Simultaneous but independent ultimatum game: strategic elasticity or social motive dependency?

Abstract

Ultimata bargaining experimentally investigates the responder behavior for multiple proposers, that is, the responder’s decision to accept or reject an offer conditional on another parallel offer. Responders’ strategies combine the two formally independent but parallel games as if inducing competition among proposers by more frequently rejecting an offer when it is the lower one of the two offers received simultaneously. Furthermore, proposers’ public offers strongly correlate, due to adapting over repetitions to the parallel offer or because of only slightly outbidding the other proposer’s offer if known before their own announcement. Social preferences of inequity aversion or of pure altruism for income distributions resulting from simultaneous ultimata cannot explain a positive dependency of own offers on other parallel offers, and joint responsibility is proposed as a reference-dependent social motive.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Notes

  1. 1.

    In the standard ultimatum game two players divide an amount of money between them in the following sequence. The first player (the proposer) offers a part of the total amount and the second player (the responder) decides whether to accept or reject this offer. If the second player accepts, the amount is divided as suggested by the proposer. If the offer is rejected both players receive nothing. In the subgame perfect Nash equilibrium the responder accepts any positive amount and the proposer offers the smallest amount possible. Experimental or field investigations of the ultimatum game show that actual players of the game make non-negligible offers and responders reject low offers (i.e., Güth et al. 1982; Thaler 1988; Roth 1995; Camerer 2003). The various behavioral investigations of the ultimatum game also include high stakes or repeated interaction. High stakes slightly decrease rejection rates which in return leads to lower offers (Slonim and Roth 1998; Cameron 1999). Repeated interaction with fixed matching partners fosters higher rejection rates, presumably for building up the reputation of a tough player and thus increase future offers (Slembeck 1999). Furthermore, subjects prefer to reject only part of the offer when this is possible, instead of choosing zero payoffs for both players (Andreoni et al. 2003).

  2. 2.

    Through backward induction, rational responders should accept all positive offers, and proposers should offer the smallest amount possible (one eurocent here). Usually, in ultimatum experiments, responders reject small positive offers, and proposers offer between 30 and 50% of the pie.

  3. 3.

    Giving by the other proposer would reduce my givings to the responder due to diminishing utility. Warm glow would remain constant and, thus, independent of the other proposer’s givings. For rivalry to hold, the unreasonable responder behavior of consistently rejecting the lower offer must be fully anticipated.

  4. 4.

    The ranges are formed around the provided options under the one-shot strategy method. The appearance of the offer on the left side versus the right side of the screen did not affect general rejection rates. Only for the same amount for both proposers the rejection rates for the right side offer increases by 3.6%. Order effects of the offers as in Galinsky and Mussweiler (2001) are not further considered, and in the analysis rejection rates were averaged for all the same offers.

  5. 5.

    Significance levels are confirmed in a Wilcoxon signed rank test, with the only difference for comparing the standard ultimatum game with the simultaneous ultimata game for offers \(<20\)% being significant with \(p<0.01\).

References

  1. Andreoni J (1990) Impure altruism and donations to public goods: a theory of warm-glow giving. Econ J 100:464–477

    Article  Google Scholar 

  2. Andreoni J, Castillo M, Petrie R (2003) What do bargainers’ preferences look like? Experiments with a convex ultimatum game. Am Econ Rev 93:672–685

    Article  Google Scholar 

  3. Batson C D, Lishner D A, Carpenter A, Dulin L, Harjusola-Webb S, Stocks E, Gale S, Hassan O, Sampat B (2003) “... As you would have them do unto you”: does imagining yourself in the other’s place stimulate moral action? Pers Soc Psychol Bull 29:1190–1201

    Article  Google Scholar 

  4. Bohnet I, Zeckhauser R (2004) Social comparisons in ultimatum bargaining. Scand J Econ 106:495–510

    Article  Google Scholar 

  5. Bolle F, Breitmoser Y, Heimel J, Vogel C (2012) Multiple motives of pro-social behavior: evidence from the solidarity game. Theory Decis 72:303–321

    Article  Google Scholar 

  6. Bolton GE, Ockenfels A (1998) Strategy and equity: an ERC-analysis of the Güth-van Damme game. J Math Psychol 42:215–226

    Article  Google Scholar 

  7. Bolton GE, Ockenfels A (2000) ERC: a theory of equity, reciprocity, and competition. Am Econ Rev 90:166–193

    Article  Google Scholar 

  8. Bosch-Rosa C (2015) Tale of two tails: preferences of neutral third-parties in three-player ultimatum games. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2261081. Accessed 29 Oct 2018

  9. Brandts J, Charness G (2011) The strategy versus the direct-response method: a first survey of experimental comparisons. Exp Econ 14:375–398

    Article  Google Scholar 

  10. Camerer C (2003) Behavioral game theory: experiments in strategic interaction. The Roundtable series in behavioral economics. Russell Sage Foundation, New York

