Discount Factors and Cooperation in a Vee

The Value of Tomorrow: Defining the Discount Factor

In repeated games that could, in principle, go on forever, we cannot simply sum the payoffs from each round, as this would lead to infinite totals. We need a way to weigh payoffs received sooner more heavily than those received later. This is accomplished using the discount factor, denoted by the Greek letter delta, $\delta$.

The discount factor is a number between 0 and 1 ($0<\delta <1$) that represents how much future payoffs are valued relative to present ones. A payoff of $x$ received in the next period is worth $\delta x$ today. A payoff of $x$ received two periods from now is worth ${\delta }^{2}x$ today, and so on.

The video below from William Spaniel’s "Game Theory 101" series provides an excellent intuitive explanation for why discount factors are necessary and what they represent.

As the video explains, a high $\delta$ (e.g., $\delta =0.95$) means a player is patient. They value future rewards almost as much as present ones. A low $\delta$ (e.g., $\delta =0.2$) means a player is impatient, heavily prioritizing immediate gratification.

In the context of a vee relationship, these interpretations are direct and powerful:

• Patience: A partner with a high $\delta$ is invested in the long-term health of the relationship. They are willing to make short-term sacrifices for long-term stability and happiness.
• Probability of Continuation: A high $\delta$ reflects a strong belief that the relationship has a future. If partners feel the relationship is stable and likely to last, their $\delta$ is high. If they feel it's on rocky ground and might end soon, their $\delta$ is low.

Your background in Operations Research will likely make you familiar with the concept of Net Present Value (NPV), where future cash flows are discounted by an interest rate $r$. The discount factor $\delta$ is directly analogous, often modeled as $\delta =1/(1+r)$.

Calculating Long-Term Payoffs

With the discount factor, we can calculate the total present value of an infinite stream of payoffs. If a player receives a constant payoff of $c$ in every period, their total discounted utility, $U$, is the sum of a geometric series:

This infinite sum has a simple, finite solution:

This formula is the engine we will use to compare the value of different long-term strategies. The following resource provides a clear refresher on this mathematical tool.

The Critical Threshold for Cooperation

Now we can address the central question: how patient do partners need to be to sustain cooperation? Let's analyze this using the Grim Trigger strategy in a standard Prisoner's Dilemma framework.

Consider a simple agreement in a vee, like being emotionally open and supportive.

• Cooperate (C): Both partners are open and supportive. Payoff = 3 for each.
• Defect (D): One partner is open, but the other is dismissive to gain a sense of control or save emotional energy. The defector gets a higher payoff (e.g., 5), and the cooperator gets a lower one (e.g., 0).
• Mutual Defection: Both are closed off and dismissive. Payoff = 1 for each. This is the punishment outcome.

Let's say both partners in a dyad of the vee (e.g., Hinge and P1) adopt a Grim Trigger strategy. If they cooperate, what is the incentive to defect?

1. Payoff from Cooperating: If a partner sticks to the agreement, they receive a payoff of 3 in every period, forever. Their total discounted utility is:

   $$U_{Cooperate}=3+3\delta +3{\delta }^2+\cdots =\frac{3}{1-\delta }$$

2. Payoff from Defecting: A partner could choose to defect now. They would get the high temptation payoff of 5 in this period. But in every subsequent period, Grim Trigger kicks in, and they are punished with the mutual defection payoff of 1, forever. Their total discounted utility is:

   $$U_{Defect}=5+1\delta +1{\delta }^2+\cdots =5+\frac{\delta }{1-\delta }$$

Cooperation is sustainable if and only if the payoff from cooperating is at least as good as the payoff from defecting.

Now, we just need to solve for $\delta$. Multiplying everything by $(1-\delta )$ to clear the denominators:

This result is the core of the lesson. Cooperation is only sustainable if the discount factor $\delta$ is greater than or equal to 0.5. If players are more patient than this threshold, the long-term benefits of cooperation outweigh the short-term temptation to defect. If they are less patient, the relationship will unravel into mutual defection.

The following resource walks through a similar calculation. Seeing the logic in a slightly different format will help solidify your understanding.

What Influences Patience in a Vee Relationship?

The critical threshold ${\delta }^{\ast }$ is a mathematical property of the game's payoffs. Whether a real relationship can sustain cooperation depends on whether the partners' actual discount factor $\delta$ is above or below this threshold. So, what factors in a vee relationship would influence a person's $\delta$?

Factors that tend to increase $\delta$ (making cooperation easier):

• Shared long-term goals: If the hinge and partners plan to build a life together (e.g., buy property, raise children, grow old together), their time horizon is long, and $\delta$ is high.
• High relationship satisfaction: When the cooperative outcome is very rewarding (high payoffs from mutual cooperation), the long-term prospect is more valuable, and people are more willing to be patient.
• Deep interdependence: Shared finances, a shared home, or intertwined social circles raise the stakes. The probability of the relationship continuing is perceived as higher because breaking up is more costly.
• Strong communication and trust: A history of successfully resolving conflicts makes partners believe the relationship can weather future storms, increasing their perceived probability of continuation.

Factors that tend to decrease $\delta$ (making cooperation harder):

• Relationship uncertainty: If the vee is new or has a history of instability and conflict, partners may doubt its longevity, lowering their $\delta$.
• Attractive outside options: If a partner believes they could easily find another, equally satisfying relationship, the current one becomes less essential to preserve. This lowers the value placed on future payoffs within this specific relationship.
• A known end-date: If one partner has a definite plan to move away in a year, the "infinite" horizon becomes finite. This has a powerful effect, which we will now examine.

The End-Game Problem

What happens if the relationship has a known, finite endpoint? Game theory provides a stark prediction through backward induction.

Imagine you both know the relationship will end in the next period. In that final period, there is no "shadow of the future" to enforce cooperation. It is rational to defect. Since you both know defection will happen in the last period, the incentive to cooperate in the second-to-last period is gone (as there's no future cooperation to look forward to), so it becomes rational to defect then as well. This logic unravels all the way back to the present.

This is why indefinite interaction is so important. As long as there is a possibility of a next period—even a small one—the threat of punishment can work. It is the ambiguity of the end date that sustains cooperation.

Conclusion

In this lesson, we have placed the discount factor under a microscope, transforming it from an abstract variable into a concrete measure of commitment, patience, and belief in a shared future.

Key Takeaways:

• The discount factor ($\delta$) quantifies how much players value future payoffs, representing both patience and the perceived probability of the relationship continuing.
• Cooperation can be sustained only if players are sufficiently patient. This is determined by comparing a player's actual $\delta$ to a critical threshold ${\delta }^{\ast }$ derived from the game's payoffs. If $\delta \ge {\delta }^{\ast }$, cooperation is a rational long-term strategy.
• In a vee, a partner's $\delta$ is influenced by tangible factors: long-term commitment, relationship satisfaction, interdependence, and the presence of outside options.
• The prospect of an indefinite future is essential. A known end-date can cause cooperation to unravel via backward induction.

Our analysis with Grim Trigger and Tit-for-Tat assumed rather black-and-white responses to defection: either infinite punishment or immediate forgiveness. Real relationships are more nuanced. People can forgive, but trust may need to be rebuilt over time. In our next lesson, we will begin to explore this by modeling forgiveness and renegotiation mechanisms that can be used to repair cooperation after a breach of trust.