China-Dutch GTA 2016, China GTA 2016: Game Theory and Applications pp 31-39

# Non-cooperative Monomino Games

• Judith Timmer
• Harry Aarts
• Peter van Dorenvanck
• Jasper Klomp
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 758)

## Abstract

In this paper we study monomino games. These are two player games played on a rectangular board with R rows and C columns. The game pieces are monominoes, which cover exactly one cell of the board. One by one each player selects a column of the board, and places a monomino in the lowest uncovered cell. This generates a payoff for the player. The game ends if all cells are covered by monominoes. The goal of each player is to place his monominoes in such a way that his total payoff is maximized. We derive the equilibrium play and corresponding payoffs for the players.

### Keywords

Monomino games Non-cooperative games Nash equilibrium Pure strategies

91A10 91A05

### References

1. 1.
Albert, M., Nowakowski, R.J., Wolfe, D.: An Introduction to Combinatorial Game Theory. A K Peters, Wellesley (2007)
2. 2.
Borm, P., Hamers, H., Hendrickx, R.: Operations research games: a survey. TOP 9(2), 139–199 (2001)
3. 3.
De Schuymer, B., De Meyer, H., De Baets, B.: Optimal strategies for equal-sum dice games. Discrete Appl. Math. 154, 2565–2576 (2006)
4. 4.
van Dorenvanck, P., Klomp, J.: Optimale strategieën in dominospelen. M.Sc. thesis, University of Twente, Enschede, The Netherlands (2010). (In Dutch)Google Scholar
5. 5.
Fraenkel, A.S.: Combinatorial games: Selected bibliography with a succinct gourmet introduction. The Electronic Journal of Combinatorics, Dynamic Surveys, DS2 (2009)Google Scholar
6. 6.
Gardner, M.: Dominono. Comput. Math. Appl. 39, 55–56 (2000)
7. 7.
Nash, J.: Non-cooperative games. Ann. Math. 54, 289–295 (1951)
8. 8.
Orman, H.K.: Pentominoes: a first player win. In: Nowakowski, R.J. (ed.) Games of No Chance, MSRI Publications, vol. 29, pp. 339–344. Cambridge University Press, Cambridge (1996)Google Scholar
9. 9.
Peters, H.: Game Theory: A Multi-leveled Approach. Springer, Heidelberg (2015). doi:
10. 10.
Rawsthorne, D.A.: Tiling complexity of small $$N$$-ominoes ($$N<10$$). Discrete Math. 70(1), 71–75 (1988)

© Springer Nature Singapore Pte Ltd. 2017

## Authors and Affiliations

• Judith Timmer
• 1
• Harry Aarts
• 1
• Peter van Dorenvanck
• 1
• Jasper Klomp
• 1
1. 1.Department of Applied Mathematics, Faculty of Electrical Engineering, Mathematics, and Computer ScienceUniversity of TwenteEnschedeThe Netherlands