Modeling Interdependent Utilities with Altruism, Compersion, and Jealousy

Beyond Material Payoffs: The Utility of Emotion

In game theory, it's common to start with "material" payoffs—the tangible outcomes of a game. In the context of relationships, this might be getting to attend a desired event, winning an argument, or having a rule decided in one's favor. However, the final utility we derive is rarely just about the material outcome. It is profoundly shaped by our feelings about the process and the other person's role in it.

The blog post "Prisoner's Dilemmas : Altruism :: Battles of the Sexes : Convention" provides excellent examples of this.

As the article illustrates, a payoff matrix showing a score of (11, 10) doesn't capture the internal monologue: "Why does he care more about X than about me?" The disutility from feeling deprioritized or the utility from feeling cherished can easily outweigh the points assigned to the material outcome. Our task is to model this "other crap."

Empathetic Games: Formalizing Interdependent Utility

To formalize this, we turn to a more advanced concept from academic game theory: the idea of interdependent utilities. In this framework, a player's utility is a function not only of the strategic outcome but also of the utility experienced by the other players. The realized utility for player $i$ can be expressed as:

$u_i=U_i(s,u_{-i})$

where $s$ is the strategy profile, and $u_{-i}$ is the vector of utilities for all other players. This creates a system of equations where each person's happiness is a variable in everyone else's happiness calculation. A paper by Vásquez and Weretka explores this concept, which they term "empathetic games."

The paper provides us with two crucial categories:

1. Sympathetic Games: A player's utility increases with the utility of others. This is the formal structure for modeling altruism and compersion.
2. Antipathetic Games: A player's utility decreases with the utility of others. This is the structure for modeling jealousy and envy.

Let's use these concepts to build more realistic payoff matrices.

Modeling Compersion: The Sympathetic Game

Compersion is the joy one feels for a partner's joy, particularly regarding their other relationships. It's a specific, powerful form of altruism within open relationships.

Let's model a simple decision for Ben (cuckold) and Anna (hotwife). Anna is deciding whether to accept a last-minute date.

1. Material Payoffs:
First, we define the payoffs based purely on the "selfish" or material outcomes.

• Anna goes on date: Anna is excited (+10). Ben feels a bit lonely and anxious (-3). Outcome: (10, -3).
• Anna stays home: They have a pleasant, standard evening together (+5 each). Outcome: (5, 5).

Based on these payoffs, it looks like a conflict. Anna's joy comes at Ben's expense.

2. Incorporating Compersion:
Now, let's say Ben has a degree of compersion. We can model his new utility, $U_B^{\prime }$, as his material payoff plus some fraction, $\gamma$, of Anna's utility.
Let $U_A$ and $U_B$ be the material payoffs. Ben's new utility is: $U_B^{\prime }=U_B+\gamma U_A$.
Let's assume Ben's compersion factor, $\gamma$, is 0.5.

• If Anna goes on date ($U_A=10,U_B=-3$):

  • Anna's utility remains 10.
  • Ben's utility becomes: $U_B^{\prime }=-3+(0.5\times 10)=-3+5=2$.
  • New Outcome: (10, 2)

• If Anna stays home ($U_A=5,U_B=5$):

  • Anna's utility remains 5.
  • Ben's utility becomes: $U_B^{\prime }=5+(0.5\times 5)=5+2.5=7.5$.
  • New Outcome: (5, 7.5)

The payoff table transforms:

Notice the transformation. The "Go on Date" option is no longer a negative outcome for Ben; his compersion for Anna's great happiness outweighs his own loneliness. However, he still achieves his highest utility when they stay home together. This nuanced result, where both outcomes are now positive for him but one is still preferred, is a much more realistic representation of the emotional trade-offs involved.

Modeling Jealousy: The Antipathetic Game and Inequality Aversion

Just as we can model positive interdependence, we can model negative interdependence like jealousy or envy. One of the most established frameworks for this is the Fehr-Schmidt Inequality Aversion model.

This model proposes that utility is derived from one's own material payoff, but is reduced by differences in payoffs between you and others. Crucially, people dislike being behind (disadvantageous inequality) more than they dislike being ahead (advantageous inequality).

The following video provides an excellent and clear explanation of the model.

The Fehr-Schmidt utility function for player $i$ in a two-player game is:
$U_i(x)=x_i-{\alpha }_i\max (x_j-x_i,0)-{\beta }_i\max (x_i-x_j,0)$
Where:

• $x_i$ and $x_j$ are the material payoffs for players $i$ and $j$.
• ${\alpha }_i$ is player $i$'s "envy" or "jealousy" parameter (disutility from having less).
• ${\beta }_i$ is player $i$'s "guilt" parameter (disutility from having more). Typically $\alpha >\beta$.

Scenario: Ben and Anna are negotiating the rules for her date with a bull, Chris. The "payoff" is the level of excitement and novelty from the experience.

