Abstract
We study the applicability of quantum algorithms in computational game theory and generalize some results related to Subtraction games, which are sometimes referred to as one-heap Nim games.
In quantum game theory, a subset of Subtraction games became the first explicitly defined class of zero-sum combinatorial games with provable separation between quantum and classical complexity of solving them. For a narrower subset of Subtraction games, an exact quantum sublinear algorithm is known that surpasses all deterministic algorithms for finding solutions with probability 1.
Typically, both Nim and Subtraction games are defined for only two players. We extend some known results to games for three or more players, while maintaining the same classical and quantum complexities: \(\varTheta \left( n^2\right) \) and \(\tilde{O}\left( n^{1.5}\right) \) respectively.
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Notes
- 1.
Note that in accordance with the chosen numbering of rows and columns of \(\varGamma \), \(\varGamma _{ji}=1 \implies i<j\), so this recursive definition of the function \(\textsc {Win}\) is valid.
- 2.
The term balanced naturally comes from the notion of balanced functions: balanced game \(\varGamma \) is such that its payoff function \(\textsc {Win}\left( \varGamma ,j\right) \) is balanced. Formally, the definition of perfect balancedness should look like \(\forall w: \#\big \{j:{\textsc {Win}\left( \varGamma ,j\right) =w}\big \}_{1 \le j \le n} = \frac{n}{k}\), but we use the little-o notation to extend our results also to almost balanced games.
- 3.
Should one feel that discarding in this step essentially destroys the uniformity of \(\textsc {Win}\left( \varGamma ,j\right) \), they can at step 2 assign each position “w stones”, \(0 \le w < k\), value \(\left( k-w\right) \bmod k\). This will make the last step obsolete, as no failure can occur, and will preserve the perfect uniformity. Our further observations are valid for either kind of picking a random balanced Subtraction game.
- 4.
Formally, these vectors in the complex Hilbert space represent equivalence classes of vectors under multiplication by non-zero complex number. We also note that, for the purposes of this paper and throughout all the algorithms which we mention and refer here, one may assume all complex values to be in \(\mathbb {R}\). However, in other important quantum algorithms the imaginary parts may play an essential role.
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Acknowledgement
The research is supported by PostDoc Latvia Program, and by the ERDF within the project 1.1.1.2/VIAA/1/16/099 “Optimal quantum-entangled behavior under unknown circumstances”. A part of the reported study was funded by RFBR according to the research project No. 19-37-80008. A part of the research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project No. 0671-2020-0065.
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Kravchenko, D., Khadiev, K., Serov, D., Kapralov, R. (2020). Quantum-over-Classical Advantage in Solving Multiplayer Games. In: Schmitz, S., Potapov, I. (eds) Reachability Problems. RP 2020. Lecture Notes in Computer Science(), vol 12448. Springer, Cham. https://doi.org/10.1007/978-3-030-61739-4_6
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