Skip to main content
Log in

The stationary equilibrium of three-person coalitional bargaining games with random proposers: a classification

  • Published:
International Journal of Game Theory Aims and scope Submit manuscript

Abstract

We present a classification of all stationary subgame perfect equilibria of the random proposer model for a three-person cooperative game according to the level of efficiency. The efficiency level is characterized by the number of “central” players who join all equilibrium coalitions. The existence of a central player guarantees asymptotic efficiency. The marginal contributions of players to the grand coalition play a critical role in their expected equilibrium payoffs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Recently, Nash (2008) considered a non-cooperative bargaining model called the agencies method for a three-person cooperative game and presented some computational results.

  2. It is also well known that an efficient allocation is guaranteed if renegotiation is allowed. See Seidmann and Winter (1998) and Okada (2000), among others.

  3. This stopping rule loses no generality of analysis for our aim to study a three-person game where, if a two-person coalition forms, then one player outside the coalition has no choice except receiving the zero value. A general rule used in Okada (1996) allows sequential formation of coalitions in an \(n\)-person game.

  4. The players’ responses depend surely on the proposal in the present round.

  5. For simplicity of notation, we assume without loss of generality that a central player \(k\) is the same in the sequence \(\{ \sigma ^\delta \}\). If not, choose such a subsequence of it. This is possible since the number of players is finite. It does not matter for the proof whether or not \(k\) is always the same.

  6. We have simplified the set notation \(\{1, 2, 3 \}\) to \(123\) in Table 1. Similar notation is used in this section.

  7. The list of 162 possible configurations is available upon request.

  8. The first inequality can be derived as follows. Let \(r_1\) and \(r_1'\) be the probabilities that player 1 chooses coalitions 12 and 13, respectively, let \(r_2\) and \(r_2'\) be the probabilities that player 2 chooses coalitions 12 and 23, respectively, and let \(r_3\) and \(r_3'\) be the probabilities that player 3 chooses coalitions 23 and 13, respectively. Since \(\theta _1 = 1 - (r_2' + r_3)/2\), \(\theta _2 = 1 - (r_1' + r_3')/2\), and \(\theta _3 = 1 - (r_1 + r_2)/2\), we have \(\theta _1 + \theta _2 + \theta _2 = 3 - ((r_1 + r_1') + (r_2 + r_2') + (r_3 + r_3'))/2 > 3/2\).

  9. Subcases (i) and (ii) are degenerate in the sense that the coalitional values \(v(S)\) satisfy some equality constraint.

  10. There is another constraint, \(\theta _2 + \theta _3 = 1\), which we omit for simplicity of exposition.

  11. We can compute \(\theta _2 = (3v(12) + \delta v(13) - \delta v(23) - (9 + 3 \delta )v_1)/(2 \delta v(12) - 6 \delta v_1)\) and \(\theta _3 = (\delta v(12) + 3v(13) - \delta v(23) - (9 + 3\delta )v_1)/(2 \delta v(13) - 6 \delta v_1)\). Since it is cumbersome to derive a general formula for the expected equilibrium payoffs in subcases (iii) and (iv), we have omitted this derivation.

References

  • Baron D, Ferejohn J (1989) Bargaining in legislatures. Am Polit Sci Rev 83:1181–1206

    Article  Google Scholar 

  • Compte O, Jehiel P (2010) The coalitional Nash bargaining solution. Econometrica 78:1593–1623

    Article  Google Scholar 

  • Montero M (2002) Non-cooperative bargaining in apex games and the kernel. Games Econ Behav 41: 309–321

    Google Scholar 

  • Montero M (2006) Noncooperative foundations of the nucleolus in majority games. Games Econ Behav 54:380–397

    Article  Google Scholar 

  • Nash JF (2008) The agencies method for modeling coalitions and cooperation in games. Int Game Theory Rev 10:539–564

    Article  Google Scholar 

  • Okada A (1996) A noncooperative coalitional bargaining game with random proposers. Games Econ Behav 16:97–108

    Article  Google Scholar 

  • Okada A (2000) The efficiency principle in non-cooperative coalitional bargaining. Jpn Econ Rev 51:34–50

    Article  Google Scholar 

  • Okada A (2010) The Nash bargaining solution in general \(n\)-person cooperative games. J Econ Theory 145:2356–2379

    Article  Google Scholar 

  • Okada A (2011) Coalitional bargaining game with random proposers: theory and application. Games Econ Behav 73:227–235

    Article  Google Scholar 

  • Ray D (2007) A game-theoretic perspective on coalition formation. Oxford University Press, Oxford

    Book  Google Scholar 

  • Seidmann DJ, Winter E (1998) A theory of gradual coalition formation. Rev Econ Stud 65:793–815

    Article  Google Scholar 

  • von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton, NJ

    Google Scholar 

  • Yan H (2002) Noncooperative selection of the core. Int J Game Theory 31:527–540

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Akira Okada.

Additional information

I am grateful to two anonymous referees for helpful comments. I would also like to thank Takeshi Nishimura for excellent assistance with this research. Financial support from the Japan Society for the Promotion of Science under Grant No. (S)20223001 is gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Okada, A. The stationary equilibrium of three-person coalitional bargaining games with random proposers: a classification. Int J Game Theory 43, 953–973 (2014). https://doi.org/10.1007/s00182-014-0413-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00182-014-0413-2

Keywords

JEL Classification

Navigation