That certainly is not the case, unless you believe in zero-sum theory.
I think what you're trying to claim, despite your confused use of the jargon of game theory, is that the addition of more people grows the metaphorical pie.
In some instances that's true, but I'm not sure it's true for modern food production.
In a world of free land and free water, every additional family might be expected to start a farm and grow their own food. In that case, yes, the pie grows.
But that isn't what happens today. Today most food comes from industrial farms using modern equipment, petroleum derived fertilizer, and strained water resources. How does the addition of the typical illegal immigrant proportionately increase any of these things?
Does every new immigrant bring more water?
Does every new immigrant bring more petroleum for fertilizer?
Does every new immigrant design more efficient farming equipment?
Does every new immigrant genetically modify some plant to be more vigorous and productive?
Probably not.
But suppose the pie does grow for some reason, if it doesn't grow in proportion to population increases, then prices still tend to go up.
The reason food production has grown so much in the USA is because a few clever people found oil, gas, water, and engineered machines, plants, and chemicals.
Unless these immigrants are engineers and scientists, it's likely they're putting an extra burden on food production.