Solving Zero-Sum Games Using Best-Response Oracles with Applications to Search Games

Hellerstein, Lisa; Lidbetter, Thomas; Pirutinsky, Daniel

doi:10.1287/opre.2019.1853

Cited by 14 publications

(10 citation statements)

References 39 publications

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“…More specifically, consider the expanding search game where the target acts as an adversary that hides in one of the graph's vertices in order to maximize the expected search time. Together with the work of Hellerstein et al (2019), our branch-and-cut procedure yields a method to compute the game's value exactly, whereas our greedy approach allows to approximate this value within factor eight. Note that these results are incomparable with the approximations of Alpern and Lidbetter (2019) because, in that article, the target can hide at any point of an edge in the network instead of being restricted to hide at vertices only.…”

Section: Discussionmentioning

confidence: 99%

Exact and Approximation Algorithms for the Expanding Search Problem

Hermans

Leus

Matuschke

2022

INFORMS Journal on Computing

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Suppose a target is hidden in one of the vertices of an edge-weighted graph according to a known probability distribution. Starting from a fixed root node, an expanding search visits the vertices sequentially until it finds the target, where the next vertex can be reached from any of the previously visited vertices. That is, the time to reach the next vertex equals the shortest-path distance from the set of all previously visited vertices. The expanding search problem then asks for a sequence of the nodes, so as to minimize the expected time to finding the target. This problem has numerous applications, such as searching for hidden explosives, mining coal, and disaster relief. In this paper, we develop exact algorithms and heuristics, including a branch-and-cut procedure, a greedy algorithm with a constant-factor approximation guarantee, and a local search procedure based on a spanning-tree neighborhood. Computational experiments show that our branch-and-cut procedure outperforms existing methods for instances with nonuniform probability distributions and that both our heuristics compute near-optimal solutions with little computational effort. Summary of Contribution: This paper studies new algorithms for the expanding search problem, which asks to search a graph for a target hidden in one of the nodes according to a known probability distribution. This problem has applications such as searching for hidden explosives, mining coal, and disaster relief. We propose several new algorithms, including a branch-and-cut procedure, a greedy algorithm, and a local search procedure; and we analyze their performance both experimentally and theoretically. Our analysis shows that the algorithms improve on the performance of existing methods and establishes the first constant-factor approximation guarantee for this problem.

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Section: Discussionmentioning

confidence: 99%

Exact and Approximation Algorithms for the Expanding Search Problem

Hermans

Leus

Matuschke

2022

INFORMS Journal on Computing

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“…The technical report Lin and Singham (2015) presents an algorithm to estimate each player's optimal strategy by successively bounding the value of the game. However, as discussed in Hellerstein et al (2019), an algorithm that guarantees convergence in polynomial time remains a challenge, mainly because the searcher's pure strategy set is infinite.…”

Section: Introductionmentioning

confidence: 99%

“…As first discovered by Blackwell and reported in Matula (1964), when the hider's mixed strategy is fixed, an optimal search strategy is simple to calculate. Hellerstein et al (2019) provides general methods for solving games where a best response of player 2 to any fixed player 1 strategy is easily obtained. However, the algorithms of Hellerstein et al (2019) require both players to have finite pure strategy sets, which is not the case for the searcher in G.…”

Section: Introductionmentioning

confidence: 99%

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A Classical Search Game in Discrete Locations

2023

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Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among n discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location i takes ti time units and detects the hider—if hidden there—independently with probability [Formula: see text] for [Formula: see text]. The hider aims to maximize the expected time until detection, whereas the searcher aims to minimize it. We prove the existence of an optimal strategy for each player. In particular, any optimal mixed hiding strategy hides in each location with a nonzero probability, and there exists an optimal mixed search strategy that can be constructed with up to n simple search sequences.

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“…The emergence of a CCE or a NE in a game between no-regret learners provides yet another strong evidence for the importance of these game-theoretic equilibrium concepts. From a practical point of view, the convergence of the expected ergodic distribution to the set of CCE or of the ergodic average to the set of NE makes no-regret algorithms an appealing way to approximate a CCE or a NE when the game matrix is unknown so only simulating the game is possible (see Hellerstein et al (2019)). When simulating an unknown game, bandit feedback is a more realistic assumption than full information (or gradient feedback).…”

Section: Introductionmentioning

confidence: 99%

No Weighted-Regret Learning in Adversarial Bandits with Delays

Bistritz¹,

Zhou²,

Chen³

et al. 2021

Preprint

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Consider a player that in each round t out of T rounds chooses an action and observes the incurred cost after a delay of dt rounds. The cost functions and the delay sequence are chosen by an adversary. We show that even if the players' algorithms lose their "no regret" property due to too large delays, the expected discounted ergodic distribution of play converges to the set of coarse correlated equilibrium (CCE) if the algorithms have "no discounted-regret". For a zero-sum game, we show that no discounted-regret is sufficient for the discounted ergodic average of play to converge to the set of Nash equilibria. We prove that the FKM algorithm with n dimensions achieves a regret of O nT 3 4 + √ nT 1 3 D 1 3 * . This is an extended version of Bistritz et al. (2019). For details, see Subsection 1.1 on previous work.

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Solving Zero-Sum Games Using Best-Response Oracles with Applications to Search Games

Cited by 14 publications

References 39 publications

Exact and Approximation Algorithms for the Expanding Search Problem

Exact and Approximation Algorithms for the Expanding Search Problem

A Classical Search Game in Discrete Locations

No Weighted-Regret Learning in Adversarial Bandits with Delays

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