I have a different take on why Katniss agrees with President Coin. I believe that Katniss connected the dots and realized the Coin was responsible for Prim's death afterall. She voted yes to feign allegiance with President Coin. As was previously said, if she wasn't with Coin then she was a threat. By agreeing with Coin, Katniss gains her trust and the chance to avenge Prim's death.
At this point of the story Katniss has some serious doubts as to the motives and methods undertaken by Coin to win the war. Did Coin bomb the children at the end of the war to frame the Capitol? Did Coin ensure Prim was killed in the explosion to sway The Mockingjay from any Capitol sympathy? Is Coin just as evil as President Snow? When President Coin proposes a final hunger games Katniss realizes the answer to her questions is yes. Coin truly is the same as Snow, and responsible for killing Prim. Coin must die. Voting yes gains Coin's trust.
Katniss votes yes to the final hunger games... "for Prim". Remember that Prim put this entire story into motion. It seems only fitting that Prim should end it. Katniss lures President Coin into believing Prim's death swayed her complete allegiance to District 13. However, it just the opposite. Katniss knows the capitol was framed for Prim's death and is now set on a mission of revenge. Katniss may just as well repeated that same sentiment when she shot Coin through the heart: "For Prim". It's always been for Prim. Not for the Capitol, not for the rebels, not for the hunger games. For Prim.