Matching markets with price bargaining - Springer Link

Exp Econ (2011) 14:322–348 DOI 10.1007/s10683-010-9270-4

Matching markets with price bargaining Philipp E. Otto · Friedel Bolle

Received: 6 October 2009 / Accepted: 19 December 2010 / Published online: 14 January 2011 © Economic Science Association 2011

Abstract The extant microeconomic literature on matching markets assumes ordinal preferences for matches, while bargaining within matches is mostly excluded. Central for this paper, however, is bargaining over joint profits from potential matches. We investigate, both theoretically and experimentally, a seemingly simple allocation task in a 2 × 2 market with repeated negotiations. When inefficiency is possible, about 1/3 of the complete matches are inefficient and, overall, more than 3/4 of the experimental allocations are unstable. These results strongly contradict existing bargaining theories requiring efficient matches. Even with regard to efficient matches, the tested theories perform poorly. Standard bargaining and behavioral concepts, such as Selten’s (1972) Equal Division Core, are outperformed by the simplistic ε-Equal Split, i.e., an equal split of the joint profit plus/minus ε. Keywords Matching market · Price negotiation · Optimal allocation · Bargaining models JEL Classification C78 · C91 · D40 · D82 · J41 1 Introduction Imagine two small firms (F1 and F2 ), both urgently in need of an accountant, and two accountants (W1 and W2 ) being available on the regional market. The productivities in terms of saved costs/additional income depend on the specific matches Electronic supplementary material The online version of this article (doi:10.1007/s10683-010-9270-4) contains supplementary material, which is available to authorized users. P.E. Otto () · F. Bolle Lehrstuhl Volkswirtschaftslehre, Insbesondere Wirtschaftstheorie (Mikroökonomie), Europa-Universität Viadrina Frankfurt (Oder), Postfach 1786, 15207 Frankfurt (Oder), Germany e-mail: [email protected]

Matching markets with price bargaining

323

Table 1 Typology of one-on-one matching markets with two distinct market sides Small numbers (game theoretic equilibria)

Large numbers (market equilibria with search)

Assignment problems (income related preferences, negotiated side-payments within matches)

Negotiations between the two partners in all possible matches

Search and negotiate, supported by internet platforms, job agencies, brokers, etc. (no clearing house)

Matching problems (ordinal preferences over “partners”, no negotiated side-payments)

The marriage problem: Strategic offers and acceptance of matches

Search and offer/accept matches, supported by internet platforms, matchmakers, etc. (sometimes clearing house)

achieved. Matches are defined as efficient if they maximize the sum of productivities. Will the market allocation be efficient? What kind of wages will the players negotiate? Will the negotiated wages make the matches stable, which requires that no unmatched pair will have the ability to make Pareto improvements by matching? How good are different bargaining concepts in describing the results of this matching game? The example and the questions posed describe pretty much exactly what the paper is about. In order to position this approach and distinguish the targeted problem from closely related but strategically and structurally different problems, we introduce a schematic classification with Table 1. The topic of this paper corresponds to its upper left cell. In large assignment problems, such as the general job market or the market for houses and apartments in a city area, it is no longer possible that participants negotiate with all potential partners from the other market side. The problem has to be supplemented by a general search model, usually based on a matching function. After two potential partners have met, prices/wages (for the case a match is formed) are determined, for example, by the Nash Bargaining Solution (Mortensen and Pissarides 1994) or by Rubinstein Bargaining (Hall and Milgrom 2008). Alternatives are wages which depend on the sector-structure of the economy (Bolle 1985) or which are determined in negotiations between a firm and its incumbent workers (Brown et al. 2009). Small matching problems are described as the marriage problem (i.e., Gale and Shapley 1962; Lundberg and Pollak 2008; Nosaka 2007; Roth 1984; Sasaki and Toda 1992). Matching is grounded on all participants’ ordinal ranking of potential partners. The question of stable and efficient matches is then investigated under different institutions. In cases of large matching problems clearing houses can be introduced in order to increase the efficiency of matches, i.e. to avoid matching structures where Pareto-improvements are possible. There are several matching problems beyond the structures described in Table 1. Several-on-one matching is described in the college admission problem (Roth 1985). Special examples of these are hospitals hiring physicians (Roth 1990) and law clerk matching (Haruvy et al. 2006). Matching problems without different market sides are roommate problems (Abdulkadiroglu and Sönmez 1999; Kamecke 1992). When only one market side has preferences we cope with the house allocation problem, e.g.,

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the allocation of apartments within a house (different from the housing market which consists of an assignment problem) or matching kidney donors and receivers (Roth et al. 2004). Let us now come back to the problem of the present study: matching markets with small number assignment problems as described in the upper left cell of Table 1. The generalized problem has Fi , i = 1, . . . , m and Wi , i = 1, . . . , n players on the two sides of the market. Small numbers assignment problems are the internal job market of firms, markets for specialists, as well as the markets for top jobs in management, academia, sports, or show business. It is assumed that every match has a certain monetary productivity which the partners split between them. Our investigation adopts this assumption and directly investigates how productivities are split in the market process, that is, in experiments the partners in a match have to bargain over the distribution of their joint profit. So far, how matches are formed in free-form bargaining and which splits apply have not been investigated empirically. Productivities are not available in field studies and experimental studies are practically nonexistent. The theoretical literature on matching with bargaining on the distribution of productivity is limited. For the general case, Koopmans and Beckmann (1957) show that market prices exist which support stable and efficient matches, and Shapley and Shubik (1971) independently came to the same conclusion. Under such prices, no other matches can be formed without making at least one of the partners of a potential other match worse off. The set of such prices is equivalent to the core which is never empty in this problem. Becker (1974) investigates the marriage market under simplifying assumptions (i.e., men and/or women are homogeneous). There is only one experimental study which is comparable with our experiment, namely Tenbrunsel et al. (1999), but they fully concentrate on personal relationships and their influence on matching efficiency, whereas we are concerned with the general matching and bargaining results. Only a few additional experimental studies have investigated other types of matching markets. Kagel and Roth (2000) investigate a situation with ordinal ranking of partners (corresponding to the lower left cell in Table 1) and reproduce a phenomenon found in many examples with small and large numbers, as for example in law clerk matching, namely premature matches. For the same problem Haruvy and Ünver (2007) find no significant differences between high-information environments (all players are completely informed) and lowinformation environments (only their own ordering is known to them). Interestingly, we find the same result in our study. Coalition experiments based on characteristic function games have been conducted by Albers (1988) as well as by Selten and Uhlich (1988), although with different payoff structures and explicit general bargaining among all group members, while our problem allows only pair-wise bargaining. Many other “matching” experiments (from Chamberlin 1948, to Cason and Noussair 2007) consider buyer-seller relationships with homogenous goods where, in principle, all information about preferences can be comprised in one market price. Entering a completely new class of experimental games, (i) we concentrate on the most simple but characteristic case, namely the 2 × 2 case; (ii) we conduct three experimental settings, each with another “extreme” frame; and (iii) we investigate six treatment variations (parameter constellations). The requirement of (i) excludes simpler 1 × 1 and 1 × 2 games, which are interesting on their own but do not offer the


