Abstract
Mastermind is a famous two-player deduction game. A Codemaker chooses a secret code and a Codebreaker tries to guess this secret code in as few guesses as possible, with feedback information after each guess. Many existing works have computed optimal worst-case and average-case strategies of the Codebreaker, assuming that the Codemaker chooses the secret code uniformly at random. However, the Codemaker can freely choose any distribution probability on the secret codes. An optimal strategy in this more general setting is known as a Nash Equilibrium. In this research, we compute such a Nash Equilibrium for all instances of Mastermind up to the most classical instance of 4 pegs and 6 colors, showing that the uniform distribution is not always the best choice for the Codemaker. We also show the direct relation between Nash Equilibrium computations and computations of worst-case and average-case strategies.
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Notes
- 1.
The straightforward proof is left to the reader.
- 2.
We use (b, w) since the original game use black and white pins to display the grade.
- 3.
A unicolor (resp. bicolor) code is a code with both pegs with same (resp. different) color. There are 3 unicolor codes and 6 bicolor codes. Note that \(3\times \frac{1}{6}+6\times \frac{1}{12}=1\).
- 4.
This formula is incorrect when \(k=1\) since there exists a single code, hence a single possible grade (n, 0). For \(k=2\) colors, there are also some unreachable grades such as \((b,w)=(0,n)\) when n is odd.
- 5.
To compute WC and AVG, it is sufficient to consider only pure strategies. However for NE, it is required to consider also mixed strategies.
- 6.
Pure strategies of the Codemaker are trivially computed. There are exactly \(k^n\) pure strategies, one for each possible code.
- 7.
- 8.
A full description will be given in a longer version of this article.
- 9.
Fortunately, \(4\times 0 + 48\times \frac{2}{345} + 36\times \frac{1}{276} + 144\times \frac{1}{276} + 24\times \frac{1}{345}=1\).
- 10.
We could not prove yet the uniqueness for instances (5, 3), (4, 5), and (4, 6), but it should be obtained very soon.
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This research was supported in part by JSPS KAKENHI Grant Number 26870228.
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A Example for Game Instance \((n,k)=(3,3)\)
A Example for Game Instance \((n,k)=(3,3)\)
Exploration tree. Squared nodes correspond to grade nodes and rounded nodes to guess nodes. Grade nodes include the list of codes still possible as the secret code. Edges from a guess node to a grade node are labeled with the corresponding grade.
Computing WC value using Min and Max operations
Computing AVG value using Min and Sum operations
Computing NE value using Union and MinkowskiSum operations
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Bonnet, F., Viennot, S. (2016). Nash Equilibrium in Mastermind. In: Plaat, A., Kosters, W., van den Herik, J. (eds) Computers and Games. CG 2016. Lecture Notes in Computer Science(), vol 10068. Springer, Cham. https://doi.org/10.1007/978-3-319-50935-8_11
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