Rubinstein Bargaining in Sequential Negotiation

The Dynamics of Negotiation: Offers, Time, and Patience

At its heart, the Rubinstein model views bargaining not as a single decision, but as a sequence of actions. Player 1 makes an offer, Player 2 can accept or reject. If she rejects, the pie shrinks slightly, and she gets to make a counter-offer. This continues until an agreement is reached. This structure is a much more realistic depiction of haggling, whether over a price at a market or over plans for the weekend.

The "shrinking pie" is formalized through a discount factor, denoted by $\delta$, where $0<\delta <1$. A payoff of $V$ received in the next period is only worth $\delta V$ today. This $\delta$ is a measure of a player's patience.

• A $\delta$ close to 1 indicates a patient player. They don't mind waiting, as the future is almost as valuable as the present.
• A $\delta$ close to 0 indicates an impatient player. They need a deal now, as future payoffs are heavily discounted.

This concept is central to the model. To get a clear intuition for this setup and its importance, the following video provides an excellent overview.

Solving the Game: Backward Induction in Finite Time

To solve this game, we use the now-familiar tool of Subgame Perfect Equilibrium (SPE), which we find through backward induction. Because an infinite game has no "end" to start from, it's easier to first build intuition with a finite-horizon version.

Let's analyze a simple two-period game.

• Period 0: Player 1 proposes a split $(x,1-x)$, where $x$ is their share. Player 2 accepts or rejects.
• Period 1: If Player 2 rejected, she gets to propose a split $(y,1-y)$. Player 1 accepts or rejects. If he rejects, both get 0.

The following video provides a clear, step-by-step walkthrough of solving this exact game. Given your background in operations research, the logic of backward induction should be second nature.

Let's summarize the logic from the video. Let the players' discount factors be ${\delta }_1$ and ${\delta }_2$.

1. Analyze Period 1: Player 2 makes a take-it-or-leave-it offer. If Player 1 rejects, he gets 0. Therefore, Player 1 will accept any non-negative offer. To maximize her own payoff, Player 2 will offer Player 1 exactly 0 and keep the entire pie. The outcome in this subgame is $(0,1)$.

2. Analyze Period 0: Player 1 is making an offer. He knows that if Player 2 rejects, the game moves to Period 1, where she will secure a payoff of 1. From Player 2's perspective in Period 0, that future payoff of 1 is worth ${\delta }_2\times 1={\delta }_2$. This is her continuation value—what she gets if she waits.

3. Player 1's Optimal Offer: To make Player 2 accept the offer in Period 0, Player 1 must offer her a share of at least ${\delta }_2$. Since Player 1 wants to keep as much as possible, he will offer her exactly ${\delta }_2$.

The SPE outcome is that Player 1 offers the split $(1-{\delta }_2,{\delta }_2)$, and Player 2 accepts immediately in Period 0. The game ends without any delay. The final payoffs are:

• Player 1 gets $1-{\delta }_2$
• Player 2 gets ${\delta }_2$

Notice two things: only Player 2's patience matters, and the player who moves last has all the power. This "last-mover advantage" is an artifact of the finite horizon.

The Infinite Horizon: Where Patience Truly Shines

The classic Rubinstein model has an infinite horizon, which removes the artificial power of a pre-determined final stage. To solve this, we use a powerful assumption: players use stationary strategies. This means a player will make the same kind of offer whenever it's their turn to propose.

The key is to find a pair of offers, $x_1^{\ast }$ (Player 1's share when he proposes) and $x_2^{\ast }$ (Player 1's share when Player 2 proposes), that are in equilibrium with each other.

This leads to a system of two equations:

1. Player 1's offer must make Player 2 indifferent between accepting her share $(1-x_1^{\ast })$ today, versus rejecting and getting her share $(1-x_2^{\ast })$ tomorrow (which is worth ${\delta }_2(1-x_2^{\ast })$ today).
2. Player 2's offer must make Player 1 indifferent between accepting his share $x_2^{\ast }$ today, versus rejecting and getting his share $x_1^{\ast }$ tomorrow (which is worth ${\delta }_1{x}_1^{\ast }$ today).

Solving the system from the reading yields the unique SPE. In the first period, Player 1 proposes, and Player 2 immediately accepts. The shares are:

The most important takeaway is what happens when we compare players' patience:

• The more patient you are (the higher your $\delta$), the larger your share of the pie.
• If Player 1 becomes more patient (${\delta }_1$ increases), his share $x_1^{\ast }$ increases.
• If Player 2 becomes more patient (${\delta }_2$ increases), Player 1's share $x_1^{\ast }$ decreases.

Patience is bargaining power. A patient player can credibly threaten to wait, forcing the more impatient player to concede a larger share to get a deal done quickly.

Application: Sequential Negotiation in a Vee

Let's apply this to a negotiation in a vee relationship. Imagine a Hinge (H) and one of their partners (P1) are deciding how to split a block of 100 hours of discretionary "relationship time" over the next month. Let's model this as H making the first offer.

The crucial step is to think about their relative patience (${\delta }_H$ vs. ${\delta }_{P1}$).

• An impatient partner might be one who feels insecure in the relationship and wants to "lock in" time and commitment now. Their $\delta$ would be low.
• A patient partner might be one who is secure and has a rich life outside the relationship, making them less anxious about immediate agreements. Their $\delta$ would be high.

Now, consider the Hinge's structural position. The Hinge has another partner, P2. If negotiations with P1 stall, H doesn't get zero utility; they can spend that time with P2. This access to a built-in alternative makes delay less costly for the Hinge. It effectively increases their patience. Therefore, it's reasonable to model the Hinge as having a higher discount factor than a non-hinge partner in many scenarios.

Let's assume ${\delta }_H=0.9$ (very patient) and ${\delta }_{P1}=0.6$ (moderately patient).

Using the formula, the Hinge (as the first proposer) would secure a share of:

The model predicts the Hinge would get approximately 87% of the time (87 hours), and P1 would get 13% (13 hours). This might seem extreme, but it demonstrates the powerful effect of asymmetric patience. The Hinge's ability to wait, backed by their other relationship, translates directly into a larger share of the negotiated resource.

This idea—that outside options influence the bargaining process itself—is a key extension to the Rubinstein model. The resource below discusses this in the context of international relations, but the logic is universal.

Conclusion

Today, you've added a dynamic and powerful tool to your analytical toolkit. Unlike the static Nash solution, the Rubinstein model shows how the process of bargaining, especially the strategic use of time, determines the outcome.

Key Takeaways:

• The Rubinstein model frames bargaining as a sequence of alternating offers over time.
• The discount factor ($\delta$) quantifies a player's patience. An agreement is always reached immediately in equilibrium, but the threat of future rounds of negotiation determines the terms of that agreement.
• Patience is power: The player who can better afford to wait (has a higher $\delta$) will secure a larger share of the resource.
• In a vee, the Hinge's structural position—having another partner to turn to—can be modeled as a higher discount factor, giving them more power in sequential negotiations over shared resources like time.

In our last two lessons, we've seen how bargaining power comes from two distinct sources: the value of your disagreement outcome (NBS) and your ability to endure delay (Rubinstein). In the next lesson, we will synthesize these ideas and analyze how explicit outside options—like a new dating prospect for one of the partners—can fundamentally shift the balance of power within the entire vee structure.