Designing a Test for Private Information

What is a Screening Game?

A screening game is a type of sequential game with incomplete information. Its defining characteristic is the order of moves: the uninformed player (the Principal) moves first by offering a menu of contracts, and the informed player (the Agent) then chooses one of those contracts, thereby revealing their type.

To get a feel for this structure, let's watch a brief video that sets up a classic screening game and contrasts it with the signaling games we've discussed.

As the video explains, the Principal's initial move is an offer designed to make different types of Agents behave differently. This is not a passive observation; it's an active strategy. To formalize this, we need to understand the core components and constraints of these models.

A Classic Example: Airline Fares

Before applying this to relationship dynamics, let's solidify the logic with a classic, intuitive example: how airlines use different ticket types to "screen" business travelers from tourist travelers. This provides a simple numerical illustration of how to formulate the constraints.

Formulating a Screening Model for Partner Vetting

Now, let's apply this framework to your goal. Imagine a couple (the Principal) wants to vet a potential new Bull (the Agent). The Bull has private information about his "type." For simplicity, let's assume there are two types:

• Type H (High-Investment): Genuinely interested in building an emotional and physical connection. He values the quality of the long-term dynamic and is willing to invest significant time and effort upfront (e.g., non-sexual dates, deep conversations).
• Type L (Low-Investment): Primarily interested in casual, physical encounters with minimal upfront time investment.

The couple (Principal) prefers a Type H Bull but cannot directly observe his type. Asking him "Are you high investment?" is cheap talk; both types have an incentive to say "yes" to get access. So, the couple decides to design a screening mechanism by offering a menu of two "contracts."

A "contract" here is a pair (q, t), where q represents the quality of the relationship offered (e.g., frequency of dates, level of intimacy, integration into their life) and t is the "price" or upfront investment required from the Bull (e.g., number of pre-play dates, time spent in conversation).

The Menu of Contracts:

1. Contract H (The Partnership Track): Offers high quality q_H (frequent, intimate, integrated encounters) for a high price t_H (e.g., "We'll see you weekly, but first, we require three non-sexual dates and regular communication over two weeks").
2. Contract L (The Casual Track): Offers low quality q_L (infrequent, purely physical encounters) for a low price t_L (e.g., "We can meet for play once a month, after one initial coffee meeting").

The couple's task is to set the terms (q_H, t_H) and (q_L, t_L) to maximize their own utility, while ensuring the Bulls self-select according to their true type.

The Optimization Problem

The couple is solving an optimization problem. Let's define the utility functions:

• Bull's Utility: U(q, t, θ) = θv(q) - c(t)

  • θ is the Bull's type, with θ_H > θ_L. This captures that the High-Investment type has a higher marginal utility for the "quality" of the relationship q. The v(q) function translates quality into utility.
  • c(t) is the cost of the upfront investment t. We assume this cost is the same for both types, but it weighs more heavily on Type L because his potential benefit θ_L v(q) is lower.

• Couple's Utility: π(q, t, θ)

  • The couple's utility depends on the quality of the interaction, the investment shown by the bull, and, crucially, his true type. They might get very high utility from a high-quality interaction q_H with a true Type H Bull, but very negative utility from the same interaction with a Type L Bull who is just going through the motions.

The couple must design the contracts subject to the following constraints:

1. Individual Rationality (IR): Both types must prefer their contract to walking away (utility of 0).

   • (IR-H): θ_H v(q_H) - c(t_H) ≥ 0
   • (IR-L): θ_L v(q_L) - c(t_L) ≥ 0

2. Incentive Compatibility (IC): Each type must prefer their designated contract over the other one.

   • (IC-H): θ_H v(q_H) - c(t_H) ≥ θ_H v(q_L) - c(t_L)
     This is crucial: The High-Investment type must not be tempted to take the "easy" casual track.
   • (IC-L): θ_L v(q_L) - c(t_L) ≥ θ_L v(q_H) - c(t_H)
     This is usually less of a problem. The Low-Investment type rarely has the incentive to undertake the high effort of the partnership track.

