On this basis, individual agents rationally decide to form new coalitions with their acquaintances or merge or leave existing coalitions. Finally, they may also form new acquaintance links to members of their own coalition or break existing ones to members of other coalitions. For subcritical adaptation rates see Methods for the description of the model and parameters , the coalition structure is dominated by very few macroscopic or even near-global coalitions. This leads to a peculiarly multimodal size distribution that can also be observed in various real-world systems 8 , 9 , 10 , 11 , 12 , 13 , not only in socio-economic contexts but also in purely physical systems such as droplets Note that some of the smaller network components consist of more than one coalition.

Each network is the equilibrium result of one model run. There are two extremes. Its multimodal nature emerges endogenously from the nonlinear dynamics of coalition formation. As a matter of fact, from empirical observations, multimodal distributions of social structures are well known, e. The reason for such a high critical value are the more macroscopic effects of the coalition formation process as compared to the network adaptation process; it may involve hundreds of agents at once whereas network adaptation only affects three agents at a time see Methods.

In Fig. It is not very distinctive but this is expected due to finite size effects Samples of up to nodes are rather small for statistical physics and an exponential progression of system sizes would be more revealing as network distances scale slowly with N. Still, we chose a linear progression in N because the coalition formation process causes high computational costs with rising N and agent numbers of up to several hundreds are quite realistic for many socio-economic systems Nevertheless, phase transitions appear, only the accompanying singularities are washed out or smoothed 18 due to finite-size effects.

Also, we expect only small finite-size corrections 19 to critical scaling with an exponential progression to larger N which are insignificant in the context of socio-economic modeling. Thus, the detailed analysis of finite-size phase transitions may be of great use for further models in this context. All variables are averaged over model runs. From these plots alone, the type of phase transition first- or second-order is hard to identify because a finite sample size will give both first- and second-order transitions a similar appearance.

However, the type of transition is revealed by the probability density function of the order parameter S. For Fig. A long tail appears, featuring bigger coalition sizes that cannot be explained by a Gaussian distribution Both arguments underline the assumption of a continuous phase transition.

## Theories of Coalition Formation - CRC Press Book

Via the maximum of the coefficient of variation of S , V S , it is possible to identify the critical region Fig. With this knowledge, it is possible to quantitatively grasp the critical dynamics. This way, in Fig. What can we infer from these results?

## Theories of Coalition Formation

If the acquaintance network in our coalition formation model adapts only slowly to the coalition structure, the formation of a grand coalition is most probable. Only for really high adaptivity, a fast transition to a heterogeneous coalition structure appears because then the effect of coalition formation is suppressed by a permanent rewiring of the acquaintance links. Before it is even possible to find a neighbor coalition to unite with, at some earlier stage the link to this coalition was already removed.

If adaptivity represents some kind of social punishment deprivation of social contact between agents from different coalitions then in this case punishment would actually be counterproductive. At high frequency it leads to the isolation of a high number of small and midsize coalitions forming independent network components. However, full cooperation provides the highest benefits to all agents in many socio-economic situations, including the common pool context of our study.

From an outside perspective, e. Of course, this phenomenon of coalition isolation is caused by our assumption that the total number of acquaintances stays constant over time which has been argued to be approximately realistic in social relations 1. As this assumption has such large effect on the model outcome and implications, it would be most interesting to study different scenarios in future work. Both our model and real-world systems may undergo non-equilibrium phase transitions in equilibrium physics an isolated system maximizes its entropy whereas non-equilibrium phase transition are driven by an external force, e.

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Less realistic is the investigation of agents in a fixed state after model convergence. However, after only a few time steps we observe the same basic appearance of coalition size distributions. Still, this aspect is part of ongoing work. In our model, we have increased the level of complexity from simple opinion adaptation processes from earlier research to more complex decision making determined by costs and benefits, and from local social interaction to mesoscopic cooperation.

Our approach gets support from empirical data. In our model, the fat-tailed and bimodal coalition size distributions develop over time, they are model-inherent. The distributions resulting from such processes of self-organization deliver measurable quantities to study such transformation processes. Future work should vary the payoff sub-model in order to represent different archetypical socio-economic situations than the common-pool setting.

One may then compare them with firm size distributions 12 , 13 , 17 from different economic sectors to identify the drivers of firm size growth. Other cases of multimodality in social systems were found in city size distributions of developing countries 11 and fish cohorts 8 , 9 possibly resulting from cooperative phenomena. However, even natural processes such as droplet growth give a similar picture: a power-law distribution for small droplets and a maximum for larger ones For the study of specific real-world systems, it might however be necessary to model heterogeneous agents.

But most importantly, the observation of phase transitions with respect to network adaptivity in our model encourages follow-up work on the role that the relative speed of processes in social feedback loops has for transformation process. Coalitions are collective decision-makers and the members of a coalition act as one player and therefore, each agent can only be a member of one coalition. Log out of Readcube.

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Log out of ReadCube. Results of three experiments showed that when the coalition payoff was such that members of each possible coalition could obtain an equitable payoff share, the outcome tended to be coalitions that also maximized the payoff of its members. Moreover, depending on the way people were allowed to interact, the final outcome was a coalition that maximized the payoff of its members or a coalition that provided them with an equitable payoff share.

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