# «IZA DP No. 1375 Dividing Justly in Bargaining Problems with Claims: Normative Judgments and Actual Negotiations Simon Gächter Arno Riedl October ...»

To our knowledge this study is the ﬁrst to combine questionnaires, experiments and diﬀerent claim points in one design. The only other experiments that explicitly investigate bargaining with claims problems we are aware of are Klemisch-Ahlert (1996), Herrero et al. (2004), and G¨chter and Riedl (2004). Klemisch-Ahlert (1996) a is interested in the evaluation of distributive principles subjects apply in bargaining environments. Speciﬁcally, she investigates how the subjects’ distributive principles depend on their bargaining position and how the subjects’ principles and the bargaining environment inﬂuence the agreements. G¨chter and Riedl (2004) use a bargaining a with claims framework to investigate the role of entitlements in the negotiation process. The paper closest to ours is Herrero et al. (2004). These authors investigate the experimental performance of non-cooperative procedures that underpin prominent bankruptcy rules and how diﬀerent framings of the experiment inﬂuence the outcomes.

Our paper is structured as follows. In the next section we formally introduce the claims problem and the three prominent solutions investigated in our study. Following Herrero and Villar (2001) we call them the “three musketeers”. Section 3 describes the details of our research design. Section 4 discusses the main results. Section 5 introduces the ‘beauty contest of normative rules’ and the results of this contest. Section 6 concludes.

**2 The claims problem and the ‘three musketeers’**

Informally, a claims problem is a distribution problem that involves the allocation of a single (perfectly divisible) good, the estate, in a situation where the available resource is not suﬃcient to meet all agents’ claims simultaneously. A solution to such a problem is a rule that prescribes how the resource should be allocated among the claimants. The idea behind such a rule is that the solution should not depend on particular circumstances of the situation where the problem occurs but only on the economically relevant variables, that is the agents’ claims and the value of the resource. Hence, a rule delivers solutions to a whole family of problems and not only to a particular problem.

In the following we provide formal deﬁnitions of a claims problem and the prominent rules for solving it. We follow the presentation of Thomson (2003) (see also, e.g., Herrero and Villar (2001)). Denote the net worth of the estate (good, resource) by E. The set of agents is N, and each agent i ∈ N has a claim or demand ci ≥ 0 on E ∈ R. The vector of claims is then denoted c = (ci )i∈N and C = i∈N ci.

these problems is denoted by B N. A solution to the claims problem is a function F, a rule, deﬁned on the class of claims problems that associates with each problem in the class a division of the estate between the claimants. It has the following properties: (i) 0 ≤ F (c, E) ≤ c, (ii) Fi (c, E) = E.

i∈N In this paper we investigate three prominent and classical solutions to the claims problem: the constraint equal-losses rule, the proportional rule, and the constraint equal-awards rule, which are the “three musketeers”, according to Herrero and Villar (2001). These rules are of particular interest because they are the ones used most often in practice. Furthermore, these rules have in common that they are the only three rules that simultaneously satisfy a intuitively reasonable set of axioms. These are the axioms of equal treatment of equals, scale invariance, composition, path-independence and consistency (see Moulin, 2000; Herrero and Villar, 2001).

The constraint equal-losses (CEL) rule distributes the estate such that all agents suﬀer the same losses, subject to the condition that no claimant ends up with a negative reward. Formally, for all (c, E) ∈ B N and all i ∈ N there exists a λ 0 such that CELi (c, E) = max{0, ci − λ}. By the deﬁnition of a rule it follows that λ is such that max{0, ci − λc} = E.

i∈N The proportional (PROP) rule awards the estate in proportion to the claims.

That is, it equalizes the ratios between claims and awards. The formal deﬁnition is as follows: For all (c, E) ∈ B N there exists a λ 0 such that P ROP (c, E) = λc. By the deﬁnition of a rule it follows that E/C = λ ∈ ] 0, 1].

Note that the idea of equality underlies each of the above rules. However, each rule applies the idea of equality to diﬀerent variables. CEL focuses on the equality of losses, the PROP rule ensures the equality of ratios, and the CEA rule puts its emphasis on the equality of awards.

We also consider the simple equal division of the estate, an outcome often observed in symmetric bargaining experiments (e.g. Nydegger and Owen, 1975), as a fourth

**possible rule and benchmark:**

The equal-awards (EQUA) rule just divides the estate equally between all agents. That is, for all (c, E) ∈ B N, EQU A(c, E) = (E/n,..., E/n).

In our experiments and vignettes (for details see the next section) we investigate two-person bargaining problems with claims where the pie to be distributed is 2050 ‘points’. We consider two diﬀerent claims points, (1980, 510) and (1640, 850), which both sum to 2490.

Figure 1 graphically depicts our bargaining problems with claims and the solutions discussed above for our parameters. In the ﬁgure we normalize the pie of 2050 to one.

3 The research design Recall that we are interested in two dimensions of the empirical performance of the four bankruptcy rules - the normative dimension and the behavioral dimension. We measure the normative dimension with the help of questionnaires that present a vignette (‘a scenario’) to the respondent. We assess the behavioral dimension by looking at actual negotiations of ﬁnancially motivated bargainers in a bargaining with claims environment. A second important feature of our design is the diﬀerence in the asymmetry of the claims points. Since we are employing two research methods and investigate two claims points we have four treatments, which we summarize in Table 1. We use the labels 1980E and 1640E for the experiments where the claims points were (1980, 510) and (1640, 850), respectively, and the labels 1980V and 1640V for the vignette studies.