1. Material Payoffs (Excitement Units):

• Flexible Rules: Anna has a highly novel and exciting experience (+10). Ben feels left out and the experience is low-novelty for him (+2). Outcome: (10, 2) (for Anna, Ben).
• Strict Rules: Anna's experience is more constrained but still fun (+6). Ben feels more involved and secure (+4). Outcome: (6, 4).

2. Incorporating Ben's Jealousy:
Let's model Ben's utility. He's sensitive to being "left behind" in terms of excitement. Let's set his envy parameter ${\alpha }_B=0.7$ and his guilt parameter ${\beta }_B=0.1$.

• With Flexible Rules ($x_A=10,x_B=2$):

  • Ben is behind by 8. This triggers the $\alpha$ term.
  • $U_B=2-{\alpha }_B\max (10-2,0)-{\beta }_B\max (2-10,0)$
  • $U_B=2-0.7(8)-0.1(0)=2-5.6=-3.6$.
  • His material payoff of 2 is swamped by the strong disutility from the inequality.

• With Strict Rules ($x_A=6,x_B=4$):

  • Ben is behind by 2. This also triggers the $\alpha$ term.
  • $U_B=4-{\alpha }_B\max (6-4,0)-{\beta }_B\max (4-6,0)$
  • $U_B=4-0.7(2)-0.1(0)=4-1.4=2.6$.
  • His utility is still reduced by envy, but the final outcome remains positive.

For Ben, the empathetic game looks like this:

The model clearly shows why Ben would strongly prefer Strict Rules, even though his material payoff is only slightly better. The reduction in emotional disutility from jealousy is the deciding factor. A negotiation that only considers the material payoffs would miss this crucial dynamic.

Reciprocity: The Role of Intentions

Our models are now more nuanced, but they still focus on outcomes. What if utility also depends on our perception of our partner's intentions? This is the domain of reciprocity. We feel good when we are kind to those who are kind to us, and we may even derive satisfaction from punishing those who are unkind.

Rabin's Reciprocity Model formalizes this. It's more mathematically intensive, but the core intuition is what's important for our purposes.

Rabin's model adds a reciprocity term to the utility function, which is essentially:
$(\text{Your Kindness})\times (\text{Perceived Partner’s Kindness})$

• If your partner is kind (+), you get positive utility from being kind (+). (+) * (+) = +
• If your partner is mean (-), you get positive utility from being mean (-). (-) * (-) = +
• If there is a mismatch (one is kind, one is mean), you get negative utility. (+) * (-) = -

Kindness is measured by comparing the payoff you give to your partner versus the range of payoffs you could have given them. Choosing an action that gives them a high potential payoff is "kind."

Consider the Trust Game from our previous lessons. Ben (Player 1) can Trust Anna by giving her freedom, or Not Trust. If trusted, Anna (Player 2) can Honor that trust or Exploit it.

• Material Payoffs (Ben, Anna): (Trust, Honor) -> (5,5); (Trust, Exploit) -> (-5, 10); (Not Trust, -) -> (1,1).
• Ben's decision to Trust is kind. He forgoes a safe payoff of 1 and gives Anna the chance to get either 5 or 10, at great risk to himself.
• If Ben trusts, Anna now decides. She knows his action was kind.
  • If she chooses to Honor, her action is kind (giving him 5 when she could have given him -5). The reciprocity term in her utility function will be (Anna's Kindness) * (Ben's Kindness) = (+) * (+), yielding positive psychological utility.
  • If she chooses to Exploit, her action is mean. The reciprocity term will be (Anna's Meanness) * (Ben's Kindness) = (-) * (+), yielding negative psychological utility.

Even if Exploit has a higher material payoff (10 vs 5), a strong aversion to being unkind to a kind partner could make Honor the utility-maximizing choice for Anna. This formalizes why trust can be sustained even when there are material incentives to betray it.

Conclusion

Today we have opened the "black box" of payoffs to reveal the complex emotional machinery within. By moving from simple material outcomes to interdependent utilities, we can create far more realistic and predictive models of relationship dynamics.

Key Takeaways:

• Payoffs in relationship games are not just material; they are heavily influenced by emotions.
• Interdependent utility functions, where one person's utility depends on another's, are essential for realistic modeling.
• Sympathetic games model altruism and compersion, where utility increases with a partner's happiness. This can be modeled by adding a fraction ($\gamma$) of your partner's utility to your own.
• Antipathetic games model jealousy and envy. The Fehr-Schmidt model is a powerful tool for this, showing how utility is diminished by unfavorable payoff inequalities.
• Reciprocity models (like Rabin's) add another layer of sophistication by considering intentions, explaining why we derive utility from being kind to the kind and returning meanness for meanness.

Preview of the Next Lesson:
We have focused on how to model the payoffs once we have them. But what happens when one player's ability to act depends on another's prior commitment? In our next lesson, we will explore "Commitment problems and the strategic value of limiting one's own future options." We will see how, paradoxically, giving up choices can sometimes be the most powerful move you can make.