The limits of protest politics have been laid bare in Egypt

As liberals, Islamists, and the military wrestle for legitimacy in Egypt, the unrolling of elections will change the dynamics for all three groups.

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The political standoff between Egypt's generals, the secular liberal political opposition and the Muslim Brotherhood that has gripped Cairo since the ouster of President Hosni Mubarak last February looks like a standard Hong Kong gangster movie scene: three men, each armed with two guns, points one at each of the other two.

But by bringing the preferences of many thousands of ordinary Egyptians into the equation, the elections that began on Monday may profoundly alter the balance within the triangle of power.

Over the past nine months, the military and the Islamists have made common cause against the liberals (over how to draw up a new constitution); the military and the liberals have joined forces against the Islamists (on "supra-constitutional" principles that would limit Islamist influence in shaping Egypt's future); and the liberals and Islamists have united against the generals (to limit military influence over the democratic process, and demand a faster transition).

Each faction has sought to maximise its own power and influence, combining with one of the others only to the extent that doing so enhanced its own agenda.

The current drama began two weeks ago when liberals and Islamists stood together to demand a more rapid transfer of power and limits on military influence, but the Muslim Brotherhood declined to support the re-occupation of Tahrir Square by liberal revolutionaries - who were viciously attacked by security forces - and their demand that elections be postponed, and power instead be immediately transferred to a caretaker government acceptable to the protestors in Tahrir Square.

Instead, the Islamists agreed with the military to start parliamentary elections on time (they run into early next year) and to bring forward to April or May the presidential election that the generals had hoped to hold only in 2013. And the polling which began on Monday could provide the Islamists with a powerful platform from which to challenge both the liberals and the generals.

But now that Egyptians have grabbed the chance to vote for those who will represent them in shaping the post-Mubarak era, the liberal demand for the installation of a hand-picked government becomes difficult to sustain.

The democratic voice of the electorate wins hands-down in any legitimacy contest with 20,000 demonstrators in Tahrir Square, no matter how passionate their commitment or how brutal the pummelling inflicted on them by the security forces. The long lines of voters queueing for hours at polling stations was an eloquent statement of the fact that Egypt's immediate political future will not be written in Tahrir Square.

Moreover, the balloting is unlikely to be kind to the secular forces that dominate the revolutionary politics of Tahrir Square.

Most of the liberal parties lack a clear political identity, much less the grass-roots presence and organisational machinery that the Brotherhood has built in working class communities despite decades of repression, maintaining a long-term presence through providing desperately needed social services.

Liberal parties may have halted their campaigns, insisting that now is not the time to hold elections and denouncing as self-interested opportunism the Brotherhood's decision to stay out of the clashes at Tahrir and press forward with the election. But the charge could just as easily be reversed- after all, skipping elections and instead demanding the immediate handover to a government acceptable to the Square would give those parties a better chance of wielding influence than if the matter is left up to voters.

While an election that has taken the wind out of the sails of the revolutionaries of Tahrir Square may have suited the Supreme Council of the Armed Forces (SCAF), it may yet confound the generals' own plans. According to SCAF's plan the parliament that Egyptians began electing on Monday will not govern, but will restrict itself to writing a new constitution, while the generals hold onto the reins. But while the generals have claimed to represent a "silent majority" and vowed not to yield to a "slogan-chanting crowd", they too can't compete with a democratically elected assembly in the legitimacy stakes.

The assembly created by the election that began on Monday, in which the Islamists are expected to emerge as the strongest party, may yet prove to be a powerful vehicle for challenging SCAF's claim to a monopoly on political power.

Despite their common rejection of the revolutionaries' demand to postpone elections and hand power to an un-elected civilian government picked by the Square rather than by the military, the Brotherhood and the SCAF hardly share an agenda - conspiracy theories notwithstanding. Like the liberals, the Brotherhood had called its supporters onto the streets to press SCAF to limit military influence.

Perhaps because of its leadership coming of age under Mr Mubarak's repression, perhaps simply because they're playing the long game, or perhaps because they have a keener sense than their rivals do of the preferences of the vast majority of Egyptians, the Brotherhood's leadership appears instinctively cautious. Still, by holding out for the electoral option, they may have once again eclipsed their liberal rivals, moving the epicentre of popular legitimacy away from street protests to democratically elected institutions.

The crowds in the square had dwindled by Sunday, the liberals' attempt at a "second revolution" having failed to ignite the mass support necessary to translate outrage into leverage. Even if they've lost support among those who took to Tahrir last week, the Brotherhood's leadership will likely eclipse its liberal rivals in the election. Once Egyptians began voting en masse, the Tahrir movement was cruelly confronted by the limits of protest politics.

Tony Karon is an analyst based in New York. Follow him on Twitter @TonyKaron