3.1. Investments and Payoffs for a Given Network
Ballester
et al. [
4] show that
given the network each actor’s
unique Nash investment is proportional to his Bonacich centrality for any
α,
β, and
λ for which
β >
λ(
n – 1). Bonacich centrality is a network centrality measure that counts all possible paths from each actor to all other actors, while path of length
k have a weight
wk. [
27]. The weight parameter
w can be chosen to fine-tune the centrality measure. The Bonacich centrality of an actor thus increases when a link is added by someone with whom the actor is connected through a path of any length. The measure reflects the feedback effects within the network: how much an actor invests does not only depend on the investments of his neighbors, but also on the investments of their neighbors, which in turn depend on the investments of their neighbors, and so on. This leads in Ballester
et al.’s model to Nash investment levels that are proportional to the Bonacich centrality measure in which paths of length
k are weighted by (
λ/
β)
k: the larger the ratio
λ/
β the more interdependent the network is, and the more beneficial it becomes to collaborate. For regular networks,
i.e., networks where each actor has the same degree
η, this boils down to a Nash investment
x* for every actor equal to
=
α/(
β −
ηλ). The earnings that relate to these investments are
= (
α2β)/(2(
β −
ηλ)
2) −
cη.
Ballester
et al. [
4] predict actors’ investment levels using Nash equilibrium strategies in case the network is exogenously given. Although the Nash equilibrium is an obvious solution concept, it ignores the possibility of cooperation by actors. Actors may cooperate by investing according to the socially optimal investment strategy instead of the Nash equilibrium strategy, which is attractive when the social optimum is Pareto efficient. The social optimum for a given structure is reached when actors choose investment levels such that the sum of all actors’ payoffs is highest. In regular networks, symmetry prescribes that the social optimum is reached for an investment profile in which all actors invest the same. In these networks, the socially optimal investment is
=
α/(
β − 2
ηλ), which leads to earnings equal to
=
α2/(2(
β − 2
ηλ)) −
cη.
We calculated all payoffs for Nash investments and socially optimal investments for the parameter configuration used in our experiment (
α = 48,
β = 16,
λ = 2) and all possible networks with four actors taking into account that subjects could only choose integer values for their investments. These Nash investments and socially optimal investments do not depend on the link costs. We present the results of these calculations in
Table 2. Because subjects can only choose integer investments, there are multiple equilibria for some networks. We chose to display the Pareto-efficient equilibrium in
Table 2, which is always unique. For some networks, there are also multiple investment combinations that are socially optimal. In that case we chose to display the most equal investment profile. Although our choices are kind of arbitrary here, note that the Pareto-optimal Nash investments are only relevant for the theoretical equilibrium analysis below and the networks related to both choice problems hardly occurred in this experiment.
When we ignore link costs (
c = 0), we see in
Table 2 that investments and payoffs quickly increase when there are more links in the network. These payoffs imply that actors are best off either in the empty network or in the complete network depending on the link costs.
Table 2 shows also that for low link costs any addition of a link becomes beneficial if actors choose Nash investments. In the intermediate cost case, actors need to reach almost the complete network, to do better than on their own, if they choose Nash investments. If they choose socially optimal investments, a triangle would already be better than investing alone. In the high cost condition, the complete network and socially optimal investment are both necessary for all actors to do better than investing alone.
3.2. Combining Link Decisions and Investments
We use the calculations above to address the link choices that actors want to make. We start with a first hypothesis based on Ballester
et al. [
4] about behavior in a given network. This hypothesis states that independent of the network structure or other conditions, Nash investments will be the most common amounts invested.
Hypothesis 1. Given the network, actors are most likely to invest according to their Nash investments.
Table 2.
Pareto-optimal Nash investments (Nash) and socially efficient investments (SO) with the related payoffs for given network structures in different cost conditions.
To hypothesize which networks can be expected to emerge, we extend the concept of “pairwise equilibrium” for our combination of investments and linking decision [
25,
26].
Definition. A strategy profile
s* = (
x*,
h*) for the game and payoffs defined above is a pairwise equilibrium constituting a pairwise equilibrium network
g if and only if
- (i)
s* is a Nash equilibrium and
- (ii)
Πi(, g+ij) > Πi(x*, g) implies Πj(, g+ij) < Πj(x*, g) for all i, j ϵ N where are the Nash investments for the network g+ij.
This definition limits the equilibria by excluding rather trivial coordination problems in linking decisions that are also easily solved in our experimental set-up. Thus, we only consider equilibria in which no pair of actors wants to add a link (because at least one actor receives lower payoffs in the network with the link than in the network without the link). All pairwise equilibrium configurations are indicated with bold figures in
Table 2. These are the complete network for the low cost condition and the empty network for the high cost condition. For the intermediate cost condition, both the empty and the complete network are pairwise equilibria. All other configurations do not constitute equilibria as can be inferred from
Table 2. Note that
c = 0 is not one of the experimental conditions, but only provided as a reference. The results in
Table 2 lead to the following three hypotheses:
Hypothesis 2. In the low cost condition (c = 10), actors are more likely to coordinate on the complete network than on any other network.
Hypothesis 3. In the intermediate cost condition (c = 30), actors are more likely to coordinate on the empty network or on the complete network than on any other network.
Hypothesis 4. In the high cost condition (c = 50), actors are more likely to coordinate on the empty network than on any other network.
