Abstract
We analyze a bargaining protocol recently proposed in the literature vis-à-vis unconstrained negotiation. This new mechanism extracts “gains from trade” inherent in the differing valuation of two parties towards various issues where conflict exists. We assess the role of incomplete vs. complete information in the efficiency achieved by this new mechanism and by unconstrained negotiation. We find that unconstrained negotiation does best under a situation of complete information where the valuations of both bargaining parties are common knowledge. Instead, the newly proposed mechanism does best in a situation with incomplete information. The sources of inefficiencies in each of the two cases arise from the different strategic use of the available information.
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Notes
Keeney and Raiffa (1991).
For simplicity, in our experiment subjects earn half their valuation whenever ties occur on an issue. That is, we do not decide on a tied issue with the toss of a fair coin; instead we assign the expected value of such randomisation.
We partitioned the subjects into three sets of six players (or two sets of ten players) so as to obtain three (two) independent observations. Throughout the paper we present non-parametric tests using the average of each independent group as a single data point.
Whenever information about preferences of the other player is incomplete (which will be the case for one of the treatments), the game played is a non-zero sum Colonel Blotto game with incomplete information. A Colonel Blotto Game can be described by a situation where two colonels are fighting over a number of regions and have to decide how to divide their forces; the one with larger forces wins the region and the winner of the battle is the one who wins the most territory. This is, unlike in our setting, a zero-sum game because both colonels value all regions equally. Recently, Robertson (2006) and Hart (2006) have provided a complete solution of the Colonel Blotto game when parties value all issues equally, but the game has yet to be solved in our case where opposing parties have differing relative intensities (so there are gains to be extracted)
A fully fletched analysis of individual behaviour under the PM can be found in the Hortala-Vallve and Llorente-Saguer (2010)—a short summary of this paper’s key findings is included in the appendix to this paper.
In the instructions we stated that revealing one’s identity implied being excluded from the lab’s recruiting list; we did not find anyone breaking this rule.
Carnevale and Isen (1986) claim that most negotiations occur under some form of time pressure. Besides it is not feasible to run a laboratory experiment without such constraints as the possibility of gridlock may freeze the experiment indefinitely. Yukl et al. (1976) and Carnevale and Isen (1986) show that early deadlines can affect negatively the payoff of integrative negotiations. Roth et al. (1988) show that tied deadlines lead to delaying bargaining into the last seconds. We tried avoiding such effects by running a pilot experiment and implementing a time constraint above the maximum time subjects declared optimal.
For example (300, 200, 100), (100, 400, 100), (100, 100, 400) or (500, 50, 50) were all equally likely.
Framing effects imply that voters may behave differently when they are assigned payments (1, 2) or (200, 400)—see the seminal reference Kahneman and Tversky (1983).
An extensive analysis of the PM sessions can be found in Hortala-Vallve and Llorente-Saguer (2010).
As one can guess from the graph, not all these differences are significantly different from zero. About one fourth of the comparisons are significant.
See the Appendix for a detailed description of the equilibrium.
See the Appendix for a detailed description of the equilibria.
Roth and Murnighan (1982) study the case in which information is asymmetric in the sense that only one player is better informed than the other player. These cases are not discussed here.
One could argue that this player is best responding to his opponent bidding truthfully (i.e. matching the ratio of points to his relative intensities). This could be interpreted as level 0 behaviour according to cognitive hierarchy models (see Nagel, 1995).
All the aforementioned differences are significant at the 1 % level (Mann Whitney test) with the only exception of difference between the time in the negotiation sessions with complete information and incomplete information that is marginally insignificant (z=1.643, p=0.1003). However, this difference becomes significant at the 10 % if we drop the initial six periods of the data (out of 18 periods). In the Appendix we include Fig. 5 in which we report average times in each of our treatments by the level of conflict.
We are indebted to Rachel Croson for bringing this fact to our attention.
