Title: Dynamic Outside Options and Optimal Negotiation Strategies
Domain: economics

Question: In a continuous-time "split-the-pie" model, a principal negotiates with an agent whose outside option, Y_t, follows a publicly observable diffusion process. The principal can commit to a negotiation strategy defined by two tools: the demand for their share of the pie (α_t) and the pressure applied to the agent (exercised via a split threshold S_t, below which the agent must accept the offer). | Design the optimal negotiation contract from the principal's perspective. Specifically, you must: | Identify the minimal set of state variables on which the optimal contract's terms (demand and pressure) depend. | Characterize the structure of the optimal contract in terms of these state variables. Explain precisely how the principal's demand and the split threshold evolve dynamically in response to changes in the agent's outside option. | Explain the economic intuition behind the dynamic relationship between the two negotiation tools (demand and pressure). Are they substitutes or complements?

Background: The problem is set within the field of dynamic bargaining theory, which studies negotiations that unfold over time rather than as single, take-it-or-leave-it offers. A central element in such models is the concept of an outside option, which represents a negotiator's best alternative to a negotiated agreement. In this context, the outside option is not static but stochastic, evolving unpredictably, which captures the real-world uncertainty of a changing environment. A key assumption is that one party, the principal, has commitment power, meaning they can credibly bind themselves to a pre-announced negotiation strategy. This creates a strategic asymmetry. The principal's challenge is to design a strategy that optimally balances the desire for a larger share of the pie against the risk that the agent's improving outside option will cause them to abandon the negotiation altogether. This involves managing the strategic use of delay, as waiting can either improve or worsen the principal's position.

Definitions: State Variables: The contract does not just depend on the current outside option Y_t. A second, crucial state variable is the historical maximum of the outside option process, M_t = max_{s≤t} Y_s. This variable tracks the agent's peak bargaining power and the history of the principal's concessions. | Contract Components: The optimal contract is defined by three key functions/values: | a) Demand Function (α*(M_t)): The principal's share of the pie, which depends only on M_t. | b) Split Threshold (S*(M_t)): The boundary for the outside option. If Y_t falls below S*(M_t), the agent must accept the offer 1 - α*(M_t). This also depends only on M_t. | c) Breakdown Threshold (R*): A constant upper bound. If Y_t exceeds R*, the negotiation ends and the agent leaves.

Methodology: The key dynamic event is when the agent's outside option reaches a new maximum (Y_t > M_{t-1}). The optimal contract remains stationary as long as M_t is constant. When a new maximum is reached, the principal is forced to concede. The central finding is that the principal optimally concedes by adjusting both tools in a complementary fashion: the demand α* is decreased (a better offer for the agent) AND the split threshold S* is also decreased (less pressure on the agent).

Answer: 1. The Principal's Problem and the Optimal Contract's Form | The principal (P) seeks to find a contract that maximizes their expected discounted utility from an agreement, J = E[e^(-rτ) * d_τ * u_P(α_τ)], subject to the agent (A) being willing to continue the negotiation at all times. This willingness is captured by the Dynamic Individual Rationality (DIR) constraint: the agent's continuation value, V_t, must always be at least as great as their current outside option, Y_t. | V(Y_t, M_t) ≥ Y_t for all t < τ | The key insight is that the optimal policy is not a function of the entire path of Y_t, but is instead governed by two state variables: Y_t itself, and M_t = max_{s≤t} Y_s, the historical maximum of the outside option. M_t acts as a sufficient statistic for the history of concessions. | The optimal contract takes a simple threshold form. It is defined by a demand function α*(m), a split threshold S*(m), and a constant breakdown threshold R*. The negotiation stops at time τ* when Y_t first exits the continuation region (S*(M_t), R*). | τ* = inf{t: Y_t ∉ (S*(M_t), R*)} | If Y_t hits the lower boundary S*(M_t), a split occurs (d_τ* = 1) with the principal's share being α*(M_t). If it hits the upper boundary R*, the agent leaves (d_τ* = 0). | 2. The Core Dynamic: Concession at New Highs | The contract terms (α* and S*) are functions of M_t only. This leads to a distinct dynamic: | Stasis Phase: As long as Y_t remains below its historical peak M_t, the contract terms α*(M_t) and S*(M_t) are constant. The principal holds firm. | Concession Trigger: A concession is only triggered when Y_t reaches a new historical maximum. At the precise moment t where Y_t = M_t = m, the agent's DIR constraint becomes binding. At these points, the agent's continuation value is exactly equal to their outside option: | V(m, m) = m | This is the mathematical expression of the agent being on the verge of leaving. To prevent this, the principal must immediately and permanently update the contract terms to a more favorable policy for the agent. It is established that both the demand and the split threshold are decreasing functions of m: | d(α*)/dm < 0 and d(S*)/dm < 0 | This means when M_t increases, the principal must lower their demand (offer the agent a better deal) and simultaneously lower the split threshold (apply less pressure). | 3. The Complementarity of Tools: A Formal Explanation | The simultaneous reduction of demand (α*) and pressure (S*) reveals they are complements, not substitutes. This can be understood by considering how the principal delivers a promised value W to the agent. For a given state m, the principal chooses a new offer α and a new split threshold S to deliver this value W. This relationship can be expressed formally through a value equation for the agent: | W = φ(S) * u_A(1 - α) + Φ(S) * V_m | where φ(S) is the discounted probability of Y_t hitting the threshold S before hitting m, and Φ(S) is the discounted probability of hitting m first. This equation can be inverted to find the required demand α for any given S and W: | u_A(1 - α(S, W)) = (W - Φ(S) * V_m) / φ(S) | The principal then chooses S to maximize their own utility, F(S,W) = φ(S) * u_P(α(S,W)) + Φ(S) * J_m. The key insight comes from the concavity of the agent's utility function u_A. | If u_A is strictly concave (risk aversion), there are diminishing returns to increasing the agent's share. To deliver a higher value W, it becomes increasingly costly for the principal to do so only by reducing α. For example, moving the agent's share from 90% to 91% provides less utility than moving it from 50% to 51%. | It is therefore more efficient for the principal to deliver part of the required value W by also lowering S. A lower S increases the duration of the negotiation and the agent's option value (φ(S) changes). By combining a moderate reduction in α with a moderate reduction in S, the principal can satisfy the agent's binding DIR constraint at a lower cost to themselves than using either tool alone. This mathematical trade-off demonstrates their complementarity. | 4. Conclusion and Implications | The optimal negotiation strategy is a sophisticated, history-dependent policy where the historical maximum of the agent's outside option, M_t, serves as the sole summary of the past. The contract features long periods of intransigence, where the principal's offer is fixed, punctuated by immediate and permanent concessions precisely when the agent's outside option reaches a new peak. At these moments, the principal optimally reduces both their demand and the pressure on the agent, using these tools as complements to efficiently provide the necessary incentives for the agent to continue negotiating.

Checklist: 1. Correctly identify the two state variables governing the optimal contract: the current outside option Y_t and its historical maximum M_t.  
2. Define the three components of the optimal contract: the demand α*(M_t), the split threshold S*(M_t), and the breakdown threshold R*.  
3. Explain that the contract terms α* and S* are functions of M_t only and remain constant as long as M_t does not change.  
4. Describe the dynamic trigger: what happens when Y_t hits a new maximum?  
5. Prove that both α*(m) and S*(m) are decreasing functions of the state variable m.  
6. Explain the complementarity of negotiation tools: why does the principal reduce both demand and pressure simultaneously when forced to concede?  
7. Discuss the real-world implications of this contract structure, such as periods of negotiation intransigence followed by sudden concessions, and the efficiency of strategic delay.