The role of information in different bargaining protocols

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.

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:

$$\mathit{CI} = \frac{2}{(\bar{\pi}-\underline{\pi})^{2}} \int_{\underline{\pi}}^{\bar{\pi}} \bigl(\bar{\pi} - G(x)\bigr)\,dx, $$

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 $\underline{\pi}=0$.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.

Fig. 3

Utility pairs associated to all possible outcomes when two players have preferences (400, 50, 150) and (50, 250, 300); G(x) denotes the possibility frontier of the convexified set of outcomes

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).

Fig. 4

Frequency map of CI in our 4 treatments

Table 8 Number of observations for each treatment and each category of conflict used

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

Table 9 Mann-Whitney test p-values for the comparison between complete and incomplete information. Although not indicated in the table, the p-values for the comparison in the case of negotiation (point mechanism) show that the different variables are significantly higher (lower) for complete information

Table 10 Mann-Whitney test p-values for the comparison between Negotiation and the Point Mechanism. Although not indicate in the table, the p-values for the comparison in the case of complete (incomplete) information show that the different variables are significantly higher (lower) for Negotiation

1.4 A.4 Time needed to reach an agreement

Fig. 5

Average time of agreement by the Conflict of Interest. The left figure compares the bargaining protocol negotiation under complete and incomplete information; the right figure does the same with the PM. In gray we display the minimum of the average time achieved by either complete or incomplete information. Above these bars we display the excess above the minimum of the relevant informational setting: in white when the average time used in complete information is higher; in black when the time with incomplete information is higher