Great post, @chrisrice.
I'm afraid I object, though, and your final picture proves my point.
Before I tell you how let's think about the requirements for something to be a "zero sum game".

Obviously, there must be a winner and a loser.
Which means that there must be a limited amount of the resource been fought over.
And it must be possible for one person to completely control that resource to the exclusion of someone else.

Virtually all physical things meet these requirements.
For example, food, clothing, time, access to people, etc.
But not all physical things fall into the category because of abundance.
No one, at this time, fights over the air, for example.
Additionally, we have created such abundance in the modern world that fewer and fewer people are fighting over access to entertainment or clothing.
That is to say, there's just so much that no one person can stop another from having it.

At the same time, virtually all spiritual things don't meet the requirements above.

By "spiritual" I don't mean "mystical", I just mean things that we can't categorize with measures.
We create rules and measures around art or music, but beauty remains spiritual.
We can quantify knowledge, but wisdom remains spiritual.

Most emotions, including happiness, remain spiritual.
Spiritual things do not, however, follow the rules of a zero-sum game.
There is no limit to spiritual things.
Increasing my wisdom does not in any way limit the amount you can increase yours.
Nor is there any way for me to limit your spiritual experience by having more of my own.
That is, by my appreciation of beauty, I can't stop you from appreciating the same beauty or any other.

So, the conditions for a zero-sum game cannot be met on the "battlefield of happiness".

When you say that "The happiness of one person always seems to come at the expense of another", what I think you're really saying is that others can control your happiness.
I can see why you might feel that way.
The truth is, that while others can do things that can trigger unhappiness in you, it remains your choice to respond with unhappiness.
As Victor Frankl taught us, we have control over our emotional state despite the worst of circumstances because we retain the ability to choose.

The last picture you posted was that of your family.
You all look happy, and I hope that your current circumstances are a direct mirror of that picture!
In that picture, your happiness in your son does not in any way keep your wife from experiencing happiness in your son.
Your happiness is not at her expense or vice versa.

So, I submit to you that the only way one person's happiness can be at the expense of another's is if they both imagine the source of their happiness is in some physical thing, rather than happiness' true spiritual source.