Abstract
We define balance games, which describe the formation of friendships and enmity in social networks. We show that if the agents give high priority to future profits over short term gains, all Pareto optimal strategies will eventually result in a balanced network. If, on the other hand, agents prioritize short term gains over the long term, every Nash equilibrium eventually results in a network that is stable but that might not be balanced.
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Notes
Neither of these restrictions is fundamentally necessary, all proofs presented in this paper can easily be adapted to mixed strategies that do use memory. But the restrictions do greatly simplify the proofs, so we assume them for reasons of clarity of presentation.
The lack of an attitude may be due to an agent’s ignorance or unawareness of the other. Occasionally we may also understand a 0-edge to be a neutral or indifference attitude.
We consider it hard for someone to go from a positive or negative attitude towards someone to be ignorant of that person, so 00 is in general not a possible output of \({+}{0}\).
The notion of balance in Harary et al. (1965) is defined for a less general concept, in the sense that our definition of balance in Sect. 5.1 (only for pairs and triads, but not longer cycles) is called 3-balance there, and the balance defined there needs to be achieved for any length of cycles. For details see (Harary et al. 1965, p. 341).
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This is an extended version of the paper van der Hoek et al. (2019) that was presented at LORI 2019. We thank the anonymous reviewers of LORI and of this special issue for useful comments and suggestions. Yì N. Wáng acknowledges funding support by the National Social Science Fund of China (18ZDA290, 20&ZD047).
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Hoek, W., Kuijer, L.B. & Wáng, Y.N. Who Should Be My Friends? Social Balance from the Perspective of Game Theory. J of Log Lang and Inf 31, 189–211 (2022). https://doi.org/10.1007/s10849-022-09356-z
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DOI: https://doi.org/10.1007/s10849-022-09356-z