Nash Bargaining

1. Introduction
A Nash bargaining is a form of cooperative game, which analyses the choice of a particular point
on the Pareto efficiency frontier (Harsanyi).
Consider the case of two individuals: i = 1, 2. Let
 ui denote the utility level of the i-th individual
 ci denote the utility obtained by the i-th individual in case of conflict (ci is often called the
"threat point").
2. Derivations
Make the following assumptions:
1/ individual rationality: ui  ci
2/ joint efficiency: Pareto optimality: h(u1, u2) = 0 as a representation of the Pareto efficiency
3/ symmetry: If the bargaining game is symmetric (i.e. if h(u1, u2) = 0 and (c1, c2) are symmetric
with respect to the line u1 = u2), then the solution of the bargaining process is on the line u1 = u2.
4/ ui represents a cardinal utility function, i.e. a function defined up to a positive linear
transformation. An example is the utility function when decision makers maximize expected utility under
5/ independence of irrelevant alternatives: The bargaining solution is invariant to changing
irrelevant restrictions on the feasible space.
Proposition: Under assumptions 1-5, the bargaining game always has a unique solution (u1*, u2*) given by:
Max(u1,u2) {(u1-c1)(u2-c2), subject to h(u1, u2) = 0, ui  ci, i = 1, 2}.
ui*/ci  0,
ui*/cj  0 for i  j.
Note: These results generalize to n agents, where the solution of the Nash bargaining game is given by:
Maxu {i (ui-ci), subject to h(u1, ..., un) = 0, ui  ci, i = 1, ..., n}.
3. Zeuthen's bargaining scheme
Consider a bargaining process where, at each stage k of the process, player 1 has proposed an
agreement A1k to player 2, while player 2 has proposed another agreement A2k to player 1. If they fail to
agree, then a conflict situation C will develop and the players will receive the conflict payoffs ui(C) = ci, i =
1, 2.
Assume that ui(C) < ui(Ajk) < ui(Aik) , i = 1, 2, i  j.
At stage k, the i-th player can:
 either accept his opponent's offer, thus yielding an agreement.
 or make a concession by proposing Aik+1, such that
uj(Aik) < uj(Aik+1) < uj(Ajk).
 or repeat his last offer, possibly leading to a conflict.
Who is going to make a concession? Let pk be the probability of a conflict at stage k of the bargaining
process. Consider:
pk ui(C) + (1-pk) ui(Aik) = ui(Ajk).
The left hand side is the expected utility obtained by the i-th agent when the probability of conflict is pk.
The right hand side is the utility obtained by the i-th agent in the case of an agreement. Solving the above
equation for pk gives pik = the probability limit of conflict = the highest probability of conflict the i-th
individual is willing to face before he agrees.
The probability limit of conflict for the i-th individual is:
pik = [ui(Aik) - ui(Ajk)]/[ui(Aik) - ui(C)], i = 1, 2.
If pik < pjk, then the i-th individual is less willing than the j-th individual to risk a conflict.
Zeuthen's principle:
 if p2k < p1k, then player 2 makes a concession.
 if p1k < p2k, then player 1 makes a concession.
 if p1k = p2k, then both players make a concessions.
4. Equivalence of Nash's and Zeuthen's theories
Proposition: If the two players follow Zeuthen's principle during the bargaining process, they will
eventually reach an agreement corresponding to the Nash solution.
Proof: Assume that ui(Ai) > ui(C)  ci, i = 1, 2.
According to Zeuthen's principle, the i-th player makes a concession if:
pi = [ui(Ai) - ui(Aj)]/[ui(Ai) - ui(C)]  pj = [uj(Aj) - uj(Ai)]/[uj(Aj) - uj(C)]
[ui(Ai) - ui(Aj)][uj(Aj) - uj(C)]  [uj(Aj) - uj(Ai)][ui(Ai) - ui(C)]
[ui(Ai) - ui(C) + ui(C) - ui(Aj)][uj(Aj) - uj(C)]  [uj(Aj) - uj(C) + uj(C) - uj(Ai)][ui(Ai) - ui(C)]
[ui(C) - ui(Aj)][uj(Aj) - uj(C)]  [uj(C) - uj(Ai)][ui(Ai) - ui(C)]
[ui(Ai) - ui(C)][uj(Ai) - uj(C)]  [ui(Aj) - ui(C)][uj(Aj) - uj(C)].
