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Balancedness. Emerson’s Theory in the Upanishads, in Cooperative Game Theory, and Beyond

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Abstract

This article builds on an important paper by Emerson (1962) and its central idea that relations that are characterized by unbalanced dependency or by differential power-over tend to balance out. It can be shown that the Shapley (1953) value from Cooperative Game Theory and Old Indian (post-Vedic, pre-classical) texts can be linked to Emerson’s paper in a fruitful manner. Emerson’s approach presupposes interpersonally comparable utilities. Adducing some vague ideas of Emerson (1987), interpersonal comparability can be defended. I then apply Emerson’s balancing approach to the standard (to microeconomists) Edgeworth exchange box. This allows to determine a particular allocation within the core. Inversely, taking the “revealed preference” approach, allocations resulting from Emerson bargaining help to determine utility functions with interpersonal comparability. Furthermore, a dynamic bargaining model is presented that is somewhat related to the Ståhl-Rubinstein model, but builds on balancedness. As an aside, I offer a few comments on the perennial problem of defining power.

Keywords

  • Emerson
  • Balancedness
  • Shapley value
  • Old Indian rank-order contests
  • Interpersonal comparability of utility

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Notes

  1. 1.

    This power-definition paradox should not be confused with the well-known paradox of power which refers to the theoretical possibility that more parliamentary seats (or agenda-setting power or …) may work against the interest of the allegedly more powerful party. See, for example, Ordeshook (1992, p. 162).

  2. 2.

    Translation by Olivelle (1998)

  3. 3.

    Translation based on Olivelle (1998), copied from Wiese (2021)

  4. 4.

    The algorithmic approach amounts to calculating the average “marginal contributions” of players along all sequences of players. Wiese (2021) manages to link the algorithmic approach to Old Indian texts that are about as old as the Upaniṣads.

  5. 5.

    For further examples see Wiese (2009) who discusses emotional dependence, centrality in a network, and armed robbery.

  6. 6.

    There is a pun here. “I see you” may be short for “I see you and your preferences. I have some idea about how much you might lose if we do not strike a bargain. I sort of know where you would be without me.”.

  7. 7.

    The paper is rather cryptic and I remain agnostic as to whether the main part of the paper (following the introduction) might deserve a modern reconstruction as Andreas Tutić suggests in a private communication.

  8. 8.

    It seems that Emerson (1987, pp. 12–13) uses “cardinal utility” and ICU interchangeably.

  9. 9.

    Malinowski (1926, p. 40).

  10. 10.

    https://www.letras.mus.br/belinda-carlisle/364810/ accessed on October 13, 2020.

  11. 11.

    https://lyricstranslate.com/en/markus-schulz-nothing-without-me-lyrics.html accessed on October 13, 2020.

  12. 12.

    Consult the survey article by Bresnahan (1989).

  13. 13.

    See, for example, Mizruchi and Potts (1998).

  14. 14.

    See the programmatic paper by Gewirtz (1956).

  15. 15.

    See the monograph by Felsenthal and Machover (1998).

  16. 16.

    I have also toyed with the term “Emerson-Walras economy”. In the Walras economy, agents possess endowments of goods. They consume these endowments or buy other bundles. The possible links between agents in the newly defined model correspond to the endowments in the Walras model, while the concrete links chosen by the dealing agents are somewhat analogous to the bundles chosen by the households. In the Walras model, the solution concept concerns the prices for all the goods. They need to be set in such manner that every agent maximizes his utility and such that supply equals demand. In our new model, the role of prices is taken over by the terms. While this analogy may be suggestive to some extend, it misses the main point of the Walras model. In that model, agents do not deal directly with each other, but optimally adjust to given prices. Thus, in contrast to the Edgeworth model, the Walras model is about anonymous exchange, not about person-to-person exchange.

  17. 17.

    If a specific pair of agents is involved in several deals, one might enlarge the player and link sets by providing copies of the players and their links, together with appropriate subindexing.

