Bargaining Experiments Research Articles

A Dynamic Level-k Model in Sequential Games

Backward induction is a widely accepted principle for predicting behavior in sequential games. In the classic example of the “centipede game,” however, players frequently violate this principle. An alternative is a “dynamic level-k” model, where players choose a rule from a rule hierarchy. The rule hierarchy is iteratively defined such that the level-k rule is a best response to the level-(k-1) rule, and the level-∞ rule corresponds to backward induction. Players choose rules based on their best guesses of others' rules and use historical plays to improve their guesses. The model captures two systematic violations of backward induction in centipede games, limited induction and repetition unraveling. Because the dynamic level-k model always converges to backward induction over repetition, the former can be considered to be a tracing procedure for the latter. We also examine the generalizability of the dynamic level-k model by applying it to explain systematic violations of backward induction in sequential bargaining games. We show that the same model is capable of capturing these violations in two separate bargaining experiments. This paper was accepted by Peter Wakker, decision analysis.

Resistance to Truthful Revelation in Bargaining: Persistent Bid Shading and the Play of Dominated Strategies

We report results from a simultaneous bilateral bargaining experiment with attention to the effects of a settlement bonus on strategic decision-making behavior. In instances with a sufficiently large settlement bonus, truthful revelation emerges as the dominant strategy. However previous work (Parco and Rapoport, 2004) has experimentally tested this ― Bonus Effect and found that although the presence of a settlement bonus improves efficiency, behavior falls drastically short of the normative predictions. This finding illustrates the persistent tendency of decision makers to bid strategically, i.e. shading their bids, even when truthful revelation is a strictly dominant strategy. Herein we investigate the influence of the framing of information and look for ways to nudge decision makers toward making better choices in these strategic environments. Additional results from an adaptive reinforcement-based learning model are discussed as they relate to a potential innate bias for strategic misrepresentation even when contrary to self-interest and collective interest.

A Dynamic Level-<i>k</i> Model in Sequential Games

Backward induction is a widely accepted principle for predicting behavior in sequential games. In the classic example of the game'', however, players frequently violate this principle. An alternative is a dynamic level-k model, where players choose a rule from a rule hierarchy. The rule hierarchy is iteratively defined such that the level-k rule is a best-response to the level-(k-1) rule and the level-infinity rule corresponds to backward induction. Players choose rules based on their best guesses of others' rules and use historical plays to improve their guesses. The model captures two systematic violations of backward induction in centipede games, limited induction and repetition unraveling. Since the dynamic level-k model always converges to backward induction over repetition, the former can be considered to be a tracing procedure for the latter. We also examine the generalizability of the dynamic level-k model by applying it to explain systematic violations of backward induction in sequential bargaining games. We show that the same model is capable of capturing these violations in two separate bargaining experiments.

When social accounts promote acceptance of unfair ultimatum offers: The role of the victim’s stress responses to uncertainty and power position

We examined which type of social account (denying responsibility versus apologizing) following an unfair offer makes recipients more likely to accept the offer in ultimatum bargaining. We identified stress responses to uncertainty as an individual difference factor that should moderate the relative effectiveness of these social accounts. A denial should make acceptance of an unfair offer more likely among recipients who respond to uncertainty with low stress. An apology should make such acceptance more likely among recipients who respond with high stress. Further, we argued that this cross-over interaction should be observed particularly among recipients interacting with a high power allocator. Two ultimatum bargaining experiments supported these ideas. Employing the perspective of victims of unfairness, the present research identifies a relevant individual difference moderator of the effectiveness of social accounts in bargaining situations and identifies power as a situational variable that promotes the expression of this factor.

Negotiation Under Possible Third Party Settlement

The effect of possible third party settlement on negotiation behaviour is studied in an economic bargaining experiment. The bargaining phase is preceded by a production phase that allows for different fairness principles to guide the division of the total production value. The experimental results show that a possible third party settlement lowers the dispute costs by reducing the number of rounds of alternating offers. In the presence of a third party, negotiators make first offers that are more strongly related to their production, which reduces the number of rounds of bargaining. The production phase has an effect on the distributional property of the settlements. In negotiations where third party settlement is an option, the negotiation outcome shifts towards a more unequal outcome, more in line with each person's contribution.

