Abstract
The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric (quota) games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending on invariants of simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts.
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Notes
 1.
More precisely, the computational problem to decide whether \(\nu ([q;w_1,\dots ,w_n])=2\) is NPhard. A proof can be obtained by a reduction to the NPhard partition problem. So, for integers \(w_1,\dots ,w_n\) we have to decide whether there exists a subset \(S\subseteq N\) such that \(\sum _{i\in S}w_i=\sum _{i\in N\backslash S} w_i\), where we use the abbreviation \(N=\{1,\dots ,n\}\). Consider the weighted game \([w(N)/2;w_1,\dots ,w_n]\). It has Nakamura number 2 if and only if a subset S with \(w(S)=w(N\backslash S)\) exists. Further complexity results for the Nakamura number can e.g. be found in Bartholdi et al. (1991) and Takamiya and Tanaka (2016).
References
Bachrach Y, Elkind E, Meir R, Pasechnik D, Zuckerman M, Rothe J, Rosenschein J (2009) The cost of stability in coalitional games. In: Proceedings of the 2nd international symposium on algorithmic game theory, SAGT ’09, pp 122–134. Springer, Berlin
Bartholdi JJ, Narasimhan LS, Tovey CA (1991) Recognizing majorityrule equilibrium in spatial voting games. Soc Choice Welf 8(3):183–197
Baum S, Trotter L Jr (1981) Integer rounding for polymatroid and branching optimization problems. SIAM J Algebraic Discret Methods 2(4):416–425
Beck M, Robins S (2007) Computing the continuous discretely. Springer, Berlin
Carreras F, Freixas J (1996) Complete simple games. Math Soc Sci 32:139–155
Deb R, Weber S, Winter E (1996) The Nakamura theorem for coalition structures of quota games. Int J Game Theory 25(2):189–198
Deǐneko VG, Woeginger GJ (2006) On the dimension of simple monotonic games. Eur J Oper Res 170(1):315–318
Eisenbrand F, Pálvölgyi D, Rothvoß T (2013) Bin packing via discrepancy of permutations. ACM Tran Algorithms 9(3):24
Ferejohn J, Grether D (1974) On a class of rational social decision procedures. J Econ Theory 8(4):471–482
Freixas J, Kurz S (2014a) On \(\alpha \)roughly weighted games. Int J Game Theory 43(3):659–692
Freixas J, Kurz S (2014b) On minimum integer representations of weighted games. Math Soc Sci 67:9–22
Freixas J, Marciniak D (2009) A minimum dimensional class of simple games. Top 17(2):407–414
Freixas J, Zwicker WS (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21(3):399–431
Gilmore P, Gomory R (1961) A linear programming approach to the cuttingstock problem. Oper Res 9(6):849–859
Greenberg J (1979) Consistent majority rule over compact sets of alternatives. Econometrica 47(3):627–636
Hof F, Kern W, Kurz S, Paulusma D (2018) In: Deng X (ed) International symposium on algorithmic game theory, SAGT 2018, vol 11059. Lecture notes in computer science, pp 69–81
Holzman R (1986) The capacity of a committee. Math Soc Sci 12(2):139–157
Kartak V, Kurz S, Ripatti A, Scheithauer G (2015) Minimal proper nonIRUP instances of the onedimensional cutting stock problem. Discret Appl Math 187:120–129
Keiding H (1984) Heights of simple games. Int J Game Theory 13(1):15–26
Kumabe M, Mihara H (2008) The Nakamura number for computeable simple games. Soc Choice Welf 31(4):621–640
Kurz S (2012) On minimum sum representations for weighted voting games. Ann Oper Res 196(1):361–369
Kurz S, Napel S (2016) Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record. Optim Lett 10(6):1245–1256
Kurz S, Nohn A, Napel S (2014) The nucleolus of large majority games. Econ Lett 123(2):139–143
Le Breton M, Salles M (1990) The stability set of voting games: classification and genericity results. Int J Game Theory 19(2):111–127
Martin M (1998) Quota games and stability set of order d. Econ Lett 59(2):145–151
Nakamura K (1979) The vetoers in a simple game with ordinal preferences. Int J Game Theory 8(1):55–61
Peleg B (1978) Consistent voting systems. Econometrica 46(1):153–161
Peleg B (1987) Cores and capacities of compound simple games. Soc Choice Welf 4(4):307–316
Peleg B (2008) Game theoretic analysis of voting in committees. Books, Cambridge
Saari D (2014) Unifying voting theory from Nakamura’s to Greenberg’s theorems. Math Soc Sci 69:1–11
Scheithauer G, Terno J (1995) The modified integer roundup property of the onedimensional cutting stock problem. Eur J Oper Res 84(3):562–571
Schofield N (1984) Social equilibrium and cycles on compact sets. J Econ Theory 33(1):59–71
Schwartz T (2001) From arrow to cycles, instability, and chaos by untying alternatives. Soc Choice Welf 18(1):1–22
Takamiya K, Tanaka A (2016) Computational complexity in the design of voting. Theory Decis 80:33–41
Tchantcho B, Lambo L, Pongou R, Moulen J (2010) On the equilibrium of voting games with abstention and several levels of approval. Soc Choice Welf 34(3):379–396
Truchon M (1996) Acyclicity and decisiveness structures. J Econ Theory 69(2):447–469
Acknowledgements
The authors thank the anonymous referees and the associate editor for their careful reading of a preliminary version of this paper. Their constructive remarks were extremely useful to improve its presentation.
