I can see her point, but it’s not necessarily valid.
First of all do YOU pay for the lunch. Or are you taking the money from the company and buying lunch.
If it’s company money you’re buying the lunches with than she has a point.
If it’s your money, then no.
Second, are the goals based on competition between teams or just the people themselves.
In other words do you say “Joe should have 100 points, he got 101 points.” That is competition amongst the person. OR is it “Joe’s goal is 100 points and he has 101 and your team Martha has 95 and her goal is 100.”
So that is your pitting people directly against each other.
The issue is when people fail too much, they quit. We had an incentive program and one of my reservationist Annie, was so much better than everyone else. She constantly won.
Now Annie was nearly 60 years old and spent her whole life in the hotel industry. All the rest were in their early 20s and had no experience. Of course she won. But then the effect was every one else quit trying. They, on the surface, were happy for Annie, she was a very friendly, popular employee, but on some level there was resentment.
So one month I took Annie aside and and I explained it to her. I said, you always win (the prize was $25.00) and this month you won’t. I gave her $25.00 out of my pocket and said, “this way you won’t be out any money,” and she was cool with it. I said “We’ll both know that you won, but this way others will think you can be beaten and they’ll try harder.”
And it worked out great. As soon as someone else won, they all started to improve.
In your case I would simply change the reward to something less public. Instead of lunch, give them all a gift card to Starbucks, so they get a reward but it’s not as public of a defeat for the other team.
There are two issues, number one, your team has a right to celebrate it’s work, but if that celebration is bringing the entire company down, whether it’s right or not, you need to revise your plan. 'Cause in the end if the other manager fails, the company fails and all the jobs are lost for the sake of a lunch.
And even though your right, do you want to risk this all for the sake of making a point?