How is a surplus divided when two parties must agree to realize it? Bargaining theory has two complementary traditions: Nash’s axiomatic approach (what properties should a solution have?) and Rubinstein’s strategic approach (model the back-and-forth explicitly and solve for equilibrium). Their eventual convergence is one of game theory’s most satisfying results.
The axiomatic approach: Nash bargaining
Bargaining problembargproblem A (two-player) bargaining problem is a pair \((S, d)\) where \(S \subset \R^2\) is a compact convex set of feasible utility pairs, and \(d = (d_1, d_2) \in S\) is the disagreement point (what each gets if talks fail), with some \(s \in S\) satisfying \(s > d\) componentwise. A bargaining solution is a map \(f(S,d) \in S\).
Nash bargaining solution (1950)nashbarg There is exactly one bargaining solution satisfying:
Pareto efficiency: no feasible point dominates \(f(S,d)\);
Symmetry: if \(S\) is symmetric and \(d_1 = d_2\), then \(f_1 = f_2\);
Invariance to affine transformations of either player’s utility;
Independence of irrelevant alternatives (IIA): if \(T \subseteq S\) and \(f(S,d) \in T\), then \(f(T,d) = f(S,d)\).
It is the point maximizing the Nash product: \[f(S, d) \;=\; \argmax_{s \in S,\; s \geq d}\; (s_1 - d_1)(s_2 - d_2).\]
Splitting a dollar with risk aversionsplitdollar Two players split $1; utilities are \(u_1(x) = x\) and \(u_2(x) = x^\alpha\) with \(\alpha \in (0,1)\) (player 2 is risk-averse), \(d = (0,0)\). Maximizing \(x \cdot (1-x)^\alpha\) gives \(x^* = \frac{1}{1+\alpha}\). As \(\alpha\) falls (more risk aversion), player 2’s share \(\frac{\alpha}{1+\alpha}\) shrinks: risk aversion is a bargaining weakness. Impatience will play the identical role in the strategic model.
The generalized (asymmetric) Nash solution maximizes \((s_1 - d_1)^\beta (s_2 - d_2)^{1-\beta}\), where \(\beta\) captures “bargaining power”—a black box that the strategic approach opens.
The strategic approach: alternating offers
Rubinstein’s alternating-offers game (1982)rubinsteingame Players alternate proposing divisions \((x, 1-x)\) of a pie of size 1: player 1 proposes in periods \(0, 2, 4, \dots\), player 2 in periods \(1, 3, 5, \dots\). A proposal accepted in period \(t\) gives discounted payoffs \((\delta_1^t x,\, \delta_2^t (1-x))\), where \(\delta_i \in (0,1)\) are discount factors. Rejection moves to the next period; perpetual disagreement gives \((0,0)\).
Rubinstein (1982)rubinstein The alternating-offers game has a unique subgame perfect equilibrium. Agreement is immediate; player 1 proposes and player 2 accepts \[x^* \;=\; \frac{1 - \delta_2}{1 - \delta_1 \delta_2}, \qquad 1 - x^* \;=\; \frac{\delta_2 (1 - \delta_1)}{1 - \delta_1 \delta_2}.\]
Proof idea (stationary equilibrium + uniqueness sketch). Look for values \((x, y)\): \(x\) is what the current proposer gets when player 1 proposes, \(y\) what player 2 gets as proposer. A responder accepts anything at least as good as her continuation value from proposing next period: \[1 - x = \delta_2\, y, \qquad 1 - y = \delta_1\, x .\] Solving: \(x = \frac{1 - \delta_2}{1 - \delta_1\delta_2}\), \(y = \frac{1-\delta_1}{1-\delta_1\delta_2}\). Uniqueness (Shaked–Sutton argument): let \(\bar{x}, \underline{x}\) be the supremum and infimum of player 1’s SPE payoffs as proposer. Optimal play bounds \(\bar{x} \le 1 - \delta_2(1 - \delta_1 \bar{x})\) and \(\underline{x} \ge 1 - \delta_2 (1 - \delta_1 \underline{x})\), which force \(\bar{x} = \underline{x} = x^*\). ◻
In Rubinstein’s model, shares are pinned down by relative patience: as \(\delta_2 \to 0\) (player 2 desperate to settle), player 1 takes almost everything; with equal discounting \(\delta_1 = \delta_2 = \delta\), player 1 gets \(\frac{1}{1+\delta} > \tfrac12\)—a vanishing first-mover advantage as \(\delta \to 1\). Outside options, risk of breakdown, and commitment tactics all map into the same calculus. “Bargaining power” is no longer a free parameter but an implication of the environment.
