Tyr was smart, this was an ingenious idea by the young warrior because he knew that the only reason Kjell or SIgny was interested in this battle was because of him.
If he took himself out of the equation, then they had no reason to join in the battle.
Sighvatur was going to take this after he heard his son had been killed, there was no reason why he would not want to avenge his son.
He was a father, one thing Tyr had learned over the years was to never underestimate the love a father has for his children.
Tyr was about to take advantage of that, Sighvatur would throw common sense out the window, Tyr had hoped that Sigvatur would make a decision emotionally driven as the sole reason for Vingulmark's contribution would be gone.
During the time he had spent addressing them, the night before, before he went into the house where a battle ensued, a man had spoken to him.