The Logic of the Solution: Distortion and Information Rents

If the couple could observe the Bull's type (full information), they would offer each type the most efficient contract to maximize their joint surplus. However, under asymmetric information, they can't. The IC-H constraint is the binding one. To prevent the Type H Bull from pretending to be Type L to get a low-effort deal, the couple must make the Type L contract less attractive.

This leads to the classic results of screening models:

1. No Distortion at the Top: The contract for the "good" type (Type H) is set at the efficient level. The couple designs (q_H, t_H) to be the best possible arrangement for a genuine High-Investment partner.
2. Distortion at the Bottom: The contract for the "bad" type (Type L) is made inefficiently worse than it would be under full information. The couple must deliberately lower the quality q_L of the casual track, not because they dislike casual arrangements, but to make it unappealing to the Type H Bull. This distortion is the cost of screening.
3. Information Rent: The Type H Bull earns a surplus. Because the couple must make the (q_H, t_H) contract attractive enough to dissuade him from choosing the (distorted) (q_L, t_L) contract, he ends up with a better deal than he would under full information. This extra utility is his information rent, a reward for being the desirable type in an environment of uncertainty.

This theoretical result is profound. It suggests that to reliably find a high-investment partner, you may need to offer a casual option that is deliberately less appealing than you might otherwise be comfortable with, precisely to make it a credible filter.

Let's reinforce this key insight with another look at the Grokipedia resource.

From Model to Dialogue

To bring this from abstract model to practical application, here is how the "menu offer" might sound in a real conversation.

Scenario: The couple is having a video call with a potential Bull.

Husband: "Thanks for talking with us. We wanted to be upfront about how we see things potentially developing, as we know everyone is different. We're open to two paths."

Hotwife: "Exactly. One path is more of a deep, ongoing connection. That would involve us all getting to know each other over a few dates—dinners, drinks, just hanging out—before we even think about playing. It's more investment from all of us, but it builds towards something very integrated and frequent."

Husband: "The other path is more straightforward and casual. We'd meet for coffee to check the vibe, and if it's good, we could schedule a playdate. It would likely be less frequent, more of a fun, occasional thing."

Hotwife: "We don't have a preference for which path someone chooses, but we want to be clear about the different trajectories. We'd love to know which of those sounds more aligned with what you're looking for."

In this dialogue, the couple isn't asking the Bull to state his type. They are presenting two distinct contracts—(q_H, t_H) and (q_L, t_L)—and asking him to choose. A genuine Type H Bull will be drawn to the first offer, seeing the high "price" t_H as a worthy investment for the high "quality" q_H. A Type L Bull will see the high price as a major cost and will be more attracted to the low-investment casual track, even with its lower quality. His choice reveals his type.

Conclusion

In this lesson, we have shifted our perspective from the informed player trying to signal, to the uninformed player actively screening. By designing a menu of options, the Principal can create a strategic environment where the Agent's self-interest leads them to reveal their private information.

Key Takeaways:

• Screening vs. Signaling: Screening is initiated by the uninformed Principal, who designs a menu of contracts to sort Agents. Signaling is initiated by the informed Agent, who chooses a costly action to prove their type.
• Core Constraints: A successful screening mechanism must satisfy Incentive Compatibility (each type prefers their own deal) and Individual Rationality (all deals are better than walking away).
• Distortion and Rents: To prevent desirable types from picking the "easy" option, the Principal must often make the option for less desirable types inefficiently worse (distortion at the bottom). This, in turn, means the desirable types earn a utility surplus, known as information rent.
• Application: In relationship negotiations, screening can be formulated as offering a menu of different relationship tracks (e.g., "partnership vs. casual"), where the required upfront investment (t) and the potential relationship quality (q) are designed to make potential partners self-select based on their true level of interest and commitment.

Preview of the Next Module:

We have now completed our module on Information, Signaling, and Beliefs. You have learned how to model signaling, cheap talk, Bayesian updating, and screening. We are now equipped to tackle more complex negotiation scenarios. In our next module, "Advanced Bargaining Models for Boundary Setting," we will move from identifying types to finding mutually agreeable outcomes. We will begin by formulating boundary negotiations as a Nash Bargaining Problem to find efficient and fair agreements when both parties' preferences are known.