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Josep Freixas’s research is partially supported by funds from the spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds) under Grant MTM201566818P (MINECO/FEDER).
Proof of Theorem 2
Proof of Theorem 2
Proof

(a)
At first we remark that the proposed exact value coincides with the lower bound from Theorem 1. Next we observe
$$\begin{aligned} q^r=\frac{\left\lceil \overline{q}(\varOmega +r)\right\rceil }{\varOmega +r} \le \frac{1+\overline{q}(\varOmega +r)}{\varOmega +r}= \overline{q}+\frac{1}{\varOmega +r}\le \overline{q}+\frac{1}{r}. \end{aligned}$$Consider the following greedy way of constructing the list \(S_1,\dots ,S_k\) of winning coalitions with empty intersection. Starting with \(i=1\) and \(h=1\) we choose an index \(h\le g\le n\) such that \(U_i=\{h,h+1,\dots ,g\}\) has a weight of at most \((1q^r)(\varOmega +r)\) and either \(g=n\) or \(U_i\cup \{g+1\}\) has a weight larger than \((1q^r)(\varOmega +r)\). Given \(U_i\) we set \(S_i=\{1,\dots ,n+r\}\backslash U_i\), \(h=g+1\), and increase i by one. If \((1q^r)(\varOmega +r)\ge w_i\) for all \(1\le i\le n\), then no player in \(\{1,\dots ,n\}\) has a too large weight to be dropped in this manner. Since we assume the weights to be ordered, it suffices to check the proposed inequality for \(w_1\). To this end we consider
$$\begin{aligned}&(1q^r)(\varOmega +r)\ge \left( 1\overline{q}\frac{1}{r}\right) \cdot (\varOmega +r)\\&\quad = (1\overline{q})\varOmega 1\frac{\varOmega }{r} +(1\overline{q})r\ge (1\overline{q})r2, \end{aligned}$$where we have used \(r\ge \varOmega \). Since \(r\ge \frac{2+w_1}{1\overline{q}}\ge \frac{2+w_i}{1\overline{q}}\) the requested inequality is satisfied.
So far the winning coalitions \(S_i\) can have weights larger than \(q^r(\varOmega +r)\) and their intersection is given by the players of weight 1, i.e. by \(\{n+1,\dots ,n+r\}\). For all \(1\le i<k\) let \(h_i\) be the player with the smallest index in \(U_i\), which is indeed one of the heaviest players in this subset. With this we conclude \(w(S_i)\le q^r(\varOmega +r)+w_{h_i}1\) since otherwise another player from \(U_{i+1}\) could have been added. In order to lower the weights of the \(S_i\) to \(q^r(\varOmega +r)\) we remove \(w(S_i)\left( q^r(\varOmega +r)\right) )\) players of \(S_i\) for all \(1\le i\le k\), starting from player \(n+1\) and removing each player exactly once. Since \(\sum _{i=1}^{k1} w_{h_i}\le \varOmega \le r\) this is indeed possible. Now we remove the remaining, if any, players of weight 1 from \(S_k\) until they reach weight \(q^r(\varOmega +r)\) and eventually start new coalitions \(S_i=\{1,\dots ,n+r\}\) removing players of weight 1. Finally we end up with \(r+l\) winning coalitions with empty intersection, where the coalitions \(1\le i\le k+l1\) have weight exactly \(q^r(\varOmega +r)\) and the sets \(\{1,\dots ,n+r\}\backslash S_i\) do contain only players of weight 1 for \(i\ge r+1\). Since each player is dropped exactly once the Nakamura number of the game equals \(k+l=\left\lceil \frac{1}{1q^r}\right\rceil \).