Strategic Ignorance in Bargaining

In his classic article An Essay on Bargaining Schelling (1956) argues that ignorance might actually be strength rather than weakness. We test and confirm Schelling's conjecture in a simple take-it-or-leave bargaining experiment where the proposer can choose between two possible offers. Option A always gives the proposer a higher payoff than option B. The payoff of the responder depends on the (randomly determined) state of nature, i.e., in state s2 payoffs of the two players are aligned while they are not in state s1. The responder is always informed about the actual state. The proposer knows the actual state in our first treatment but not in the second. We find that proposers indeed benefit from ignorance because the responders accept almost all offers (even the unfavorable ones) if the payoffs of the responder have not been transparent for the proposer. In additional treatments we investigate situations where the proposer can deliberately remain ignorant. One could assume that remaining ignorant on purpose would be punished by the responder at least if an unfavorable outcome results. Surprisingly, we find that strategically remaining ignorant tends to be beneficial for the proposer particularly if the responder does not know with certainty whether it was the proposer's intention to remain ignorant or whether it was not her intention.

The culture of private negotiation: Endogenous price anchors in simple bilateral bargaining experiments

Using bilateral bargaining data from laboratory experimental markets, this paper demonstrates how common knowledge about price averages from past bargains impacts current negotiated prices. This common information creates an endogenous and declining anchor for bargaining agents. The culture of private negotiation leads parties to agreements below the anchor. Currently negotiated prices below the anchor cause reported averages in the next period to fall. The relation between the endogenous anchor and the negotiating culture leads to a downward drift in negotiated prices. Through at least 20 periods of bargaining, relative earnings are distributed toward buyers and away from sellers.

Inequality Aversion and Efficiency with Ordinal and Cardinal Social Preferences – An Experimental Study

In this paper, we report on a series of free-form bargaining experiments in which two players have to distribute four indivisible goods among themselves. In one treatment, players are informed about the monetary payoffs associated with each bundle of goods; in a second treatment only the ordinal ranking of the bundles is given. We find that in both cases, inequality aversion plays a prominent role. In the ordinal treatment, individuals apparently use the ranks in the respective preference orderings over bundles of goods as a substitute for the unknown monetary value. Allocations that distribute the value (money or ranks, respectively) most equally serve as natural “reference points” for the bargaining processes. Frequently, such “equal split” allocations are chosen by our subjects even though they are Pareto dominated; but also if they are rejected for that reason they matter in a specific way: whether a Pareto optimal allocation is chosen or not depends on whether or not it is a Pareto improvement relative to the equal split. Interestingly, we find much less Pareto-damaging behavior due to inequality aversion in the ordinal treatment.

Inequality aversion and efficiency with ordinal and cardinal social preferences—An experimental study

In this paper, we report on a series of free-form bargaining experiments in which two players have to distribute four indivisible goods among themselves. In one treatment, players are informed about the monetary payoffs associated with each bundle of goods; in a second treatment only the ordinal ranking of the bundles is given. We find that in both cases, inequality aversion plays a prominent role. In the ordinal treatment, individuals apparently use the ranks in the respective preference orderings over bundles of goods as a substitute for the unknown monetary value. Allocations that distribute the value (money or ranks, respectively) most equally serve as natural “reference points” for the bargaining processes. Frequently, such “equal split” allocations are chosen by our subjects even though they are Pareto dominated. Whether a Pareto optimal allocation is chosen or not depends on whether or not it is a Pareto improvement relative to the “equal split” reference allocation. We find less Pareto-damaging behavior due to inequality aversion in the ordinal than in the cardinal treatment.

Loyalty, protocol, bargainer characteristics, and rationality in an experimental investigation of bilateral bargaining in dyads

This article analyses data from a bargaining experiment where dyads (i.e. two players) bargain over the chances (in terms of lottery tickets) to win a shrinking monetary reward. The analysis models both time-independent and time-dependent measures of rationality, which are derived from the negotiated distribution of lottery tickets, as a function of bargainers' stated loyalty, individual characteristics, and bargaining protocol. Statistical results suggest that constrained self-interest dominates both measures of rationality, which are significantly affected by pre-bargain cheap talk, first-offer restrictions, loyalty, gender, and academic major.

