Monte Carlo Simulation of Repeated Bargaining for Relationship Stability

From a Single Interaction to a Dynamic System

A long-term relationship can be viewed as a dynamic system whose state evolves over time. Let's define a central state variable, which we'll call Relationship Health ($H_t$). This variable is an abstract measure of trust, emotional connection, and satisfaction in the relationship at time $t$. It's analogous to the "feeling" variable, $x(t)$, used in dynamic models of love.

This Health metric is not static. It is subject to two main forces:

1. Natural Decay: Left to itself, a relationship can suffer from a kind of entropy. Complacency and lack of effort can cause a slow decay in Relationship Health.
2. Actions and Events: The choices partners make at each stage either bolster or damage the relationship. Cooperative, trusting actions increase Health, while defections and betrayals cause significant damage.

At discrete time steps (e.g., every week or month), the couple engages in a "stage game" where their actions determine the change in $H_t$. This is where our game theory model becomes the engine of our simulation.

The Probabilistic Nature of Choice

In our previous lesson's deterministic model, we solved for the optimal actions. In reality, human behavior is probabilistic. A person doesn't always defect when the expected utility is higher; rather, the probability of them defecting increases. To model this, we can turn to the concept of a Mixed-Strategy Nash Equilibrium (MSNE).

Let's consider a simplified stage game for our established couple, Clara and David. In each period, they face an opportunity that tests their agreements. Each can choose to Cooperate (uphold the agreed-upon rules and boundaries) or Defect (violate a rule for personal gain). The crucial step is to define the probability of defection.

A formal game-theoretic model of infidelity provides a powerful template for this. Such models derive the equilibrium probability of a player choosing to cheat based on the parameters of the game.

Inspired by this approach, we can define the probability of a partner defecting, $P(\text{Defect})$, as a function of key variables:

• $\gamma$: The perceived benefit from the defection (e.g., pleasure, ego boost).
• $\zeta$: The perceived cost of being caught, which is a function of the other partner's monitoring and the agreed-upon consequences.

So, in each stage of our simulation, a partner doesn't make a deterministic choice, but instead "flips a weighted coin," with the weighting determined by the strategic realities of their relationship rules.

Building the Monte Carlo Simulation

A Monte Carlo simulation allows us to project the evolution of Relationship Health ($H_t$) by repeatedly simulating the relationship's trajectory, one possible future at a time. Each individual simulation is called a "run."

Here is the blueprint for a single run of our simulation:

1. Initialization:

   • Set the initial Relationship Health, $H_0$. A new, excited relationship might start at $H_0=100$.
   • Define the parameters of the "ruleset" (we'll detail this shortly). This sets the values for $\gamma$, $\zeta$, and the impact of outcomes.
   • Set the "death" threshold, $H_{min}$. If $H_t$ drops below this (e.g., $H_{min}=20$), the relationship is considered to have failed.
   • Set the total simulation time, $T$ (e.g., 520 weeks, for 10 years).

2. Iteration Loop (for $t=1$ to $T$):
   a. Natural Decay: Apply a small decay to the health: $H_t=H_{t-1}-\delta$.
   b. Partner Choices: For each partner (Clara, David), determine their action.
   * Calculate their current probability of defection, $P(\text{Defect})$, based on the ruleset parameters ($\gamma ,\zeta$).
   * Draw a random number from a uniform distribution $U[0,1]$. If the number is less than $P(\text{Defect})$, the partner defects; otherwise, they cooperate.
   c. Determine Outcome: The combination of choices yields one of four outcomes: (Cooperate, Cooperate), (Cooperate, Defect), (Defect, Cooperate), or (Defect, Defect).
   d. Update Health: Adjust $H_t$ based on the outcome.
   * $H_t=H_t+{\text{Bonus}}_{CC}$ if the outcome is (C, C). Mutual trust builds the relationship.
   * $H_t=H_t-{\text{Penalty}}_{CD}$ if one partner defects. Betrayal causes damage.
   * $H_t=H_t-{\text{Penalty}}_{DD}$ if both defect. Mutual betrayal is highly corrosive.
   e. Check for Failure: If $H_t<H_{min}$, end this simulation run and record the failure time $t$.

3. End of Run: If the loop completes without failure, record it as a success.

By running this simulation thousands of times, we move from a single story to a statistical forecast. The collection of thousands of individual $H_t$ trajectories gives us a probability distribution of the relationship's future.

Forecasting with Different Rulesets

The true power of this approach is its ability to test the long-term viability of different rulesets. A "ruleset" is not just a list of permissions and prohibitions; it's a strategic environment that we can translate into the parameters of our model.

Let's compare two hypothetical rulesets for Clara and David:

Ruleset A: "Transparent and Strict"

• Description: Open communication is required, GPS location sharing is on, and any unapproved contact with a third party leads to a major confrontation.
• Model Parameters:
  • High monitoring leads to a high cost/probability of being caught ($\zeta$ is high).
  • This makes the equilibrium probability of defection, $P(\text{Defect})$, very low.
  • The penalty for defection (${\text{Penalty}}_{CD}$) is very large.
  • The bonus for cooperation (${\text{Bonus}}_{CC}$) might be moderate, as the environment is more about enforcement than enthusiastic collaboration.

Ruleset B: "Laissez-Faire"

• Description: "Don't ask, don't tell." Partners have wide latitude, and monitoring is seen as a breach of trust in itself.
• Model Parameters:
  • Low monitoring means the cost of being caught ($\zeta$) is low.
  • This leads to a higher equilibrium probability of defection, $P(\text{Defect})$.
  • The penalty for a discovered defection (${\text{Penalty}}_{CD}$) might be smaller, as the expectation of fidelity was weaker.
  • The bonus for cooperation (${\text{Bonus}}_{CC}$) might also be small, as there's less shared investment in the boundaries.

By running 10,000 simulations for Ruleset A and 10,000 for Ruleset B, we can generate forecasts. We might find that Ruleset A leads to a very low probability of failure but also a lower average Relationship Health (a stable but not thriving state), while Ruleset B results in a much higher failure rate over 10 years. This allows for a data-driven discussion about the trade-offs inherent in different relationship agreements.

This method of simulating dynamic trajectories based on game-theoretic principles is used at the frontiers of social science modeling.

Conclusion

Today we have made a significant leap from analyzing a single, static game to forecasting long-term, dynamic outcomes. By embedding a repeated bargaining game within a Monte Carlo simulation, we can move beyond simple equilibrium analysis and explore the resilience and stability of relationships under the genuine uncertainty they face.

Key Takeaways:

• Long-term relationships can be modeled as dynamic systems with a state variable like "Relationship Health" that evolves over time.
• The choices partners make at each stage can be modeled as a probabilistic game, where probabilities of defection are determined by the strategic environment (e.g., benefits of cheating, costs of detection).
• Monte Carlo simulation allows us to project thousands of possible future trajectories for the relationship's health, providing a statistical forecast of stability.
• "Rulesets" for a relationship can be translated into the parameters of the simulation model, allowing us to compare the long-term viability of different strategic agreements.

Preview of the Next Lesson:

Our Monte Carlo simulation analyzed a single couple (or many independent couples) in isolation. But real couples exist within a social context; they observe each other, copy successful strategies, and contribute to evolving community norms. In our next lesson, we will explore agent-based models (ABM). This powerful simulation technique will allow us to model an entire community of interacting couples, enabling us to see how different relationship strategies spread, compete, and lead to the emergence of large-scale social norms.