Three-Player Vee Games and Characteristic Functions

From Individual Strategies to Coalition Values

In the non-cooperative games we've studied so far, the fundamental unit of analysis is the individual player and their strategy. We looked for outcomes, like Nash equilibria, where no single player has an incentive to change their strategy, given the strategies of others.

Cooperative game theory starts from a different premise. It assumes players can communicate and, crucially, make binding agreements. Think of it as moving from a world of informal understandings and potential betrayals to one where enforceable contracts are possible. Because of this, the focus shifts away from individual actions and onto the value that coalitions—subsets of players acting in concert—can generate.

A key simplifying assumption we'll make in this module is that of Transferable Utility (TU). This means that the total value created by a coalition can be freely divided among its members using side payments. In a relationship context, this "utility" might be a quantifiable resource like time or money, or it could be a more abstract representation of shared happiness or well-being that can, in principle, be transferred (e.g., one partner does an unpleasant chore, "paying" the other with the benefit of free time).

The Characteristic Function: Defining the Game

With this cooperative mindset, the game is no longer defined by strategies and payoff matrices, but by something much more compact: a pair $(N,v)$.

• $N$ is simply the set of players. For our vee relationship, we will define the players as $N=\{H,P_1,P_2\}$, where $H$ is the hinge partner, and $P_1$ and $P_2$ are the non-hinge partners.
• $v$ is the characteristic function. This function is the heart of the model. It takes any coalition $C$ (which is just a subset of $N$) as input and outputs a single real number, $v(C)$, which represents the total value that the members of $C$ can guarantee for themselves by working together, regardless of what players outside the coalition do.

To see this in action, let's watch a short, intuitive explanation.

As the video shows, the characteristic function is simply a complete description of the economic potential of every possible subgroup. For our three-player vee, there are $2^3=8$ possible coalitions:

1. The empty coalition: $\varnothing$
2. The single-player coalitions: $\{H\}$, $\{P_1\}$, $\{P_2\}$
3. The two-player coalitions (dyads): $\{H,P_1\}$, $\{H,P_2\}$, $\{P_1,P_2\}$
4. The three-player "grand" coalition: $\{H,P_1,P_2\}$

Our task is to define the value $v(C)$ for each of these eight subsets. To do this formally, let's consult a more detailed definition.

Building the Characteristic Function for a Vee

Now, let's apply this framework to our vee relationship. We need to assign a value to each of the 8 coalitions. The specific numbers are less important than what they represent. Let's imagine "utility" is measured in abstract units of "well-being" generated per month.

1. The Empty and Singleton Coalitions:

• $v(\varnothing )=0$. This is by definition.
• $v(\{H\}),v(\{P_1\}),v(\{P_2\})$: This is the baseline utility each person has on their own, outside of the relationship structure. It's their "outside option." For simplicity, we can normalize these values to zero, which means we are measuring only the additional value created by the relationships. Let's set:
  • $v(\{H\})=0$
  • $v(\{P_1\})=0$
  • $v(\{P_2\})=0$

2. The Dyadic Coalitions:
These represent the value created by pairs of individuals.

• $v(\{H,P_1\})$: The value generated by the relationship between the hinge and partner 1. This could include shared joy, mutual support, etc. Let's say this dyad creates 10 units of well-being: $v(\{H,P_1\})=10$.
• $v(\{H,P_2\})$: Similarly, the value from the hinge's relationship with partner 2. Perhaps this relationship dynamic is slightly different. Let's set $v(\{H,P_2\})=12$.
• $v(\{P_1,P_2\})$: This is a fascinating coalition in a vee structure. By definition, $P_1$ and $P_2$ are not in a primary relationship. The value they can create together, as a unit, without H depends on their "metamour" relationship. If they are friends and provide mutual support, this value could be positive. If they are antagonistic, it might be zero or even negative in some models. If they simply don't interact, their joint value is just the sum of their individual values. Given our normalization, a simple starting point is: $v(\{P_1,P_2\})=v(\{P_1\})+v(\{P_2\})=0+0=0$.

3. The Grand Coalition:

• $v(\{H,P_1,P_2\})$: This is the total value generated when all three partners cooperate as a single unit. This value captures any synergies that arise from the full vee structure. For example, efficient scheduling, shared household projects, or the joy the hinge derives from seeing their partners get along (compersion). The sum of the two distinct dyadic relationships is $v(\{H,P_1\})+v(\{H,P_2\})=10+12=22$. If the whole is greater than the sum of its parts, the value of the grand coalition will be higher. Let's say the total synergy and cooperation creates a value of 25: $v(\{H,P_1,P_2\})=25$.

The difference $v(\{H,P_1,P_2\})-(v(\{H,P_1\})+v(\{H,P_2\}))=25-22=3$ represents the positive externality or synergy of the fully cooperative vee. This is the cooperative surplus that this framework allows us to identify and quantify.

Summary of our Example Vee Game:

Putting it all together, we have now fully defined a characteristic function for a hypothetical vee.

This complete description, the pair $(N,v)$, is the cooperative game model of our vee relationship.

Conclusion

In this lesson, we have taken the first crucial step into cooperative game theory. We transitioned from analyzing individual strategies to evaluating the collective power of groups.

• Key Takeaways:
  • Cooperative game theory analyzes situations where players can form binding agreements, shifting the focus to coalitions.
  • A cooperative game with transferable utility (TU) is defined by a set of players $N$ and a characteristic function $v$.
  • The characteristic function $v(C)$ assigns a numerical value to every possible coalition, representing the total worth that coalition can generate on its own.
  • By defining the players, listing all possible coalitions, and assigning a value to each, we can successfully model a vee relationship as a formal three-player cooperative game. This model allows us to quantify concepts like relationship value and synergy.

We have now established the "what"—what value can each potential coalition create? In our next lesson, we will begin to explore the "so what?". Given these potential values, which coalitions are likely to form? We will analyze the different coalition formation possibilities within a vee and begin our inquiry into the stability and fairness of cooperative outcomes.