JOURNAL
OF ECONOMIC
THEORY
52, 232-236 (1990)
A Letter to the Editor on Wage Bargaining*
HANS HALLER
Department of Economics, Virginia Polytechnic Institute and
State University, Blacksburg, Virginia 24061
AND
STEINAR HOLDEN
Department of Economics, University of Oslo, PB 1095 Blindern,
0317 Oslo 3, Norway
Received May 27, 1989; revised November 9, 1989
The Rubinstein perfect information alternating offers bargaining model is
problematic when applied to wage negotiations. A strike or any other industrial
action is not an automatic consequence of a delay in reaching an agreement,
because production can continue normally also when negotiations take place. This
paper extends the Rubinstein model to incorporate the choice of calling a strike,
and it is shown that in this model there is no longer a unique subgame perfect
equilibrium, and that strikes with a length in real time can occur in equilibrium.
Journal of Economic Literature Classification Number: 026.
0 1990 Academic
Press, Inc.
This paper extends the Rubinstein [4] perfect information bargaining
model by adding the feature that in the period that elapses between offers,
one of the players (the union) can take an action (a strike) which affects
* The basic model in this letter was developed independently by the authors; cf. Haller [2 3
and Holden [3]. The present exposition is based on Holden [3], where a more complete
presentation is given, including~ all proofs. We are grateful to Michael Heel, Karl Ove Moene,
Abhinay Muthoo, Ariel Rubinstein, and Asbjorn Rgdseth for helpful discussions and to a
referee for helpful comments on an earlier draft. Part of Haller’s work was done while the
author participated in the Research Project “Game Theory in the Behavioral Sciences” at the
Center for Interdisciplinary Research (ZF), University of Bielefeld. Holden’s work was done
at the Centre for Labour Economics, LSE. His work is part of the research project “Wage
formation and unemployment” at SAF Center for Applied Research at the Department of
Economics, University of Oslo.
232
0022-0531/90 $3.00
Copyright 0 1990 by Academic Press, Inc.
All rights of reproduction in any form reserved.
NON-COOPERATIVE
WAGE
BARGAINING
233
that period’s payoff to both players.’ This feature represents an important
aspect of wage negotiations, namely that a strike or any other industrial
action is not an automatic consequence of a delay in reaching an agreement, because production can continue normally also when negotiations
take place. It is shown that in this model tkere is no longer a unique
subgame perfect equilibrium, and that strikes with a length in real time can
occur in equilibrium.
THE BARGAINING
MODEL
The setup of the model is the following. There is an infinite number of
periods, and in each period of normal production the firm has a value
added of one unit of a good which the firm and the union can
e
between them. The union’s share is WE [O, 11, the firm’s share is Ii- W. If
the union calls a strike, then both parties get zero in that period, but it is
assumed not to affect the value added in later periods. initially the union’s
share is W, r 0 (the wage level in the previous contract). W, is assumed to
be the prevailing division until a new agreement is reached. For simpli
both parties are assumed to have linear utility functions, so their pay
can be represented by the discounted sum of future shares. For the union
this is
where U, = 0 if there is a strike in period t, ~1,= W, if there is no strike and
an agreement has yet to be reached, and u, = W if an agreement is reached.
6 is the discount factor.
For the firm we have correspondingly
where v, = 0 if there is a strike in period t, v, = I - W, if there is no strike
and an agreement has yet to be reached, and v, = 1 - W if an agreement is
reached.
The parties are assumed to make offers alternately, one offer per
and without loss of generality the firm is assumed to make an offer in the
beginning of period 1. The union can then accept or reject this offer. If t
union accepts, the bargaining ends; if it rejects the union will have to
’ Later
and firm
Fernandez
and Glazer
using different
discount
[ 1] have
factors.
analysed
essentially
the same
model-with
union
234
HALLERANDHOLDEN
decide whether to strike in this period or not. If the firm’s offer is rejected
the union makes a new offer in the next period, which the firm accepts or
rejects. If the union’s offer is accepted the game ends; if it is rejected the
union decides whether to strike in this period, and it is the firm’s turn to
make an offer in the following period, etc. Both parties are assumed to
have perfect information.
The structure of the game is illustrated in Fig. 1.
A (subgame) perfect equilibrium of this bargaining game is a pair of
strategies which constitute a Nash equilibrium in every subgame of this
game. Before proceeding to find a perfect equilibrium in this game, it is
convenient to investigate two other games where the union strike decision
is taken as exogenous. One is the original Rubinstein game, where the
union is assumed to strike in every period until an agreement is reached.
The unique perfect equilibrium outcomes W, (W,) (cf. Rubinstein [4] or
Shaked and Sutton [IS]), when the firm (union) makes the first offer, are
W,=6/(1+6),
w,=
l/(1 +6).
