By sending its top three leaders to the Annapolis peace summit, Israel is hoping to make a statement about the seriousness of its approach to peacemaking with the Palestinians.

But a more complex reality lies under the surface of this diplomatic show.

The big three -- Prime Minister Ehud Olmert, Defense Minister Ehud Barak and Foreign Minister Tzipi Livni -- have much different notions about what can be achieved with the Palestinians and how best to go about it.

Furthermore, they are bitter political rivals: All three want to be prime minister and hope to use Annapolis as a steppingstone toward keeping, regaining or winning the top job.

Olmert's peacemaking ideology is based on a view of a Middle East that is growing more dangerous by the day. He argues that although Israel faces serious risks if the peace process in which it is engaging fails, doing nothing would be far worse.

In Olmert’s view, simply maintaining the status quo would result in a Hamas takeover of the West Bank, the elimination of moderate Palestinians as a political factor and increased calls for a one-state solution. Based on the principle of one man, one vote in a single state comprising Israel, the West Bank and Gaza Strip, eventually this would bring a Palestinian majority to power.

Olmert argues further that by going to Annapolis and showing good will, Israel will not be blamed for failure. Instead, Israel will be in a better position to press for stronger international action on Iran.

The prime minister has domestic considerations, too. In a best-case scenario, progress on the Palestinian front would enable him to call an election some time next year from a position of strength. However, if the two sides make little headway, Olmert is counting on keeping the process going for as long as he feasibly can to buy more time as prime minister.

Either way, Olmert hopes an ongoing Palestinian process will help him through his domestic troubles: the upcoming Winograd Commission report on his performance in the 2006 Lebanon War and a string of corruption allegations currently under investigation.

Ironically, of the three leaders, Barak takes the hardest line, even though he heads the traditionally peace-minded Labor Party. He sees virtually no chance of a peace deal with the Palestinians, and believes the likely outcome of the current process will be more violence and terrorism.

That was the outcome the last time Israel engaged in a major peace push with the Palestinians, when Barak was prime minister and U.S. President Bill Clinton’s proposals at Camp David in 2000 were followed by the launching of the second Palestinian intifada.

Now, Barak says, as far as he is concerned, the "Clinton parameters" of December 2000 -- Clinton's peace proposal after the failure of the 2000 Camp David negotiations -- are no longer on the table.

At one point in recent months Barak warned against going to Annapolis, but he later came around to the view that Israel should do all it can to avoid being blamed for failure.

Barak also takes a tough line on dismantling unauthorized Israeli outposts in the West Bank, a move Israel is supposed to make early on in the process.

"I also admire the settlers in the illegal outposts," he declared recently. "There, too, we will have to provide for their everyday needs."

Left-wing leader Yossi Sarid charges that Barak is out to prove no one can succeed where he failed at Camp David in 2000.

"He is going to Annapolis not to save the conference but to bury it," Sarid said.

Barak anticipates a future contest for prime minister against the Likud Party's Benjamin Netanyahu, and pundits say his hawkish posturing is calculated partly to attract votes from the center and center-right away from Netanyahu, also a former prime minister.

Ideologically, Livni takes the middle ground. She strongly advocates "two states for two peoples" on the condition that the Palestinians have responsible leadership.

"We can't just throw the keys over the fence and hope for the best," she says.

In Livni's view, establishing a Palestinian state must be seen as the fulfillment of the national aspirations of all Palestinians; in other words, Palestinian refugees return to Palestine, not Israel. She sees Jerusalem as a major negotiating card, possibly to be used in a trade-off on the refugee issue, but  recently was critical of Olmert confidant Haim Ramon for playing it too early.

Livni clearly hopes to emerge from Annapolis with her prime ministerial credentials enhanced, having shown signs of negotiating smarts and a coherent worldview.

The foreign minister made a major political mistake in April when she called on Olmert to resign after a preliminary Winograd Commission report in the spring. Since then, Livni has been carefully repositioning herself to take over as leader should Olmert slip up again.

For now the polls are on her side.

One in Yediot Achronot’s Friday edition put her ahead of Olmert and Barak. Twenty-four percent of those surveyed said they had confidence in Olmert, 29 percent in Barak and 52 percent in Livni.

No doubt the way Annapolis plays out will affect the pecking order.