325

interrelated strategic options of matching games. Our three experimental settings (ii) are characterized by, first, face-to-face bargaining with completely free communication (but limited to two potential partners) and with information only about the joint profits in one’s own matches (Class); second, anonymous bargaining between potential partners with the same information (Lab); and, third, anonymous bargaining with information about all joint profits (Lab-info). The first frame is the most realistic (single applicants visit firms in order to negotiate contracts, with or without a provisional result) while the laboratory frames offer the most control. It turned out that (iii) is rather important, because some theories deliver good predictions if we restrict the test to efficient matches in one or two treatments, but across all treatments and when considering efficient as well as inefficient matches all theories fail. The following concepts were considered and also tested if they turned out to be applicable and if they offered sensible predictions. Normative concepts Stability is the most fundamental and convincing requirement in the theoretical investigations of matching. It guarantees that the distribution of incomes is in the core of the matching game. (Details are discussed below.) In the investigated experimental 2 × 2 markets, however, core theory turns out to be a poor descriptive model. Other concepts of cooperative game theory either coincide with the core (the Neumann-Morgenstern Solution) or are difficult to apply, because the matching game does not allow arbitrary transfers among all players. Two versions of the Nash Bargaining Solution are investigated. Rubinstein Bargaining was also considered, but we did not apply this concept because of necessary adaptations with an unavoidably controversial introduction of additional assumptions. Non-transferable utility coalitional games are discussed, but rejected as non-applicable. Behavioral concepts In particular Equal Split and Selten’s (1972) Equal Share Analysis are applied. Another concept known from Cooperative Game Theory, the ε-core (Shapley and Shubik 1966), which is sometimes introduced in games where the core does not exist, is interpreted behaviorally. Here, ε is a “just noticeable difference” (Georgescu-Roegen 1958) or a measure of inertia or loyalty,1 that is a tendency to remain in preliminary matches, except one can realize income increases of more than ε. Of course one can also introduce such an ε to other solution concepts, like the Nash Bargaining Solution or simple equal split (ε-ES). Applying Selten’s (1991) measure of predictive success of area theories for the evaluation of such “ε-theories” does prevent the introduction of inflated models. It turns out that the simplistic ε-ES is far ahead of all other theories when it comes to explaining the experimental results of our study. The second best explanation is offered by the Equal Division Core, which is one requirement of Selten’s (1972) Equal Share Analysis (for an axiomatization, see Bhattacharya 2004). But even this theory, which is rather successful in explaining the results of coalitional games (see Selten 1972), delivers unsatisfactory hit rates in comparison to ε-ES. 1 In an application to our experimental results the necessary magnitude of ε excludes its interpretation as

just noticeable difference as it becomes unconvincingly large.

326 Table 2 The productivities of matches

P.E. Otto, F. Bolle F1

F2

W1

α

β

W2

γ

δ

The next section derives the theoretical bargaining results for the 2 × 2 market concerning the above described normative and behavioral concepts. Section 3 describes the 2 × 2 matching experiments with repeated negotiations. In Sect. 4, the results are presented and compared to the theoretical predictions. Sect. 5 concludes the investigation.

2 Matching theory A match of Wi and Fj results in a productivity (joint profit) of aij = wi + fj , which Wi and Fj can distribute amongst themselves, with wi and fj denoting the respective payoffs for Wi and Fj . In the case of two workers and two firms, we denote the productivities as in Table 2. The most important theoretical concept for assignment markets is the core which rests on the stability requirement and implies efficient matches. However, as the empirical evidence for core solutions is weak, we also discuss a number of alternatives. In the following we explain why most solution concepts of cooperative game theory (except the core) cannot be applied, which other normative or behavioral theories can be applied (under which assumptions about information), and what their predictions are. 2.1 Non-transferable utility In matching markets, the transfer of utility is restricted to pairs of matched subjects. Players within a match can freely distribute the respective productivity between themselves, but transfers to other actors are excluded. Therefore, we have a coalitional game with non-transferable utility (NTU). Contrary to its name, however, this class also includes coalitional games with restricted transfers. For such games, the characteristic function is substituted by a set-valued function V (S) ⊂ R S (where R = real numbers) which describes the utilities that the members of coalition S ⊂ N (=set of all players) can guarantee for themselves (see Myerson 1991, p. 456). A basic assumption of NTU coalitional games is that the sets V (S) are convex. Unfortunately, in the Matching Market game V (N ) = {(w2 , w2 , f1 , f2 )|w1 , w2 , f1 , f2 ≥ 0; w1 + f1 = α, w2 + f2 = δ or w1 + f2 = β, w2 + f1 = γ } is not convex. (α, δ, 0, 0) and (β, γ , 0, 0) are in this set, but (1/2(α + β), 1/2(γ + δ), 0, 0) is not. Sets V (S) with three-player coalitions S are not convex either. Because of these non-convexities, value approaches, such as (a generalization of) the Shapley value need not exist. 2.2 Information What is the informational status of the market participants? The “natural” assumption about information is that a worker knows the productivity of matches in which


