The Syriza leader has said he would try to renegotiate Greece's debt agreements, focusing the attention of markets, euro zone leaders and not least of all, the troika of organizations overseeing Greece's bailout – the European Commission, European Central Bank and international Monetary Fund (IMF).
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Tsipras is opposed to austerity measures that were part of Greece's financial bailout agreed with its international lenders in return for a 240 billion euro ($270 billion) bailout. The cuts and reforms have drastically put the brakes on the country's economy and helped send unemployment levels to nearly 28 percent.
The IMF's managing director, Christine Lagarde, ruled out special treatment for Greece's debt situation, telling Le Monde newspaper on Monday that Greece had to respect euro zone rules and still needed to carry out key reforms. In addition, the German government said that debt restructuring was out of the question although an extension to the bailout program was an option.
Now, all eyes are on the party's unlikely coalition and what this could mean for its radical policies -- and whether it can turn those into reality.
The government's policy strategy is still unknown, according to one former government minister.
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"It's ultimately going to be much more difficult to figure out exactly what policies they will ultimately agree on," Petros Doukas, former deputy finance minister of Greece, told CNBC Monday morning.
He said that it would likely be difficult for Syriza and the Independent Greeks – coming from the left and right of the political spectrum – to agree on policy.
"It's going to be an uphill battle between Syriza and their left-wing promises, which they will not be able to deliver absolutely because they have some major problems to tackle immediately (such as) the banking problem and social security problems – there's seven billion euros (worth of debt) maturing in March and they need European liquidity support for that," Doukas said.
European markets fluctuated on Monday following the result, with the Athens Stock Exchange trading down almost 5 percent in early morning trade before paring losses to trade down 2.3 percent.
Konstantinos Botopoulos, chairman of the Hellenic Capital Market Commission, told CNBC that both markets and the public needed to know how Syriza would proceed. "We know what the pre-election promises are but we need to see some action," he said, adding that many things were "vague."
"The big problem is what the reality of government will be for a radical left party which has never been in government and has promised a complete change in politics that is very difficult to achieve.