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Sustainable solutions for dynamic bargaining problems
Economics Letters North-Holland
SUSTAINABLE PROBLEMS Chaim
SOLUTIONS
FOR DYNAMIC
BARGAINING
FERSHTMAN
Northwestern Received
147
13 (1983) 147-151
University. Evunston.
21 March
IL 60201, USA
1983
This paper investigates the dynamic bargaining problem in which players have to agree on a time path of actions. The axiomatic approach for such problems may yield a solution which is not sustainable. For every given partition of the time interval however, we can construct a sustainable solution.
1. Introduction
By refering to the Bargaining Problem the economic literature means a situation in which a group of individuals seek to reach an agreement on a partition of a pie. Following Nash (1950) the two players bargaining problem is described using only two components (S, d) when d is a point in the plane which can be interpreted as the outcome when no agreement is reached and S is a compact convex subset of the plane which contains d and describes the set of all feasible utility payoffs that can be achieved by cooperation. Consequently two different bargaining problems that can be characterized by the same (S, d) will yield the same solution [see for example Nash (1950), Kalai and Smorodinsky (1975), and Kalai (1977)]. One approach to solve the bargaining problem is the axiomatic approach [for survey see Roth (1979)] in which the bargaining process is not specified and the assumptions are directly made on the solution itself. A different way of discussing the bargaining problem is the strategic approach in which the bargaining process itself is discussed [see Rubinstein (1982)]. In daily bargaining we deal quite often with a situation in which the 0165-1765/83/$3.00
0 1983, Elsevier Science Publishers
B.V. (North-Holland)
C. Fershtman
148
/ Solutionsfor +numic bargainingproblems
players have to agree on a path of actions which takes place over time. (Players in this case can also receive their payoff along time.) As an example we can think of a firm in which activities are controlled by a group of individuals each wanting to maximize a different objective function (profit, sales, growth and so on). The firm has to decide on a production path for the next several periods. Since a production path that maximizes simultaneously profits sales and growth is not likely to exist, the path will be decided by bargaining between the executives. Given the objective functions of the executives and the set of all possible production paths, the problem can again be characterized by the two components (S, d). We denote the one stage bargaining problem (partitions of a pie) as the static problem and the problem discussed above (bargaining on a path) as the dynamic bargaining problem. As we show in this work the dynamic bargaining problem cannot be characterized just by two components. Two different problems which can be described by the same pair (S, d) may have different solutions even in the axiomatic approach.
2. Formulation
of the dynamic bargaining problem
Suppose that a group of players denoted by i = 1.. , n have to choose a control function u(r) from a set U of admissible control functions on [0, T]. Let x(t) E R, be a vector of state variables such that j;(t)
= dx,‘dt
=f(x(
The payoff of the ith players q(x,,u(t))
x(0) = xg.
t),u( t)),
is according
=ire~“R,(x(t),
(1)
to i=
u(t))dr,
l,...,n.
(2)
where x(t) is defined by (I), r denotes the discount rate and g, E C2 are all non-negative. Assuming that x0 is given, we can define for every the payoff vector y(x(,,u( I)) = possible control function u(t), (w,(X”,U(t)),. . .3 W,(+nu(t)>. Thus the set of all feasible outcomes S is
s = CYh
u(t))
which is assumed
E
R"lx(0)
=
x,,u( t)
to be a compact
E
U}
convex subset of R”. Let uN( t) denote
C. Fershtman
/ Solutions for dynamic bargaintng problems
149
the control function when no agreement is reached, then the threat point d can be written as d = y( x0, u N( t )). An utopia is a control function C(t) which simultaneously maximizes all the functionals q. If C(t) exists it will clearly be the solution of the bargaining problem. In general, it is very rare to find a control function which optimizes several functionals simultaneously. Adopting the axiomatic approach, a solution is a function p: B + R” such that B denotes the class of all n-players bargaining problems and r_l(S, d) is an element in S that has to satisfy some given axioms. Thus for the above bargaining problem we can find a solution y* = p( S, d). (Different sets of axioms will yield of course different solutions.) Let us define a function I#J: S + L?(U) such that for everyy E S, G(J) define all the control functions in U, which yield the payoff y. Thus for the solution _y* = p(S, d) of the bargaining problem, the players are free to choose any u*(t) E $( y*). The main problem that distinguishes the dynamic bargaining problem is whether u*(t) is sustainable or not. At every time 7
150
C. Fershtman
/ Solutions for dynamrc bargaining problems
= (~(x(T), u,(t))~ R”[u,(~)E UT} be the set of all feasible expected payoff from time T until the terminal time T. We assume that for every r and X(T), S( T, x( 7)) is a compact convex set. Similarily let d( 7, x( 7)) be the threat point at time T; a sustainable path with respect to the solution p is a control function C(t) and its induced state variables path _C(t) such that for every T E [0, T], ii,(t) E $[p(S( 7, iZ(T)), d(-r, a( T)))]. A sustainable path for the non-myopic case is discussed in the next section.
