The solution to an open problem for a caching game

Csóka, Endre; Lidbetter, Thomas

doi:10.1002/nav.21674

Cited by 4 publications

(3 citation statements)

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“…This problem was introduced by Alpern, Fokkink, Lidbetter and Clayton in [2], a summary about the results and related questions of Alpern's Caching Game can be found in the survey book by Alpern, Fokkink, Leszek, Lindelauf [1]. More recent results are presented in [4].…”

Section: Previous Resultsmentioning

confidence: 99%

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Limit theory of discrete mathematics problems

Csóka

2015

Preprint

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We show a general problem-solving tool called limit theory. This is an advanced version of asymptotic analysis of discrete problems when some finite parameter tends to infinity. We will apply it on three closely related problems.Alpern's Caching Game (for 2 nuts) is defined as follows. The hider caches 2 nuts into one or two of n potential holes by digging at most 1 depth in total. The goal of the searcher is to find both nuts in a limited time h, otherwise the hider wins. We will show that if h and n/h are large enough, then very counterintuitively, any optimal hiding strategy should dig less than 1 in total, with positive probability. We will prove it by defining and analyzing a limit problem. Then we will partially solve the entire problem. We will also have significant progress with two other problems: the Manickam-Miklós-Singhi Conjecture and the Kikuta-Ruckle Conjecture.

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Section: Previous Resultsmentioning

confidence: 99%

“…The problem is solved for k = j = 2, n ≤ 4. [2,4] The solution for k = j = 2, n = 4 is the stepfunction shown in Figure 1. We show the nature of these optimal strategies by the following few cases.…”

Section: Previous Resultsmentioning

confidence: 99%

Limit theory of discrete mathematics problems

Csóka

2015

Preprint

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“…The value of Alpern's Caching Game, v A (n, d, k), was determined for some small values in [1,4,5]. For example, it is easy to see that v A (n, 1, k) = ⌊k⌋ n , but the problem is open in general already for d = 2.…”

Section: History Of Alpern's Caching Gamementioning

confidence: 99%

All or Nothing Caching Games with Bounded Queries

Pálvölgyi

2018

Int. Game Theory Rev.

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We determine the value of some search games where our goal is to find all of some hidden treasures using queries of bounded size. The answer to a query is either empty, in which case we lose, or a location, which contains a treasure. We prove that if we need to find d treasures at n possible locations with queries of size at most k, then our chance of winning is kif each treasure is at a different location andif each location might hide several treasures for large enough n. Our work builds on some results by Csóka who has studied a continuous version of this problem, known as Alpern's Caching Game; we also prove that the value of Alpern's Caching Game isfor integer k and large enough n.

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A search problem on a bipartite network

Baston

Kikuta

2019

European Journal of Operational Research

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The solution to an open problem for a caching game

Cited by 4 publications

References 20 publications

Limit theory of discrete mathematics problems

Limit theory of discrete mathematics problems

All or Nothing Caching Games with Bounded Queries

A search problem on a bipartite network

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