Rubinstein Bargaining Model

Rubinstein bargaining model is the canonical non-cooperative model of strategic bilateral negotiation in game theory and business and economics theory, formulated by Israeli economic theorist Ariel Rubinstein in his landmark 1982 Econometrica paper "Perfect Equilibrium in a Bargaining Model". Prior to Rubinstein's contribution, bilateral bargaining over a cooperative surplus was either treated axiomatically—as in John Nash's (1950) cooperative bargaining solution—or viewed as fundamentally indeterminate under non-cooperative Nash equilibrium, because a one-shot "ultimatum" demand game admits every possible division (x, 1 − x) ∈ [0, 1]² as a Nash equilibrium supported by non-credible threats of perpetual disagreement. By modeling negotiation explicitly as an infinite-horizon sequential game of alternating offers in which waiting is costly due to time preference (δ₁, δ₂ ∈ (0, 1)), Rubinstein proved that imposing Reinhard Selten's refinement of Subgame Perfect Equilibrium (SPE) eliminates all non-credible threats and selects a unique equilibrium partition of the pie, achieved without delay in period 0.

Rules of the Infinite-Horizon Negotiation

Consider two rational players, Player 1 and Player 2, who must agree on how to divide a unit surplus ("a pie of size 1") across discrete time periods t = 0, 1, 2, 3, … extending potentially to infinity:

1. Alternating Proposals: In every even period (t = 0, 2, 4, …), Player 1 proposes a division (x, 1 − x) with x ∈ [0, 1] denoting Player 1's share and 1 − x denoting Player 2's share. Player 2 observes the offer and chooses in that same period either to Accept (Y)—ending the game with payoffs (x, 1 − x) at date t—or to Reject (N), advancing the negotiation to period t + 1. In every odd period (t = 1, 3, 5, …), the roles reverse: Player 2 proposes a division (y, 1 − y) (where y is Player 1's share and 1 − y is Player 2's share), and Player 1 either accepts or rejects.
2. Impatience and Exponential Discounting: Both players are impatient and discount future consumption at constant per-period discount factors δ₁ ∈ (0, 1) and δ₂ ∈ (0, 1). If agreement on division (x, 1 − x) is reached in period t, the present-value payoff evaluated at period 0 is δ₁^t · x for Player 1 and δ₂^t · (1 − x) for Player 2. If the players reject every offer forever (perpetual disagreement), both receive a payoff of zero (0, 0).

While any share s ∈ [0, 1] can be sustained as an unrefined Nash equilibrium (where both players follow the stationary strategy "always demand s for Player 1 and reject any smaller share"), such strategies violate sequential rationality off the equilibrium path: if Player 1 slightly shades the demand to s − ε, rejecting it forces Player 2 to wait at least one period, which is irrational if δ₂(1 − s) < 1 − s + ε.

Stationary Indifference Conditions

Because the subgame starting at period t = 2 is structurally identical (isomorphic up to discounting) to the original game starting at t = 0, Avner Shaked and John Sutton (1984) developed an elegant recursive bounding argument demonstrating that the supremum and infimum of Player 1's SPE continuation payoffs must coincide. In this unique Subgame Perfect Equilibrium, the equilibrium offer (x*, 1 − x*) made by Player 1 in even periods and the equilibrium offer (y*, 1 − y*) made by Player 2 in odd periods satisfy two interlocking no-delay indifference conditions:

1. Player 2's Acceptance Threshold at t = 0: When it is Player 1's turn to propose, Player 1 offers Player 2 the smallest share 1 − x* that leaves Player 2 indifferent between accepting today and rejecting to become the proposer tomorrow (obtaining 1 − y* discounted by one period at factor δ₂):

1. Player 1's Acceptance Threshold at t = 1: Symmetrically, when it is Player 2's turn to propose, Player 2 offers Player 1 the smallest share y* that leaves Player 1 indifferent between accepting on the spot and rejecting to propose in the next period (obtaining x* discounted by δ₁):

Closed-Form Equilibrium Shares

Substituting y* = δ₁ x* into the first indifference equation yields 1 − x* = δ₂(1 − δ₁ x*), which solves linearly for the unique Subgame Perfect Equilibrium partition at t = 0:

In equilibrium, Player 1 proposes (x*, 1 − x*) at t = 0, and Player 2 accepts in period 0, attaining Pareto efficiency with zero deadweight delay loss.