In the case of negotiation, \(\bar{\pi}=600\) and \(\underline{\pi}=0\). In the case of the PM the lower and upper bounds change, given that no player can win anything (it can never be the case that a single player has fewer points than his opponent in all issues). In order to make the analysis of both mechanisms comparable, we assume throughout the paper that \(\bar{\pi}=600\) and \(\underline{\pi}=0\).
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Acknowledgements
We thank Antonio Cabrales, Alessandra Casella, Gary Charness, Rachel Croson, Andy Eggers, Christoph Engel, Ido Erev, Guillaume Frechette, Simon Hix, Robin Hogarth, Nagore Iriberri, Marco Kleine, Matthias Lang, Tomás Lejarraga, Tom Palfrey, Andy Schotter and seminar participants at a number of conferences for helpful comments and discussions. We specially want to thank two anonymous referees for encouraging us analyzing the role of information in the previously circulated paper “Experimental Comparison between Free Negotiation and a Multi-issue Point Mechanism”. We would also like to thank Benedikt Herz, Pablo Lopez, Aguilar Beltran and Gwendolin Sajons for excellent assistance at running the experiments. The first author acknowledges financial support from the British Academy; the third author acknowledges financial support from ECO2008-01768 and the Generalitat de Catalunya and the CREA program.
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Appendix
Appendix
1.1 A.1 A measure of the Conflict of Interest (Axelrod, 1967)
Given a two-player game, we construct the utility possibility frontier and we denote it G(x); G(x) is the maximum utility player 2 can reach when player 1 gets x. This function is defined for x greater than the minimum possible payoff (\(\underline{\pi}\)) and smaller than the highest possible payoff (\(\bar{\pi}\)). However, the games our subjects play only have a finite set of outcomes, thus G(x) is not defined in all its range. In order to simplify our analysis we convexify the set of outcomes so that the utility possibility frontier is now continuous. Note that convexifying the set of outcomes is analogous to allowing lotteries when subjects have von Neumann-Morgenstern utilities.
A game that allows both players reach their maximum utility simultaneously should be viewed as a game of very little conflict; instead, a game where the gains for one player are losses to his opponent should be viewed as a game of high conflict. The former corresponds to a game of coordination and the latter to a constant sum game. Our measure of conflict captures how close we are from a constant-sum game where no gains from trading issues are possible. This is calculated by computing the area between the maximum utility player 1 can reach and the utility possibility frontier G(x). Formally it reads as follows:
where the term before the integral is simply a normalization so that our index is between 0 and 1. A higher CI indicates that there is a large amount of conflict and a lower CI indicates that there is low level of conflict between the two players.
In Fig. 3 we depict a situation where player 1 has preferences (400, 50, 150) and player 2 has preferences (50, 250, 300)—note that \(\bar{\pi}=600\) and \(\underline{\pi}=0\).Footnote 20 The points show the pair of utilities associated to each outcome. For instance, the outcome where player 1 decides on the first issue and player 2 decides on the second and third issues is depicted as the utility pair (400, 550). The gray area is, once normalized, the CI of this particular game. This area is bounded above and to the right by the utility possibility frontier of a pure coordination game where both players can achieve the maximum utility thus the conflict is minimum, CI = 0; the dashed line indicates utility possibility frontier of a constant sum game where conflict is maximum, CI=1.
Figure 3 shows that CI captures how far the set of utility pairs are from the dashed line where the sum of the subjects’ payoffs is constant. However CI fails to capture equitability in the achieved outcomes. Compare for instance a situation where both subjects have preferences (301, 299) with a situation where both subjects have preferences (599, 1). Both situations have the same index (CI = 1), however the conflict differs. In the first case any subject is almost indifferent between winning any of the two issues (and losing the remaining one). Instead, in the second case both subjects have a strong preference to win the first issue. This simple example highlights the fact that Axelrod’s index of conflict captures the possibilities of trade in a 2-player game but remains silent about the inequality among players. Despite its limitation, CI is extremely helpful in order to characterize different situations by their level of conflict. In Fig. 4 and Table 8 we report the distribution of CI in our experimental sessions (see histograms with relative frequencies and table with the absolute number of observations).