What ethics can learn from experimental economics — If anything

Relying on the specific example of ultimatum bargaining experiments this paper explores the possible role of empirical knowledge of behavioural “norm(ative) facts” within the search for (W)RE — (Wide) Reflective Equilibria on normative issues. Assuming that “pro-social” behaviour “reveals” moral orientations, it is argued that these “norm-facts” can and should be used along with stated preferences in justificatory arguments of normative ethics and economics of the “means to given ends” variety. At the same time behavioural norm-facts are so heterogeneous that any hopes to reach an inter-personally agreed (W)RE in matters moral seem futile.

Learning in a multilateral bargaining experiment

This paper analyzes data from an investigation of a majoritarian bargaining experiment. A learning model is proposed to account for the evolution of play in this experiment. It is also suggested that an adjustment must be made to account for the panel structure of the data. Such adjustments have been used in other fields and are known to be important as unadjusted standard errors may be severely biased downward. These results indicate that this adjustment also has an important effect in this application. Furthermore, an efficient estimator that takes into account heterogeneity across players is proposed. A unique learning model to account for the paths of play under two different amendment rules cannot be rejected with the standard estimator with adjusted standard errors, however it can be rejected using the efficient estimator. The data and the estimated learning model suggest that after proposing “fair” divisions, subjects adapt and their proposals change rapidly in the treatment where uneven proposals are almost always accepted. Their beliefs in the estimated learning model are influenced by more than just the most recent outcomes.

Shaping cooperation behavior: The role of accessibility experiences and uncertainty

AbstractThe present research investigates the influence on cooperative behavior of accessibility experiences associated with the retrieval of fairness‐relevant information from memory. We argue that the decision whether to cooperate in negotiations depends not only on information about the appropriateness of the negotiation procedure, but also on the experience of how difficult or easy it is to come up with this information. Supporting this hypothesis, it is shown that in the context of a bargaining experiment, participants' experiences of ease or difficulty in retrieving unfair aspects of the respective negotiation procedure strongly influence their cooperation behavior. In addition, we hypothesize and empirically substantiate that the influence of accessibility experiences on cooperation behavior occurs particularly in situations of certainty salience. Implications for future research on cooperation and on accessibility experiences are discussed. Copyright © 2009 John Wiley & Sons, Ltd.

Cultural differences between Tibetans and ethnic Han Chinese in ultimatum bargaining experiments

We compare ultimatum bargaining behaviour between Tibetans in Lhasa and ethnic Han Chinese in Xiamen with a focus on identifying how cultural traits and religious beliefs affect behaviour. A control group in Singapore was used to account for possible non-cultural explanations for the observed behavioural differences. It was found that, compared to Han Chinese, Tibetans are more likely to accept offers in the ultimatum game and that their decisions are unrelated to the actual offer size.

Giving in Dictator Games: Regard for Others or Regard by Others?

Recent bargaining experiments demonstrate an effect of anonymity and incomplete information on behavior. This has rekindled the question whether “fair” behavior is inspired by regard for others or driven by external forces. To test this, we compare a dictator game treatment that provides receivers with information about the source of offers with one that does not, controlling for anonymity in a double‐blind setting. Combined with extant results, our findings suggest that about half of dictator giving observed in experiments is internally motivated, and the other half is driven by external factors, such as experimenter observability or regard by receivers.

Inequity Aversion, Reciprocity, and Appropriateness in the Ultimatum-Revenge Game

This article reports the results of a simple bargaining experiment on the ultimatum-revenge game. The game enables to differentiate between fairness that is stimulated by intentional based motives, distributional motives, and fairness considerations that mix both motives. The laboratory experiments indicate considerable heterogeneity of motives. A majority of subjects seem to combine both motives. However, the composition of the mix is subject to a transition, which can be formalized by the principle of appropriateness. In contrast to contemporary reciprocity models, this approach suggests that mildly unkind treatments are responded mildly unkindly, while strong unkindness leads to harsh reactions.