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Acknowledgments

The author thanks members of the sociological research seminar at the University of Leipzig, in particular Andreas Tutić and Thomas Voss, and also members of the network “Private Macht und privatrechtliche Gestaltungsfreiheit” (sponsored by Deutsche Forschungsgemeinschaft) for very insightful discussions. Katharina Zalewski provided able research assistance as did Johanna Freistühler and Alexander Singer. Andreas Tutić helped to improve the second-to-last version of this paper.

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Appendix: Emerson-Edgeworth Network

Appendix: Emerson-Edgeworth Network

Consider a set of agents \(N\) and the full undirected network \(PB\) on \(N\), that is, \(PB\) consists of all subsets \(\left\{ {i,j} \right\}\) with \(i,j \in N\) and \(i \ne j\). A subset \(\left\{ {i,j} \right\}\) is called a link and \(PB\) is called the set of possible bargains.Footnote 17 Let \(NW\) (short for “network”) be a subset of \(PB\). Agents \(i\) and \(j\) with \(\left\{ {i,j} \right\} \in NW\) are said to “deal with each other”. For agent \(i \in N\), \(NW\left( i \right) \subseteq NW\) denotes the set of links in \(NW\) that contain agent \(i\). For any pair of dealing agents \(i,j\), let \(t_{ij} \) denote the terms for \(i\) in his dealing with \(j\), while \(t_{ji}\) denotes the terms for \(j\) when dealing with \(i\). Let \(t = (t_{ij} )_{i \in N,j \in N}\) be a tuple of terms. Let \(t \oplus \hat{t}_{ij} \oplus \hat{t}_{ji}\) be the tuple where, on the basis of \(t\), only the entries for the deal between the players \(i\) and \(j\) are replaced, by \(\hat{t}_{ij}\) and \(\hat{t}_{ji}\), respectively. Each agent \(i \in N\) has a utility function \(U^{i} \left( {NW,t} \right)\).

I now present two definitions, the first defining an Emerson-Edgeworth network and the second an appropriate equilibrium for such a network.

Definition: A tuple \(\left( {N,PB,\left( {U^{i} } \right)_{i \in N} } \right)\) is called an Emerson-Edgeworth network where \(N\) is a set of agents, \(PB\) the full undirected network on \( N\), and, for agent \(i \in N\), \(U^{i}\) is a utility function with two arguments, a partial undirected network and a tuple of terms.

Definition: Let \(\left( {N,PB,\left( {U^{i} } \right)_{i \in N} } \right)\) be an Emerson-Edgeworth network. The tuple \(\left( {NW,t} \right)\) is an Emerson-Edgeworth equilibrium (or a balanced-bargaining network) if

  • \(U^{i} \left( {NW,t} \right) \ge U^{i} \left( {NW\backslash NW^{{{\text{trim}}}} ,t} \right)\) for all agents \(i\) and all \(NW^{{{\text{trim}}}} \subseteq NW\left( i \right)\),

  • \(U^{i} \left( {NW,t} \right) - U^{i} \left( {NW\backslash \left\{ {i,j} \right\},t} \right) = U^{j} \left( {NW,t} \right) - U^{j} \left( {NW\backslash \left\{ {i,j} \right\},t} \right)\) for all pairs of agents \(i\) and \(j\), and

  • \(U^{i} \left( {NW \cup \left\{ {i,j} \right\},t \oplus \hat{t}_{ij} \oplus \hat{t}_{ji} } \right) < U^{i} \left( {NW,t} \right)\) or \(U^{j} \left( {NW \cup \left\{ {i,j} \right\},t \oplus \hat{t}_{ij} \oplus \hat{t}_{ji} } \right) < U^{j} \left( {NW,t} \right)\) for any pairs of agents \(i,j\) with \(\left\{ {i,j} \right\} \notin NW\) and any terms \(\hat{t}_{ij}\) and \(\hat{t}_{ji}\)

hold.

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Wiese, H. (2021). Balancedness. Emerson’s Theory in the Upanishads, in Cooperative Game Theory, and Beyond. In: Krumpal, I., Raub, W., Tutić, A. (eds) Rationality in Social Science. Springer VS, Wiesbaden. https://doi.org/10.1007/978-3-658-33536-6_9

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