BARGAINING Jess BENHABIB

24 (1987) 107-111

107

AND THE EVOLUTION and Giovanni

OF COOPERATION

IN A DYNAMIC

GAME *

FERRI

New York University, New York, NY 10003, USA

Received Accepted

4 March 1987 18 May 1987

An example of a dynamic game between a union and a firm is constructed in which an equilibrium non-cooperative strategies and switches to cooperative strategies as the level of employment increases.

path

starts

out with

It is well known that in repeated games it may be possible to sustain a cooperative equilibrium with the discount rate sufficiently small, if players adopt strategies that threaten to revert to non-cooperative actions when their opponents deviate from cooperative behavior. A similar situation arises in dynamic games where a cooperative solution can be maintained as a subgame perfect equilibrium under threats to revert to other credible strategies, such as, for example, stationary Markov (feedback) strategies. However, in dynamic games, whether a cooperative solution can or cannot be sustained depends not only on the discount rate, but also on the value of the state variable at the particular point in the game. Since the state variable evolves with each play, it is possible to have an equilibrium in which agents play their stationary (feedback) strategies below a critical level of the state variable where a cooperative solution cannot be sustained. Then, after crossing the critical level, they can switch to cooperative strategies which are supported by threats of reverting to stationary (feedback) strategies. It is also possible to have situations where cooperative equilibria can be sustained only for initial conditions above a critical value of the state variable, and where initial conditions below that critical value result in the use of non-cooperative stationary strategies that lead to a non-cooperative steady state. Recently some authors explored the enforceability of cooperative solutions in dynamic games. Cave (1984) has developed and extended the early work of Levhari and Mirman (1980) on the great fish war. Oudiz and Sachs (1986) have discussed related issues with respect to macroeconomic and international policy coordination. Nevertheless, the relation between the state of the system and the enforceability of a cooperative solution or the possibility of the eventual (together with the impossibility of the immediate) emergence of cooperation along an equilibrium path are issues that have not yet been explored in detail. ’ Below, we give examples of dynamic games in which the situations described above will arise. * The authors thank Rabah Amir, Andy Schotter, Roy Radner and Chuck Wilson for numerous valuable conversations. support of the C.V. Starr Center for Applied Economics is gratefully acknowledged. 1 Note that switching from cooperative to non-cooperative strategies without ever returning to cooperative ones cannot along an equilibrium trajectory because players will have the incentive to deviate earlier and therefore immediately.

016%1765/87/$3.50

0 1987, Elsevier Science Publishers

B.V. (North-Holland)

The occur

108

J. Benhabib,

G. Ferri / Bargaining,

euolutron

of cooperation in a dynamic game

We consider a simple bargaining game between a labor union and a firm. The firm is subject to adjustment costs in changing its labor force and has a linear production technology. Its profits are given by

for I_, given, where al, is the output at time t, I, is employment, w, is the wage, b(l, - IT_,)~ is the adjustment cost and /I is the discount factor. The employment level is confined to the interval [0, i], where I is the maximum labor available. The union would like to choose wages to maximize its discounted stream of wage bills, given by cc

CP’WA

(2)

0

The structure of bargaining is such that in every period, the union first announces a wage and then the firm announces an employment level. We first consider an equilibrium with stationary strategies. The union chooses a policy function IV: [0, i] -+ [0, co) which selects a wage w as a function of the amount of employment at t - 1: wt = W( l,_ 1). The firm chooses a policy function I, = L( wt, ltpl) where L: [0, co) x [0, i] + [0, i). A pair of functions (IV, L) is an equilibrium if W maximizes (2) given L, and L maximizes (1) given W. Given values of specific parameters, a subgame perfect equilibrium can be obtained using dynamic programming methods. The first-order conditions associated with the firm’s and union’s optimization problems (after eliminating the value functions for the firm and the union) are

a - w, - 2b(l, - 1,-l) = P[2b(l,+, - 1,)- I,+, . dW(l,),‘dl,]> I,+ w;dL(wt,

k,)/dw,=

P(dL(w,,

(3)

k&dwt)

x (W(wt+17l,)/‘dl,)/(dLb,+,> l,)/‘W+,))l,+,.

