Abstract
We define proper strong-Fibonacci (PSF) games as the subset of proper homogeneous weighted majority games which admit a Fibonacci representation. This is a homogeneous, type-preserving representation whose ordered sequence of type weights and winning quota is the initial string of Fibonacci numbers of the one-step delayed Fibonacci sequence. We show that for a PSF game, the Fibonacci representation coincides with the natural representation of the game. A characterization of PSF games is given in terms of their profile. This opens the way up to a straightforward formula which gives the number \(\varPsi (t)\) of such games as a function of t, number of non-dummy players’ types. It turns out that the growth rate of \(\varPsi (t)\) is exponential. The main result of our paper is that, for two consecutive t values of the same parity, the ratio \(\varPsi (t+2)/\varPsi (t)\) converges toward the golden ratio \({\varPhi }\).
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Notes
We can signal that there are a few significant papers adopting, like here, the reverse convention. See for instance Isbell (1959), p. 25.
Some authors consider weighted games which can be non-proper.
Here, we use the notation \(\mathbb {N}_0\) to denote the set of all positive integers: \(\mathbb {N}_0=\{1,2,3\ldots \}\), while \(\mathbb {N}=\mathbb {N}_{0}\cup \{0\}\).
Note that there are non-homogeneous games with a unique minimal integral representation (obviously non-homogeneous). For instance, \((\mathbf{w};q)=(1,1,2,2,3,4;7)\) is a non-homogeneous game. Moreover, this game is constant-sum and, in particular, \(w(\varOmega )=2q-1\). Isbell was the first to provide an example of a non-homogeneous game with two minimal integral representations (Isbell 1959, p. 27).
More generally, in the NR of a homogeneous game, all players of type \(j=1\) have the same individual weight \(w_1=1\).
For the sake of simplicity, we denote the profile component \({\varPhi }_j(t,z)={\varPhi }_j\).
For the sake of simplicity, we denote the profile component \(\varUpsilon _j(t,z,p)=\varUpsilon _j\).
Note that actually there are now two different coalitions, \(S_1(1)\) and \(S_2(2)\), associated respectively to the two players \(i_1\) and \(i_2\) of type \(j=1\). Such coalitions share the same profile. For this reason, we denote them by the same symbol S(1).
Also here there are two different coalitions, \(S_1(j^{*})\) and \(S_2(j^{*})\), associated respectively to the two players \(i_{1}^{*}\) and \(i_{2}^{*}\) of type \(j=j^{*}\). Such coalitions share the same profile. For this reason, we denote them by the same symbol \(S(j^{*})\).
It is in order to make a notational warning: while in Proposition 4, \(j^{\bullet }\) was necessarily a member of the set \(J^{'}\), playing (in the one-to-one correspondence with the minimal winning coalition) the role of the weakest player, in the additional part of Proposition 7, the bullet notation is extended to some players of the set \(J^{''}\), because in some minimal winning coalitions, they get the role of weakest player.
Here, let us explain the troublesome notation used for T. \(S_1(j^{*})\) is the coalition whose weakest player is the one denoted by \(i^{*}_1\) of type \(j^{*}\) (while \(i^{*}_2\) denotes the other one of the same type). We subtract from such a coalition the player of type \(j^{*}+1\), where the notation \(\{j^{*}+1\}\) denotes not the player with individual label \(j^{*}+1\), but his type index \(j^{*}+1\).
We can signal that the literature adopts the left-right lexicographic order.
Despite being a homogeneous representation involving all the first 7 components of the Fibonacci sequence, including the winning quota. See Remark 4.
This is an example of a game with a veto player; for details, see Sect. 10.
A fundamental property of SSP is that the players of \(G_m\) preserve their weights in the \(G_{m+1}\).
We exploit definition 24 to use indexes of \(g_{i}\) coherent with the individual labeling of the incidence matrix. In particular, in these games, the player with individual labeling i has Fibonacci weight \(g_{i-1}\).
It is easy to check that the two weakest players are both or none in each \(S\in W^{m}\) hence \(i_0=2\) can not be replaced by the smallest final step.
This notation turns out to be useful for the formal proof.
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We wish to thank the editor and three anonymous reviewers for their helpful comments and suggestions.
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In order to avoid misunderstandings, in the title of this paper, the word “strong” should be intended as the strong connection between the class of games we study here and the Fibonacci numbers. Hence, it does not mean strong in the well-known terminology used in the theory of simple games. A fortiori, the same remark applies to our previous paper (Pressacco and Ziani 2015), titled Constant sum strong Fibonacci games, which is often recalled here. Coherently in this paper we will use “strong-Fibonacci” to remark on this connection.
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Pressacco, F., Ziani, L. Proper strong-Fibonacci games. Decisions Econ Finan 41, 489–529 (2018). https://doi.org/10.1007/s10203-018-0212-5
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DOI: https://doi.org/10.1007/s10203-018-0212-5
Keywords
- Weighted majority games
- Natural representation
- Homogeneous representation
- Profile vector
- Fibonacci numbers
- Golden ratio