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An Experiment on Demand Commitment Bargaining

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Abstract

In this experiment, we compare three implementations of the Winter demand commitment bargaining mechanism: a one-period implementation, a two-period implementation with low delay costs, and a two-period implementation with high delay costs. Despite the different theoretical predictions, our results show that the three different implementations result in similar outcomes in all our investigation domains: namely, coalition formation, alignment with the Shapley value prediction, and satisfaction of the axioms. Our results suggest that a lighter bargaining implementation with only one period is often sufficient in providing allocations that sustain the Shapley value as an appropriate cooperative solution concept, while saving unnecessary time and resource costs.

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Notes

  1. In the formulation of this theorem, it is assumed for simplicity that the delay cost c is equal to the smallest money unit, but the result continues to hold when the delay cost is equal to some integer multiple of the smallest money unit. As stated by [22], the existence of a smallest money unit is both motivated by the necessity of reducing players’ sets of actions to finite sets and of making the exact implementation of the theoretical model possible by experiments.

  2. n is then replaced by the number of players still bargaining at that period.

  3. As the mechanism is defined in a discrete version, the first player demands a smallest unit less in order to leave an extra smallest unit more for the last player and to break his/her indifference between accepting formation of the grand coalition or receiving their individual value. However, this extra money unit becomes negligible as it approaches zero.

  4. The ex post equilibrium of the T-period implementation with \(T>2\) resembles the case where \(T=2\).

  5. We refer to the original paper by [22] for the detailed description of the assumptions under which such results hold. As already mentioned, the delay cost must not be too large. Moreover, the results about the ex ante and the ex post equilibria hold for a discrete version of the mechanism, and when the smallest money unit approaches zero.

  6. All data are available on reasonable request.

  7. The difference in the number of participants between the two mechanisms is a result of variations in the show-up rate among experimental sessions. The data of the 1p treatment is the same as the one used in [2].

  8. Participants received a copy of instruction slides, and prerecorded instruction movies were played. See Appendix A for the English translations of the instruction slides and the comprehension quiz.

  9. The figure is created based on the estimated coefficients of the following linear regressions: \(gc_i = \beta _1 1p_i + \beta _2 2pL_i + \beta _3 2pH_i + \mu _i\), where \(gc_i\) is a dummy variable that takes the value of 1 if the grand coalition is formed, and 0 otherwise, in group i; \(1p_i\), \(2pL_i\), and \(2pH_i\) are dummy variables that take a value of 1 if the 1p, 2pL, and 2pH implementations, respectively, are used, and 0 otherwise. The standard errors are corrected for within-session clustering effects. The statistical tests are based on the Wald test for the equality of the estimated coefficients of the two treatment dummies.

  10. Remember that the Winter mechanism is theoretically defined for strictly convex games. In this game, Player 1 always has a zero marginal contribution and, as such, can be left out of any coalition at no cost for them or the other players.

  11. The figure is created based on the estimated coefficients of the following linear regressions: \(Eff_i = \beta _1 1p_i + \beta _2 2pL_i + \beta _3 2pH_i + \mu _i\) where \(Eff_i \equiv \frac{\sum _i \pi _i}{v(N)}\) is the efficiency measure for group i; and \(1p_i\), \(2pL_i\), and \(2pH_i\) are dummy variables that take a value of 1 for the 1p, 2pL, and 2pH treatment, respectively, and 0 otherwise. The standard errors are corrected for within-session clustering effects. The statistical tests are based on the Wald test for the equality of the estimated coefficients of two treatment dummies.

  12. The error bars are based on the standard errors that are corrected for within-session clustering effects. These standard errors are obtained by running the system of linear regressions described in Sect. 5.4. The statistical tests are based on these regressions.

  13. The figure is created based on the estimated coefficients of the following linear regressions: \(Dis_i = \beta _1 1p_i + \beta _2 2pL_i + \beta _3 2pH_i + \mu _i\), where \(Dis_i\) is the relevant distance measure for groups i. \(1p_i\), \(2pL_i\), and 2pH are dummy variables that take a value of 1 if the 1p, 2pL, or 2pH treatments are used, respectively, and 0 otherwise. The standard errors are corrected for within-session clustering effects. The statistical tests are based on the Wald test for the equality of the estimated coefficients of the two treatment dummies.

  14. Unlike the original decomposition by [1], which ensures orthogonal components, in our decomposition, in general, vectors \(e^{null}\) and \(e^{add}\) are not orthogonal, so that \(<e^{add},e^{null}>\) is not equal to zero. However, in our data, \(2<e^{add},e^{null}>\) are very small (on average, they are \(-\)0.08, \(-\)0.08, and \(-\)0.09 in 1p, 2pL, and 2pH, respectively, based on the estimation results reported below) compared with other components.

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Acknowledgements

The experiments reported in this paper were approved by the Independent Review Board of Yamaguchi University (No. 5). We gratefully acknowledge financial support from the Joint Usage/Research Center at ISER, Osaka University, and Grants-in-aid for Scientific Research from the Japan Society for the Promotion of Science (15K01180, 15H05728, 18K19954, and 20H05631), the Fund for the Promotion of Joint International Research (Fostering Joint International Research) (15KK0123), and the French government-managed l’Agence Nationale de la Recherche under Investissements d’Avenir \(UCA^{JEDI}\) (ANR-15-IDEX-01). In particular, we thank the UCAinACTION project.

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Authors and Affiliations

  1. Université Côte d’Azur, CNRS, GREDEG, Valbonne, France

    Michela Chessa

  2. Institute of Social and Economic Research, Osaka University, Osaka, Japan

    Nobuyuki Hanaki

  3. CNRS, GATE LSE UMR 5824, Université de Saint-Etienne, Saint-Etienne, France

    Aymeric Lardon

  4. Faculty of Global and Science Studies, Yamaguchi University, Yamaguchi, Japan

    Takashi Yamada

Authors
  1. Michela Chessa
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  2. Nobuyuki Hanaki
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  3. Aymeric Lardon
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  4. Takashi Yamada
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Correspondence to Michela Chessa.

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This article is part of the topical collection “Group Formation and Farsightedness” edited by Francis Bloch, Ana Mauleon and Vincent Vannetelbosch.

Appendices

Translated Instructions and Comprehension Quiz

An English translation of the instruction materials as well as the quiz (shown on the screen) can be downloaded from

Verification of Axioms

See Table 6.

Table 6 Winter mechanisms’ Wald tests for the verification of the symmetry, additivity, homogeneity, null player, strong monotonicity, and fairness axioms
Full size table

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Chessa, M., Hanaki, N., Lardon, A. et al. An Experiment on Demand Commitment Bargaining. Dyn Games Appl 13, 589–609 (2023). https://doi.org/10.1007/s13235-022-00463-x

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