nach oben

Decisions in Economics and Finance

Erschienen in:

06.09.2018

Proper strong-Fibonacci games

verfasst von: Flavio Pressacco, Laura Ziani

Erschienen in: Decisions in Economics and Finance | Ausgabe 2/2018

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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 \({\varPhi }\).

Vorheriger Artikel Advertising a product to face a competitor entry: a differential game approach

Nächster Artikel Publisher Correction: Classic rational bubbles and representativeness

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.

Baron, D.P., Ferejohn, J.A.: Bargaining in legislatures. Am. Political Sci. Rev. 83, 1181–1206 (1989)CrossRef

Fragnelli, V., Gambarelli, G., Gnocchi, N., Pressacco, F., Ziani, L.: Fibonacci representations of homogeneous weighted majority games. In: Nguyen, N.T., Kowalczyk, R., Mercik, J. (eds.) Transactions on computational collective intelligence XXIII. Lecture Notes in Computer Science, vol. 9760, pp. 162–171. Springer, Berlin (2016)CrossRef

Freixas, J., Kurz, S.: The golden number and Fibonacci sequences in the design of voting structures. Eur. J. Oper. Res. 226, 246–257 (2013)CrossRef

Freixas, J., Kurz, S.: On minimum integer representations of weighted games. Math. Soc. Sci. 67, 9–22 (2014)CrossRef

Freixas, J., Molinero, X.: Weighted games without a unique minimal representation in integers. Optim. Methods Softw. 25, 203–215 (2010)CrossRef

Freixas, J., Molinero, X., Roura, S.: Complete voting systems with two classes of voters: weightedness and counting. Ann. Oper. Res. 193–1, 273–289 (2012)CrossRef

Gurk, H.M., Isbell, J.R.: Simple Solutions, Contributions to the Theory of Games, Vol. IV, Annals of Mathematics Studies, vol. 40, pp. 247–265. Princeton University Press, Princeton (1959)

Isbell, R.: A class of majority games. Q J Math 7, 183–187 (1956)CrossRef

Isbell, R.: On the enumeration of majority games. Math. Tables Aids Comput. 13, 21–28 (1959)CrossRef

Kalandrakis, T.: Proposal rights and political power. Am. J. Political Sci. 50–2, 441–448 (2006)CrossRef

Krohn, I., Sudhölter, P.: Directed and weighted majority games. ZOR - Math. Methods Oper. Res. 42, 189–216 (1995)CrossRef

Le Breton, M., Montero, M., Zaporozhets, V.: Voting power in the EU council of ministers and fair decision making in distributive politics. Math. Soc. Sci. 63, 159–173 (2012)CrossRef

Maschler, M., Peleg, B.: A characterization, existence proof and dimension bounds for the kernel of a game. Pacific J. Math. 18–2, 289–328 (1966)CrossRef

Montero, M.: Proportional Payoffs in Majority Games (2008). Available at SSRN: http://ssrn.com/abstract=1112626

Ostmann, A.: On the minimal representation of homogeneous games. Int. J. Game Theory 16, 69–81 (1987)CrossRef

Peleg, B.: On weights of constant-sum majority games. SIAM J. Appl. Math. 16–3, 527–532 (1968)CrossRef

Philippou, A., Bergum, G.E., Horadam, A.F.: Fibonacci Numbers and Their Applications, Mathematics and its Applications, vol. 18. Springer, Dordrecht (1986)CrossRef

Pressacco, F., Plazzotta, G., Ziani, L.: Bilateral symmetry and modified Pascal triangles in Parsimonious games. (2013). http://hal.archives-ouvertes.fr/ with validation no.: hal-00948123

Pressacco, F., Ziani, L.: A Fibonacci approach to weighted majority games. J Game Theory 4, 36–44 (2015)

Rosenmüller, J.: Weighted majority games and the matrix of homogeneity. Oper. Res. 28, 123–141 (1984)

Rosenmüller, J.: Homogeneous games: recursive structure and computation. Math. Oper. Res. 12–2, 309–330 (1987)CrossRef

Rosenmüller, J., Sudhölter, P.: The nucleolus of homogeneous games with steps. Discrete Appl. Math. 50, 53–76 (1994)CrossRef

Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17–6, 1163–1170 (1969)CrossRef

Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behaviour. Princeton University Press, Princeton (1944)

Titel: Proper strong-Fibonacci games
verfasst von: Flavio Pressacco
Laura Ziani
Publikationsdatum: 06.09.2018
Verlag: Springer International Publishing
Erschienen in: Decisions in Economics and Finance / Ausgabe 2/2018
Print ISSN: 1593-8883
Elektronische ISSN: 1129-6569
DOI: https://doi.org/10.1007/s10203-018-0212-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 2/2018

A piecewise linear model of credit traps and credit cycles: a complete characterization

A continuous-time heterogeneous duopoly model with delays

Environmental depletion, defensive consumption and negative externalities

Poverty trap, boom and bust periods and growth. A nonlinear model for non-developed and developing countries

Publisher Correction: Real options signaling game models for dynamic acquisition under information asymmetry

Oligopoly models with different learning and production time scales