Banzhaf vs. Shapley: Power in a Vee

Power vs. Payoff: A New Perspective

The Shapley value is fundamentally about allocating a divisible good (like money, or shared time). The Banzhaf power index, on the other hand, is designed for situations that are more like a vote—making a joint decision where the outcome is "yes" or "no."

The central idea of the Banzhaf index is to measure power by counting how many times a player is a swing voter—that is, a member whose vote is critical to a coalition achieving its goal. A player is powerful if their presence frequently turns a losing coalition into a winning one.

Let's begin by exploring these core concepts.

Calculating the Banzhaf Power Index for a Vee Decision

Now, let's apply this method to a concrete scenario within a vee relationship. Imagine a hinge (H) and two partners (P1 and P2) need to make a significant joint decision, such as relocating to a new city. All three agree that for the decision to pass, there must be a strong consensus. They formalize this by assigning "voting weights" to each person's preference and setting a required quota.

Let's define our weighted voting game:

• Players: $N=\{H,P_1,P_2\}$
• Weights: The hinge's buy-in is considered most crucial, followed by P1, and then P2. Let's assign weights $w_H=4$, $w_{P1}=2$, and $w_{P2}=1$.
• Quota: The decision requires a supermajority. Let's set the quota $q=6$.

The following video demonstrates the step-by-step process for calculating the Banzhaf index, which we will follow for our scenario.

Step 1: List all winning coalitions.
First, we list all possible coalitions and check if their combined weight meets the quota of $q=6$.

We have two winning coalitions: $\{H,P_1\}$ and $\{H,P_1,P_2\}$.

Step 2: Identify critical players in each winning coalition.
A player is critical if the coalition becomes losing without them.

• For coalition $\{H,P_1\}$ (Weight = 6):

  • If H leaves, weight becomes 2 (< 6). So, H is critical.
  • If P1 leaves, weight becomes 4 (< 6). So, P1 is critical.

• For coalition $\{H,P_1,P_2\}$ (Weight = 7):

  • If H leaves, weight becomes 3 (< 6). So, H is critical.
  • If P1 leaves, weight becomes 5 (< 6). So, P1 is critical.
  • If P2 leaves, weight becomes 6 ($\ge$ 6). The coalition still wins. So, P2 is not critical.

Step 3: Count how many times each player is critical.
This is the raw Banzhaf count, ${\beta }_i$.

• ${\beta }_H=2$ (critical in both winning coalitions)
• ${\beta }_{P1}=2$ (critical in both winning coalitions)
• ${\beta }_{P2}=0$ (critical in no winning coalitions)

Step 4: Normalize the counts.
The total number of critical swings is $2+2+0=4$. The Banzhaf power index, ${\varphi }_i$, is each player's share of these total swings.

• ${\varphi }_H=2/4=50\%$
• ${\varphi }_{P1}=2/4=50\%$
• ${\varphi }_{P2}=0/4=0\%$

Interpreting the Results

This result is striking. Despite having different voting weights (4, 2, and 1), the hinge (H) and partner P1 have equal power. Both control 50% of the decision-making power in this specific game. Furthermore, partner P2 is a dummy player. Even though they have a vote, their vote never changes an outcome. They have weight, but no power. This is a classic demonstration of the core Banzhaf principle: power is not proportional to weight.

Banzhaf Index vs. Shapley Value: Two Sides of Importance

We now have two indices that measure a player's importance. How do they differ, and when would you use one over the other?

The fundamental difference lies in their methodology and what they are designed to measure.

1. What they count:

   • Banzhaf: Considers all possible coalitions (subsets). It asks: "In how many coalitions is this player a swing voter?" This treats all coalitions as equally likely to form.
   • Shapley: Considers all possible permutations (ordered sequences) of players joining. It asks: "In how many sequences is this player the pivotal player who turns a losing sequence into a winning one?" This frames cooperation as a sequential process.

2. What they measure:

   • Banzhaf: Measures a player's power to be decisive in a yes/no vote. It's about influence over outcomes.
   • Shapley: Measures a player's average marginal contribution to the value created by a coalition. It's about a fair share of a divisible surplus.

The application of the Shapley value to voting games is known as the Shapley-Shubik Power Index. This video explains that concept and explicitly contrasts it with the Banzhaf index.

The difference is subtle but profound. Banzhaf looks at a coalition as a static group, while Shapley-Shubik models it as a dynamic formation process.

• Banzhaf: "If we form a group, what's the chance my presence is essential?"
• Shapley-Shubik: "If we line up to form a group one by one, what's the chance I'm the one who gets us over the finish line?"

This distinction is also noted in academic literature. The paper on network visualization we briefly saw before makes this point clearly.

Conclusion

In this lesson, we have added a second major tool to our cooperative game theory toolkit. Where the Shapley value gave us a way to negotiate fair payoffs, the Banzhaf power index gives us a way to analyze influence in decision-making.

• Key Takeaways:
  • The Banzhaf power index measures a player's influence by counting how often they are a swing voter—a critical member whose departure would cause a winning coalition to fail.
  • Power is not the same as weight. As we saw in our vee example, players with different weights can have equal power, and a player with weight can have zero power (a dummy player).
  • The Shapley value (and its voting game equivalent, the Shapley-Shubik index) calculates importance based on being the pivotal player in an ordered sequence of joining. It focuses on fair allocation of value.
  • The Banzhaf index is about combinations (which group forms?), while the Shapley value is about permutations (in what order do they form it?). Choosing between them depends on whether you are analyzing a fair division of surplus or the distribution of influence over a discrete decision.

We now have sophisticated methods for proposing how to divide value (Shapley) and how to measure decision-making power (Banzhaf). However, just because a division is "fair" or power is distributed in a certain way doesn't mean the arrangement is stable. Players might still have an incentive to break off and form their own coalitions if they can do better.

In our next lesson, we will confront this question of stability directly by introducing the Core, a solution concept that identifies which payoff allocations are immune to defections.