327

she may be involved, but not the productivities of matches of her competitor (i.e., W1 knows α and β but not γ and δ). The same applies for firms. In two of our three experiments, we provided the subjects with exactly this type of information.2 In a third experiment, subjects were given full information about all productivities. However, we did not directly deal with whether the bargaining process makes estimates of the necessary information available if it is not given explicitly. For the solution concepts “core”, “Nash Bargaining with the outside option no match” (Appendix B), Selten’s (1972) “equal division core”, and “ε-equal split” the information about one’s partner’s income is sufficient. The concept “Nash Bargaining with implicit threats” (NBIT, Appendix B and Fig. 1) and the “order of strength” assumption in Selten’s (1972) “equal share analysis” requires better information. 2.3 The core Koopmans and Beckmann (1957) showed that there is a set of vectors (allocations) C = {(w1 , . . . , wm ; f1 , . . . , fn ), with wi + fj ≥ aij for all i, j and wi + fj = aij , if (i, j ) belongs to the optimal matches}. C is equal to the core. In other words, Koopmans and Beckmann (1957) showed that the core (defined later by Gillies 1959) of the Matching Game is not empty. There are no difficulties with NTU because transfers occur only within matches. C is also equal to the unique Stable Set = NeumannMorgenstern solution.3 When matches A = {(W1 , F1 ), (W2 , F2 )} are efficient, i.e., α + δ ≥ β + γ , the core solutions are called Core A and are described by w 1 + δ − β ≥ w2 ≥ w 1 + γ − α w1 + f1 = α,

w2 + f2 = δ,

(1) wi , fi ≥ 0.

(2)

If the matches B = {(W1 , F2 ), (W2 , F1 )} are efficient, the core allocations are called Core B, the inequalities in (1) are reverted and (2) is reformulated accordingly. The core is illustrated in Fig. 1. If the efficient match A is formed but with wages outside the core, exactly one alternative match which could be formed profitably exists. When inefficient matches result, i.e., A is efficient but B is chosen, (1) with inverted inequality signs cannot be fulfilled and then (1) describes Anticore B as the “most unstable situation”. When wages are in the anticore, all players could form a more profitable match. (Outside Anticore B, there is again only one such match.) Therefore, one resulting hypothesis regarding stability is that we should find allocations in the core for efficient matches. But if inefficient matches are formed, we should at least find no bargaining result in the anticore. If both matches are efficient, i.e., α + δ = β + γ , then the core shrinks to a line and no anticore exists. The behavioral extension of the core, namely ε-core, is simple. The core conditions, with wi + fj ≥ aij for all (i, j ), need to be fulfilled only with a precision of ε > 0, i.e., wi + fj ≥ aij − ε. The requirement wi + fj = aij , if (i, j ) belongs to the optimal matches, still has to be fulfilled. Therefore, ε-core A is described by 2 Kagel and Roth’s (2000) experiments were conducted under the same information structure. 3 Note that in the case of ordinal preferences for partners there may be several Stable Sets, all containing

the core (Ehlers 2007). See Appendix A.

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Fig. 1 A is efficient and has occurred (top), with w1 + f1 = α, w2 + f2 = δ. A is efficient but B has occurred (bottom), with w1 + f2 = β, w2 + f1 = γ . NBIT = Nash Bargaining with Implicit Threats. ES = equal split

w1 + δ − β + ε ≥ w2 ≥ w1 + γ − α − ε and w1 + f1 = α, w2 + f2 = δ, wi , fi ≥ 0. Note that, even if the match A is inefficient, i.e., if α + δ < β + γ , ε-core A exists for large enough ε. ε-core B is defined accordingly. Other normative solution concepts such as Nash and Rubinstein Bargaining could not usefully describe the results of this investigation. Two versions of the Nash Bargaining Solution are discussed in Appendix B. While the prediction of one of these versions does not meet experimental results the other (NBIT) predicts bargaining results “in the middle” of the core (see Fig. 1). Rubinstein bargaining (Rubinstein 1982) assumes two players will make alternating offers about dividing a cake. Its application is difficult, however, because there is no given order of proposals in our matching game and time preferences, bargaining costs, and outside options are difficult to determine.


329

2.4 Equal share analysis Equal Share Analysis (Selten 1972) is formulated for transferable utility and with the requirement of having efficient results, but it is possible to adapt the concept to the current matching problem. Its three principles are the following: Efficiency Exhausted coalition structure in the original formulation. Equal Division Core (EDC) Fewer strategic alternatives than in the core concept are utilized here, namely only the alternative of equal splits. EDC assumes that the set of income vectors is restricted by the requirement that the equal splits of non-realized matches do not dominate the respective two players’ incomes. Because of the nontransferability of incomes between matches, we restrict this requirement to matches (coalitions between one worker and one firm). If A matches are realized, EDC for A matches is described by (w1 , w2 ) ∈ [0, α] × [0, δ] − {[0, β/2] × [δ − β/2, δ] ∪ [α − γ /2, α] × [0, γ /2]}. If B matches are realized, EDC for B matches is described by (w1 , w2 ) ∈ [0, β] × [0, γ ] − {[0, α/2] × [γ − α/2, γ ] ∪ [β − δ/2, β] × [0, δ/2]}. One may criticize EDC because, on the one hand, it employs only equal sharing threats (potential outcomes) while, on the other hand, it allows for (real) unequal outcomes. Order of Strength Selten’s (1972) general definition of the strength order of the players is not applicable here, because it relies on transferable utility. But as Selten (1972, p. 131) mentions, “In most characteristic function games which have been used for experimental purposes it is perfectly clear who is stronger and who is weaker.” Selten (1972) assumes stronger players to earn more than weaker players. The relations α ≤ β ≤ γ < δ mean strategic advantages of W2 and F2 in B matches, because their alternative option δ is larger than W1 and F1 ’s alternative option α. This should result in w1 < f2 and f1 < w2 , i.e., w1 < β/2 and w2 > γ /2. In A matches, F1 and W2 are advantaged if and only if β < γ (with the consequence w1 < α/2 and w2 > δ/2 which applies in our treatments T2, T4, and T6). We apply Equal Share Analysis without requirement (i). In Fig. 2, we see which areas of bargaining results are implied by the two other requirements. The area of EDC, and its restriction by Order of Strength, becomes apparent. 2.5 Equal split As a rather simple behavioral alternative we introduce equal split (ES), which is an equal division of productivities in the matches chosen. Which means here w1 = f1 = α/2, w2 = f2 = δ/2 for A matches and w1 = f2 = β/2, w2 = f1 = γ /2 for B matches, with ε-ES being the 2ε × 2ε square around these points. The interpretation of ε is the same as in the other ε-theories: as weakening its precision. Any acceptable descriptive theory should be at least as good as ε-ES. 3 Experiments The three experiments we conducted simulate a simplified labor market situation with two workers and two firms in repeated one-on-one price negotiations. One experiment