The Effects of Beliefs Versus Risk Attitude on Bargaining Outcomes

In bargaining environments with uncertain disagreement or “impasse” outcomes (e.g., litigation or labor strike outcomes), there is an identification problem that confounds data interpretation. Specifically, the minimally acceptable settlement value from a risk-averse (risk-loving) but unbiased-belief bargainer is empirically indistinguishable from what one could get with risk-neutrality and pessimistically (optimistically) biased beliefs. This article reports results from a controlled bargaining experiment where data on both risk attitude and beliefs under uncertainty are generated in order to assess their relative importance in bargaining experiment outcomes. The average lab subject is risk-averse, yet optimistic with respect to uncertainty, which is consistent with existing studies that examine each in isolation. I also find that the effects of optimism dominate those of risk-aversion. Optimistic bargainers are significantly more likely to dispute and have aggressive final bargaining positions. Dispute rates are not statistically affected by risk attitude, but there is some evidence that risk aversion leads to a weakened bargaining position. Though additional research is needed to understand the limits of extending these results, a key implication follows. In uncertainty environments where optimism dominates, increased settlement rates are more likely achieved by minimizing impasse uncertainty (to limit the potential for optimism) rather than maximizing uncertainty (to weaken the reservation point of risk-averse bargainers), as has been argued in the dispute resolution literature.

Envy Freeness in Experimental Fair Division Problems

Envy is sometimes suggested as an underlying motive in the assessment of different economic allocations. In the theoretical literature on fair division, following Foley [Foley, D. (1967), Yale Economic Essays, 7, 45–98], the term “envy” refers to an intrapersonal comparison of different consumption bundles. By contrast, in its everyday use “envy” involves interpersonal comparisons of well-being. We present, discuss results from free-form bargaining experiments on fair division problems in which inter-and intrapersonal criteria can be distinguished. We find that interpersonal comparisons play the dominant role. The effect of the intrapersonal criterion of envy freeness is limited to situations in which other fairness criteria are not applicable.

Who Wins in Settlement Negotiations?

When lawsuits are resolved out of court, what determines the settlement amount? As a first approximation, the legal merits of the lawsuit matter, of course. Settlements are negotiated in the shadow of the law. But there is much more to settlement negotiations than the facts of the dispute and the relevant legal rules. Within certain boundaries determined by the nature and strength of the plaintiff's claim, the plaintiff's lawyer attempts to obtain every dollar that the defendant will pay, and the defendant's lawyer attempts to avoid paying all but the most minimal amount. This article uses a bargaining experiment to attempt to identify the factors that can be assessed prior to the beginning of bargaining that determines who wins and who loses this battle. The analysis reveals that, in the conditions of the simulated negotiation, the negotiators' estimate of the opposing negotiator's reservation price is the best predictor of outcomes, follow by the gender of the negotiator and the size of the first offer made. The negotiator's target or aspiration, the negotiator's relative enthusiasm for litigating the case if need be, and the negotiator's relative confidence in his negotiating ability were also predictive of a successful outcome, although less important. The negotiators' pre-negotiation perception of what would constitute a fair outcome had a very small indirect effect on outcomes. Together these seven variables explain more than half of the variation in settlement outcomes achieved by the subjects who participated in the simulation. The specific context of this article is settlement bargaining, but the insights generates are applicable to any two-party negotiation. Thus, the results should be of interest not only to litigators and legal scholars, but to anyone who negotiates as part of his or her personal or professional life or studies negotiation in any context.

Bargaining Outside the Lab – A Newspaper Experiment of a Three‐Person Ultimatum Game

5,132 readers of the German weekly, Die Zeit, participated in a three-person bargaining experiment. In our data analysis we focus on (1) the influence of age, gender, profession and medium chosen for participation and (2) the external validity of student behaviour (inside and outside the lab). We find that older participants and women care more about equal distributions and that Internet users are more self-regarding than those using mail or fax. Decisions made by students in the lab are rather similar to those made by participants in the newspaper experiment, indicating a high degree of external validity of student data.