Signaling Commitment: An Application of Spence's Model

Introduction

In our last lesson, we established the core distinction between separating and pooling equilibria in signaling games, determining whether a partner's actions credibly reveal their true intentions or create strategic ambiguity. We saw that for a signal to be credible, it often needs to be costly in a way that only certain "types" of individuals are willing to bear.

Today, we will formalize this crucial idea using Michael Spence's Nobel Prize-winning job-market signaling model. We will adapt this economic framework to a relationship context to evaluate how costly actions can serve as powerful, hard-to-fake demonstrations of commitment. You will learn to calculate the precise level of "investment" required for a signal to successfully separate a genuinely committed potential partner from a casual one. Such a signal functions as a commitment device, a concept visually represented in the diagram below.

This diagram illustrates how commitment devices, such as costly signals, can cut through the "fog" of uncertainty about romantic alternatives and increase the credibility of one's intentions.

The Spence Model: From the Job Market to the Relationship Market

The classic Spence model describes a scenario with asymmetric information: workers (the Senders) know their own productivity (their "type"), but firms (the Receivers) do not. To get a higher wage, high-productivity workers can invest in education (the signal). Education is costly for everyone, but it is less costly for high-productivity workers. This differential cost is the key that allows education to function as a credible signal.

Let's translate this into a cuckold/hotwife dynamic that you often analyze:

Sender: A potential new partner, the "Bull."
Receiver: The established couple (Husband and Wife).
Private Information (Type): The Bull's true intentions. For simplicity, let's assume two types:
- High-Commitment Type ( $T_{H}$ ): Genuinely seeks a respectful, ongoing connection that honors the couple's boundaries and emotional well-being.
- Low-Commitment Type ( $T_{L}$ ): Is primarily interested in a short-term, casual sexual encounter and is less concerned with the couple's long-term dynamic.
Signal (Education, $e$ ): A costly, observable action taken by the Bull to demonstrate his interest. This isn't academic education, but rather an "investment" in the potential relationship.
"Wage" ( $w$ ): The level of integration, trust, and intimacy the couple offers the Bull in response to his signal. For example, a high "wage" could be deep emotional connection and inclusion in their social life, while a low "wage" might be a single, highly constrained encounter.

The crucial assumption, parallel to Spence's model, is that the cost of the signal is lower for the High-Commitment type. Why? Because a $T_{H}$ Bull intrinsically values the process of connection—long conversations, understanding boundaries, building rapport. For him, these activities are less of a "cost" and more of a benefit in themselves. For the $T_{L}$ Bull, these activities are purely instrumental—a hurdle to overcome—making the perceived cost much higher.

Solving for a Separating Equilibrium

Our goal is to find a separating equilibrium, where the Bull's actions reveal his type. In such an equilibrium:

The $T_{H}$ Bull chooses a specific, high level of investment, $e^{*}$ .
The $T_{L}$ Bull chooses zero investment, $e = 0$ .
The couple observes the investment and updates their beliefs accordingly, offering a high "wage" $w_{H}$ for $e^{*}$ and a low "wage" $w_{L}$ for $e = 0$ .

The video below provides a rigorous, step-by-step walkthrough of how to solve for such an equilibrium in the classic job-market context. The logic is identical to what we will apply.

18. Spence's Job Market Signaling Game: Solving for Separating Equilibrium (Game Theory Playlist 10)

To understand the formal structure of the Spence model, watch the introductory sections of this video by Selcuk Ozyurt.

Pay close attention to the initial setup of the model, where the story, player types, and payoff functions are explained. Also, focus on how the firms (the receivers) determine wages based on their beliefs about the worker's type, which is inferred from the education signal.

Let's build our own scenario and solve it using this method.

Scenario: The "Get-to-Know-You" Investment

A couple, Sarah and Tom, are considering a new partner, Mark. Mark can be a High-Commitment type ( $T_{H}$ ) or a Low-Commitment type ( $T_{L}$ ). They believe each type is equally likely ( $P (T_{H}) = P (T_{L}) = 0.5$ ).

Signal ( $e$ ): The number of hours Mark invests in non-sexual "get-to-know-you" activities (dinners, conversations about boundaries, etc.) before any sexual encounter.
"Wage" ( $w$ ):
- High Wage ( $w_{H} = 10$ ): If they believe he is $T_{H}$ , they offer a relationship built on deep trust and emotional intimacy.
- Low Wage ( $w_{L} = 4$ ): If they believe he is $T_{L}$ , they offer a single, strictly defined encounter.
Payoff Functions: Mark's utility is $U (w, e) = w - c (e)$ , where the cost of investment $c (e)$ depends on his type.
- For $T_{H}$ : $c_{H} (e) = e /4$ (He finds the investment less costly).
- For $T_{L}$ : $c_{L} (e) = e$ (He finds it four times as costly).

For a separating equilibrium to hold, two incentive compatibility (IC) constraints must be satisfied.