Elimination of First-Mover Advantage as Δt → 0

When both players share an identical per-period discount factor δ₁ = δ₂ = δ, the formula simplifies to x* = 1 / (1 + δ) and 1 − x* = δ / (1 + δ). Because δ < 1, Player 1 enjoys a strict first-mover advantage (x* > 1/2) simply by making the initial offer at t = 0. However, suppose the real-time interval between successive offers and counteroffers is Δt > 0, and each player i ∈ {1, 2} discounts continuous time at an instantaneous subjective interest rate r_i > 0 (as in the Ramsey–Cass–Koopmans model), so that δ_i = exp(−r_i Δt). Applying L'Hôpital's rule as the response lag shrinks to zero (Δt → 0) yields:

In the frictionless continuous-time limit, the artificial first-mover advantage disappears completely, and each player's share is inversely proportional to their impatience rate r_i. When r₁ = r₂, the surplus is divided 50–50.

Equivalence with Asymmetric Nash Bargaining and Risk of Breakdown

As Ken Binmore, Ariel Rubinstein, and Asher Wolinsky (1986) proved in their seminal unification of the Nash Program, the Rubinstein game provides the non-cooperative foundation for John Nash's (1950) axiomatic bargaining solution. Specifically, if players have concave utility functions u₁(x) and u₂(1 − x) and face either discount rates (r₁, r₂) or exogenous probabilities (p₁ Δt, p₂ Δt) that negotiations irreversibly break down following a rejection, the Δt → 0 limit of the Rubinstein SPE maximizes the asymmetric Nash product:

where α = r₂ / (r₁ + r₂). Crucially, Binmore, Shaked, and Sutton (1985) established the Outside Option Principle: how an alternative payoff affects the equilibrium division depends on whether it represents an involuntary breakdown payoff (d_i, which shifts the disagreement point inside the Nash product) or a voluntary outside option (w₂, such as a worker quitting wage negotiations to take another job). A voluntary outside option w₂ has zero effect on the negotiated split whenever w₂ ≤ δ₂(1 − y*)—because the threat to walk away is non-credible—and acts as a hard lower bound (1 − x* = w₂) only when w₂ > δ₂(1 − y*). This distinction underpins wage determination in Diamond–Mortensen–Pissarides search-and-matching labor macroeconomics, sovereign debt restructuring, and vertical supply-chain contracting alongside Cournot competition and price discrimination.

If the Rubinstein model predicts immediate agreement in period 0, why do real-world strikes and wars occur?

In the baseline Rubinstein model, immediate agreement at t = 0 relies on complete information: both players know each other's discount factors (δ₁, δ₂) and outside options with certainty, so Player 1 can calculate the exact minimum acceptable offer 1 − x* and propose it immediately. When players possess incomplete (asymmetric) information—for example, when a firm does not know a labor union's true strike fund reserve or discount factor—costly delay (strikes, litigation, or prolonged stalemates) emerges in equilibrium as a credible screening and signaling mechanism by which patient types separate themselves from impatient types (Ausubel, Cramton, and Deneckere, 2002).

Why does backward induction fail to select a unique equilibrium without subgame perfection in infinite-horizon games?

In a finite-horizon alternating-offers game (Ståhl, 1972) with terminal period T, standard backward induction starts from the final ultimatum offer at t = T and steps backward to t = 0. In an infinite-horizon game (T = ∞), there is no final period from which to start backward induction; instead, Shaked and Sutton (1984) showed that the recursive self-similarity of every two-period cycle (t → t + 2) defines a contraction mapping with modulus δ₁δ₂ < 1, which forces the upper and lower bounds of all subgame perfect continuation values to collapse to the unique point x* = (1 − δ₂)/(1 − δ₁δ₂).

1. Rubinstein, A. (1982). "Perfect Equilibrium in a Bargaining Model." Econometrica, 50(1), 97–109. https://doi.org/10.2307/1912531
2. Shaked, A., & Sutton, J. (1984). "Involuntary Unemployment as a Perfect Equilibrium in a Bargaining Model." Econometrica, 52(6), 1351–1364. https://doi.org/10.2307/1913509
3. Binmore, K., Rubinstein, A., & Wolinsky, A. (1986). "The Nash Bargaining Solution in Economic Modelling." The RAND Journal of Economics, 17(2), 176–188. https://doi.org/10.2307/2555382
4. Nash, J. F. (1950). "The Bargaining Problem." Econometrica, 18(2), 155–162. https://doi.org/10.2307/1907266
5. Osborne, M. J., & Rubinstein, A. (1990). Bargaining and Markets. San Diego: Academic Press. https://doi.org/10.1016/B978-0-12-528630-5.50001-X