Convergence (Binmore–Rubinstein–Wolinsky 1986)brw As the time between offers shrinks to zero, the unique SPE outcome of the alternating-offers game converges to the (generalized) Nash bargaining solution, with disagreement point given by the payoffs during delay/breakdown and bargaining weights determined by relative discount rates.
This “Nash program”—grounding axiomatic solutions in explicit non-cooperative games—is a template used throughout cooperative theory (Chapter [ch:cooperative]).
Bargaining with outside options
Suppose player 2 can leave and take an outside option worth \(w\) at any response node.
Outside option principleoutsideoption If \(w \le \delta_2 y^*\) (the option is worse than 2’s equilibrium continuation), the equilibrium is unchanged—the option is an empty threat. If \(w > \delta_2 y^*\), player 2 receives exactly \(w\): the option binds but confers no extra rent.
The lesson—threats affect outcomes only when credible, and then only at their exact value—recurs in wage negotiation, litigation-vs-settlement, and M&A practice. It also warns against the naïve “split the difference plus my outside option” intuition.
Applications
Union–firm wage bargainingwagebarg A firm and a union bargain over the wage \(w\); production yields revenue \(R\) per period, the union’s strike payoff is \(s\) per period. With alternating offers and a common discount factor near 1, the outcome approximates the Nash solution with disagreement \((0, s)\): \(w^* \approx \frac{R + s}{2}\). The union’s strike fund \(s\) raises the wage with weight \(\tfrac12\) even though no strike ever occurs in equilibrium. Strength that is never used still shapes the deal.
Pre-trial settlementsettlement Litigation will cost each party \(c\) and transfer \(J\) from defendant to plaintiff with probability \(p\) (common knowledge). The settlement surplus relative to trial is \(2c\); Nash bargaining predicts settlement at \(pJ\) plus/minus the split of costs saved. Settlement fails in practice mainly through asymmetric information (each side optimistic about \(p\))—the subject of Chapters [ch:bayesian] and [ch:dynamicincomplete], and a leading explanation for costly disputes, strikes, and wars.
Exercises
Nash product computationnashproduct \(S\) is the triangle with vertices \((0,0)\), \((4,0)\), \((0,2)\); \(d = (1, \tfrac12)\). Compute the Nash bargaining solution. Verify IIA by re-solving on the smaller set \(S' = S \cap \{s_1 \le 3\}\) and checking the solution is unchanged (or explain if it moved).
Three rounds of alternating offersthreeround Solve by backward induction the three-period alternating-offers game (player 1 proposes at \(t=0\) and \(t=2\), player 2 at \(t=1\); if the \(t=2\) proposal is rejected both get 0), with common discount factor \(\delta\). Verify that as the horizon grows, proposer shares approach \(\frac{1}{1+\delta}\).
Impatience vs. risk of breakdownbreakdown In the alternating-offers game, replace discounting with an exogenous breakdown risk: after each rejection, with probability \(1-\pi\) negotiations collapse and players receive \((b_1, b_2)\). Derive the analogue of the Rubinstein shares and interpret the limit \(\pi \to 1\).
Deadline effectdeadline Modify the alternating-offers game so the pie disappears after period \(T\) (no discounting). Show the last proposer takes (almost) everything, and explain why real negotiations so often settle “on the courthouse steps.”