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Equity dynamics in bargaining without information exchange

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Abstract

In this paper, completely uncoupled dynamics for n-player bargaining are proposed that mirror key behavioral elements of early bargaining and aspiration adjustment models (Zeuthen, 1930; Sauermann and Selten, 118:577–597 1962). Individual adjustment dynamics are based on directional learning adjustments, solely driven by histories of own realized payoffs. Bargaining this way, all possible splits have positive probability in the stationary distribution of the process, but players will split the pie almost equally most of the time. The expected waiting time for almost equal splits to be played is quadratic.

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Notes

  1. Axiomatic bargaining solutions such as Nash’s (1950) explicitly consider relative bargaining strengths (“outside options”). Harsanyi (1956) shows that the solutions obtained by Zeuthen’s dynamic model and Nash’s axiomatization coincide.

  2. The number of adjustments needed to reach such outcomes may not be the same for all players, hence there is ground to think of some inequality in terms of bargaining efforts or more general concepts of social exchange equity (Adams, 1965) depending on initial states.

  3. Other iterative bargaining procedures such as Raiffa (1953), Luce and Raiffa (1957), Kalai (1977), and John and Raith (1999) start from inside the bargaining set. The differences between these approaches and ours is similar in spirit to the differences with Zeuthen that are discussed in detail here.

  4. In Raiffa (1953), Luce and Raiffa (1957), Kalai (1977), and John and Raith (1999), the process moves the other way around and iterative steps towards the Pareto frontier are negotiated.

  5. See Babichenko (2010, 2012) for convergence comparisons of uncoupled and completely uncoupled dynamics.

  6. See also Tietz and Weber (1972), Tietz (1975), Weber (1976), and Tietz and Weber (1978), Tietz and Bartos (1983), Crössmann and Tietz (1983), and Tietz et al. (1978). Roth (1995) discusses subsequent experiments.

  7. Actually, such directional adjustments may turn out to be strategically rationalizable in these higher information environments. (I thank an anonymous referee for pointing this out.)

  8. See also Gale et al. (1995), Nowak et al. (2000), Konrad & Morath (2015) for evolutionary models of “ultimatum bargaining” (Güth et al. 1982), or Binmore et al. (1998) for an evolutionary analysis of alternating-offer “Rubinstein bargaining” (Rubinstein 1982).

  9. “Stochastic stability” is an equilibrium refinement that is different from “evolutionary stability” based on replicator arguments (Maynard Smith and Price, 1973; Maynard Smith, 1974) or from “evolutionary stability” in finite populations (Schaffer, 1988; Nowak et al., 2004).

  10. The difference between these convergence concepts is addressed in more detail in Young (2009), see also Babichenko (2012)

  11. It will be convenient to have set up the process with these Poisson clocks when we turn to convergence times. For the meantime, it is also possible to think of agents being activated uniformly at random in discrete time.

  12. The linear function is an approximation for more general functions or a lower bound for functions that first-order dominate the linear bound (e.g. more convex or step functions). Using a d i with \(a=\frac {f(\delta )}{\delta }\) for any convex function f(⋅) with f(0)=0, f ′(x)>0 and f ″(x)≥0 for all x>0, for example, “understates” the stickiness and works in the opposite direction in terms of our results.

  13. Assuming r<a δ guarantees that this assumption holds for any current \({d_{i}^{t}}>0\) of any player.

  14. Note that we may drop 2a r from this last expression because 2a r<2r<2δ<1.

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Acknowledgments

Much of this work is part of the author’s D.Phil. (University of Oxford, 2011; supported by the Economic and Social Research Council [grant number PTA-031-2005-00118] and Postdoc under the Office of Naval Research Grant [grant number N00014-09-1-0751]. I also acknowledge current support by the European Commission through the ERC Advanced Investigator Grant ‘Momentum’ [grant number 324247]. I am particularly thankful to Peyton Young and Steffen Issleib, with whom I worked on this topic, and to Françoise Forges, Chris Wallace and Francis Dennig for comments on earlier versions of this model. The proof technique used is due to previous, joint unpublished work. I would also like to thank anonymous referees and a helpful editor for valuable comments on earlier versions of the paper.

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  1. Department of Social Sciences, ETH Zürich, Clausiusstrasse 37-C3, 8092, Zürich, Switzerland

    Heinrich H. Nax

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Correspondence to Heinrich H. Nax.

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Nax acknowledges support by the European Commission through the ERC Advanced Investigator Grant ‘Momentum’ (Grant No. 324247).

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Nax, H.H. Equity dynamics in bargaining without information exchange. J Evol Econ 25, 1011–1026 (2015). https://doi.org/10.1007/s00191-015-0405-9

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