The most difficult case to address is the intermediate cost condition. Because the subjects in the experiment start in the empty network, it is not to be taken for granted that they reach the complete network with Nash investments, because there are many situations in between in which they are worse off than in the empty network. As mentioned in the description of the computerized game, while “virtual” earnings are updated instantly on the screen, the actual payoffs are not calculated in continuous time but only at the end of the round. This means that subjects can display a great deal of costless trial-and-error in changing investments and links. Therefore, in our design, subjects are strictly speaking not worse off in intermediate networks as only the end result counts. However, since subjects do not know when a round would end, there is the threat of ending up in an intermediate network configuration in which a subject are worse off than in the empty network. We surmise that this threat of being worse off was enough to create a real coordination problem. Indeed, observing the behavior in our experiment shows that subjects are rather sensitive to the changes in their earnings (even though they are virtual) and that it was not necessarily easy to coordinate actions. If subjects plan ahead enough steps they might realize the benefits of creating the full network, but if they only do one or a few steps of thinking, they remain stuck in the empty network. However, even if subjects realize that the full network is advantageous, they still have to form beliefs about whether the other subjects do so as well. A subject who is fully aware of the benefits of the full network might not try and create the full network if he believes that the other subjects are not aware of the benefits of the full network. In other words, subjects with limited strategic sophistication will frustrate the coordination of the full network. We thus expect that if a group consists of strategically more sophisticated actors, this group is more likely to solve the coordination problem in the intermediate cost condition. In addition, actors can gain strategic sophistication over time through several learning mechanisms [
28], which would make solving the coordination problem more likely [
29]. In our experiment, subjects play the intermediate cost condition for multiple rounds. Subjects are likely to gain strategic sophistication as they play more rounds, which should also facilitate reaching the complete network. Moreover, half of the subjects play the intermediate cost condition after they played the low cost condition (low-to-high ordering) and the other half after they played the high cost condition (high-to-low ordering). Subjects are more likely to gain insight into the benefits of collaborating in the complete network in the low cost condition than in the high cost condition, because the complete network will emerge more often in the low cost condition. This insight might be subsequently used in the intermediate cost condition. Finally, actors with lower strategic sophistication themselves can learn from, and imitate those, who understand the situation better [
29]. We compare our experimental condition in which subjects can observe the investments of others with the condition in which subjects cannot see these investments. When actors can see how much others invest, it is easier to learn from each other than when actors do not have this information. Moreover, actors can attract others to link to them by increasing their investments. Thus, coordination is more likely if subjects can observe others’ investments. The arguments above lead to the following extensions of Hypothesis 3.
Hypothesis 3a. In the intermediate cost condition, the complete network will be reached more often relative to the empty network, when the group consists of strategically more sophisticated actors.
Hypothesis 3b. In the intermediate cost condition, the complete network will be reached more often relative to the empty network, the more rounds are played.
Hypothesis 3c. In the intermediate cost condition, the complete network will be reached more often relative to the empty network, in the low-to-high ordering, as compared to the high-to-low ordering.
Hypothesis 3d. In the intermediate cost condition, the complete network will be reached more often relative to the empty network, when actors have information about the investments of others than when they do not have this information.
Linked actors face a public good type of problem, where their Nash investments can be seen as “defecting” and investing more can be interpreted as contributing to the “network good.” The maximal investment that makes sense is where the social optimum is reached. In a one-shot situation, defecting is the dominant strategy. However, subjects in the experiment play a game in which they have some time to indicate their intentions for a specific investment and observe the intentions of others. Because the experimental round ends at an unknown point in time, this resembles a repeated game setting with an unknown end. Subjects can show their intentions to cooperate by investing more than the Nash investments and if they observe that others reciprocate and invest more as well, they can continue to increase until they reach the social optimum. If partners start to decrease their investments, the focal actor can reduce own investments again or can even sever the link with an uncooperative other. Therefore, our experimental setting clearly provides opportunities for conditional cooperation even though the intentions are not actually paid. We conjecture that if one would analyze the continuous time game as a dynamic game, the socially optimal investments can also be part of an equilibrium strategy. One probably needs the assumption that actors can instantaneously react on decisions of others, which is however doubtful in the experimental setting. The mechanism described above is only likely to work if subjects can observe the investments of the others. Therefore, we expect that cooperative investments above the Nash investments are more likely if subjects are in the experimental condition in which they have information about investments of other subjects. Note that we still consider Nash investments as the baseline prediction (see Hypothesis 1), but because the experiment provides opportunities for further cooperation as well, we investigate these potential deviations of the Nash investments in more detail.
Still, for similar reasons as formulated for the coordination part of the game, we expect that not all subjects are equally able to establish cooperation based on conditional cooperation. It is not automatic for subjects to realize that jointly investing more than the Nash investments can lead to better outcomes for all. Therefore, subjects really had to anticipate themselves what the consequences of their own actions would be in combination with those of the others. It takes quite some thinking to answer questions such as: “In the complete network with everybody choosing Nash investments, might it be beneficial for two, three, or four actors, if each of them increases their investment by one point?” Since being aware of such potential is a prerequisite for establishing cooperation, we expect that strategically sophisticated actors are more likely to cooperate. In addition, if actors gain experience they might learn to establish higher payoffs by cooperating because they understand the interaction situation better. This leads to the two last hypotheses.
Hypothesis 5. The level of cooperation will be higher when subjects have information about the investments of others than when they do not have this information.
Hypothesis 6. The level of cooperation will be higher when the group consists of strategically more sophisticated actors.
Hypothesis 7. The level of cooperation will be higher the more rounds are played.