3. The axiomatic approach in the dynamic bargaining problem For simplicity lets assume that the opportunities to break the original agreement and to start a new bargaining are restricted to times t,; j= 1, . . , , k, where (2,) is strictly increasing t, = 0, t, < T. Alternatively we can think about the discrete time version of this problem. Let p be a solution function according to the axiomatic approach. For every level of the state variables at t, we can define (as in the previous section) the pair (S(t x), d( t,, x)) that describe the set of all feasible expected payoff from time t, until the terminal time T and the threat point at t, respectively, when x(f,) = x. Since t, is the last opportunity to discuss the path, the bargaining problem at time t, can be treated as a static problem. Let pk be a function from the state variables space to R” such that
Since s(t,, x) is we are now able t,_, denoted by let Y[x,~_ ,, u(t)]
a compact convex set, ph( ) is well defined. Knowing ph to define p k-’ . For every level of state variables at time x,~_ I and for every admissible control u(t) on [th_ ,, tA] be a n tuple vector, such that
~,(x,,_,~dt)) = U/;(X,~ ,.u> +P~(x,~), XfA= x,i-l + / ” f(x(t),u(t))dt, II I S(~h_,,~,,_,)={y(~,,
,,u(t))~R”Ju(t)
when and let
isadmissiblecontrolon
C. Fershtman
Since S(t,_,, defined as
x,,_,>
is
/ Solutions a
compact
for dynamzc bargaining problems convex
subset
of R”, p”-’
151
can
be
From the way pkp’ is defined it is clear that there is a sustainable path u(t) for t E [t,_,, T]. In the same way for every j = 1,. , k - 1 we can define the function k-J. Since x0 is given, p’(x,) defines a solution for the dynamic P bargaining problem which is sustainable with respect to p and the partition of the time interval {t,, . . . , tk). Thus, although the dynamic bargaining problem cannot be characterized just by two components as the static problem. We can adopt the axiomatic approach and find a solution which will be sustainable for a given partition of the interval [0, T]. For every partition of the interval we will of course get a different solution. Finally, the solution we have investigated in this paper is based on the axiomatic approach. Before one tries to implement the strategic approach, it is important to note that the strategy space in the dynamic problem is much larger than in the static problem. Players can suggest different partitions of the time interval, or different punishment schemes. They can even agree just on a partial solution [a path u(t) for t E [0, T,]; T, -C T] and while the partial solution is executed they can continue the negotiation on the rest of the solution. Using this kind of strategy, players can avoid the problem of shrinking pie which is discussed by Rubinstein (1982).
References Kalai, E., 1977, Proportional solutions to bargaining situations: Interpersonal utility comparison, Econometrica 45, 1623- 1630. Kalai, E. and M. Smorodinsky, 1975, Other solutions to Nash’s bargaining problem, Econometrica 43, 513-518. Nash, John, F., 1950, The bargaining problem, Econometrica 28, 155-162. Roth, A.E., 1979, Axiomatir models of bargaining, Lecture Notes in Economics and Mathematical Systems no. 170 (Springer, Berlin). Rubinstein, Ariel, 1982, Perfect equilibrium in a bargaining model, Econometrica 50, no. 1 97- 109.
