Abstract
Agents endowed with power compete for a divisible resource by forming coalitions with other agents. The coalition with the greatest power wins the resource and divides it among its members. The agents’ power increases according to their share of the resource.We study two models of coalition formation where winning agents accumulate power and losing agents may participate in further coalition formation processes. An axiomatic approach is provided by focusing on variations of two main axioms: self-enforcement, which requires that no further deviation happens after a coalition has formed, and rationality, which requires that agents pick the coalition that gives them their highest payoff. For these alternative models, we determine the existence of stable coalitions that are self-enforcing and rational for two traditional sharing rules. The models presented in this paper illustrate how power accumulation, the sharing rule, and whether losing agents participate in future coalition formation processes, shape the way coalitions will be stable throughout time.
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Notes
The canonical example of power accumulation comes from non-democratic societies, where typically a ruling coalition can perpetuate itself in authority because it can use the state’s resources to consolidate and accumulate power and wealth. Although data is sparse, there is some evidence that some world leaders amassed personal wealth either due to connections or investments made possible by (sometimes corrupt practices) using state resources. For instance, there is some speculation that Teodoro Obiang Nguema Mbasogo of Equitorial Guinea has amassed a fortune of over $600 million in his oil-rich country since ascending to power in 1979 (see http://www.huffingtonpost.com/2013/11/29/richest-world-leaders_n_4178514.html). Power accumulation is natural in many coalition formation settings ranging from labor unions to military alliances between countries, where the players generate benefits by being together and become more powerful as time passes.
There are valid reasons to believe that the decision to kill agents in the society may be endogenous, especially in certain political contests or military wars. For instance, Bueno de Mesquita et al. (2003) asserts that “[w]hen the private benefits of office or coalition membership are large, people are more prepared to engage in horrendous acts of cruelty against others to ensure that their personal privileges are not lost”. The size of these private benefits can be reflected by resources to be won in a coalition formation game. On the other hand, most of the biggest purges in the \(20{\mathrm{th}}\) century are identified with specific personalities, e.g. Saddam Hussein, Adolf Hitler and Josef Stalin (Goode 1998, 2003). To the extent that these purges are sometimes personality-driven, the decision to kill opponents is exogenous. While the case where the decision to kill agents is endogenous is certainly an appealing study, it is beyond the scope of this paper and we leave it for future research. In this paper, we assume that the environment where agents are killed or are able to survive is exogenous.
A transition correspondence satisfies scale invariance if any scale in the power vector would not change the coalition chosen by the transition correspondence.
A transition correspondence satisfies independence of zeros if agents with zero power do not affect the coalition chosen by the transition correspondence.
This was first articulated by Tullock (1987), where he argues that since formal institutions are weak or absent regarding distribution and sharing of power, succession of leaders, and generating consensus, a ruling coalition (“junta”) will degenerate into a dictatorship, that is, there will be power accumulation by one of the junta members until he becomes the sole ruler.
In another paper,Acemoglu et al. (2009) show that the results in the AES case extend even to situations where agents are not killed. The stable ruling coalition in this case is the “minimally winning coalition” which is the coalition with the smallest power among all winning coalitions. In their dynamic game, this corresponds to the case when all agents are eligible to vote on the proposed coalition (i.e., the democratic case). In the case where the voting power is limited to the ruling coalition but agents are not killed, the mapping in AES is modified to include all alternative coalitions not just the subset of the ruling coalition.
Juarez (2016b) considers a more general version where power can be any arbitrary monotonic function.
This definition requires winning coalitions to have relative power larger than 50 %. Our results below can be easily adapted to require winning coalitions to have relative power larger than \(\alpha \), where \(\alpha \ge 50 \%\). This is discussed in Sect. 5.1.
A correspondence is continuous if for any sequence of power vectors \(\pi ^{1}, \pi ^{2}, \ldots \rightarrow \pi ^{*}\) where \(S\in \phi (N, \pi ^{i}) \ \forall i\) and S is winning in \(\pi ^{*}\), then \(S\in \phi (N, \pi ^{*})\).
First, observe that in the case where agents are killed and \(\phi (S^{t-1}, \pi ^{t-1})\) contains more than one element, we impose no restriction in which coalition from \(\phi (S^{t-1}, \pi ^{t-1})\) will equal \(S^{t}\). This allows our results to be more robust, since the evolution of the game includes any potential path of coalitions such that \(S^{t}\in \phi (S^{t-1}, \pi ^{t-1})\) for all t.
As we will see below, the axiom of self-enforcement guarantees that this limit always exists whether agents are killed or survive.
Indeed, note that \(S^0\supseteq S^1\supseteq S^2 \supseteq S^3 \supseteq \ldots \), thus the sequence \(S^0, S^1,S^2,\ldots \) converges in finite time.
DIR is weaker than the requirement that at every round the coalition chosen be the most preferred among ISE-coalitions at that time as opposed to our axiom that only requires for the coalition to be preferred at the limit. The results of this section under equal or proportional sharing will not change whether we use this alternative definition of rationality.
This is because if \((S,\pi )\in \tilde{G}\), then \((S,\pi + \frac{I}{|S|} 1_S)\in \tilde{G}\) in order for ISE and DIR to be well-defined under ES. Moreover, in order for scale invariance to be well-defined, \((S,\lambda \pi + \lambda \frac{I}{|S|} 1_S)\in \tilde{G}\) for any \(\lambda >0\).
Note this limit exists by ESE because the coalition that is chosen at time 0 is also chosen at any time in the future.
