settings. It is also a key problem in multiagent systems, but has received little attention here so far. In this setting, the agents may have different preferences that often must be aggregated using voting. This leads to interesting issues because what, if any, information should be elicited from an agent depends on what other agents have revealed about their preferences so far. In this paper we study effective elicitation, and its impediments, for the most common voting protocols. It turns out that in the Single Transferable Vote protocol, even knowing when to terminate elicitation is mathcal NP-complete, while this is easy for all the other protocols under study. Even for these protocols, determining how to elicit effectively is NP-complete, even with perfect suspicions about how the agents will vote. The exception is the Plurality protocol where such effective elicitation is easy. We also show that elicitation introduces additional opportunities for strategic manipulation by the voters. We demonstrate how to curtail the space of elicitation schemes so that no such additional strategic issues arise.

cs.GT↗

arXiv2002

Complexity of Manipulating Elections with Few Candidates

In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard. Especially among computational agents, it is reasonable to measure this hardness by computational complexity. Some earlier work has been done in this area, but it was assumed that the number of voters and candidates is unbounded. We derive hardness results for practical multiagent settings where the number of candidates is small but the number of voters can be large. We show that with complete information about the others' votes, individual manipulation is easy, and coalitional manipulation is easy with unweighted voters. However, constructive coalitional manipulation with weighted voters is intractable for all of the voting protocols under study, except for the nonrandomized Cup. Destructive manipulation tends to be easier. Randomizing over instantiations of the protocols (such as schedules of the Cup protocol) can be used to make manipulation hard. Finally, we show that under weak assumptions, if weighted coalitional manipulation with complete information about the others' votes is hard in some voting protocol, then individual and unweighted manipulation is hard when there is uncertainty about the others' votes.

cs.GT↗

arXiv2002

Complexity Results about Nash Equilibria

Noncooperative game theory provides a normative framework for analyzing strategic interactions. However, for the toolbox to be operational, the solutions it defines will have to be computed. In this paper, we provide a single reduction that 1) demonstrates NP-hardness of determining whether Nash equilibria with certain natural properties exist, and 2) demonstrates the #P-hardness of counting Nash equilibria (or connected sets of Nash equilibria). We also show that 3) determining whether a pure-strategy Bayes-Nash equilibrium exists is NP-hard, and that 4) determining whether a pure-strategy Nash equilibrium exists in a stochastic (Markov) game is PSPACE-hard even if the game is invisible (this remains NP-hard if the game is finite). All of our hardness results hold even if there are only two players and the game is symmetric. Keywords: Nash equilibrium; game theory; computational complexity; noncooperative game theory; normal form game; stochastic game; Markov game; Bayes-Nash equilibrium; multiagent systems.

cs.GT↗

arXiv2002

Complexity of Mechanism Design

The aggregation of conflicting preferences is a central problem in multiagent systems. The key difficulty is that the agents may report their preferences insincerely. Mechanism design is the art of designing the rules of the game so that the agents are motivated to report their preferences truthfully and a (socially) desirable outcome is chosen. We propose an approach where a mechanism is automatically created for the preference aggregation setting at hand. This has several advantages, but the downside is that the mechanism design optimization problem needs to be solved anew each time. Focusing on settings where side payments are not possible, we show that the mechanism design problem is NP-complete for deterministic mechanisms. This holds both for dominant-strategy implementation and for Bayes-Nash implementation. We then show that if we allow randomized mechanisms, the mechanism design problem becomes tractable. In other words, the coordinator can tackle the computational complexity introduced by its uncertainty about the agents' preferences by making the agents face additional uncertainty. This comes at no loss, and in some cases at a gain, in the (social) objective.

cs.GT↗

arXiv2002

Effectiveness of Preference Elicitation in Combinatorial Auctions

Combinatorial auctions where agents can bid on bundles of items are desirable because they allow the agents to express complementarity and substitutability between the items. However, expressing one's preferences can require bidding on all bundles. Selective incremental preference elicitation by the auctioneer was recently proposed to address this problem (Conen & Sandholm 2001), but the idea was not evaluated. In this paper we show, experimentally and theoretically, that automated elicitation provides a drastic benefit. In all of the elicitation schemes under study, as the number of items for sale increases, the amount of information elicited is a vanishing fraction of the information collected in traditional ``direct revelation mechanisms'' where bidders reveal all their valuation information. Most of the elicitation schemes also maintain the benefit as the number of agents increases. We develop more effective elicitation policies for existing query types. We also present a new query type that takes the incremental nature of elicitation to a new level by allowing agents to give approximate answers that are refined only on an as-needed basis. In the process, we present methods for evaluating different types of elicitation policies.

cs.GT↗

arXiv1998

Anytime Coalition Structure Generation with Worst Case Guarantees

Coalition formation is a key topic in multiagent systems. One would prefer a coalition structure that maximizes the sum of the values of the coalitions, but often the number of coalition structures is too large to allow exhaustive search for the optimal one. But then, can the coalition structure found via a partial search be guaranteed to be within a bound from optimum? We show that none of the previous coalition structure generation algorithms can establish any bound because they search fewer nodes than a threshold that we show necessary for establishing a bound. We present an algorithm that establishes a tight bound within this minimal amount of search, and show that any other algorithm would have to search strictly more. The fraction of nodes needed to be searched approaches zero as the number of agents grows. If additional time remains, our anytime algorithm searches further, and establishes a progressively lower tight bound. Surprisingly, just searching one more node drops the bound in half. As desired, our algorithm lowers the bound rapidly early on, and exhibits diminishing returns to computation. It also drastically outperforms its obvious contenders. Finally, we show how to distribute the desired search across self-interested manipulative agents.

cs.MA↗