Let's say you are being alpha in a group situation. You decide on a plan, could be as simple as where to eat or what movie to see, or a more complex course of action, and initially the group agrees with your leadership. However, an amog or aspiring beta in the group challenges you by proposing an alternate plan. The two scenarios are he either proposes a better plan, or he has more social proof so the group follows him. Either way, it's clear your plan has just been trashed. How do you maintain your dominance and not be seen as submitting to a challenger while refraining from a petty argument about what restaurant is better (which doesn't seem very alpha either).

Thanks