References
Acemoglu D, Egorov G, Sonin K (2008) Coalition formation in non-democracies. Rev Econ Stud 75(4):987–1009
Acemoglu D, Egorov G, Sonin K (2009) Do Juntas Lead to Personal Rule? Am Econ Rev 99(2):298–303
Acemoglu D, Golosov M, Tsyvinski A (2011) Power fluctuations and political economy. J Econ Theory 146(3):1009–1041
Baik KH (1994) Winner-Help-loser group formation in rent-seeking contests. Econ Politics 6(2):147–162
Barro R (1973) The Control of politicians: an economic model. Public Choice 14(1):19–42
Bartling B, von Siemens FA (2010) Equal sharing rules in partnerships. J Inst Theor Econ 166(2):299–320
Bloch F, Dutta B (2010)Formation of networks and coalitions. Mimeo University of Warwick, UK
Bogomolnaia A, Jackson M (2002) The stability of hedonic coalition structures. Games Econ Behav 38(2):201–230
Bueno de Mesquita B, Smith A, Siverson R, Morrow J (2003) The logic of political survival. The MIT Press, Cambridge
Chwe M (1994) Farsighted coalitional stability. J Econ Theory 63(2):299–325
Diamantoudi E, Xue L (2003) Farsighted stability in Hedonic games. Soc Choice Welf 21(1):39–61
Ehlers L (2002) Coalitional strategy-proof house allocation. J Econ Theory 105(2):298–317
Ferejohn J (1986) Incumbent performance and electoral control. Public Choice 50(1):5–25
Goode E (1998) Insane or Just Evil? A Psychiatrist takes a new look at Hitler. The New York Times. http://www.nytimes.com/1998/11/17/science/insane-or-just-evil-a-psychiatrist-takes-a-new-look-at-hitler.html. Accessed on 14 Aug 2015
Goode E (2003) The World; Stalin to Saddam: So Much for the Madman Theory. The New York Times. http://www.nytimes.com/2003/05/04/weekinreview/the-world-stalin-to-saddam-so-much-for-the-madman-theory.html. Accessed on 14 Aug 2015
Jandoc K, Juarez R,(2015) Self-enforcing coalitions under a general class of sharing rules. Mimeo University of Hawai‘i, USA
Jordan JS (2006) Pillage and property. J Econ Theory 131(1):26–44
Juarez R (2013) Group strategyproof cost sharing: the role of indifferences. Games Econ Behav 82:218–239
Juarez R (2016a) Three equilibria in tournament games with externalities. Mimeo University of Hawaii, Manoa
Juarez R (2016b) The no-threat equilibrium in tournament games with externalities. Mimeo University of Hawaii, Manoa
Konishi H, Ray D (2003) Coalition formation as a dynamic process. J Econ Theory 110(1):1–41
Lee S (1993) Inter-group competition for a pure private rent. Q Rev Econ Finance 33(3):251–266
Lee S (1995) Endogenous sharing rules in collective-group rent seeking. Public Choice 85(1):31–44
McGuire M, Olson M (1996) The economics of autocracy and majority rule: the invisible hand and the use of force. J Econ Lit 34(1):72–96
Nitzan S (1991) Collective rent dissipation. Econ J 101(409):1522–1534
Nitzan S (1994) Modelling rent seeking contests. Eur J Polit Econ 10(1):41–60
Olson M (1965) The logic of collective action. Harvard University Press, Cambridge
Olson M (1993) Dictatorship, democracy, and development. Am Polit Sci Rev 87(3):567–576
Papai S (2004) Unique stability in simple coalition formation games. Games Econ Behav 48(2):337–354
Roland G, Persson T, Tabellini G (1997) Separation of powers and political accountability. Q J Econ 112(4):1163–1202
Roland G, Persson T, Tabellini G (2000) Comparative politics and public finance. J Polit Econ 108(6):1121–1161
Piccione M, Razin R (2009) Coalition formation under power relations. Theor Econ 4(1):1–15
Ray D (2007) A game-theoretic perspective on coalition formation. Oxford University Press, New York
Ray D, Vohra R (2013) Coalition formation. North-Holland
Ray D, Vohra R (2015) The Farsighted stable set. Econometrica 83(3):977–1011
Sekeris P (2011) Endogenous elites: power structure and patron-client relationships. Econ Gov 12(3):237–258
Thomson W (2003) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Math Soc Sci 45(3):249–297
Thomson W (2007) On the existence of consistent rules to adjudicate conflicting claims: a constructive geometric approach. Rev Econ Design 11(3):225–251
Tullock G (1987) Autocracy. Kluwer Academic Publishers, Dorderecht
Xue L (1998) Coalitional stability under perfect foresight. Econ Theor 11(3):603–627
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Financial support from the AFOSR Young Investigator Program is greatly appreciated. Any errors or omissions are the authors’ own. We are grateful for helpful comments by two outstanding referees, the associate editor, Daron Acemoglu, Anna Bogomolnaia, Vikram Manjunath, Hervé Moulin and seminar participants at the Econometric Society Conference in Mexico, Conference on Economic Design in Lund, Public Economic Theory Conference in Lisbon and the International Conference on Game Theory in Stony Brook.
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Jandoc, K., Juarez, R. Self-enforcing coalitions with power accumulation. Int J Game Theory 46, 327–355 (2017). https://doi.org/10.1007/s00182-016-0538-6
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DOI: https://doi.org/10.1007/s00182-016-0538-6