Sharp thresholds for half-random games I
| Autor: | Tibor Szabó, Jonas Groschwitz |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Computer Science::Computer Science and Game Theory
Applied Mathematics General Mathematics Probability (math.PR) 010102 general mathematics Probabilistic logic ComputingMilieux_PERSONALCOMPUTING Randomized strategy 0102 computer and information sciences 01 natural sciences Computer Graphics and Computer-Aided Design Bottleneck Graph Combinatorics 010201 computation theory & mathematics Department Linguistik FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Software Connectivity Mathematics - Probability Mathematics |
| Popis: | We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-$k$. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly more tilted towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph. This paper has been split in two parts, the second part can be found at http://arxiv.org/abs/1602.04628 . A full version from before the split is submission v2 of this arXiv entry, available at http://arxiv.org/abs/1507.06688v2 |
| Databáze: | OpenAIRE |
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