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P.E. Otto, F. Bolle

Fig. 2 Bargaining results implied by the requirement of EDC (grey and black) and combined with Order of Strength (only black), shown separately for A matches on the left side (but note that, for β = γ , the required Order of Strength has no bite so that the relevant grey area remains) and B matches on the right side Table 3 Productivities of matches in the six experimental treatments (rows = workers, columns = firms; compare Table 2), with efficient matches underlined Treatment

T1

T2

T3

T4

T5

T6

Productivities

280

400

280

280

280

460

160

400

160

460

280

400

400

640

520

640

460

640

520

640

460

640

520

640

took place in a classroom setting with face-to-face interactions and the other two in a laboratory setting with anonymous computer based interactions. Each participant took part in one condition, i.e. one informational setting only. In every experimental session eight participants were confronted with six different treatments (see Table 3). First, participants were separated into two groups of four subjects and assigned the role of Worker 1 (W1 ), Worker 2 (W2 ), Firm 1 (F1 ), or Firm 2 (F2 ). The order of the treatments and the individual allocation to roles was randomized over the sessions. Each subject was allocated three times to a worker position and three times to a firm position,4 playing each role (W1 , W2 , F1 , and F2 ) at least once. Furthermore, no subject was to interact with the same person more than three times.5 Interactions took part either in one of the laboratory settings (Lab or Lab-info) or in the classroom settings (Class). Each session provided us with one in4 In treatments T1, T3, and T5, W and F as well as W and F were in the same strategic position. Do 1 1 2 2

they earn the same amount? In the laboratory as well as in the classroom setting the differences are rather small and insignificant (t -tests show a minimal p-value of .18). Therefore, the mere labeling of players as “workers” or “firms” does not seem to influence the results. 5 The experimental setting of eight players negotiating in two different groups, offers 70 possibilities to

form pairs of groups. Choosing six of these for our experiment, we could not avoid that player i meets some players k up to three times (but not more often). Thereby the possibility for building up trust over treatments is limited and only possible in the face-to-face bargaining situation where the partner is known.


331

dependent observation. In all sessions, the following sequence of actions was carried out. 1. Participants drew a subject number, which determined the sequence of their roles in the six treatments. 2. Then instructions were handed out (for the instructions in the laboratory and the classroom setting see Web Appendix E). After questions from participants had been answered and everyone had confirmed their understanding of the task, the first matching phase began. 3. In the experiment Lab-info, the participants were informed about all productivities; in the other two experiments, the subjects were informed only about the productivities of the two matches they could participate in, for instance, W1 was informed about α and β but not about γ and δ. 4. A ten minute negotiation phase began which is described in more detail below. 5. At the end of the negotiation phase, the subjects were asked to evaluate their “satisfaction” and “fairness” levels with regard to the results, both rated on a five-point scale. 6. If the six treatments had not been exhausted, the information about the productivities of a new treatment was given as in 3. and the experiment proceeded with 4. and 5. 7. At the end of a session, the subjects answered several demographic questions before they received the income earned from all six treatments. The laboratory settings (Lab, Lab-info) were run on z-Tree (Fischbacher 2007). First, individual allocation to a role with the corresponding productivities in eurocents was displayed on the start screen. On the next screen, the matching process of submitting and accepting offers took place. An example screen of the Lab setting is shown in Web Appendix D. During the negotiation phase, the top box shows an individual’s allocation and the two boxes below are for negotiating with the two partners. Here offers could be made or accepted. Once an offer was accepted, meaning a provisional contract had been made with this partner, then this box became inactive, so that it was only possible to interact with the other potential partner. If a subject or her partner cancelled their contract by reaching an agreement with the other partner, new offers could be made to the former partner. This phase continued until the allotted negotiation time of ten minutes for one treatment expired. The last contract resulted in the individual payment for this particular treatment. In the classroom settings (Class), all four players could see all others but only two potential partners were allowed to talk to one another. The firms of the two groups were seated at separate tables and were not allowed to move. One worker at a time could negotiate with a firm. They could talk completely freely and not even the experimenters knew the content of their talk except the possible negotiation result. The firms wrote a protocol which contained every provisional match they formed, together with the distribution of the respective profitability. Once an agreement had been settled, this was marked on the wall behind the table. The distribution of profits Due to the uncertainty concerning the possibility of later matching, cooperation over the treatments is not assumed though it might result in the classroom settings.

