A huge thanks to @Agustín Covarrubias 🔸 for his feedback and support on the following article:

Shapley values are an extremely popular tool in both economics and explainable AI.

In this article, we use the concept of “synergy” to build intuition for why Shapley values are fair. There are four unique properties to Shapley values, and all of them can be justified visually. Let’s dive in!

A figure from Bloch et al., 2021 using the Python package SHAP

The Game

On a sunny summer day, you and your two best friends decide to run a lemonade stand. Everyone contributes something special: Emma shares her family’s secret recipe, Liam finds premium-quality sugar, and you draw colorful posters.

The stand is a big hit! The group ends up making $280. But how best to split the profits? Each person contributed in a different way, and the success was clearly due to teamwork…

Luckily, Emma has a time machine. She goes back in time — redoing the day with different combinations of team members and recording the profits. This is how each simulation went:

Who was involved?	Total profits
Only Emma	$20
Only you	$30
Only Liam	$10
Emma and you	$90
You and Liam	$100
Liam and Emma	$30
Emma, Liam, and you	$280

Individually, Emma makes 20 dollars running the lemonade stand, and you make 30 dollars. But working together, the team makes 90 dollars.

The sum of individual profits is 20 + 30 = 50 dollars, which is clearly less than 90 dollars. That extra 90 - 50 = 40 dollars can be attributed to team dynamics. In game theory, this bonus is called the “synergy” of you and Emma. Let’s visualize our scenario as a Venn diagram.

**Note:** Synergies can also be negative (e.g., if Liam and Emma fought it could hurt profits).

The synergy bonuses in the Venn diagram are “unlocked” when the intersecting people are part of the team. To calculate total profit, we add up all areas relevant to that team.

For example, when the team consists of just you and Liam, three portions of the Venn diagram are unlocked: the area exclusive to you (30 dollars), the area exclusive to Liam (10 dollars), and the area exclusively shared by you and Liam (60 dollars). Adding these areas together, the total profit for team “You and Liam” comes out to 30 + 10 + 60 = 100 dollars.

Referring to our Venn diagram, the same formula holds true for every other team:

Team members	Sum of synergies	Total profits
Emma	$20	$20
You	$30	$30
Liam	$10	$10
Emma, You	$20 + $30 + $40	$90

...

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