Rubinstein Bargaining Model: Structure and Rules

From a Single Offer to a Negotiation

To understand the logic of alternating-offers bargaining, it's helpful to start with the simplest possible bargaining game: a one-shot, take-it-or-leave-it offer. This is known as the Ultimatum Game.

The Ultimatum Game is static. Real negotiations, however, often involve a back-and-forth exchange of counter-offers. What happens if, after a rejection, the roles are reversed and the other player gets to make an offer? This introduces the element of time and the strategic incentive to wait for a better deal.

To build this intuition, let's look at a simple two-period bargaining game. This is a direct stepping stone to the Rubinstein model.

This two-period model captures the essence of strategic negotiation. Player 1, when making their initial offer, must anticipate what Player 2 could get by rejecting the offer and moving to the second period. This kind of forward-thinking is the basis of the solution we will explore in the next lesson.

The Rubinstein Alternating-Offers Model

Ariel Rubinstein, in a seminal 1982 paper, generalized this structure to an infinite horizon. This was a major breakthrough because it avoids imposing an arbitrary deadline on the negotiation. The game can, in principle, go on forever. This structure forces us to find a solution that is stable regardless of how many rounds have passed.

Let's formally define the structure and rules of the game:

1. Players: Two players, Player 1 and Player 2.
2. The Pie: They are bargaining over the division of a pie of size 1. An agreement is a pair $(x,1-x)$, where $x$ is Player 1's share and $1-x$ is Player 2's share, with $x\in [0,1]$.
3. Time: The game takes place over a sequence of discrete periods, $t=0,1,2,\dots$.
4. Alternating Offers: In even periods ($t=0,2,\dots$), Player 1 makes an offer. In odd periods ($t=1,3,\dots$), Player 2 makes an offer.
5. Responses: When one player makes an offer, the other player can either:
   • Accept (Y): The game ends. The pie is divided as proposed.
   • Reject (N): The game moves to the next period, and the player who just rejected the offer gets to make a counter-offer.
6. Payoffs: If players agree on a split $(x,1-x)$ in period $t$, their payoffs are discounted to present value.
   • Player 1's payoff: $U_1={\delta }_1^t\cdot x$
   • Player 2's payoff: $U_2={\delta }_2^t\cdot (1-x)$
     If they never agree, both players receive a payoff of 0.
7. Discount Factors: The players have discount factors ${\delta }_1,{\delta }_2\in (0,1)$. A higher $\delta$ means a player is more patient. A $\delta$ close to 0 means they are very impatient and strongly prefer an agreement now rather than later.

This sequential process is visualized in the diagram below.

To solidify your understanding of these rules and the underlying assumptions, let's turn to a more formal text.

The discount factor $\delta$ is a powerful and flexible concept. As noted in the resources, it can be interpreted in several ways:

• Impatience: Players simply prefer to receive a payoff sooner rather than later.
• Interest Rate: Money received today can be invested to earn interest, making it more valuable than the same amount of money received tomorrow.
• Breakdown Probability: There might be an exogenous probability $1-\delta$ that the negotiations break down after each period. So, $\delta$ is the probability the game continues. This connects well to your goal of understanding planning in a "chaotic world" where outside events can suddenly end an interaction.

Describing Strategies in the Bargaining Game

Now that we have the rules, how do we describe a player's strategy? In this dynamic game, a strategy is not just a single action. It must be a complete plan of action that specifies what a player will do in every possible contingency.

A player's strategy must specify:

1. The offer to make whenever it is their turn to propose.
2. The decision to accept or reject for every possible offer they might receive from the opponent.

Crucially, this plan must cover all possible histories of the game, even histories that would not occur if the player followed their own strategy. This is essential for evaluating the credibility of threats and for applying the concept of subgame perfection, which we will use to solve the game.

The PDF resource you just reviewed provides a formal, set-theoretic definition of these strategies.

This level of formal detail is exactly what's needed to analyze the game with mathematical precision. It ensures that our concept of a "strategy" is robust enough to handle the infinite, dynamic nature of the game.

Conclusion

In this lesson, we have constructed the Rubinstein alternating-offers bargaining model from the ground up. We started with a simple one-shot ultimatum and gradually added the key features of a real-world negotiation: alternating offers, the passage of time, and the discounting of future payoffs.

Key Takeaways:

• The Rubinstein model describes a negotiation between two players over a divisible surplus (a "pie") across an infinite time horizon.
• The core structure is alternating offers: Player 1 proposes, Player 2 responds; if Player 2 rejects, they propose, and Player 1 responds, and so on.
• The cost of delay is captured by a discount factor ($\delta$), where $0<\delta <1$. A payoff of $x$ received in period $t$ is worth only ${\delta }^{t}x$ today.
• A player's strategy is a complete contingent plan, specifying their action at every possible turn and for every possible history of prior offers.

We have now meticulously set up the rules of the game. We have the players, the actions they can take, and how they value the outcomes. However, we have not yet determined how rational players would actually play.

In our next lesson, we will do just that. We will apply the concept of backward induction to identify the recursive relationship between offers in the Rubinstein model. This will be the first step toward finding the model's unique, and perhaps surprising, solution.