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.
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Notes
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).
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.
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.
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.
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 \(p<0.01\).
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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”.
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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
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DOI: https://doi.org/10.1007/s00182-018-0646-6
Keywords
- Multiple ultimatum game
- Context dependency
- Responder rejection rate
- Joint responsibility
- Comparative dependency