Given what we've seen from Jack, I think it's safe to say that Kaido is the unreasonable type too. If we think about it he wants to start a whole war, just so that he can get what he wants. It shows what kind of person he is. So I doubt he can be reasoned with.The only way to stop him is by making him submit by cheer force.
If Shengao's theory turns out to be true, that Kidd is targetting Shanks because he's one of the stronger Yonko, then I really doubt that he'd have such a hard time dealing with Kaido. Remember he's just hard to kill, not invincible. He lost a couple of times to the other Yonko.
So here's what could have happened. Kaido hearing that WB is going to war, wants to get his last chance at suicide by having WB kill him. After all he is the strongest man, and it makes sense that he'd be able to take on the strongest creature. Shanks learning of Kaido's intentions decides to intervene, since Kaido going after WB could seriously delay Ace's rescue. So he decides to intercept Kaido, but talking with him won't work. So they both clash, and my guess is that Shanks won.He already faced WB and he seems to have come out of it unscathed, so that should be the proof that Shanks is indeed pretty strong.
After the clash, Shanks makes it to Marineford. His mere presence should dissuade the WG, and even BB himself. At that point the Marines were already exhausted from the battle with the WB pirates, so they couldn't simply face another Yonko with his crew. And BB just got what he wanted, so I guess he wasn't willing to take the risk of losing everything by doing something reckless, which would be taking on Shanks and his crew. It's not just about himself, it's also about BB's crew, since I doubt they'll be enough to deal with the red hair pirates.
This is my final prediction about what could have happened between Shanks and Kaido.I think we're getting a lot of hints about Shanks being possibly one of the strongest Yonko(if not the strongest). And Kaido simply isn't invincible, and given how the Kaido pirates act, I really doubt talking with him would solve anything.