Noise Bargaining: A New Perspective on Single Proposers, Negotiation and Delays

Many real-world negotiations feature a single proposer and delays in reaching agreement. For example, budgets or debt ceiling agreements fit this structure, with formal proposals coming from an executive or committee that legislators must vote to accept or delay, potentially at large economic or political cost. Mergers and acquisitions provide another example, as potential acquirers make offers that have to be accepted by shareholders. In these cases, one side proposes, and the other must accept the proposal or wait, making delay a central feature of bargaining.

Literature on Bargaining and Negotiation

The study of bargaining in economics goes back to at least 1950 and the paper "The Bargaining Problem" by John Nash. The early approach by Nash featured a set of axioms (or mathematical assumptions) that dictated what a reasonable allocation between two bargaining parties could look like. In Nash bargaining, who proposes is ambiguous, and settlement happens immediately (unless one party cannot be made better off). Bargaining terms are controlled by an exogenous bargaining power parameter.

The 1982 paper "Perfect Equilibrium in a Bargaining Model" by Ariel Rubinstein moved bargaining from an axiomatic approach to a game-theoretic one where parties would take turns proposing offers. A surprising result here is that settlement also happens immediately. Bargaining power in this model is endogenously determined by the threat of delay (even though delay never happens in equilibrium).

These two approaches have sparked a large and fruitful body of research. But neither naturally fits the context of the real-world examples above, and delay in negotiations has proven surprisingly difficult to generate in the literature, usually requiring some degree of asymmetric information.

Noise bargaining — a new approach introduced in my 2025 working paper "Sovereign Default Intensity and Noise Bargaining," co-authored with Pablo Guerron-Quintana — provides a tractable way to generate delay and endogenous bargaining power with a single proposer. Like in Rubinstein's paper, the approach is game theoretic. But unlike either Rubinstein or Nash's paper, there is a single proposer, and bargaining usually entails delays.

The single proposer makes an offer while considering that the other party's decision is affected by noise. When the proposer improves the offer, the probability the other party accepts is greater. Thus, the marginal cost of sweetening an offer must be balanced by the marginal benefit of faster settlement, as delay is costly. A large degree of noise means sweetening offers does not quicken settlement by much, which on balance makes the proposer more aggressive and delays settlement. So in noise bargaining, there is generally delay in equilibrium, and bargaining power is determined by the noisiness of acceptance decisions.

Bargaining and Cake

To be precise, consider a canonical cake-eating problem. There are two parties, which we can call A and B. There is one cake, which A and B have to decide how to split. Both parties are impatient, so consuming a certain share right now is as good as consuming somewhat more than that share later.

In Nash bargaining, a failure to bargain means both parties get nothing. Under the axioms Nash proposes, settlement occurs immediately, and the share A gets is determined by an exogenous bargaining weight.

In Rubinstein bargaining, the solution comes from a noncooperative game between A and B. Suppose that A proposes first. If the offer is rejected, then B will subsequently propose. If that offer is rejected, then A will propose, and so on indefinitely. Rubinstein showed there is a unique subgame perfect equilibrium of that game where the initial proposer A gets a larger share of the cake. Because waiting is costly, B finds it optimal to accept less than what could be received by waiting to be the proposer next period. Like in Nash bargaining, settlement occurs immediately. The bargaining power in this case is not some exogenously fixed number but endogenously depends on how costly waiting is. The threat of delay impacts the outcome, even though there is no delay in equilibrium.

In noise bargaining, there is a single proposer (again, let's assume A). In evaluating the initial offer, B has fundamental values associated with accepting and rejecting the offer. The fundamental value of accepting is the share B gets. The fundamental value of rejecting depends on expected future offers that A will make as well as B's probability of accepting those offers. Crucially, B also has random noisy valuation shock attached to accepting or rejecting the offer. These valuation shocks occur after A makes the offer but before B decides to accept or reject. So when considering what offer to make, A knows there is a probability of acceptance that will increase if he offers B more.

Bargaining and Elasticity

The Markov perfect equilibrium (which is a type of subgame perfect equilibrium) corresponding to this game gives A's share as a function of the acceptance probability elasticity. Importantly, the greater this elasticity is in equilibrium, the smaller A's share is. To see the economics of this result, it is useful to walk through two examples.

If the acceptance probability is highly elastic, that means slightly sweetening the offer increases the probability of acceptance by a large amount. In this case, the proposing party A has large incentives to offer a good deal to party B, because doing so will result in a substantially larger acceptance probability. This higher acceptance probability avoids delay in negotiations, which is good for A (and B). For example, suppose that if A reduced his own share from 50 percent of the cake to 40 percent, acceptance probabilities would increase from 1 percent to 99 percent. This is a highly elastic acceptance rate. And so, the cost to A of sacrificing one-tenth of the cake is likely offset by the sharp increase in the probability of settling today rather than waiting (which, again, is costly because of discounting future receipts). Given these incentives, the share going to A in equilibrium is small, and settlement is faster all else equal.

If instead the acceptance probabilities were inelastic, increasing from 50 percent to 50.01 percent, the incremental benefit associated with a slightly higher acceptance rate would likely be less than the additional one-tenth cost. In that case, the share going to A in equilibrium will be large, and settlement will take longer all else equal.

In other words, the noisier bargaining is — in the sense of reducing the elasticity between offered terms and acceptance rates — the more incentive the proposer has to be aggressive in negotiations, causing longer delays all else equal. While the equilibrium amount of delay is affected by many factors, noise bargaining generally features delays (and necessarily so if the valuation shocks have unbounded support). Noise bargaining endogenously links bargaining noisiness to offers and delay, capturing dynamics the bargaining literature has found challenging.

Table 1 summarizes the differences between the bargaining approaches.

Conclusion

This article describes how noise bargaining works in a canonical cake-eating problem. In a much more complicated but conceptually similar setting, my paper uses noise bargaining to model sovereign debt restructurings and shows that the model can generate realistic dispersion in recovery rates and delays. But noise bargaining is a general concept that should prove useful in the analysis of many other economic contexts that feature a single proposer and endogenous delay.

Grey Gordon is a senior economist in the Research Department at the Federal Reserve Bank of Richmond.

To cite this Economic Brief, please use the following format: Gordon, Grey. (November 2025) "Noise Bargaining: A New Perspective on Single Proposers, Negotiation and Delays." Federal Reserve Bank of Richmond Economic Brief, No. 25-42.

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