332

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in this match was private knowledge of the partners. The worker could now (if she wanted to and if there were no ongoing negotiations between the other two members of the group) start negotiations with the other firm. Negotiations were required to be one-on-one and worker/worker and firm/firm communication was interdicted. Some rather limited general communication,6 however, could not be prevented. As in the laboratory experiment, a new agreement implied the cancellation of the old agreement and the negotiation phase of every treatment expired after ten minutes. In total, 224 subjects took part in the study: 80 students were allocated to the Class experiment (average age 22.8 years; 37.5% male; 70% German, 16% Polish, 14% other nationalities), 80 to the Lab experiment (average age 21.9 years; 47.5% male; 70% German, 25% Polish, 5% other nationalities) and 64 to the Lab-info experiment (average age 23.2 years; 18.8% male; 70% German, 20% Polish, 10% other nationalities). Class as well as Lab had ten sessions and Lab-info had eight sessions, with eight participants in each session. The average duration of a session was 1.5 hours in all experiments and participants received on average payments of €13.41 in Class, €11.75 in Lab, and €11.48 in Lab-info.7 Why so many variations in our experiments? We believe that, in the case of bargaining, the question of whether face-to-face interactions yield different results than anonymous interactions is of particular interest. In important cases, bargaining is a face-to-face issue. Therefore, we should find out which differences are caused by the laboratory situation, which is usually preferred for its better control of parameters. The difference between full and limited information about productivities may be decisive or negligible—the comparison between the results of Lab and Lab-info should decide this question. In addition, we expected that the performance of bargaining theories depends on the structure of productivities. In T1 and T2 efficiency requires matching with a partner who is in a similar strategic situation. In T4 and T5 efficiency requires matching between a strategically advantaged and a disadvantaged partner, which should result in large income differences within matches. In T3 and T6, where all matches are efficient, most theories give the subjects the choice of whether to match with an equal or unequal partner. This increase of discretion, however, causes the shrinking of the core from an area to a line. Requiring a theory to perform well under all these different conditions is a particularly strong test.

4 Experimental results A graphical overview concerning matches and distributions of joint profits for all six treatments is given in Appendix C. The average payoffs for W1 and W2 are reported in Table 5, excluding the cases of zero income if no match was formed. Each of the latter cases generates a point on one of the axes in the figures of Appendix C. 6 Communication in the classroom could remain nonverbal or, for example, include exclamations of disap-

pointment when a match had been terminated by the partner. 7 The total payment is purely based on the achieved results in the final matches and no show-up fee was

implemented. As we will see in the results section, the differences between the Lab and the Class settings mainly results from the higher number of unmatched subjects in the Lab.


333

Table 4 Relative frequencies of “no matches” out of forty (thirty-two in Lab-info) possible matches in each treatment and relative (with respect to matches) frequencies of A and B matches. Frequencies of efficient matches are in bold type Lab

Class

Info % No

Non-info

Non-info

%A

%B

% No

%A

%B

% No

%A

%B 35

T1

13

89

11

18

61

39

0

65

T2

28

39*

61

18

67*

33

0

70*

30

T3

19

54

46

13

77

23

3

64

36

T4

22

48+

52

20

53+

47

3

28+

72

T5

19

38

62

25

50

50

0

35

65

T6

13

71

29

10

56

44

0

75

25

* Indicates a difference between info and non-info (Class alone or Class plus Lab non-info) at the .05 level

in a two-sided Fisher test + Indicates a difference between Lab (info plus non-info) and Class at the .05 level in a two-sided Fisher

test

4.1 Anonymous vs. face-to-face bargaining and complete vs. incomplete information The most striking difference is the low number of incomplete matches ( γ /2 and in A matches, if β < γ , we should find w1 < α/2 and w2 > δ/2) considerably reduces the prediction area in many cases, but also leads to a shrinkage in prediction rates. Table 8 provides the results of EDC and of EDC combined with OS for the different treatments and matches. Table 9 shows that the Selten’s measure of predictive success for area theories (Selten 1991) is higher for EDC than the core in five out of the eight cases where the core is defined. In most cases, except A matches in T1, Selten’s measure for the core is far lower and not significantly different from zero. EDC also performs

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P.E. Otto, F. Bolle

Table 8 Results (%) compared with predictions by EDC and combined with OS when it applies. Area = percentage of points in the grid with a width of 1 T1 A

T2 B

Area EDC

55.3

Results EDC

96.8*

Area EDC∩OS

55.3

Results EDC∩OS (96.8)

T3

A

B

26.0 51.3

T4

A

25.0

B

41.0

42.3

B

28.1

0.1 92.6* 46.2

89.8* 27.6 100*

6.1 15.6

41.0

0

14.8

11.5

7.7 (89.8)

T5

A

T6

A

B

A

48.6 40.0

B

47.7

28.1

61.5

78.6* 57.7 81.5* 54.8 28.1

21.9

41.3

15.8

9.4

21.9

9.4

15.6

6.9

7.1

42.3 (78.6) 30.7 21.5

39.0

* Significantly (p < .001 according to a two sided binomial test) higher proportion of choices within pre-

diction area than percentage of prediction area Table 9 Selten’s measure for comparing the performance of the different theories in the treatments T1 A

T2 B

A

T3 B

A

T4 B

A

T5 B

T6

A

B

–

A

B

Corea

61.3

–

−0.2

–

17.0

6.5

10.6

6.0

−0.2

EDC

41.4

−25.9

41.3

21.2

48.8

−14.7

71.9

13.8

50.5

9.1

41.5

13.5

EDC∩OS

41.4

−6.1

−0.8

−3.8

48.8

−8.9

−2.3

20.4

50.5

8.8

12.1

23.4

ε-ESb

74.5

44.2

66.6

57.1

70.8

61.1

60.9

30.8

46.7

38.8

65.2

55.8

–

6.5

a The core only applies for efficient matches and provides a “line” in T3 and T6 b Assuming a uniform 100 for ε in all treatments and matches

slightly better than ε-ES for A matches in treatments T4 and T5. In all other matches ε-ES strongly outperforms EDC (as well as the core), resulting in an average Selten’s measure of 57.4% compared to 28.9% and 15.4%, respectively, for EDC and the core (the latter only for efficient matches). EDC∩OS provides no improvement, and neither have ε-core nor ε-NBIT a chance to come close to the values of ε-ES. Note that this evaluation does not change if we analyze the three experiments (Class, Lab, and Lab-info) separately. 5 Conclusion and discussion We have investigated pioneering experiments in the area of matching markets with price bargaining. The results strongly contradict established concepts of theoretical and behavioral economics. The main result from the 2 × 2 experimental matching markets is that standard bargaining concepts as the core (strategic considerations), Nash Bargaining Solution (normative requirements), or Equal Share Analysis (behavioral concept) do not provide a satisfactory explanation of the data. Other concepts from Cooperative Game Theory are shown to be inapplicable because matching with price bargaining is a NTU game with non-convex sets describing the possible transfers, or as evidently not fitting the data as the ε-core. Far better than these approaches is ε-ES, i.e., the “naïve” explanation of an equal split of joint profits +/− 100 euro-cents. Yet the latter is not really satisfactory either because the prediction