    Google Scholar 

  11. Cameron LA (1999) Raising the stakes in the ultimatum game: experimental evidence from Indonesia. Econ Inq 37:47–59

    Article  Google Scholar 

  12. Cason TN, Mui V-L (1998) Social influence in the sequential dictator game. J Math Psychol 42:248–265

    Article  Google Scholar 

  13. Coats JC, Hannan RL, Rankin FW, Towry KL (2013) Does reciprocity have a dark side: behavior in ultimatum games with multiple proposers. https://pdfs.semanticscholar.org/e0f1/e1556294367f3203ab004a502469b705bbce.pdf. Accessed 29 Oct 2018

  14. Cooper DJ, Dutcher EG (2011) The dynamics of responder behavior in ultimatum games: a meta-study. Exp Econ 14:519–546

    Article  Google Scholar 

  15. Croson R, Fatas E, Neugebauer T (2005) Reciprocity, matching and conditional cooperation in two public goods games. Econ Lett 87:95–101

    Article  Google Scholar 

  16. Duffy J, Feltovich N (1999) Does observation of others affect learning in strategic environments? An experimental study. Int J Game Theory 28:131–152

    Article  Google Scholar 

  17. Dufwenberg M, Kirchsteiger G (2004) A theory of sequential reciprocity. Games Econ Behav 47:268–298

    Article  Google Scholar 

  18. Engelmann D, Strobel M (2004) Inequality aversion, efficiency, and maximin preferences in simple distribution experiments. Am Econ Rev 94:857–869

    Article  Google Scholar 

  19. Engelmann D, Strobel M (2012) Deconstruction and reconstruction of an anomaly. Games Econ Behav 76:678–689

    Article  Google Scholar 

  20. Falk A, Fehr E, Fischbacher U (2003) On the nature of fair behavior. Econ Inq 41:20–26

    Article  Google Scholar 

  21. Falk A, Fischbacher U (2006) A theory of reciprocity. Games Econ Behav 54:293–315

    Article  Google Scholar 

  22. Fehr E, Schmidt KM (1999) A theory of fairness, competition, and cooperation. Q J Econ 114:817–868

    Article  Google Scholar 

  23. Fischbacher U (2007) z-Tree: Zurich toolbox for ready-made economic experiments. Exp Econ 10:171–178

    Article  Google Scholar 

  24. Fischbacher U, Gächter S, Fehr E (2001) Are people conditionally cooperative? Evidence from a public goods experiment. Econ Lett 71:397–404

    Article  Google Scholar 

  25. Galinsky AD, Maddux WW, Gilin D, White JB (2008) Why it pays to get inside the head of your opponent: the differential effects of perspective taking and empathy in negotiations. Psychol Sci 19:378–384

    Article  Google Scholar 

  26. Galinsky A D, Mussweiler T (2001) First offers as anchors: the role of perspective-taking and negotiator focus. J Pers Soc Psychol 81:657

    Article  Google Scholar 

  27. Grosskopf B (2003) Reinforcement and directional learning in the ultimatum game with responder competition. Exp Econ 6:141–158

    Article  Google Scholar 

  28. Güth W, Huck S, Müller W (2001) The relevance of equal splits in ultimatum games. Games Econ Behav 37:161–169

    Article  Google Scholar 

  29. Güth W, Huck S, Ockenfels P (1996) Two-level ultimatum bargaining with incomplete information: an experimental study. Econ J 106:593–604

    Article  Google Scholar 

  30. Güth W, Marchand N, Rulliere J-L (1997) On the reliability of reciprocal fairness: an experimental study. Wirtschaftswissenschaftliche Fakultät, Berlin

    Google Scholar 

  31. Güth W, Schmittberger R, Schwarze B (1982) An experimental analysis of ultimatum bargaining. J Econ Behav Organ 3:367–388

    Article  Google Scholar 

  32. Güth W, van Damme E (1998) Information, strategic behavior, and fairness in ultimatum bargaining: an experimental study. J Math Psychol 42:227–247

    Article  Google Scholar 

  33. Halko M-L, Seppälä T (2006) Ultimatum game experiments. Unpublished. Helda-Helsingin yliopiston digitaalinen arkisto. https://helda.helsinki.fi/bitstream/handle/10138/16631/ultimatu.pdf?sequence=1. Accessed 29 Oct 2018 (Finland)

  34. Harrison G, McCabe K (1996) Expectations and fairness in a simple bargaining experiment. Int J Game Theory 25:303–327. https://doi.org/10.1007/BF02425260

    Article  Google Scholar 

  35. Hennig-Schmidt H, Irlenbusch B, Rilke RM, Walkowitz G (2017) Asymmetric outside options in ultimatum bargaining: a systematic analysis. Int J Game Theory 47(1):301–329

