The Revelation Principle and Vee Agreements

From Analyzing Games to Designing Them

So far, we have largely taken the "game" of a relationship as given and analyzed the strategic choices within it. But what if we could design the game itself? This is the central question of mechanism design. Given your background in Operations Research, you can think of this as moving from analyzing a given system's performance to designing an optimal system from the ground up.

This short video provides an excellent introduction to this "engineering" mindset.

As the video explains, the core challenge of mechanism design is to create rules that produce good outcomes (e.g., efficiency, fairness, stability) even when the designer doesn't know the private information of the players—such as their true valuations, preferences, or feelings.

A relationship agreement in a polyamorous vee is a perfect example of a mechanism. The partners are the "designers," and their goal is to create a set of rules for communication, time allocation, and decision-making that leads to a stable, happy, and fulfilling relationship for all three people. The challenge is that each person has private information: their deepest desires, insecurities, and changing emotional states. The goal is to design an agreement that works well regardless of these unknown private details, by encouraging everyone to share them.

The Revelation Principle: A Powerful Simplification

The task of designing a game seems daunting. One could imagine infinitely complex rules, with multiple rounds of communication, strategic bluffs, and convoluted actions. How could we ever hope to find the "best" one?

This is where the Revelation Principle comes in. It offers a radical simplification. In essence, it states that almost anything we could hope to achieve with any complicated mechanism, we can also achieve with a much simpler one: a direct, truthful mechanism.

Let's break that down:

• A direct mechanism is one where players simply report their private information (their "type") to the mechanism. For example, "I need more quality time this week," or "I am feeling insecure about your other relationship."
• A truthful mechanism is a direct mechanism where every player's best strategy is to report their type honestly. Lying or misrepresenting their feelings would lead to a worse outcome for them.

The following video gives a brilliant and intuitive explanation for why this is true.

This "without loss of generality" is a powerful phrase you've surely encountered in mathematical proofs. Here, it means that when we are designing a relationship agreement, we don't need to waste time brainstorming elaborate schemes to outsmart each other. We can confine our attention to designing a straightforward system where everyone is simply asked to state their needs and preferences, and where doing so truthfully is the best course of action for everyone.

Formalizing the Principle

Let's look at a more formal statement of the principle. Your background in statistics and game theory will make the notation here straightforward.

The proof sketch in the notes illustrates the same logic as the video: you can construct a new direct mechanism that takes the players' reported types, feeds them into the equilibrium strategies of the old, complex mechanism, and implements the resulting outcome. Because the old strategies were an equilibrium, no player has an incentive to lie to the new mechanism.

Implications for Designing Vee Agreements

The Revelation Principle is not just an abstract theorem; it has profound practical implications for how partners in a vee can design their relationship agreements.

1. The Goal is Truthfulness, Not Lie-Detection

The principle tells us that instead of trying to design a system that can catch lies or withstand strategic manipulation, we should focus on designing a system where no one wants to lie in the first place.

This shifts the entire objective of creating an agreement. The goal is to make honesty the path of least resistance and greatest reward for everyone involved. The "Relationship Anarchy Smörgåsbord" is a great conceptual tool that embodies this.

This tool encourages partners to go through different "platters" (e.g., Domestic, Financial, Sexual) and directly state what "items" they want in the relationship. It is an attempt to create a direct mechanism for preference revelation. The challenge, which we'll address in future lessons, is ensuring it's truthful (incentive-compatible).

2. The Twin Constraints: Incentive Compatibility and Individual Rationality

For a direct mechanism (an agreement) to be truthful and functional, it must satisfy two key conditions.

Let's apply these to a vee agreement:

• Incentive Compatibility (IC): The agreement must be structured such that each partner's payoff is maximized when they are honest about their type (their feelings, needs, and desires). If a partner feels they can get a better outcome by misrepresenting their needs—for example, by pretending to be "low-maintenance" to seem more attractive, when they actually need a lot of reassurance—the agreement is not incentive-compatible. The challenge is to design rules where this kind of deception is self-defeating.

• Individual Rationality (IR): The agreement must provide an expected utility to every partner (hinge, partner A, partner B) that is at least as good as their outside option (their "disagreement outcome," which could be leaving the vee). Simply put, the deal has to be good enough for everyone to want to participate. No one should feel that the agreement makes them worse off than they would be on their own.

3. The Agreement as an Impartial Mediator

Nobel laureate Roger Myerson, a key figure in developing these ideas, helps frame this through the idea of a central mediator.

In a vee relationship, the agreement itself serves as this "trustworthy mediator." It's an impartial system that the partners design together.

1. Partners "report" their needs and preferences to the "mediator" (i.e., they communicate according to the rules of the agreement).
2. The "mediator" (the agreement's rules) then processes these inputs and "recommends actions" (e.g., this is how the calendar is set for the week, this is the decision on the shared expense).

The Revelation Principle guarantees that if we design this "mediator" (our agreement) correctly, we can ensure that honesty and adherence to the plan is the equilibrium behavior.

Conclusion

Today's lesson introduced a fundamental shift in perspective: from merely playing a game to designing its rules. This is the essence of mechanism design.

• Key Takeaways:
  • Mechanism Design is the engineering of institutions or games to achieve specific goals, like fairness or stability, in the face of private information.
  • The Revelation Principle is a foundational result stating that any outcome achievable with a complex, potentially deceptive mechanism can also be achieved with a simple direct and truthful one.
  • This principle's main implication is that we can simplify the problem of designing relationship agreements by focusing only on those where partners are asked to state their true preferences directly.
  • For such an agreement to work, it must satisfy two conditions: Incentive Compatibility (honesty is the best policy for everyone) and Individual Rationality (the agreement is better for everyone than their outside option).

By leveraging the Revelation Principle, the daunting task of creating a fair and stable vee agreement becomes more manageable. It directs our focus away from anticipating deception and toward creating a system that incentivizes truth.

In our next lesson, we will get more concrete. Building on this principle, we will learn how to formulate a specific vee relationship agreement as a mechanism design problem by defining agents, actions, and outcome functions. We will begin to build the actual "machinery" of a good agreement.