(3)
The union can, however, use its strike decision to obtain a larger share
than in the original Rubinstein game. By its. calling a strike only after its
Period:
1
firm:
union:
x
P
game
ends,
(W,,
union:
l-WI)
strike/no
(O,O)
strike
wo, l-W01
strike/no
(OtO)
strike
PO, l-W01
&
union:
propose
firm:
acce
W2
t t/reject
H
game
ends,
(W,,
x
l-W,)
union:
)r
firm:
propose
W,
etc.
FIG. 1. Structure
in parentheses.
of the game.
Per period
payoffs
for the union
and the firm,
respectively,
NON-COOPERATIVE
WAGE
235
BARGAINING
own proposals, and not after the firm’s, the costs of rejecting the union’s
offers will be larger than the costs of rejecting the firm%.
The unique perfect equilibrium outcomes IV’ (IV”) in this game, where
the union is assumed to strike only after rejections of own proposals, an
when the firm (union) makes the first offer, are
lV’=(IV,+6)/(1+6),
WU=(l
+SWo)/(l
+6).
(43
This can be proved by the Shaked and Sutton [S] method. This method
can also be used to prove that this is the best the union can achieve by any
strike strategy.
We now return to the more general bargaining model, where the union
strike decision is endogenous.
PROPOSITION
1. For all W* E [ W,,, W’],
W* is a perfect equilibrium
partition.”
Proof (Sketch). The equilibrium path is an immediate proposal and
acceptance of W*. These actions are supported by the convention that a
deviation will be punished by one of the extreme equilibria below.
No-strike equilibrium (union is forced down to Wo).
Firm: offer W,, reject all W > W,, and accept a
Union:
offer W,, reject all W < W,, accept all
strike.
Disrupted strike equilibrium (firm is forced down to I - Wf).
Firm: offer W”, reject all W> W”, and accept all W< W”.
Unisn:
offer W”, reject all W < Wf, accept all W 2 W’, strike in all
even-numbered periods, and never strike in all odd-numbered periods, until
agreement is reached.
0th players: switch to no-strike equilibrium if the union fails to
strike in an even-numbered period without agreement.
Remarks. Wage levels below W, or above Wf cannot be supported as
perfect equilibria, as the union can never be forced below Wo, nor can the
firm be forced above Wf.
If disrupted strikes are not possible, the union cannot get a higher wage
level than W, in any perfect equilibrium (this can be proved by the Shaked
and Sutton method). In this case one couId define a strike equi~~b~~~~~
(where the union strikes in every period until an agreement is rea
which can be substituted for the disrupted strike equilibrium in pr
tion 1, and the only change would be that the range of wage levels which
be supported as perfect equilibria would shri
’ We assume that W, -=c6Wf, otherwise it will never be optimal for the union to str&c after
the firm has rejected an offer, and W’, is the unique perfect equilibrium.
236
HALLER AND HOLDEN
PROPOSITION 2. For any integer N > 0 and all W* E [ W,/?iN, 1 (1 - W’)/S”], there is a discountfactor 6 < 1 such that a perfect equilibrium
exists where the union strikes in N periods, whereupon an agreement is
reached on W*. These strikes can have a length in real time, also when the
length of each bargaining goes to zero. (Proox see Holden [3]).
In such an equilibrium, as long as there is a strike both players propose
an outcome which is very favourable to themselves (the union proposes
W”, the firm proposes W,). Thus both players prefer a continuation of the
strike to an agreement on the other player’s offer. If one of the players
deviates from this path, for example by suggesting an early compromise,
then this player is punished by a switch to one of the extreme equilibria in
the proof of Proposition 1, which is worse for him than the one with the
strike. The strike can have a significant length also when the length of each
bargaining period goes to zero, as the length of the strike can be bounded
away from zero by increasing the number of periods of strike at the same
rate.
REFERENCES
1. R. FERNANDEZAND J. GLAZER, “Striking for a Bargain between Two Completely Informed
Agents,” Working Paper, Boston University, 1989.
2. H. HALLER, “Wage Bargaining as a Strategic Game,” Virginia Polytechnic Institute and
State University, Working Paper E-88-09-02, 1988.
3. S. HOLDEN, “Non-cooperative Wage Bargaining,” Centre for Labour Economics, LSE,
Discussion Paper 349, 1989.
4. A. RUBINSTEIN,Perfect equilibrium in a bargaining model, Econometrica
50 (1982), 97-109.
5. A. SHAKED AND J. SUTTON, Involuntary unemployment as a perfect equilibrium in a
bargaining model, Econometrica
52 (1984), 1351-1364.