Bargaining, Disagreement States, and Choice-Law Consistency

Conceptual map

1. IV-07.01The bargaining object
2. IV-07.02Nash bargaining
3. IV-07.03Choice kernels
4. IV-07.04Target-specific computation
5. IV-07.05Welfare consistency

1. Bargaining starts from agreement and disagreement surpluses

A Nash formula becomes an economic model only after the feasible agreement set, disagreement payoffs, bargaining weights, transfers, and demand responses are fixed.

Let a distributor have pre-transfer agreement surplus \(A>0\) and a channel have surplus \(B>0\) relative to disagreement. A fee \(f\) paid by the distributor to the channel gives surpluses \(A-f\) and \(B+f\). With distributor bargaining weight \(\beta \in(0,1)\), the weighted Nash solution maximizes \((A-f)^{\beta}(B+f)^{1-\beta}\) over \(-B<f<A\).

2. A fixed-surplus numerical solution

Let \(A=8\), \(B=2\), and \(\beta=0.6\). Equation (1) gives fee two, distributor surplus six, and channel surplus four. The two agreement surpluses sum to ten, the fixed total surplus.

A nonnegative-fee restriction would bind if (1) were negative. Fee caps, limited liability, and discrete carriage terms similarly replace the interior equality with KKT or finite-choice conditions.

3. Demand can make total surplus fee dependent

If the negotiated fee changes retail price, carriage quality, promotion, or channel investment, \(A\) and \(B\) become functions of \(f\). The derivative of the Nash objective then contains \(A^{\prime}(f)\) and \(B^{\prime}(f)\) computed through the same consumer choice probabilities used in agreement and disagreement states. Formula (1) no longer applies as a closed form.

The disagreement calculation removes carriage or changes the bundle according to a declared fallback. Consumer substitution, distributor profit, channel advertising value, and welfare must all use that same choice law.

4. Target-specific kernels create incompatible models

Zero total agreement surplus places the Nash product at its participation boundary. Negative surplus means agreement is infeasible under individual rationality. Multiple fee roots can arise with endogenous demand; global objective comparison and participation constraints must accompany stationarity.

5. Implementation, exercises, and sources

Build agreement and disagreement states with one demand routine, calculate each party's surplus and derivative, enforce participation and fee bounds, and solve the log Nash objective. Report the fee, surplus split, objective, derivative residual, active constraints, and alternative roots.

Exercises

1. Reproduce the fee and surplus split in Section 2.
2. Find the \(\beta\) values for which a nonnegative fee binds when \(A=2\) and \(B=8\).
3. Let retail demand be linear in a fee pass-through price and derive the additional first-order terms.

1. John F. Nash (1950), “The Bargaining Problem,” Econometrica 18, 155–162.Axiomatic bargaining solution.
2. Martin J. Osborne and Ariel Rubinstein (1990), Bargaining and Markets.Agreement, disagreement, and strategic bargaining foundations.
3. Gregory S. Crawford, Robin S. Lee, Michael D. Whinston, and Ali Yurukoglu (2018), “The Welfare Effects of Vertical Integration in Multichannel Television Markets,” Econometrica 86, 891–954.Demand, carriage, and affiliate-fee bargaining.