SUSTAINABLE PROBLEMS Chaim
SOLUTIONS
FOR DYNAMIC
BARGAINING
FERSHTMAN
Northwestern Received
147
13 (1983) 147-151
University. Evunston.
21 March
IL 60201, USA
1983
This paper investigates the dynamic bargaining problem in which players have to agree on a time path of actions. The axiomatic approach for such problems may yield a solution which is not sustainable. For every given partition of the time interval however, we can construct a sustainable solution.
1. Introduction
By refering to the Bargaining Problem the economic literature means a situation in which a group of individuals seek to reach an agreement on a partition of a pie. Following Nash (1950) the two players bargaining problem is described using only two components (S, d) when d is a point in the plane which can be interpreted as the outcome when no agreement is reached and S is a compact convex subset of the plane which contains d and describes the set of all feasible utility payoffs that can be achieved by cooperation. Consequently two different bargaining problems that can be characterized by the same (S, d) will yield the same solution [see for example Nash (1950), Kalai and Smorodinsky (1975), and Kalai (1977)]. One approach to solve the bargaining problem is the axiomatic approach [for survey see Roth (1979)] in which the bargaining process is not specified and the assumptions are directly made on the solution itself. A different way of discussing the bargaining problem is the strategic approach in which the bargaining process itself is discussed [see Rubinstein (1982)]. In daily bargaining we deal quite often with a situation in which the 0165-1765/83/$3.00
0 1983, Elsevier Science Publishers
B.V. (North-Holland)
C. Fershtman
148
/ Solutionsfor +numic bargainingproblems
players have to agree on a path of actions which takes place over time. (Players in this case can also receive their payoff along time.) As an example we can think of a firm in which activities are controlled by a group of individuals each wanting to maximize a different objective function (profit, sales, growth and so on). The firm has to decide on a production path for the next several periods. Since a production path that maximizes simultaneously profits sales and growth is not likely to exist, the path will be decided by bargaining between the executives. Given the objective functions of the executives and the set of all possible production paths, the problem can again be characterized by the two components (S, d). We denote the one stage bargaining problem (partitions of a pie) as the static problem and the problem discussed above (bargaining on a path) as the dynamic bargaining problem. As we show in this work the dynamic bargaining problem cannot be characterized just by two components. Two different problems which can be described by the same pair (S, d) may have different solutions even in the axiomatic approach.
2. Formulation
of the dynamic bargaining problem
Suppose that a group of players denoted by i = 1.. , n have to choose a control function u(r) from a set U of admissible control functions on [0, T]. Let x(t) E R, be a vector of state variables such that j;(t)
= dx,‘dt
=f(x(
The payoff of the ith players q(x,,u(t))
x(0) = xg.
t),u( t)),
is according
=ire~“R,(x(t),
(1)
to i=
u(t))dr,
l,...,n.
(2)
where x(t) is defined by (I), r denotes the discount rate and g, E C2 are all non-negative. Assuming that x0 is given, we can define for every the payoff vector y(x(,,u( I)) = possible control function u(t), (w,(X”,U(t)),. . .3 W,(+nu(t)>. Thus the set of all feasible outcomes S is
s = CYh
u(t))
which is assumed
E
R"lx(0)
=
x,,u( t)
to be a compact
E
U}
convex subset of R”. Let uN( t) denote
C. Fershtman
/ Solutions for dynamic bargaintng problems
149
the control function when no agreement is reached, then the threat point d can be written as d = y( x0, u N( t )). An utopia is a control function C(t) which simultaneously maximizes all the functionals q. If C(t) exists it will clearly be the solution of the bargaining problem. In general, it is very rare to find a control function which optimizes several functionals simultaneously. Adopting the axiomatic approach, a solution is a function p: B + R” such that B denotes the class of all n-players bargaining problems and r_l(S, d) is an element in S that has to satisfy some given axioms. Thus for the above bargaining problem we can find a solution y* = p( S, d). (Different sets of axioms will yield of course different solutions.) Let us define a function I#J: S + L?(U) such that for everyy E S, G(J) define all the control functions in U, which yield the payoff y. Thus for the solution _y* = p(S, d) of the bargaining problem, the players are free to choose any u*(t) E $( y*). The main problem that distinguishes the dynamic bargaining problem is whether u*(t) is sustainable or not. At every time 7
150
C. Fershtman
/ Solutions for dynamrc bargaining problems
= (~(x(T), u,(t))~ R”[u,(~)E UT} be the set of all feasible expected payoff from time T until the terminal time T. We assume that for every r and X(T), S( T, x( 7)) is a compact convex set. Similarily let d( 7, x( 7)) be the threat point at time T; a sustainable path with respect to the solution p is a control function C(t) and its induced state variables path _C(t) such that for every T E [0, T], ii,(t) E $[p(S( 7, iZ(T)), d(-r, a( T)))]. A sustainable path for the non-myopic case is discussed in the next section.