We start by describing the survey study and will then explain the experimental setup and the procedures of our study.

The ‘bargaining with claims’ environments. The decision setup of both the vignette study as well as the experiments is a ‘bargaining with claims’ environment as it is graphically described in Figure 1.

In the surveys subjects were given the following vignette (translated from German),

**either with the claims (1980, 510), or (1640, 850):**

Please imagine the following situation. You and your bargaining partner have to negotiate over the division of a total budget of 2490 money units. Historically, the total budget has always been split according to performance. The bargaining partner who has shown the better performance, has so far received an amount of 1980 [1640] money units and the bargaining partner with the lower performance has received 510 [850] money units. Take it for granted that the performance (i.e., who has shown the higher or lower performance) can be objectively determined. It now turns out that the hitherto valid claims cannot be satisﬁed anymore. The new total budget amounts now to 2050 money units (i.e., is 440 money units lower than the old budget). According to your opinion, what would be a ‘fair’ new division from the vantage point of a non-involved, neutral arbitrator? (Please give exact amounts and no intervals! The amounts have to add up to 2050 money units) [emphasis in original].

**Your opinion on the division of the arbitrator:**

Amount for the bargaining partner with the better performance:..........

Amount for the bargaining partner with the lower performance:..........

2050.

The experiment used the same decision problem as in the survey studies. However, while in the vignette the claims point can just be given as part of the description of the problem, in the experiment this might by considered as cheap talk and is therefore not a useful thing to do. In our experiment, therefore, subjects ﬁrst earn the claims in a competitive task. With a certain probability these claims are actually paid out to the subjects. With the remaining probability subjects are told that the claims are infeasible and that they have to negotiate an agreement in a symmetric free-form bargaining. In case they fail to reach an agreement, they earn nothing. We will now describe the experiment in more detail.

The experiments were conducted as follows.3 At the beginning of the experiment, subjects were randomly allocated to computer booths, which were located in two different rooms. After subjects had ﬁnished reading the instructions the ﬁrst part of the experiment started, in which subjects earned claims in a general knowledge quiz. In particular, subjects had to answer 24 questions from a variety of ﬁelds, including astronomy, history, sports, music, politics, etc. We were very careful to select questions that students with a high school degree should in principle be able to answer, and that subjects would recognize as testing their high school knowledge. The knowledge quiz was a multiple choice test with ﬁve possible choices and only one correct answer. All The experimental instructions can be downloaded from ‘http://www.fee.uva.nl/creed/pdﬃles/instr 2bankruptcy.pdf’.

subjects had to answer the same questions. They had twelve minutes to answer all questions. Unanswered questions were counted as wrong answers.

After the quiz we told the subjects which of the two bargaining partners did better in the knowledge quiz. We only informed them about the rank of their performance (i.e., whether they did better or worse than their bargaining partner) and not about the actual number of correct answers. Apart from simplicity reasons, we wanted to hold the claims constant across subjects and between bargaining pairs.

A chance move then determined whether the budget shrunk to 2050 points or remained at 2490 points.4 The former implied that subjects had to negotiate over the division of the smaller pie of 2050 points. If the latter outcome occurred the claims according to the knowledge quiz were actually paid out.5 By making the claims a potential payment in the experiment, we gave the subjects an incentive to see the knowledge quiz as an important part of the experiment. Moreover, previous research shows that a knowledge quiz is indeed viewed as representative of true desert (e.g., Hoﬀman et al., 1994; Clark, 1998; Ball et al., 2001). Thus, given the previous evidence on the eﬀectiveness of non-paid knowledge quizzes and given that in our case the claims could actually be paid out, we believe that our procedure is an eﬀective way to implement the claims.

To examine whether the claims aﬀect bargaining behavior at all we conducted a control experiment without claims. This experiment was set up in exactly the same way as the one just described, except that there was no knowledge quiz and no claims to be earned. Twenty-four subjects participated in this experiment.

The bargaining was free-form, i.e., there was no ﬁxed bargaining protocol (see, e.g., Roth and Murnighan, 1982). We implemented a free form bargaining protocol because (i) it is naturally occurring and (ii) because the cooperative bargaining theories do not In the experiment the chance move was implemented as follows. After subjects were informed about the rank of their performance, each bargaining partner in a dyad had to roll a six-sided die. It was explained that the claims would be actually paid out if the sum of the numbers of both dice was greater or equal to eleven. If the sum of the dice numbers was smaller than 11, the bargaining partners had to bargain over how to split the smaller pie of 2050 points.

In case the dice determined that the claims will be paid out, we told the pairs to bargain hypothetically over the sharing of 2050 points. We ensured the subjects that they will receive their claims regardless of the outcome in the hypothetical bargaining. This procedure ensured that no bargaining pair left earlier than the others, which would have been technically diﬃcult and disturbed the experiment. We only observed three pairs that had to bargain hypothetically.