339

area is rather large, because results are not centered round equal splits, and because ES does not provide us with any reason—beyond general bounded rationality—for the occurrence of inefficient matches. Our conclusion is that a different theoretical approach is necessary (i.e., social preferences) and, under the impression generated by the large variety of bargaining results (see Appendix C), that real progress in explaining the results can only come from taking further individual or situational variations9 into account. In order to gain substantial insight into the matching process, it is necessary to further vary the market and expand its frame. Similar to oligopoly experiments (i.e., Huck et al. 2004), an increasing number of competitors may reduce the deviations from the competitive equilibrium (= the core in our case), but on the other hand the strongly increasing complexity of larger matching markets with price bargaining may prevent such a convergence.10 Another type of results does not concern the explanation of the bargaining process, but rather the question of how important frames and information are. We found strong influences of face-to-face versus anonymous bargaining on the completeness of matches but no influences on the efficiency of complete matches. There seem to be certain differences in the bargaining process, but hardly any differences in the bargaining results. Thus, the efficiency enhancing effect of face-toface bargaining (Radner and Schotter 1989; Valley et al. 1998, 2002) is only supported as far as the completion of matches is concerned. Full information about productivities does not result in different average bargaining results than in treatments with information only about one’s own options. The only observed difference concerns the significantly smaller number of equal splits under complete information. What might a new theoretical approach for matching markets with bargaining about prices, including wages, rents, etc., look like? One class of candidates is the merging of strategic concepts with social preferences while taking into account situational and individual attributes. Another research strategy may explain and further develop the surprisingly successful relaxed equal split (ε-ES). Matching markets with price bargaining are certainly important enough to justify further research in order to improve their understanding. Acknowledgements We would like to thank Werner Güth and Reinhard Selten as well as the participants of the Sozialwissenschaftlicher Ausschuss (Koblenz, April 2009) for their helpful comments. Also the suggestions of two anonymous reviewers helped to improve the paper considerably. This work has been funded by the Deutsche Forschungsgemeinschaft (BO 747/11-1). 9 The introduction of social preferences (e.g., altruism or inequity aversion) into strategic concepts (e.g.,

the core) requires a lengthy derivation of the respective solution area plus a numerically tedious estimation of parameters. Results of such attempts can be found in Bolle et al. (2009), where the altruistic core with role-dependent (not individually varying) preferences turned out to provide the best fit with the data. 10 Meanwhile 6 × 6 and 2 × 3 experiments have also been conducted. Results (in particular in comparison

to the 2 × 2 case) will be described in a separate paper. The problem of comparing the bargaining results with different numbers of workers and firms lie in the necessarily different matrices of productivities. This makes it far more difficult to compare results than, say, in the case of Cournot markets with two and six firms. In addition, we have to disentangle the effects of a more complicated environment with more random deviations from systematic effects of increasing the number of workers and firms, i.e., the number of strategic options.

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Appendix A C = core = unique Stable Set (von Neumann–Morgenstern solution): First, let inequality. From wi + fk ≥ aik for all i, k fol us derive a characteristic lows that i∈Bw wi + fk(i) ≥ i∈Bw aik(i) for all injective functions (= assignments) k(i) : Bw → BF , with BW ⊂ NW = {1, 2, . . . , m}, BF ⊂ NF = {1, 2, . . . , n}, |Bw | < |BF |. Second, assume that a payoff vector (wi , fk ) from C were dominated by another vector (from inside or from outside C) via the coalition (Bw , BF ). Without restriction of generality we can assume |Bw | ≤ |BF |, otherwise we exchange the roles of workers and firms. As all values stem from pair-wise assignments, the value of the coalition is described by i∈BW aik ∗ (i) with an optimal assignment k ∗ (i). However, the characteristic inequality shows that this value is too low to make the coalition better off than under any core allocation. Therefore all payoff vectors in C are undominated. A payoff vector is outside C if one of the inequalities wi + fk ≥ aik is violated. Apparently this vector is dominated by every core allocation. So C is equal to the core and it is also a stable set. Is it the only stable set? The first part shows that we cannot remove any imputation from the core because it could not be dominated. The second part shows that we cannot add any imputation to the core because it would be dominated. Therefore the core is equal to the unique Stable Set.

Appendix B Nash Bargaining is concerned only with V (N ) and V ({i}), but not with V (S) of intermediate coalitions S. Therefore, we only have to “amend” the non-convexity of V (N). We adopt a standard proposal, namely to use the convex hull of V (N ) with the interpretation of convex combinations as probability mixtures. Note that this is different from the TU assumption that max{α + δ, β + γ } can be arbitrarily distributed among the four players. Let (w1 , w2 , f1 , f2 ) be an allocation with A matches, (w1 , w2 , f1 , f2 ) be an allocation with B matches, and let p be the probability that A matches are formed. Then the expected utility of the players is Ew1 = pw1 + (1 − p)w1 ,

Ef1 = pf1 + (1 − p)f1 ,

Ew2 = pw2 + (1 − p)w2 ,

Ef2 = pf2 + (1 − p)f2

(3)

with w1 + f1 = α,

w2 + f2 = δ,

w1 + f2 = β,

w2 + f1 = γ .

