Abstract
This paper is concerned with an old question: Will oligopolistic firms have incentives to merge to monopoly and will the monopoly, if the firms indeed merge, be stable? To answer this question, I motivate and introduce a new core concept for a general partition function game and prove stability of the merger-to-monopoly by applying the new core concept, labelled the strong-core, to Cournot oligopoly modelled as a partition function game. The paper shows that the Cournot oligopoly with any finite number of homogeneous firms without capacity constraints admits a non-empty strong-core and so does the Cournot oligopoly of not necessarily homogeneous firms with capacity constraints that are equal to their “historical” outputs. These results imply that oligopolistic firms will have incentives to merge to monopoly both in the long- and short-run and if the firms indeed merge to monopoly, the merger-to-monopoly will be stable.
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Notes
See Kóczy (2018: Ch. 11) for an excellent review of the literature on applications of partition function games to Cournot oligopoly.
While for coalitional games there exists an adequately and widely accepted core concept, this is not true for partition function games.
See Corchòn and Marini (2018) for more recent applications of cooperative game theory and the core to oligopolistic markets.
See Hafalir (2007) for formal definitions of both c- and r-cores.
This argument is reminiscent of the premise in Salant et al. (1983) that a market structure in which some cartel (i.e. a coalition with two or more firms) is worse-off cannot be an equilibrium market structure.
Although the \(\gamma\)-cores of these games are known to be non-empty (Chander 2018a), it does not follow that their strong-cores are also non-empty, because these games may neither be partially super-additive nor exhibit negative externalities.
Some authors consider constant marginal costs a relatively less interesting case (see e.g. Perry and Porter 1985).
The oligopolistic firms may not have the same incentives to merge to monopoly if the distribution of monopoly profits is not a strong-core payoff vector, but instead is, say, a \(\delta\)-core payoff vector.
In public good models this phenomenon is known as “free riding” and in climate change models as “leakages”.
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An earlier version of this paper was completed during my visit to Nanyang Technological University (NTU) in 2016–17. I am thankful to the Department of Economics, NTU, for its hospitality and stimulating environment. I am also thankful to two anonymous referees of this journals for their comments which have helped me greatly improve the paper.
Appendix
Appendix
Proof of the Lemma
Since each \({\pi }_{i}\left(.\right)\) is concave and continuous in \({q}_{1},\dots ,{q}_{n}\) and each \({A}_{i}\) is compact and convex, the game \(\left(N,A,\pi \right)\) admits a Nash equilibrium \(({\bar{q}}_{1},\dots ,{\bar{q}}_{n})\). Suppose contrary to the assertion that the game has another Nash equilibrium, say \(({\overline{\overline q}}_{1},\dots ,{\overline{\overline q}}_{n})\), and \(({\bar{q}}_{1},\dots ,{\bar{q}}_{n})\ne ({\overline{\overline q}}_{1},\dots ,{\overline{\overline q}}_{n}).\) Without loss of generality, let \(\bar{q}=\sum_{i\in N}{\bar{q}}_{i}\ge {\sum }_{i\in N}{\overline{\overline q}}_{i}=\overline{\overline q}.\) Since \(({\bar{q}}_{1},\dots ,{\bar{q}}_{n})\ne ({\overline{\overline q}}_{1},\dots ,{\overline{\overline q}}_{n}),\) \({\bar{q}}_{i}>{\overline{\overline q}}_{i}\) for at least one \(i.\) Furthermore, \({p}^{{\prime}}\left(\overline{\overline q}\right){\overline{\overline q}}_{i}+p\left(\overline{\overline q}\right)>{p}^{{\prime}}\left(\overline{\overline q}\right){\bar{q}}_{i}+p\left(\overline{\overline q}\right)\ge {p}^{{\prime}}\left(\bar{q}\right){\bar{q}}_{i}+p\left(\bar{q}\right)\), since \(\bar{q}\ge \overline{\overline q}\) and by assumption the marginal revenue of each firm is non-increasing with total demand \(q.\) From the first order conditions for a Nash equilibrium \({c}_{i}^{{\prime}}\left({\overline{\overline q}}_{i}\right)={p}^{{\prime}}\left(\overline{\overline q}\right){\overline{\overline q}}_{i}+p\left(\overline{\overline q}\right)>{p}^{{\prime}}\left(\bar{q}\right){\bar{q}}_{i}+p\left(\bar{q}\right)={c}_{i}^{{\prime}}({\bar{q}}_{i})\) implying \({\bar{q}}_{i}<{\overline{\overline q}}_{i},\) which is a contradiction. \(\square\)
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Chander, P. Stability of the merger-to-monopoly and a core concept for partition function games. Int J Game Theory 49, 953–973 (2020). https://doi.org/10.1007/s00182-020-00721-5
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DOI: https://doi.org/10.1007/s00182-020-00721-5