Core Existence in Vee Coalition Structures

In our last two lessons, we explored methods for assigning value and power to individuals within a cooperative group. The Shapley value provided a "fair" way to divide a surplus, while the Banzhaf index measured a player's power to influence a decision. These are prescriptive tools—they suggest what should happen based on axioms of fairness or definitions of power. Today, we shift from prescription to prediction and analysis. We will ask a more fundamental question: regardless of fairness, which outcomes are actually stable?

An arrangement might seem fair on paper, but if a subgroup of partners can break away and achieve a better outcome for themselves, the original arrangement is unlikely to last. This lesson introduces the Core, a central concept in cooperative game theory that identifies the set of all such "unshakeable" or stable payoff distributions. Our goal is to determine whether the core is non-empty for a given vee coalition structure. Understanding this allows us to diagnose the inherent stability of a relationship's cooperative framework.

From Possible Payoffs to Stable Allocations

Before we can talk about stable outcomes, we need to define the set of all reasonable outcomes. In game theory, a proposed distribution of payoffs, represented by a vector $x$ , is called an imputation if it meets two basic criteria:

Individual Rationality: Every player receives at least as much as they could get on their own. Formally, for each player $i$ , their payoff $x_{i}$ must be greater than or equal to the value of their singleton coalition, $v ({i})$ . No one would agree to a deal that leaves them worse off than they would be by themselves.
Collective Rationality (or Efficiency): The total payoff distributed to all players equals the total value created by the grand coalition. Formally, $\sum x_{i} = v (N)$ . The entire pie is divided up, no more and no less.

An imputation is a possible deal, but is it a stable one? The key threat to stability comes from sub-coalitions. If a subgroup of players can get more by forming their own coalition than they are receiving under the current imputation, they have an incentive to leave the grand coalition and do so. This leads to the idea of dominance.

[PDF] Introduction to Game Theory: Cooperative Games - CUHK CSE

These lecture notes from CUHK provide a formal and clear introduction to the concepts that build up to the Core.

Please read the following two sections: Start at the section titled "Imputations" and continue through the definitions of Individual and Collective Rationality. Then, read the section on Dominance, which explains how one imputation can be considered "better" than another from the perspective of a specific coalition.

With the concepts of imputation and dominance in hand, we can now formally define the Core.

The Core is the set of all imputations that are not dominated by any other imputation through any coalition.

In simpler terms, an outcome is in the Core if there is no subgroup of players who, by leaving the grand coalition, can make all of its members better off. These are the truly stable agreements. The following video provides a concise definition.

(AGT4E7) [Game Theory] Core

This video from the channel selcuk ozyurt offers a clear, mathematical definition of the Core.

Watch the first minute of the video (00:00-01:04) to see the formal definition, which synthesizes the ideas of individual rationality, feasibility, and stability against deviations by coalitions.

The Core Condition: Checking for Stability

The definition of the Core gives us a direct method for testing whether a payoff vector $x = (x_{1}, x_{2}, ..., x_{n})$ is a stable outcome. An allocation is in the core if and only if it satisfies a system of linear inequalities. Given your background in Operations Research, you'll recognize this as a feasibility problem.

For a three-player vee with Hinge (H), Partner 1 (P1), and Partner 2 (P2), an allocation $(x_{H}, x_{P 1}, x_{P 2})$ is in the core if it satisfies:

Efficiency: $x_{H} + x_{P 1} + x_{P 2} = v ({H, P 1, P 2})$
Individual Rationality:
- $x_{H} \geq v ({H})$
- $x_{P 1} \geq v ({P 1})$
- $x_{P 2} \geq v ({P 2})$
Coalitional (or Group) Rationality:
- $x_{H} + x_{P 1} \geq v ({H, P 1})$
- $x_{H} + x_{P 2} \geq v ({H, P 2})$
- $x_{P 1} + x_{P 2} \geq v ({P 1, P 2})$

If we can find at least one allocation $(x_{H}, x_{P 1}, x_{P 2})$ that satisfies all these conditions simultaneously, the core is non-empty. If the system of inequalities leads to a contradiction, the core is empty.

The formal theorem stating this is straightforward and is the primary tool for this analysis.

[PDF] Introduction to Game Theory: Cooperative Games - CUHK CSE

The same lecture notes now present the main theorem for identifying the core.

Please read the section titled "The Core", focusing on the Theorem and the Corollary that provide the precise conditions an imputation must satisfy to be in the core.

An Example: Analyzing a Vee Relationship

Let's apply this to a hypothetical vee. Imagine H, P1, and P2 are deciding how to allocate a shared pool of 10 "units" of weekend time. Their ability to generate value (enjoyment, productivity) from this time, alone and together, is given by the following characteristic function:

$v ({H}) = 0$ , $v ({P 1}) = 1$ , $v ({P 2}) = 1$ (Each person gets some minimal value from having time to themselves).
$v ({H, P 1}) = 6$ (Hinge and P1 have good synergy).
$v ({H, P 2}) = 7$ (Hinge and P2 have even better synergy).
$v ({P 1, P 2}) = 2$ (The partners have low synergy without the hinge).
$v ({H, P 1, P 2}) = 10$ (The value of everyone coordinating their weekend).

