Glossary

10.2 Rubinstein bargaining model

4 min read•august 7, 2024

The is a game theory concept that explores how two negotiate to divide a surplus. It involves , an , and considers factors like and .

The model uses and to predict outcomes. As the time between offers shrinks, it converges to the , linking non-cooperative and theories.

Bargaining Process

Alternating Offers and Infinite Horizon

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  • Players make offers in an alternating fashion, with one player proposing a division of the surplus and the other player either accepting or rejecting the offer
  • If the offer is accepted, the game ends and the surplus is divided according to the agreed-upon terms
  • If the offer is rejected, the game continues to the next period, with the roles of the players switching (the proposer becomes the responder and vice versa)
  • The game has an infinite horizon, meaning that the bargaining process can potentially continue indefinitely until an agreement is reached
  • The infinite horizon setup captures the idea that players have the option to continue bargaining as long as they have not reached an agreement

First-Mover Advantage

  • The player who makes the first offer in the bargaining game has a strategic advantage known as the first-mover advantage
  • The first mover can propose a division of the surplus that is more favorable to themselves, knowing that the other player may be willing to accept a less favorable offer to avoid the cost of delay
  • The first-mover advantage arises because the first proposer can anticipate the other player's response and adjust their offer accordingly
  • The extent of the first-mover advantage depends on factors such as the players' time preferences and the cost of delay (captured by the )

Equilibrium Concepts

Subgame Perfect Equilibrium and Backward Induction

  • The Rubinstein bargaining model is solved using the concept of subgame perfect equilibrium (SPE)
  • SPE is a refinement of Nash equilibrium that requires players' strategies to be optimal not only at the beginning of the game but also at every decision point (subgame) throughout the game
  • To find the SPE, the game is solved using backward induction, a process of reasoning backward from the end of the game to determine the optimal strategies at each stage
  • In backward induction, players anticipate the future actions of their opponents and make decisions based on this anticipation
  • The SPE of the Rubinstein bargaining model involves the proposer offering a division of the surplus that makes the responder indifferent between accepting and rejecting, and the responder accepting this offer

Convergence to Nash Bargaining Solution

  • As the time interval between offers becomes arbitrarily small (i.e., the discount factor approaches 1), the SPE of the Rubinstein bargaining model converges to the Nash bargaining solution
  • The Nash bargaining solution is a cooperative bargaining concept that satisfies a set of desirable properties, such as , symmetry, and independence of irrelevant alternatives
  • In the limit, the Rubinstein bargaining model predicts that the players will agree on a division of the surplus that is consistent with the Nash bargaining solution
  • The convergence to the Nash bargaining solution highlights the connection between non-cooperative (Rubinstein) and cooperative (Nash) bargaining theories

Time and Patience

Discount Factor and Time Preference

  • The discount factor (δ\delta) represents the players' time preferences and captures the cost of delay in reaching an agreement
  • A higher discount factor implies that players are more patient and place a higher value on future payoffs relative to present payoffs
  • Conversely, a lower discount factor implies that players are less patient and prefer to receive payoffs sooner rather than later
  • The discount factor is typically assumed to be between 0 and 1, with δ=1\delta = 1 representing no discounting (i.e., future payoffs are valued the same as present payoffs)
  • The discount factor can be derived from the players' time preferences, which are often modeled using exponential discounting (e.g., δ=e−rΔt\delta = e^{-r\Delta t}, where rr is the discount rate and Δt\Delta t is the time interval between offers)

Patience in Bargaining

  • Patience plays a crucial role in determining the outcome of the Rubinstein bargaining game
  • The more patient player (i.e., the player with a higher discount factor) has a strategic advantage in the bargaining process
  • A more patient player is willing to wait longer to reach an agreement and can credibly threaten to reject offers that are not sufficiently favorable
  • In equilibrium, the more patient player receives a larger share of the surplus compared to the less patient player
  • The relationship between patience and can be seen in the equilibrium division of the surplus, which is proportional to the players' discount factors (e.g., 1−δ21−δ1δ2\frac{1-\delta_2}{1-\delta_1\delta_2} for player 1 and δ2(1−δ1)1−δ1δ2\frac{\delta_2(1-\delta_1)}{1-\delta_1\delta_2} for player 2, where δ1\delta_1 and δ2\delta_2 are the discount factors of player 1 and player 2, respectively)

Key Terms to Review (20)