1.2 A.2 Equilibrium strategies under PM
The following table lists all the possible valuation profiles that were induced (ordered in decreasing order) together with the symmetric Bayesian Nash equilibrium action for each type of subject in the incomplete information game (taken from Hortala-Vallve and Llorente-Saguer, 2010):
Profile # | θ 1 | θ 2 | θ 3 | v 1 | v 2 | v 3 |
---|---|---|---|---|---|---|
1 | 200 | 200 | 200 | 2 | 2 | 2 |
2 | 250 | 200 | 150 | 3 | 2 | 1 |
3 | 250 | 250 | 100 | 3 | 3 | 0 |
4 | 300 | 150 | 150 | 4 | 1 | 1 |
5 | 300 | 200 | 100 | 4 | 2 | 0 |
6 | 300 | 250 | 50 | 3 | 3 | 0 |
7 | 350 | 150 | 100 | 5 | 1 | 0 |
8 | 350 | 200 | 50 | 5 | 1 | 0 |
9 | 400 | 100 | 100 | 6 | 0 | 0 |
10 | 400 | 150 | 50 | 5 | 1 | 0 |
11 | 450 | 100 | 50 | 6 | 0 | 0 |
12 | 500 | 50 | 50 | 6 | 0 | 0 |
Under incomplete information the equilibrium has very appealing properties that seem intuitive: indifferent subjects (those that care about all the issues with equal intensity) distribute their voting power evenly and extreme subjects concentrate their voting power on their most preferred issue. As explained in the main text, the prescribed strategies are monotonic and this is the key factor in generating outcomes that are close to the efficient ones.
One of the nice features of this setting is that actions can be categorized by level of intensity. Let us define truthful strategy by the voting profile that minimizes the angle with the voting profile. By considering the angle between each voting profile and the voting profile that assigns equal voting power to all issues. This measure captures the degree of intensification of each voting profile or, in other words, the extent to which our subjects concentrate their voting power on a few issues. A feature of the equilibrium is that the equilibrium action is always more intense than the truthful action. Therefore, whenever truthful behavior and equilibrium behavior differ, we can classify all actions as belonging to one of the following 6 (self-explanatory) equivalence classes: 28.10 % of the observations are less intense than truth, 10.79 % coincide with the truthful action, 30.49 % are less intense than theory but more than truthful, 24 % coincide with the equilibrium prediction and 6.52 % play more intense than theory.
The set of games with equilibrium in pure strategies in the case of complete information is generally characterized in Hortala-Vallve and Llorente-Saguer (2012). In the context of the experiment, there exist two potential types of equilibrium. In the first type of equilibrium both voters concentrate all the votes on their preferred issue. This equilibrium can be sustained whenever both subjects’ preferred issue coincides and the valuation of this issue is equal or higher than 300 (profile numbers 4–12). In the second type of equilibria, both subjects invest a single vote in their preferred issue and five votes in their second preferred one. This equilibrium can be sustained whenever subjects’ preferred issue does not coincide, but the second most preferred issue does and their second valuation must be at least twice their third valuation. Hence, this equilibrium can only be sustained with valuations profiles 3, 5, 6, 8, 10 and 11. All games that can’t sustain these two types of equilibrium have an equilibrium in mixed strategies.
1.3 A.3 Non-parametric tests summary
1.4 A.4 Time needed to reach an agreement
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Hortala-Vallve, R., Llorente-Saguer, A. & Nagel, R. The role of information in different bargaining protocols. Exp Econ 16, 88–113 (2013). https://doi.org/10.1007/s10683-012-9328-6
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DOI: https://doi.org/10.1007/s10683-012-9328-6