In America, a price war is threatened. In Europe, combatants in a bloody conflict hope for peace. Both struggles have the same cause: that the number of big mobile-phone operators might be cut from four, the norm in many countries, to three. Regulators have been wary of this, fearing higher prices. In America the regulators remain cautious. But in Europe, where markets are smaller and operators are struggling, they may budge.

The talk of war came on Tuesday from Masayoshi Son, the boss of Softbank, a Japanese firm that controls Sprint, America's third-largest mobile operator. Son has his eye on T-Mobile USA, which ranks fourth: he wants to create a bigger rival to the two dominant carriers, AT&T and Verizon. Regulators have received the idea coolly. So has T-Mobile USA.

Lately T-Mobile USA — which bought a smallish operator, MetroPCS, last year — has waged a guerrilla campaign against AT&T, which has fought back with price cuts of its own. Son promised no letup: there would be a "massive price war" if he got his way. He added that a strong third mobile operator also could in time compete with fixed-line broadband, where choice is limited, prices are high and speeds slow.

Hope of peace comes from France, where Free, a cut-price entrant, has been harrying the established operators for two years. According to James Barford of Enders Analysis, a research firm, mobile-service revenue in France fell by 11 percent last year and by 6 percent in 2012. Profits have been squeezed hard. Among big European markets, only in Italy and Spain, where recession has been deep and long, have operators suffered more.

Hostilities may become less intense in France if Bouygues Telecom, the third-biggest operator, buys the second, SFR, from Vivendi, an ex-conglomerate that intends to concentrate on media. On March 5 Bouygues and Numericable, a cable operator, both made offers for SFR. Last week, Bouygues raised the cash portion of its bid. It is now offering Vivendi $15.7 billion in cash and 43 percent of the merged entity. Numericable has bid slightly less in cash and would give Vivendi 32 percent of the new company. Vivendi also has a third option, floating SFR on the stock market.

A takeover of SFR by Bouygues looks comfiest for France's mobile operators — including Orange, the leader — if not for customers. To forestall their worries, Bouygues says that if it won SFR it would sell its existing transmission network and some spectrum to Free. Arnaud Montebourg, the industry minister, made his preference plain to Le Parisien, a newspaper: a return to three operators would end "competition through destruction."

The European Commission decides on most telecoms mergers in the E.U., because companies usually do business in several member countries. But because SFR and Bouygues Telecom serve only France, and Numericable has little revenue from abroad, in this case the national competition authority would decide. Its chairman has said this might take nine months.

Copyright 2013 The Economist Newspaper Limited, London. All Rights Reserved. Reprinted with permission.