Determinants of Equilibrium Selection in Network Formation: An Experiment
- for all and and
- for all then .
3. Setup of the Experiment
3.1. Experimental Game
3.2. Multiple Equilibria
3.3. Factors Influencing the Network Formation Process
- The need for players’ farsightedness to reach the equilibrium
- The perceived riskiness of equilibria
3.4. Influences on Equilibrium Selection without Network Disruption
3.5. The Effect of Network Disruption on Equilibrium Selection
5. Experimental Procedures
6.1. Description of Subject and Group Variables
6.2. Equilibrium Selection across Treatments
- it has not been in any other equilibrium in the course of the game, and
- only one link is missing to reach the equilibrium and
- this link has not been suggested since reaching the quasi-equilibrium network.
6.3. The Effect of Network Disruption
The Effect of a Network Disruptor on Farsighted Behavior and Perceived Risk
Conflicts of Interest
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Although field research seems more natural, given the abundance of networks in the real world, it is notoriously hard to investigate network formation in the field. This is due to many possibly confounding factors such as imperfect knowledge of the existing network structure, the sheer size of most networks, agents’ motivation, as well as possible asymmetries in information, payoff or costs. In laboratory experiments, these factors can be controlled for, and other concepts relevant to network formation models can be influenced in a systematic way, such as linking costs and the value of links to players themselves and to others.
The theoretical models on network formation under the threat of disruption can be classified into two categories: centralized and decentralized models. In centralized models, such as Dziubiński and Goyal , Dziubiński and Goyal , Goyal and Vigier , Hoyer and De Jaegher  or Haller , a network designer forms the network, anticipating a subsequent attack. In decentralized models, such as Hoyer and De Jaegher  or Haller and Hoyer , the nodes are individual players that build the network.
See Hoyer and De Jaegher  for a more detailed description of the model with a strategic network disruptor.
Note that calculates the expected benefits of a player, as those may be stochastic if network disruption targets several links with equal probability. We take the expected benefits to calculate the payoffs to player i, as he/she has to decide on links before he/she knows which links will be disrupted.
In the complete network, there are 6 links to connect the 4 nodes.
We added to the values and multiplied them with 30.
Thereby, we took into account that we, of course, inherently changed the game without network disruption in such a way that it is no longer equivalent to the basic network formation model introduced by Jackson and Wolinsky . This can also be seen in the possible stable networks. Whereas in the model introduced in Jackson and Wolinsky , only the empty network is pairwise stable, in our treatment with the adjusted payoffs, we have the same stable networks as in the case with network disruption: the empty network, the separated network and the circle network.
An end-player is any player connected by degree one.
Strictly speaking, the separated network is a set of four equilibria with isomorphic structures.
In the following analysis, we focus on the circle and the empty network, as these two equilibrium networks can be reached using the same amount of linking decisions independently of the starting network. The separated network can be reached with less linking decisions when the starting network is the line than when it is the dyads. As in the experiment, it was reached in only very few cases, we will focus our analysis on the circle and the empty network.
An improving path is a sequence of networks that can emerge when players can form or sever links based on the increase in payoff they will receive in the emerging network as compared to the current one. In such a sequence, each network differs from its predecessor by exactly one link. Whether links are added or severed is determined by the conditions of pairwise stability.
For a short discussion on the lack of analysis of cognitive abilities in standard economic theory, see Frederick .
It is easy to see that when network disruption is introduced to the game, nothing changes in the robustness of the equilibria. It remains that while the empty network is robust against errors, the circle network is not.
Unlike for the case of farsightedness, we do not have a revealed measure of risk. We can control for the general risk attitude of subjects, using the Holt and Laury measure, but we do not find a revealed measure for risk in the network formation game. Therefore, we focus on the outcome of the game as our variable of interest.
The instructions used during the experiment are available upon request.
As is done in Holt and Laury , we simply use the number of safe choices that subjects take to categorize them on the individual level in this table, ignoring any inconsistencies in choices. In total, we observe 10 inconsistent choices, which we take into account for the median values per group, as used in the further analysis. Subjects with inconsistent choices were treated as though there were missing data on their risk attitude for this purpose.
Conducting the analysis without including the three groups does not change the results qualitatively, however.
While the test statistic is reported, also a Fisher’s exact test is performed to correct for the fact that some of our cells included only less than five observations. The test statistic, however, remained insignificant. The same holds for every other test statistic in this paper. If there are cells with less than five observations, we always also performed a Fisher’s exact test. We will only specifically report the results if they differ between the two tests.
We consider relative frequencies here instead of absolute frequencies to control for the different number of groups we had per setting.
The result on the separated equilibrium can easily be explained. The separated equilibrium is reached by adding one link when starting from the line network and adding this link is a myopic best response. As opposed to this, it is highly unlikely to reach the separated equilibrium when starting from the dyads.
For the correlation analysis, we dropped all the ambiguous and irrational decisions.
This observation is inline with observations stated in Arad and Rubinstein .
We performed a Fisher’s exact test instead of a test, as we had two cells with less than five entries.
Again, here, we also view the groups that are in quasi-equilibrium as groups that coordinate on an equilibrium network.
The separated network is reached only in a very small minority of the cases.
|No. of Decisions total||11.420||4.408||2||22||148|
|No. of Decisions R1||5.122||3.388||0||14||148|
|No. of Decisions R2||6.297||3.193||1||14||148|
|No. of Decisions ND||5.311||3.19||0||14||148|
|No. of Decisions D||6.108||3.447||0||14||148|
|Individual||Median per Group|
|Number of Steps||Frequency||Percent||Frequency||Percent|
|Individual||Median per Group|
|highly risk loving||2||1.35||0||0|
|very risk loving||0||0||0||0|
|slightly risk averse||20||13.51||6||8.11|
|very risk averse||35||23.65||23||31.08|
|highly risk averse||13||8.78||8||10.81|
|very highly risk averse||13||8.78||0||0|
|Not in Equilibrium||9||12.16|
|Final Network||Starting Network Line||Starting Network Dyads|
|Median of Revealed Farsighted Players Per Group||1.909 **||6.743||−2.827||0.059|
|Disruption||−1.488 **||0.226||−3.599 **||0.027|
|Median Risk Lovingness per Group||−0.055||0.946||0.338||1.402|
|Average No. of Females per Group||−0.503||1.654||−1.230||0.292|
|Average No. of Economics Students per Group||−1.717||0.179||−3.508||0.029|
|0 Players Who Know Game Theory||0.099||1.104||1.976||7.214|
|1 Player Who Knows Game Theory||−0.838||0.433||−0.264||0.768|
|2 Players Who Know Game Theory||0.874||2.396||16.730||1.84 × 10|
|3 Players Who Know Game Theory||1.717||5.567||15.856||7.690|
|Not in Equilibrium||5||13.50||1||2.70|
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Hoyer, B.; Rosenkranz, S. Determinants of Equilibrium Selection in Network Formation: An Experiment. Games 2018, 9, 89. https://doi.org/10.3390/g9040089
Hoyer B, Rosenkranz S. Determinants of Equilibrium Selection in Network Formation: An Experiment. Games. 2018; 9(4):89. https://doi.org/10.3390/g9040089Chicago/Turabian Style
Hoyer, Britta, and Stephanie Rosenkranz. 2018. "Determinants of Equilibrium Selection in Network Formation: An Experiment" Games 9, no. 4: 89. https://doi.org/10.3390/g9040089