(4)

Let tW 1 , tW 2 , tF 1 , tF 2 be the respective threat values or “outside options” for the workers and firms. A single player cannot earn any income; therefore, we assume (in this adaptation of the Nash Bargaining Solution) all outside options to be zero. The Nash Bargaining Solution results from the maximization of the Nash product: P = Ew1 Ew2 Ef1 Ef2

(5)


341

In order to increase P , transfers may be considered which increase the low expected incomes and decrease the high ones. Transfers with p > 0, however, imply losses of aggregate income if α + δ = β + γ . To keep these losses minimal, i.e., to keep p minimal, extreme income distributions are chosen in the case of inefficiency. Consequently, inefficient matches should be accompanied by the giving of all income to one of the matching partners. This is, however, a rather implausible prediction. Thus, we do not give this version of Nash Bargaining further attention. The real threat in our bargaining game is not the option to remain unmatched but to try to form a match with the other possible partner. This approach is called Nash Bargaining with Implicit Threats (NBIT). In our illustration, NBIT is a line in the middle of the core (as shown in Fig. 1). The connected threat value is only implicitly given because it is derived from (has to be consistent with) the bargaining result. To be precise, we now consider Nash Bargaining between the partners in a match (not in the whole group) where the threat value is derived from the possibility to enter the alternative match with a competitive bid. For general threat values we find 1 w1∗ = (α − tF 1 + tW 1 ) 2

(6a)

or 1 w1∗∗ = (β − tF 2 + tW 1 ), (6b) 2 depending on whether A or B matches are formed. Assume that matches A result. The implicit threat tW 1 of Worker 1 (who is in a match indifferent) offers the profit f2∗ = β − tW 1 . If B matches result, Worker 1 threads to offer Firm 1 f1∗∗ = α − tW 1 . In the same way we can determine the other implicit threats. We thus get a system of eight equations for the four threat values and the four payoffs. This system is linearly dependent but not contradictory. We therefore find only a linear condition for (w1 , w2 ) 1 w2 = w1 + [δ − β + γ − α]. 2

(7)

Outside options are not smaller than zero if and only if (7) is restricted by α+β +γ −δ δ+α−γ −β ≤ w1 ≤ . 2 2

(8)

Equations (7) and (8) describe all possible Nash Bargaining Solutions with implicit threats. In Fig. 1 this line is indicated as NBIT. With implicit threats, the Nash product is maximized with efficient matches. Condition (7) remains unchanged if the B matches are formed and are efficient. In both cases it describes the middle of the restrictions (1). Condition (8) is substituted by δ − β ≤ w1 ≤

α+β +γ −δ 2

(9)

which is empty under the parameters of our treatment T4, but not so in treatments T5, T6. T3 and T6 are degenerate cases but can be derived from α + δ → β + γ .

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For general threat values we find 1 w1∗ = (α − tF 1 + tW 1 ) 2

(10a)

1 w1∗∗ = (β − tF 2 + tW 1 ); 2

(10b)

1 w2∗ = (δ − tF 2 + tW 2 ) 2

(11a)

1 w2∗∗ = (γ − tF 1 + tW 2 ); 2

(11b)

1 f1∗ = (α − tW 1 + tF 1 ) 2

(12a)

1 f1∗∗ = (β − tW 2 + tF 1 ); 2

(12b)

1 f2∗ = (δ − tW 2 + tF 2 ) 2

(13a)

or

or

or

or 1 f2∗∗ = (γ − tW 1 + tF 2 ). (13b) 2 Assume that the matches A result. The implicit threat tW 1 of Worker 1 (who is in a match with Firm 1) is to offer Firm 2 a match (which results in the productivity β and) which makes Firm 2 indifferent, i.e., Worker 1 offers the profit f2∗ = β − tW 1 . If B matches result, Worker 1 offers Firm 1 f1∗∗ = α − tW 1 . In the same way we can determine the other implicit threats f2∗ = β − tW 1

(14a)

f1∗∗ = α − tW 1 ;

(14b)

w2∗ = γ − tF 1

(15a)

w1∗∗ = α − tF 1 ;

(15b)

f1∗ = γ − tW 2

(16a)

f2∗∗ = δ − tW 2 ;

(16b)

or

or

or


343

w1∗ = β − tF 2

(17a)

w2∗∗ = δ − tF 2 .

(17b)

or

Let us first regard the case where matches A are formed. Unfortunately, the system of the eight equations (10a) to (17a) is linearly dependent (but not contradictory). So we can determine only a linear condition for (w1 , w2 ): 1 w2 = w1 + [δ − β + γ − α]. 2

(18)

The respective outside options are 1 tW 1 = w1 + [β + γ − α − δ] 2 1 tF 1 = −w1 + [α + β + γ − δ] 2 tW 2 = w 1 + γ − α

(21)

tF 2 = −w1 + β.

(22)

(19) (20)

The outside options tW 1 and tF 1 are not smaller than 0 if α+β +γ −δ δ+α−γ −β ≤ w1 ≤ . 2 2

(23)

tW 2 , tF 2 ≥ 0 is implied by (23). If the first inequality of (23) is not fulfilled, then we have to set tW 1 = 0 which leads to w1 = α+δ−β−γ . If the second inequality is 2 −δ not fulfilled, then we have to set tF 1 = 0 which leads to w1 = α+β+γ . We do not 2 analyze the case δ + α − γ − β > α + β + γ − δ because, for the parameters in our experiment, this relation never occurs. Therefore (18) and (23) indicate all possible Nash Bargaining Solutions with implicit threats.

Appendix C Results for Worker 1 (w1 , horizontal) and Worker 2 (w2 , vertical). For illustration purposes the results are changed randomly by +/− 5 euro-cents. The legend is the same in all treatments/matches. ES – – – NBIT —— Core × Results Class (non-info) Results Lab (non-info) + Results Lab-info · · · · · · Maximum