It follows that the i-th player makes a concession whenever the Nash product associated with his offer Ai is
smaller than the Nash product associated with his opponent offer Aj. Thus, at each stage of the bargaining
process, the concessions made increase the Nash product. This continues until the Nash product is
maximized, at which point a final agreement is reached.
5. Applications
5.1. Public R&D
Consider a market with price-dependent aggregate supply function ps(Q,T) and aggregate demand
function pd(Q), where Q denotes aggregate quantity and T is a technology index. Assume that ps/Q > 0,
ps/T < 0 and pd/Q < 0. Let the technology index T be positively influenced by public research and
development (public R&D). Denote the cost of R&D by C(T), where C(0) = 0 (corresponding to "no R&D
investment": T = 0) and C/T > 0.
producer surplus (PS)
consumer surplus (CS)
no R&D (T=0)
R&D (T>0)
Change due to R&D
Neglecting the cost of R&D for the moment, it follows from the above table that public R&D investments
benefit consumers and generate a positive net surplus (as measured by PS+CS). However, as the result of
technical progress, the producers can:
. either gain (if +F+G > B)
. or loose (if +F+G < B, e.g. under an inelastic demand).
It follows that technical progress created by public R&D policy:
. can generate a net welfare gain (as long as the benefits just identified (PS+CS) dominate the
R&D costs C(T)).
. can redistribute income from the producers to consumers.
. is Pareto optimal.
. is a Pareto improvement if appropriate compensations are paid to the loosers.
. can imply a more unequal welfare distribution if such income compensations are not paid.
5.2. Example: (Homework #6)
Consider the case where direct income compensations are not possible (e.g. because of the
difficulty in identifying precisely the target groups). A Nash bargaining game can provide a basis for
obtaining a constrained Pareto optimal allocation of resources (the constraint being the infeasibility of
direct income transfers).
Consider the situation represented in the graph where:
p1 = price paid by consumers
p2 = price received by producers (="target price")
p2 - p1 = price subsidy paid by the taxpayers (= direct payment per unit of output)
Q2 = quantity produced and consumed.
In this context, the problem is: how to choose Q2, p1 , p2 and T?
Competitive market: The competitive market solution can be obtained by maximizing net social
MaxQ,T {  p (q) dq   p (q, T) dq  C(T)}
producer surplus (PS) + consumer surplus (CS) - cost of R&D.
The solution to this optimization problem gives: Q* = Q1 (the optimal quantity produced and consumed)
and T* (the optimal technology level generated by public R&D policy). The corresponding optimal price is
p* = ps(Q*, T*) = pd(Q*). This is a Pareto optimal allocation.
Nash bargaining: Assume that direct income transfers are not possible. The Nash bargaining solution
then is obtained by maximizing the Nash product:
Max Q,T {PS(Q, T)  PS( Q0 ,0)][CS(Q)  C(T)  (p (Q, T)  p (Q)) Q  CS( Q0)}
where PS(Q, T) = producer surplus = ps (Q, T) Q   ps (q, T) dq
CS(Q) = consumer surplus =  pd (q) dq  p (Q) Q
PS(Q0,0) is the "threat point" for producers (where Q0 is the competitive solution obtained in the
absence of R&D policy, corresponding to T = 0)
CS(Q0) is the "threat point" for consumers (again evaluated at Q0)
C(T) is the cost of R&D policy, paid by the consumers-tax payers.
(ps-pd)Q is the total subsidy paid to producers by the consumers-taxpayers.
The solution of this optimization problem generates the Nash bargaining solution denoted by Qb, Tb. The
associated pricing scheme is given by: p2b = ps(Qb, Tb) and p1b = pd(Qb). In general, Qb  Q*, Tb  T*, and
p2b p1b  p*. Since Nash bargaining generates a constrained Pareto optimal solution, it follows that a price
subsidy (= p2b - p1b > 0) can be constrained Pareto optimal. The reason is that, in the absence of direct
income transfers, a price subsidy paid by consumers-tax payers is a means for consumers to share the
benefits of technical progress (generated by R&D policy) with the producers.
Note: In this case, public R&D is treated as a public good. This is an example of a more general situation
where, in the presence of public goods, there can be motivations to "distort" competitive market
equilibrium in an attempt to redistribute the "unequal benefits" generated by these public goods.