3. The axiomatic approach in the dynamic bargaining problem For simplicity lets assume that the opportunities to break the original agreement and to start a new bargaining are restricted to times t,; j= 1, . . , , k, where (2,) is strictly increasing t, = 0, t, < T. Alternatively we can think about the discrete time version of this problem. Let p be a solution function according to the axiomatic approach. For every level of the state variables at t, we can define (as in the previous section) the pair (S(t x), d( t,, x)) that describe the set of all feasible expected payoff from time t, until the terminal time T and the threat point at t, respectively, when x(f,) = x. Since t, is the last opportunity to discuss the path, the bargaining problem at time t, can be treated as a static problem. Let pk be a function from the state variables space to R” such that
Since s(t,, x) is we are now able t,_, denoted by let Y[x,~_ ,, u(t)]
a compact convex set, ph( ) is well defined. Knowing ph to define p k-’ . For every level of state variables at time x,~_ I and for every admissible control u(t) on [th_ ,, tA] be a n tuple vector, such that
~,(x,,_,~dt)) = U/;(X,~ ,.u
,,u(t))~R”Ju(t)
when and let
isadmissiblecontrolon
C. Fershtman
Since S(t,_,, defined as
x,,_,>
is
/ Solutions a
compact
for dynamzc bargaining problems convex
subset
of R”, p”-’
151
can
be
From the way pkp’ is defined it is clear that there is a sustainable path u(t) for t E [t,_,, T]. In the same way for every j = 1,. , k - 1 we can define the function k-J. Since x0 is given, p’(x,) defines a solution for the dynamic P bargaining problem which is sustainable with respect to p and the partition of the time interval {t,, . . . , tk). Thus, although the dynamic bargaining problem cannot be characterized just by two components as the static problem. We can adopt the axiomatic approach and find a solution which will be sustainable for a given partition of the interval [0, T]. For every partition of the interval we will of course get a different solution. Finally, the solution we have investigated in this paper is based on the axiomatic approach. Before one tries to implement the strategic approach, it is important to note that the strategy space in the dynamic problem is much larger than in the static problem. Players can suggest different partitions of the time interval, or different punishment schemes. They can even agree just on a partial solution [a path u(t) for t E [0, T,]; T, -C T] and while the partial solution is executed they can continue the negotiation on the rest of the solution. Using this kind of strategy, players can avoid the problem of shrinking pie which is discussed by Rubinstein (1982).
References Kalai, E., 1977, Proportional solutions to bargaining situations: Interpersonal utility comparison, Econometrica 45, 1623- 1630. Kalai, E. and M. Smorodinsky, 1975, Other solutions to Nash’s bargaining problem, Econometrica 43, 513-518. Nash, John, F., 1950, The bargaining problem, Econometrica 28, 155-162. Roth, A.E., 1979, Axiomatir models of bargaining, Lecture Notes in Economics and Mathematical Systems no. 170 (Springer, Berlin). Rubinstein, Ariel, 1982, Perfect equilibrium in a bargaining model, Econometrica 50, no. 1 97- 109.