Is there a stable way to divide this time? We need to check if the core is non-empty by solving the system of inequalities:

Efficiency: $x_{H} + x_{P 1} + x_{P 2} = 10$
Individual Rationality: $x_{H} \geq 0$ , $x_{P 1} \geq 1$ , $x_{P 2} \geq 1$
Coalitional Rationality:
- $x_{H} + x_{P 1} \geq 6$
- $x_{H} + x_{P 2} \geq 7$
- $x_{P 1} + x_{P 2} \geq 2$

Let's test for a contradiction. One common technique is to sum the coalitional inequalities for the two-person groups:

(x_{H} + x_{P 1}) + (x_{H} + x_{P 2}) + (x_{P 1} + x_{P 2}) \geq 6 + 7 + 2

2 (x_{H} + x_{P 1} + x_{P 2}) \geq 15

Since we know from the efficiency condition that $x_{H} + x_{P 1} + x_{P 2} = 10$ , we substitute this in:

2 (10) \geq 15 ⟹ 20 \geq 15

This statement is true. It does not lead to a contradiction, which suggests the core is likely non-empty. Let's find a point within it.
Let's try to find an allocation $(x_{H}, x_{P 1}, x_{P 2})$ that satisfies the constraints.
From $x_{H} + x_{P 1} + x_{P 2} = 10$ and $x_{H} + x_{P 1} \geq 6$ , we can deduce that $10 - x_{P 2} \geq 6$ , which means $x_{P 2} \leq 4$ .
Similarly, from $x_{H} + x_{P 2} \geq 7$ , we get $10 - x_{P 1} \geq 7$ , which means $x_{P 1} \leq 3$ .

So we need $x_{P 1} \leq 3$ and $x_{P 2} \leq 4$ . Let's pick a point at these boundaries: $x_{P 1} = 3$ and $x_{P 2} = 4$ .
From the efficiency condition, $x_{H} = 10 - 3 - 4 = 3$ .
Let's check if the allocation $(3, 3, 4)$ is in the core:

Efficiency: $3 + 3 + 4 = 10$ . (Checks out)
Individual Rationality: $x_{H} = 3 \geq 0$ , $x_{P 1} = 3 \geq 1$ , $x_{P 2} = 4 \geq 1$ . (All check out)
Coalitional Rationality:
- $x_{H} + x_{P 1} = 3 + 3 = 6 \geq 6$ . (Checks out)
- $x_{H} + x_{P 2} = 3 + 4 = 7 \geq 7$ . (Checks out)
- $x_{P 1} + x_{P 2} = 3 + 4 = 7 \geq 2$ . (Checks out)

Since the allocation $(3, 3, 4)$ satisfies all conditions, it is in the core. Therefore, for this vee game, the core is non-empty. Stable agreements are possible.

The Empty Core: When No Agreement is Stable

What happens when the core is empty? It signifies a game with inherent instability. No matter how the total value is distributed, there will always be a subgroup with a powerful incentive to defect.

This often happens in games where two-person coalitions are disproportionately powerful. A classic example is a "majority rules" voting game where any two of three players can win.

(AGT4E17) [Game Theory] Solving for Core: Cooperative Voting Game

This video provides an excellent, intuitive, and then formal walkthrough of a game with an empty core. The political voting analogy is very effective for understanding decisions in a vee.

Please watch the following segments: The setup: The video defines a 3-player voting game and its characteristic function. Note that v(S)=1 if a coalition is winning and v(S)=0 if it is losing. Intuitive probing: The presenter tests potential payoff vectors and shows why they are unstable, demonstrating the logic of coalitional deviation. Formal conditions: Here, the core conditions are translated into a specific system of inequalities for this game. The proof: Finally, the video demonstrates how adding the inequalities leads to a mathematical contradiction, proving the core is empty.

The logic demonstrated in the video is key: by summing the inequalities for the powerful sub-coalitions ( $a + b \geq 1$ , $a + c \geq 1$ , $b + c \geq 1$ ), you arrive at a condition ( $2 (a + b + c) \geq 3$ ) that contradicts the efficiency condition ( $a + b + c = 1$ ). This proves that no allocation can simultaneously satisfy all parties. An empty core tells you the structure of the game itself is prone to conflict and instability.

Visualizing the Core

For three-player games, we can visualize the set of imputations as a triangle. The core appears as a geometric shape within this triangle.

This image shows the core for a specific 3-player game. The core is the central hexagon. Any point outside this region represents an unstable allocation. For instance, any allocation in the top-right region labeled "Unstable through {2,3}" would be rejected by players 2 and 3 in favor of forming their own coalition.

The existence and size of the core give you a visual sense of the stability of the game. A large core means there are many stable agreements possible. A small or empty core suggests the relationship is on shakier ground, with fewer (or no) arrangements that can prevent subgroups from becoming dissatisfied.

Conclusion

In this lesson, we moved from discussing "fair" outcomes to "stable" ones. The core provides a powerful analytical tool for understanding the fundamental stability of a cooperative arrangement in a vee relationship.

Key Takeaways:
- The Core is the set of all payoff distributions that are stable against defections by any possible sub-coalition.
- An allocation is in the core if it is efficient (the whole pie is divided) and group rational (no coalition can get more on its own than the allocation gives them).
- We can determine if the core is non-empty by solving a system of linear inequalities. A feasible solution means the core is non-empty; a contradiction means it is empty.
- An empty core is a significant finding: it indicates the game has an inherently unstable structure where no agreement can prevent a subgroup from having a rational incentive to break away.

We now have a method to test if any stable grand coalition is possible. In our next and final lesson of this module, we will use this concept to assess the stability of different potential coalition structures. For example, is it more stable for all three partners to cooperate, or for the hinge to form a lasting dyad with just one partner, leaving the other out? This will allow us to compare the resilience of various relationship configurations within the vee.

Recommended resources