    Article  Google Scholar 

  36. Heyes C (2011) Automatic imitation. Psychol Bull 137:463–483

    Article  Google Scholar 

  37. Hopkins E (2008) Inequality, happiness and relative concerns: what actually is their relationship? J Econ Inequal 6:351–372

    Article  Google Scholar 

  38. Kagel J, Wolfe K (2001) Tests of fairness models based on equity considerations in a three-person ultimatum game. Exp Econ 4:203–219

    Article  Google Scholar 

  39. Keser C, Van Winden F (2000) Conditional cooperation and voluntary contributions to public goods. Scand J Econ 102:23–39

    Article  Google Scholar 

  40. Knez MJ, Camerer CF (1995) Outside options and social comparison in three-player ultimatum game experiments. Games Econ Behav 10:65–94

    Article  Google Scholar 

  41. Oxoby RJ, McLeish KN (2004) Sequential decision and strategy vector methods in ultimatum bargaining: evidence on the strength of other-regarding behavior. Econ Lett 84:399–405

    Article  Google Scholar 

  42. Rabin M (1993) Incorporating fairness into game theory and economics. Am Econ Rev 83:1281–1302

    Google Scholar 

  43. Roth A (1995) Bargaining experiments. Handb Exp Econ 195:243–348

    Google Scholar 

  44. Roth AE, Prasnikar V, Okuno-Fujiwara M, Zamir S (1991) Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh, and Tokyo: an experimental study. Am Econ Rev 81:1068–95

    Google Scholar 

  45. Selten R (1967) Die Strategiemethode zur Erforschung des engeschränkt rationalen Verhaltens im Rahmen eines Oligopolexperiments. In: Sauermann H (ed) Beiträge zur Experimentellen Wirtschaftsforschung. J. C. B. Mohr, Tübingen, pp 136–168

    Google Scholar 

  46. Selten R, Ockenfels A (1998) An experimental solidarity game. J Econ Behav Organ 34:517–539

    Article  Google Scholar 

  47. Simonson I (1989) Choice based on reasons: the case of attraction and compromise effects. J Consum Res 16:158–174

    Article  Google Scholar 

  48. Slembeck T (1999) Reputations and fairness in bargaining—experimental evidence from a repeated ultimatum game with fixed opponents. Experimental 9905002, EconWPA, May 1999

  49. Slonim R, Roth AE (1998) Learning in high stakes ultimatum games: an experiment in the Slovak Republic. Econometrica 66:569–596

    Article  Google Scholar 

  50. Stewart N, Chater N, Brown G (2006) Decision by sampling. Cogn Psychol 53:1–26

    Article  Google Scholar 

  51. Thaler RH (1988) The ultimatum game. J Econ Perspect 2:195–206

    Article  Google Scholar 

  52. Tversky A, Kahneman D (1974) Judgment under uncertainty: heuristics and biases. Science 185:1124–1131

    Article  Google Scholar 

  53. Tversky A, Kahneman D (1991) Loss aversion in riskless choice: a reference-dependent model. Q J Econ 106:1039–61

    Article  Google Scholar 

  54. Van Baaren RB, Holland RW, Kawakami K, Van Knippenberg A (2004) Mimicry and prosocial behavior. Psychol Sci 15:71–74

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Philipp E. Otto.

Electronic supplementary material

Below is the link to the electronic supplementary material.

182_2018_646_MOESM1_ESM.pdf

Supplementary material 1 (pdf 118 KB)

Appendices

Appendix

Regression Analysis

The rejection rates are influenced not only by the current offer, but also by the parallel offer received simultaneously. The restricted maximum likelihood regression analysis for the repeated (1120 observations for 28 responders) and the one-shot (4200 observations of 84 responders) ultimata games clearly illustrates this dependency. Note that the relative comparison, as being the lower offer of the two, has a stronger influence on the rejection rate than the absolute amount of the parallel offer. The absolute amount is important as of the previous offer in the repeated ultimata game. Here also the period (with 20 repetitions) remains influential, while the information treatment does not have any significant effect. In both the repeated and the one-shot ultimata game the best model, according to the Akaike information criterion (AIC), results when the amount of the parallel offer is replaced by its relative differences (i.e., the current offer being the lower offer) capturing a kind of “comparative dependency”.

Table 3 Liner mixed effects regression model with the rejection rate as the dependent variable. Estimated regression coefficients of the fixed effects are shown with random effects for subjects (28 in the repeated and 84 in the one-shot ultimatum game) and p-values in brackets. Akaike information criterion (AIC) provided for model comparison.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Otto, P.E., Dittmer, D. Simultaneous but independent ultimatum game: strategic elasticity or social motive dependency?. Int J Game Theory 48, 61–80 (2019). https://doi.org/10.1007/s00182-018-0646-6

Download citation

Keywords

  • Multiple ultimatum game
  • Context dependency
  • Responder rejection rate
  • Joint responsibility
  • Comparative dependency

JEL Classification

  • C78
  • C90
  • C91