The $T_{L}$ Bull must not want to mimic the $T_{H}$ Bull.
The $T_{L}$ type's payoff from choosing $e = 0$ (his equilibrium action) must be greater than or equal to his payoff from choosing $e = e^{*}$ (deviating).
$Payoff(No Deviation) \geq Payoff(Deviation)$ $w_{L} - c_{L} (0) \geq w_{H} - c_{L} (e^{*})$ $4 - 0 \geq 10 - e^{*}$ $e^{*} \geq 6$
This constraint tells us that the required investment $e^{*}$ must be at least 6 hours. If it were any less, the Low-Commitment type would find it worthwhile to "fake" commitment to get the higher payoff. This sets the lower bound for a credible signal.
The $T_{H}$ Bull must prefer to send the signal.
The $T_{H}$ type's payoff from choosing $e = e^{*}$ (his equilibrium action) must be greater than or equal to his payoff from choosing $e = 0$ (and being mistaken for a $T_{L}$ type).
$Payoff(Signaling) \geq Payoff(Not Signaling)$ $w_{H} - c_{H} (e^{*}) \geq w_{L} - c_{H} (0)$ $10 - \frac{e ^{*}}{4} \geq 4 - 0$ $6 \geq \frac{e ^{*}}{4}$ $24 \geq e^{*}$
This constraint shows that the required investment $e^{*}$ cannot be more than 24 hours. If it were, the cost would be so high that even the High-Commitment type wouldn't find it worthwhile. This sets the upper bound.

Conclusion: A separating equilibrium exists if and only if the required level of investment $e^{*}$ is in the range $6 \leq e^{*} \leq 24$ . Any investment level in this range is costly enough to deter the mimic but not so costly that it deters the genuine type.

The following segment of the same video demonstrates this exact process of using incentive compatibility constraints to find the equilibrium range.

18. Spence's Job Market Signaling Game: Solving for Separating Equilibrium (Game Theory Playlist 10)

Now, watch how the presenter solves for the equilibrium education levels.

Follow the analysis of the low-productivity worker's incentive constraints. This corresponds to our T_L type. Then, observe the same process for the medium-productivity worker and the high-productivity worker. Notice how checking each type's potential deviations establishes a range of possible equilibrium values, just as we did.

From Theory to Practice: Identifying Costly Signals

The Spence model provides the mathematical structure, but identifying credible signals in the real world is key. The following research paper by Mogilski et al. empirically identifies practices people use to maintain consensual non-monogamous relationships. Many of these can be interpreted as costly signals.

[PDF] How Do People Maintain Consensual Non-Monogamy? An ...

This paper provides a fantastic inventory of real-world behaviors that can function as costly signals of commitment.

First, read the section Partner Loss and Resource Diversion, which discusses the core challenges that necessitate commitment signals. Next, focus on the section Intra-pair Positive Inducements (in Table 1), as it explicitly theorizes about costly signals. Finally, review Table 4 and the subsequent paragraph describing the nine factors, such as attraction disclosure, which lists the empirically identified maintenance practices. As you read, think about how each practice imposes a differential cost.

The practices identified in the paper serve as excellent real-world examples of the "education" variable in our model. Consider:

Thoughtful Resource Distribution: The act of consistently and fairly sharing time, attention, and money is a costly signal. A $T_{L}$ type, focused on minimizing effort for a short-term gain, would find this far more "costly" than a $T_{H}$ type who sees it as part of building a genuine connection.
Open Communication about Jealousy: Engaging in emotionally vulnerable conversations about jealousy requires significant time and psychological effort. This is a high cost for someone looking for a simple hookup ( $T_{L}$ ), but a worthwhile investment for someone building trust ( $T_{H}$ ).
Willingness to Care for a Partner's Children: This is an extremely high-cost signal. The investment of time, resources, and emotional energy is substantial, making it a highly credible indicator of long-term commitment.

Deeper Dive: Beyond Perfect Separation

The separating equilibrium is a powerful concept, but it assumes perfect honesty from high types and no bluffing from low types. A more advanced perspective considers hybrid equilibria, where communication is only partially informative.

[PDF] Between cheap and costly signals: the evolution of partially honest ...

This paper by Zollman et al. explores a fascinating middle ground between fully revealing and fully pooling equilibria. Given your background in stochastic models, you may find this particularly interesting.

Read the Introduction to understand the motivation for looking beyond classic costly signaling. Then, review the section Hybrid equilibria. Don't worry about deriving the math; focus on the concept: the high type always signals, but the low type sometimes signals (bluffs), and the receiver, in turn, only sometimes believes the signal.

In a relationship context, a hybrid equilibrium would mean that the $T_{H}$ Bull always makes the high investment ( $e^{*}$ ), but the $T_{L}$ Bull also chooses to make that investment with some small probability. Consequently, the couple, upon seeing $e^{*}$ , doesn't become 100% certain he is $T_{H}$ ; they just significantly update their belief in that direction. This often reflects reality more closely, where signals increase confidence but rarely eliminate all doubt.

Conclusion

This lesson has equipped you with a formal model to analyze and quantify the credibility of commitment signals. By applying Spence's job-market model to relationship dynamics, we can see with mathematical precision why certain actions speak louder than words.

Key Takeaways:

Spence's model formalizes the idea of costly signaling, where a signal's credibility comes from its differential cost across sender types.
A separating equilibrium is sustained when the signal cost is high enough to deter low-commitment types from mimicking, but not so high that it deters high-commitment types from signaling.
We can solve for the range of signal "investments" that support a separating equilibrium using incentive compatibility constraints.
Many real-world relationship maintenance practices, such as resource sharing and open communication about difficult emotions, can be modeled as costly signals that demonstrate commitment.

Preview of the Next Lesson:

We have focused on "costly talk"—signals that involve tangible investments of time, effort, or money. But what about signals that are free to send, like making a promise or stating a preference? In our next lesson, we will explore the strategic use of "cheap talk" and investigate the conditions under which words alone can be enough to align expectations and build trust.

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