(b)
We write \(\overline{q}=\frac{p}{q}\) with positive comprime integers p, q. If \(p\ne q1\), then
$$\begin{aligned} \left\lceil \frac{1}{1\overline{q}}\right\rceil =\left\lceil \frac{q}{qp}\right\rceil >\frac{1}{1\overline{q}}, \end{aligned}$$i.e., we always round up. Obviously \(\lim _{r\rightarrow \infty } q^r=\overline{q}\) (and \(q^r\ge \overline{q}\)). Since also
$$\begin{aligned} \lim _{r\rightarrow \infty } \frac{w(N^r)}{w(N^r)q^r w(N^r)w_1+1}= \lim _{r\rightarrow \infty } \frac{w(N^r)}{w(N^r)q^r w(N^r)}=\frac{1}{1\overline{q}}, \end{aligned}$$we can apply the upper bound of Theorem 1 to deduce that the lower bound is attained with equality for sufficiently large replication factors r.
In the remaining part we assume \(p=q1\), i.e., \(1\overline{q}=\frac{1}{q}\). If \(\varOmega \cdot r\) is not divisible by q, i.e. \(q^r>\overline{q}\), we can apply a similar argument as before, so that we restrict ourselves to the case \(q\varOmega \cdot r\), i.e. \(\overline{q}=q^r\). Here we have to show that the Nakamura number exactly equals q (in the previous case it equals \(q+1\)). This is possible if we can partition the grand coalition N into q subsets \(U_1,\dots ,U_q\) all having a weight of exactly \(\frac{\varOmega \cdot r}{q}\). (The list of winning coalitions with empty intersection is then given by \(S_i=N\backslash U_i\) for \(1\le i\le q\).) This boils down to a purely theoretical question of number theory, which is solved in the next lemma.
\(\square \)
Lemma 6
Let \(g\ge 2\) and \(w_1,\dots ,w_n\) be positive integers with \(\sum \limits _{i=1}^n w_i=\varOmega \) and greatest common divisor 1.
There exists an integer K such that for all \(k\ge K\), where \(\frac{k\cdot \varOmega }{q}\in \mathbb {N}\), there exist nonnegative integers \(u_j^i\) with
for all \(1\le i\le q\), and
for all \(1\le j\le n\).
Proof
For \(k=1\), setting \(u_j^i=\frac{1}{q}\) is an inner point of the polyhedron
so that is has nonzero volume.
For general \(k\in \mathbb {N}_{>0}\) we are looking for lattice points in the dilation \(k\cdot P\). If q is a divisor of \(k\cdot \varOmega \), then \(\mathbb {Z}^{nq}\cap k\cdot P\) is a lattice of maximal rank in the affine space spanned by \(k\cdot P\). Let \(k_0\) the minimal positive integer such that q divides \(k_0\cdot \varOmega \). Using Erhart Theory one can count the number of lattice points in the parametric rational polytope in \(m\cdot k_0\cdot P\), where \(m\in \mathbb {N}_{>0}\), see e.g. Beck and Robins (2007). To be more precise, the number of (integer) lattice points in \(m\cdot k_0\cdot P\) grows asymptotically as \(m^d {\text {vol}}_d(k_0 P)\), where d is the dimension of the affine space A spanned by \(k_0\cdot P\) and \({\text {vol}}_d (k_0 P)\) is the (normalized)
volume of \(k_0\cdot P\) within A. Due to the existence of an inner point we have \({\text {vol}}_d (k_0 P)>0\), so that the number of integer solutions is at least 1 for \(m\gg 0\). \(\square \)
There is a relation between the problem of Lemma 6 and the Frobenius number, which asks for the largest integer which can not be expressed as a nonnegative integer linear combination of the \(w_i\). Recently this type of problem occurs in the context on minimum sum integer representations, see Freixas and Kurz (2014b). According to the Frobenius theorem every sufficiently large number can be expressed as such a sum. Here we ask for several such representations which are balanced, i.e., each coin is taken equally often.
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Freixas, J., Kurz, S. Bounds for the Nakamura number. Soc Choice Welf 52, 607–634 (2019). https://doi.org/10.1007/s003550181164y
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Keywords
 Nakamura number
 Stability
 Simple games
 Complete simple games
 Weighted games
 Bounds
Mathematics Subject Classification
 91A12
 91B14
 91B12