Here are some excerpts from Robert Gates' Duty: Memoirs of a Secretary at War that look at some of the decision making aspects under Bush and Obama.
Obama was the most deliberative president I worked for. His approach to problem solving reminded me of Lincoln’s comment on his approach to decision making: “I am never easy when I am handling a thought, till I have bounded it north, and bounded it south, and bounded it east, and bounded it west.” As Obama would tell me on more than one occasion, “I can’t defend it unless I understand it.” I rarely saw him rush to a decision when circumstances allowed him time to gather information , analyze, and reflect. He would sometimes be criticized for his “dilatory” decision making, but I found it refreshing and reassuring, especially since so many pundits and critics seem to think a problem discovered in the morning should be solved by evening. As a participant in that decision-making process, I always felt more confident about the outcome after thorough deliberation. When the occasion demanded it, though, Obama could make a big decision— a life-and-death decision— very fast.
Keep in mind that Gates worked for 8 different presidents. So when he says Obama was the most deliberative president he worked for, he's coming from a place of experience. But how do you foster deliberation? Obama made some decisions that were controversial with his top senior appointments. Here's an example,
The president wanted Jim Steinberg, who had been deputy national security adviser under President Clinton, to become deputy secretary of state. Having been a deputy twice myself, I suspect Jim did not want to return to government as a deputy anything. In order to persuade Steinberg to accept the offer, Obama agreed to his request that he be made a member of the Principals Committee and have a seat in National Security Council meetings as well as one on the Deputies Committee. As far as I know, no deputy had ever been given an independent chair at the principals’ table.
Steinberg’s presence on the Principals Committee gave State two voices at the table— two voices that often disagreed. Steinberg would often stake out a position in the Deputies Committee that was at odds with what Hillary believed, then express that position in meetings of the principals and even with the president. Let’s just say that having two State Department positions on an issue was an unnecessary complication in the decision-making process …
Obama also took a “team of rivals” approach, (a play on Lincoln selecting 3 key rivals for Cabinet appointments) for example, by appointing Hillary Clinton as Secretary of State.
Obama also actively encouraged bad news and disagreement.
Less than two weeks after the inaugural, at the end of his weekly meeting with Mullen and me, the president asked me to remain behind for a private conversation. He asked me whether everything was going okay. I told him I thought the team was off to a good start, the chemistry was good, and the principals were working well together. … As Obama had done before on several occasions with all the principals, he encouraged me always to speak up and to be sure to give him bad news or to express disagreement. He concluded with what I thought was a very insightful observation twelve days into his presidency: “What I know concerns me. What I don’t know concerns me even more. What people aren’t telling me worries me the most.” It takes many officials in Washington years to figure that out; some never do.
While Gates isn't as explicit on the decision process under Bush, we get some insights. At one point, talking about the U.S. role of the Israeli attack on the reactor in Syria (under Bush), he writes:
On our side, a very sensitive and difficult security challenge had been debated openly with no pulled punches. The president heard directly from his senior advisers on a number of occasions and had made a tough decision based on what he heard and on his own instincts. And there had been no leaks. Although I was unhappy with the path we had taken, I told Hadley the episode had been a model of national security decision making. In the end, a big problem was solved and none of my fears were realized. It is hard to criticize success. But we had condoned reaching for a gun before diplomacy could be brought to bear, and we had condoned another preventive act of war. This made me all the more nervous about an even bigger looming national security problem.