Alternating offers: Alternating offers refer to a bargaining process where two parties take turns proposing offers to each other in an effort to reach an agreement. This method is commonly associated with the Rubinstein bargaining model, which provides a formal framework for understanding how parties negotiate and make concessions over time, highlighting the strategic aspects of timing and patience in negotiations.
Asymmetric information: Asymmetric information refers to a situation where one party in a transaction has more or better information than the other party, leading to an imbalance in decision-making. This imbalance can create problems such as adverse selection and moral hazard, affecting how parties interact in strategic scenarios. Understanding asymmetric information is crucial for analyzing behavior in situations with incomplete information, negotiation dynamics, and designing mechanisms that ensure efficient outcomes.
Backward induction: Backward induction is a method used in game theory to determine optimal strategies by analyzing a game from the end to the beginning. It involves looking at the last possible moves of players and determining their best responses, then moving sequentially backward through the game tree to deduce the optimal actions of earlier moves. This technique is particularly relevant in analyzing strategic interactions in sequential games and helps in identifying subgame perfect equilibria.
Bargaining Power: Bargaining power refers to the ability of one party to influence the terms and conditions of a negotiation, often determining how favorable the outcome will be for them. This power can be derived from various factors, including resources, information, and alternatives available to the parties involved. Understanding bargaining power is crucial for analyzing interactions in both competitive and cooperative settings, such as market negotiations, labor disputes, and international trade deals.
Cooperative bargaining: Cooperative bargaining is a negotiation process where parties work together to achieve mutually beneficial outcomes rather than competing against each other. In this approach, the focus is on collaboration, communication, and finding solutions that satisfy the interests of all involved, leading to win-win scenarios. This method contrasts with adversarial bargaining, where the parties see each other as opponents.
Discount factor: The discount factor is a numerical value between 0 and 1 that represents how future payoffs are valued in comparison to present payoffs. It captures the idea that people generally prefer receiving money or benefits sooner rather than later, reflecting time preference. In economic models, this factor plays a crucial role in determining the present value of future cash flows or benefits, impacting decision-making processes in various scenarios, including negotiations and strategic interactions.
Equilibrium Outcome: An equilibrium outcome refers to a stable state in a game or negotiation where all players have chosen their strategies and no one has an incentive to deviate from their chosen strategy. This concept indicates that each player's strategy is optimal, given the strategies chosen by others, leading to a situation where all parties are satisfied with the outcome, and further changes would not improve their individual payoffs.
First-mover advantage: First-mover advantage refers to the competitive edge that a company or individual gains by being the first to enter a market or adopt a new product or technology. This can lead to brand loyalty, control over resources, and the ability to establish standards, creating barriers for later entrants. It is often associated with strategic decisions in business and economic contexts, influencing negotiations, market dynamics, and competitive positioning.
Incomplete Information: Incomplete information refers to a situation in a game where players do not have perfect knowledge about the other players' characteristics, strategies, or payoffs. This lack of information influences how players form strategies and make decisions, leading to uncertainties in predictions about opponents' behavior and outcomes in various interactions.
Infinite horizon: Infinite horizon refers to a modeling framework in which decision-making extends indefinitely into the future. In this context, agents consider not only immediate payoffs but also the long-term consequences of their actions across an unbounded time frame, leading to strategic interactions that can significantly impact bargaining outcomes and stability.
Nash bargaining solution: The Nash bargaining solution is a concept in game theory that provides a way to determine how two or more parties can reach an agreement that maximizes their joint utility. This solution is based on the idea that rational players will negotiate and choose outcomes that are mutually beneficial, while also ensuring fairness in how the benefits are distributed. It connects to various applications in economics, as well as other fields like political science and negotiation theory, highlighting the importance of cooperative strategies in competitive situations.
Non-cooperative games: Non-cooperative games are a type of game in which players make decisions independently, without collaboration or communication with one another. Each player's strategy is designed to maximize their own payoff, often leading to competitive behavior as individuals seek to outsmart their opponents. These games are fundamental in understanding strategic interactions where cooperation isn't possible or is not enforced.
Pareto Efficiency: Pareto efficiency is an economic state where resources are allocated in a way that no individual can be made better off without making someone else worse off. This concept emphasizes the idea of optimal distribution of resources among players in a game, relating closely to strategies, payoffs, and the rational behavior of individuals involved.
Patience: Patience, in the context of negotiation and decision-making, refers to the willingness of a player to wait for an optimal outcome rather than settling for a less favorable agreement quickly. This concept is crucial in understanding strategic interactions, particularly when parties have to choose between immediate gains and potentially better long-term benefits. In bargaining scenarios, higher levels of patience can lead to more favorable outcomes, as players may be willing to endure delays for a more satisfying agreement.
Payoff Matrix: A payoff matrix is a table that shows the payoffs or outcomes for each player in a game, given all possible combinations of strategies chosen by the players. It visually represents the choices available to players and their potential results, making it essential for analyzing strategic interactions in various types of games.
Players: In game theory, players are the decision-makers in a strategic interaction, each with their own preferences and goals. Players can be individuals, groups, or entities that make choices to maximize their own utility based on the strategies available to them. Understanding players is essential as it connects directly to concepts like strategies, payoffs, and rationality, which all influence how decisions are made and outcomes are achieved in various game formats.
Rubinstein Bargaining Model: The Rubinstein bargaining model is a theoretical framework used to analyze negotiations between two parties over a fixed pie of resources, where each party makes alternating offers. It highlights how the timing of offers and the patience of the players can affect the outcome of the bargaining process, illustrating important dynamics in negotiation strategies and equilibrium concepts.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games where players make decisions at various stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that the strategies are credible and optimal, even if the game is played out from any point along the decision path.
Time preference: Time preference refers to the relative valuation individuals place on receiving goods or services at different points in time, reflecting a preference for immediate rewards over future ones. This concept is crucial in understanding decision-making processes, as it influences savings, investments, and consumption patterns. A higher time preference means that an individual is more inclined to favor present benefits over future gains.
Utility Functions: Utility functions are mathematical representations that describe how individuals or agents prioritize and rank their preferences over different outcomes or bundles of goods. These functions help to quantify satisfaction or happiness derived from consuming goods and services, which is essential in analyzing decision-making processes in various contexts, including strategic interactions and bargaining scenarios.
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