Michael,Vito, and even Tom knew that once Michael made it appear he was not going to war that either Tessio or Clemenza would betray him. The only diference was Tom thought it would be Clemenza. My question here is was Michael's false appearance of weakness in the meeting with Tessio and Clemenza present designed to have one of them go over to the Barzini side?

There are several reasons this would have been advantageous to Michael.

First it would cement in Barzini's mind that Michael was weak, and stupid enough to fall for a fake peace meeting where he would be assassinated.
If you look at the exchange of glances between Michael and Barz at Vito's funeral you see mutual contempt, and a smugness that Barzini knew he had Tessio in his pocket.

Second it cleared the way for Michael to have completely loyal people around him and allowed him to finally show the Clemenza people and other loyalists his real plan as well as the existence and strength of Rocco's regime.

This takes us back to the discussion about the Corleone Family having just two capos which seems almost absurd. However, based soley on that with which the film presnets us, Michael was seeking to root out the traitor which would have been either capo or Carlo.

All good points.

I've posted before that Tessio's betrayal was a terrific stroke of good luck for Michael:

1. There never was, and never will be, a dual Donship. Sooner rather than later, Tess and Clem would have fought to the death over sole control, or split the family in two. Either scenario would have fatally weakened the Corleones.

2. Just as Vito's pledge not to be the first to break the peace wasn't binding on Michael, so was Vito's pledge to let Tess and Clem form their own families. Michael needed to run NY for the money that came his way, and for the synergy with his legitimate gaming interests in Nevada. He knew Clem was pliant enough to play second-fiddle to Michael, but Tess would never have accepted being Michael's subordinate--he'd want the family as his own enterprise.

3. Michael needed NY as his muscle to scare away potential Mob incursions on his Nevada interests. Without that deterrent, he'd be just another legitimate casino owner, a potential patsy for stronger interests--not excluding Tessio.

Honestly, the provoked betrayal has never made much sense to me. Barzini could have approached at all kinds of levels. Rocco, Neri or some rising soldier. Vito's prophecy sounds like an ancient oracle. In a situation so complex it's impossible to foresee how things are going. Remember how wrong he was when he tried to handle Sollozzo/Tattaglia.
Also, there could have been the danger that more than one capo or important person might betray Michael. What if Clemenza AND Tessio had betrayed him?
I think, Michael should have been honest to Tessio and Clemenza or at least let them know a little more.
Tessio was the smart one. Michael was left with the old fat, naive but loyal Clemenza.

Ironically I think part of the problem might have been in the fact that Tessio had more autonomy than Clemenza; recall that Tessio's group was made out to be its own family, independent of the Corleones, to fool enemies into thinking the Corleones were weaker than they were.

Could that autonomy have been the root of Tessio's desire for more? We don't know how he ran things under Vito before the novel began; perhaps he was given a lot of power and felt that Michael would have kept him closer under his thumb? It might have seemed like a demotion to him.

I don't understand this. Why would Michael want Tessio to betray him. Michael played along even when he knew it was a trap to leave Tessio and Barzini non-suspicious.

Tessio had been loyal to Don Vito for decades and as I know, Mike did like Sal Tessio.

This isn't rocket science. As Puzo decribed it, Vito and Michael feined weakness to deceive Barzini so that they had the time to gather their resources and plan for Barzini's undoing. Tessio was fooled also as I am sure Clemenza was fooled. That was the intent. Tessio did what he could to secure his life and fortune; Clemenza didn't. As Michael stated: "It's a smart move. Tessio was always smarter."

I don't think Michael would want anyone to betray him, but Vito knew once the power will be shifted to Michael, members in the family won't have faith in his abilities to run the Family. Loyal or not, they will seek other opprutunities to benefit themselves. Michael & Vito sitting back waiting to fish out the traitor.