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References Abdulkadiroglu, A., & Sönmez, T. (1999). House allocation with existing tenants. Journal of Economic Theory, 88(2), 233–260. Albers, W. (1988). Revealed aspirations and reciprocal loyalty in apex games. In R. Tietz, W. Albers, & R. Selten (Eds.), Lecture notes in economics and mathematical systems: Vol. 314. Bounded rational behavior in experimental games and markets (pp. 303–316). Berlin: Springer. Becker, G. S. (1974). A theory of marriage. In T. W. Schulz (Ed.), Economics of the family (pp. 299–344). The University of Chicago and London. Bhattacharya, A. (2004). On the equal division core. Social Choice & Welfare, 22(2), 391–399. Bohnet, I., & Frey, B. (1999). The sound of silence in prisoner’s dilemma and dictator games. Journal of Economic Behavior & Organization, 38(1), 43–57. Bolle, F. (1985). Natural and optimal unemployment. Zeitschrift für die gesamte Staatswissenschaft (JITE), 141, 256–268. Bolle, F., Breitmoser, Y., & Otto, P. E. (2009). Individual versus role dependent social preferences: evidence from bargaining experiments. Manuscript, EUV. Brown, A. J. G., Merkl, C., & Snower, D. J. (2009). An incentive theory of matching. Working Paper (4145, IZA). Cason, T. N., & Noussair, C. (2007). A market with frictions in the matching process: an experimental study. International Economic Review, 48(2), 665–691. Chamberlin, E. H. (1948). An experimental imperfect market. Journal of Political Economy, 56(2), 95– 108. Drolet, A. L., & Morris, M. W. (2000). Rapport in conflict resolution: accounting for face-to-face contact fosters mutual cooperation in mixed-motive conflicts. Journal of Experimental Social Psychology, 36, 26–50. Ehlers, L. (2007). Von Neumann-Morgenstern stable sets in matching problems. Journal of Economic Theory, 134(1), 537–547. Fischbacher, U. (2007). z-tree: zurich toolbox for ready-made economic experiments. Experimental Economics, 10(2), 171–178. Gale, D., & Shapley, L. S. (1962). College admission and the stability of marriage. American Mathematical Monthly, 69, 9–15. Georgescu-Roegen, N. (1958). Threshold in choice and the theory of demand. Econometrica, 26, 157–168. Gillies, D. B. (1959). Solutions to general non-zero-sum games. In A. W. Tucker & R. D. Luce (Eds.), Annals of mathematics studies: Vol. 40. Contributions to the theory of games IV (pp. 47–85). Princeton: Princeton University Press. Hall, R. E., & Milgrom, P. R. (2008). The limited influence of unemployment on the wage bargain. American Economic Review, 98(4), 1653–1674. Haruvy, E., & Ünver, M. U. (2007). Equilibrium selection and the role of information in repeated matching markets. Economics Letters, 94(2), 284–289. Haruvy, E., Roth, A., & Ünver, M. U. (2006). The dynamics of law clerk matching: an experimental and computational investigation of proposals for reform of the market. Journal of Economic Dynamics & Control, 30(3), 457–486. Huck, S., Normann, H.-T., & Oechssler, J. (2004). Two are few and four are many: number effects in experimental oligopolies. Journal of Economic Behavior & Organization, 53(4), 435–446. Kagel, J. H., & Roth, A. E. (2000). The dynamics of reorganization in matching markets: a laboratory experiment motivated by a natural experiment. Quarterly Journal of Economics, 115(1), 201–235. Kamecke, U. (1992). On the uniqueness of the solution to a large linear assignment problem. Journal of Mathematical Economics, 21(6), 509–521. Koopmans, T. J., & Beckmann, M. (1957). Assignment problems and the location of economic activity. Econometrica, 25(1), 53–76. Lundberg, S., & Pollak, R. A. (2008). Marriage market equilibrium and bargaining in marriage. Working Paper (December 2008, University of Washington). Mortensen, D. T., & Pissarides, C. A. (1994). Job creation and job destruction in the theory of unemployment. Review of Economic Studies, 61(3), 397–415. Myerson, R. B. (1991). Game theory—analysis of conflict. Cambridge/Massachusetts/London: Harvard University Press. Nosaka, H. (2007). Specialization and competition in marriage models. Journal of Economic Behavior & Organization, 63(1), 104–119.

348

P.E. Otto, F. Bolle

Radner, R., & Schotter, A. (1989). The sealed-bit mechanism: an experimental study. Journal of Economic Theory, 48, 179–220. Roth, A. E. (1984). Misrepresentation and stability in the marriage problem. Journal of Economic Theory, 34(2), 383–388. Roth, A. E. (1985). The college admissions problem is not equivalent to the marriage problem. Journal of Economic Theory, 36(2), 277–289. Roth, A. E. (1990). New physicians: a natural experiment in market organization. Science, 250, 1524– 1528. Roth, A. E., Sönmez, T., & Ünver, M. U. (2004). Kidney exchange. Quarterly Journal of Economics, 119(2), 457–488. Rubinstein, A. (1982). Perfect equilibrium in a bargaining model. Econometrica, 50(1), 97–109. Sasaki, H., & Toda, M. (1992). Two-sided matching problems with externalities. Journal of Economic Theory, 70(1), 93–110. Sell, J., & Wilson, R. (1991). Levels of information and contributions to public goods. Social Forces, 70(1), 107–124. Selten, R. (1972). Equal share analysis of characteristic function experiments. In H. Sauermann (Ed.), Contributions to experimental economics: Vol. 111. Beiträge zur experimentellen Wirtschaftsforschung (pp. 130–165). Tübingen: Mohr. Selten, R. (1991). Properties of a measure of predictive success. Mathematical Social Sciences, 21, 153– 167. Selten, R., & Uhlich, G. R. (1988). Order of strength and exhaustivity as additional hypotheses in theories for three-person characteristic function games. In R. Tietz, W. Albers, & R. Selten (Eds.), Lecture notes in economics and mathematical systems: Vol. 314. Bounded rational behavior in experimental games and markets (pp. 235–250). Berlin: Springer. Shapley, L. S., & Shubik, M. (1966). Quasi-cores in a monetary economy with non-convex preferences. Econometrica, 34, 805–827. Shapley, L. S., & Shubik, M. (1971). The assignment game, I: The core. International Journal of Game Theory, 1(1), 111–130. Tenbrunsel, A. E., Wade-Benzoni, K. A., Moag, J., & Bazerman, M. H. (1999). The negotiation matching process: relationships and partner selection. Organizational Behavior & Human Decision Processes, 80(3), 252–283. Valley, K. L., Moag, J., & Bazerman, M. H. (1998). ‘A matter of trust’: effects of communication on the efficiency and distribution of outcomes. Journal of Economic Behavior & Organization, 34, 211– 238. Valley, K. L., Thompson, L., Gibbons, R., & Bazerman, M. H. (2002). How communication improves efficiency in bargaining. Games & Economic Behavior, 38, 127–155.