(4)

Given

a, b, p and i, the parameters of the optimal linear policy function of the form l,,, = k, + kll, using the undetermined coefficients method. (Of + k2wr+1 and wt+l = c1 + c21r can be computed course, concavity restrictions and transversality conditions can also be shown to hold for the union’s and the firm’s optimization problems.) We will denote the value functions of the firm and the union by F”(1,) and U’(l,,). Both U”(1,) and F”(1,) are quadratic functions, satisfying the dynamic programming equations below:

F”( 1,) = b, + b,l, + b,l,2 = , z?i, 1 lJs(lo)

= a0 + a,l, + a,1,2 =

w (1,) . 1, - b( 1, - 1,)2 + j3FS(l,),

(5)

9

max w,L(w,, W,E[O1m)

1,) +Pu”(L(w,,

Substituting the policy functions into (5) and (6) and equating value functions can then be computed.

lo)).

coefficients,

(6)

the parameters

of the

‘{I-*‘“’ “03 (00 ‘01 3’M

‘[j

‘01 s!II

‘Z‘~=.rpue~-r‘...‘~‘O=~ ‘1~ ‘I-’ I ‘...ct,

(I-‘M““‘Z&,

s”r)}

=‘H

la? ‘sa!8alr?.1ls d~?uo~~~ls yl~~ wnuqglnba uv 01 1.1a~a.101 SlsaIyl a~-[oau! siua8v ayi 30 sa$!tale.~~s aye 31 pauysns aq uv3 uognIos aayxadoo3 B JaylayM Lpnls MOU uw a~ -(e fOl),R Pue (8 f01)3d icq uo!un ayl puv wn3 ayi 01 an133e ieyi + yp~ paiyDosse uoynIos agxadoog v 30 sanp2A ayl aiouap 111~ ah xu.u3 ayl pue uorun ayl uaaMiaq [email protected] IE?IO~ayi [email protected] y3y~ uoyyos agxadoo3 B II! s1Insa.I uayi [& ‘01 3 8 a&M iuwsuo:, e 30 aD!oy:, v ‘1 li;.Iepunoq ayl $“I shop 111~ [email protected] 1uawiCoIduIa ayl UoyI~os aAy?_Iadoo3 he lo3 leyl aloN z .piooy suog~puo~ lClgeslaasueJ1 leyl aJnsse 01 suoprpuo~ Sysoduy Icq (8) 30 uopn~os ayl 01 Shupuodsallo~ suoy~puo:, laplo-wu3 ayy u10.13 pauyqo si uoptyos sg~ .( d - 1)42/o = z a.cayM ‘as;rM_~ayio

(6)

!=

Aq [email protected] s! (8) 30 aprs [email protected] ayl sazp?vw ley~ holDa[wl v_rawdoIdwa pxugdo ayL ‘[email protected] a&M luwuo:, e qir~ pauywe aq uv3 (Ol)s [email protected] ayl 30 uoyle~o~p~? pal3!3adsald he axIts sa~oi~a[t?Ji a&M luelsuo:, glen suqynba aAy?.Iadoo~ 01 uoguall~ mo l~,nlsa.I aM TIoyI ayl puk? u1x3 ayl uaamlaq [email protected] aql 30 UO~~I?~O~~E? leIrwy.Icd E ut s1Insa.I uayl h0l~afe.U a8vM .wIngp.wd v ‘(8) SaAlos ~3g~ auo ayl SF ]uawdoIdtua 103 aDuanbas IEurydo ayl pun sauy 111~le alvuruIla]apur s! awl aSeM ayl wyl suoy~puo3 laplo-lsq3 woq y3ayD 01 dwa st if [email protected] pxnba yltfi XIo~n 10 ~1x3 ayl 01 [email protected] algua ayl alEDoIIB ~3g~ pue Suo~lnIos lauI103 alelaua8 $~:y] [email protected] [enbaun p!o~r! aM ‘L$!xIdcu~s 10~ urnwyw

ayl so (Ol)s

pug (8) 30 aprs [email protected]

(8)

ayl saz?FeuI

‘(?)sd

ieyi

1~ 30 anlEA ayl sr II” alayM

[! ‘013 ‘I + z(“l - ‘I)4 - ‘I” xwu = (9)s

= t]iM

a&M ayi aq ue3 IO Lv2.wq.w aq ue3 E alayM ‘I 11~ 103 [_M‘01 01 ‘M a%M ayi pgsaJ aM a3uayuaAuoD 10~ ‘[email protected] ,@x2ay alow so YDFYMlua% ayi 01 [email protected] ayi 30 11e saleDoIIr! y3q~ uopnIos I? 01 spx?aI ‘uraIqold ayl uy @eauy[ ayl [email protected] ‘[email protected]~M IBnbaun %.I~u~!ss~ -‘I put? ‘M 30 saDuanbas 01 IDadsal YJ~M (z) pue (I) 30 tuns [email protected] e %~z~u_~yxeur 6q pauyqo ST UognIos aApwadooD v

uail?

aq 111~ sa$aleqs

LILWO~~WSIJI~M cunpqglnba

sql -1031uawhoIdwa

30 wyureuhp

ayl wyl mN

J. Benhabib, G. Ferri / Bargaining, evolution of cooperation in a dynamic game

110

I, = L,(h,>

if

= I,_1 + z

wt>

= k, + k,l,_,

+ k2w,

w,=G

forallist,

otherwise.

Let the strategy of the union at time t be given by the function W,: H, + [0, co) such that w,= W(h,)

=D

if

1,=I,_,+z

= ci + $1,-i

otherwise.

foralliSt--1.

Under these strategies, which punish the opponent for deviating from cooperative behavior by reverting forever to an equilibrium with stationary (feedback) policies, the firm and the union must compute whether it pays to deviate. The value to the firm from deviating is given by

while the value to the union from deviating, since it moves first, is simply the value of the equilibrium associated with the stationary strategies, that is, uD(l,_,) = U”(l,_,). Therefore, to be enforced, a cooperative equilibrium starting from I, and described by (9) must have U’(Z, G) 2 uD(Z) and F”( I,G) 2 FD(I) for all 1 along its path. However, the possibility that a cooperative equilibrium is only enforceable for some interval (I*, i) which contains [ the steady state of the equilibrium associated with stationary strategies, raises the possibility of an equilibrium sequence where agents use stationary (feedback) strategies up to I * and then switch to cooperative strategies. For such a possibility, f must be stable under stationary strategies. If a critical I * exists, that is, if uD(I) 5 U”(I, S) and FD(I) I F”(1, G) for some G and any ZE [I*, i], but not for I E [0, 1*) and any 6 E [0, W], we can consider the following equilibrium pairs of strategies for the firm and the union. Let the strategy of the firm at t be given by the function t,: H, x [0, cc) + [0, i] such that

Z,=i,(h,, Wr)

if

=ltplfz

I,_, E [I*,

i]

whenever I,_, E [I*, = k, + k,f,_,

+ k2w,

if

= G

1,-i E [I*,

i]

and

whenever I,_, E [[*, = Cl + C21,-*

w,=C

i] for i < t,

otherwise.

Let the strategy of the union at t be given by the function q: w,= *t(h,)

and

H, -+ [0, 00) such that

l,=l,_,+z

i] for i < t - 1,

otherwise.

We now give parameter sequence of strategy pairs until employment reaches For a = 1.5, b = 1, j3 =

values a, b, /3 and i which generate a subgame perfect equilibrium for the (R, i,), t = 0,. . . with the property that stationary strategies are used 1* and cooperative strategies are played from then on. 0.2 and i = 0.75, the equilibrium stationary strategies are given by

I, = 0.393231 + 0.451301,_i,

(10)

w, = 0.67695 +0.959261,-i.

(II)

The non-cooperative strategies are:

steady state is r= 0.7150.

The value functions

corresponding

to stationary

J. Benhabib,

G. Ferri / Bargaining,

F”( 1,-i)

= 0.22183 + 0.4082961,_1

U’(I,pi)

= 0.426296 + 0.7846301,_1

For the cooperative

solution,

evolution

- 0.7651161;_,

111

of cooperation in a dynamic game

,

- 0.4513041;_,.

the path of employment

is given by

(12)

lI = I,_, + 0.9375.

If we set the constant cooperative wage at S = 1.362332, the critical value of I is given by I * = 0.67. F” - FD -c 0 and the cooperative equilibrium cannot be supported. For I> I*, For ICI*, F’ - FD > 0 and UC - UD > 0 so that a cooperative equilibrium can be supported. For f0 = 0.64, the firm and the union will at first play their stationary strategies given by (10) and (11). In the second period, I, = 0.681 > I* so that the firm and union will switch forever to cooperative strategies, given by (12). Of course, as the discount rate is increased (discount factor decreased), the range of cooperative wages G that are sustainable shrinks. For sufficiently high discounting there does not exist a 6 and a corresponding cooperative equilibrium that can be sustained from any value of 1. It is also possible to construct examples where the steady state under stationary strategies, 17 is stable but is below I*, Then for f0 E [0, I *), the equilibrium strategies lead to convergence to r but for 1, E [I*, i], a cooperative solution can be implemented as an equilibrium for the industry. Therefore, whether efficient allocation and production can or cannot be sustained depends critically on initial conditions. If we slightly alter the previous example so that i = 1 and G = 1.45, we obtain r= 0.715 < 0.87377 = I*. For 1, -C I*, F” < F D and a cooperative equilibrium cannot be sustained. equilibrium is Under stationary strategies employment converges to L For [,, 2 I*, the cooperative implemented and employment converges to 1.

References Cave, J., 1984, The cold fish war: Long-term competition in a dynamic game, Mimeo. Levhari, D. and L. Mirman, 1980, The great fish war: An example using a dynamic Coumot-Nash solution, Economics, 322-344. Oudiz, G. and J. Sachs, 1986, International policy coordination in dynamic macroeconomic models, NBER series no. 1417.

Bell Journal working

of

paper