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3Bargaining Powers, a Surface of Weights, and Implementation of the Nash Bargaining Solution *
Vladimir D. Matveenko
St. Petersburg Institute for Economics and Mathematics RAS, and National Research University Higher School of Economics at St. Petersburg Tchaikovskogo Str., 1, St. Petersburg, 191187, Russia E-mail: [email protected]
Abstract. In the present paper a new approach to the Nash bargaining solution (N.b.s.) is proposed. (Shapley, 1969) introduced weights of individual utilities and linked the N.b.s. with utilitarian and egalitarian solutions.
This equivalence leaves open a positive question of a possible mechanism of weights formation. Can the weights be constructed in result of a recurrent procedure of reconciliation of utilitarian and egalitarian interests? Can a set of feasible bundles of weights be a result of a procedure or a game independent on a concrete bargaining situation? We answer these questions in the paper. A two-stage n-person game is considered, where on the first stage the players on base of their bargaining powers elaborate a set of all possible bundles of weights A = {(Ai, ...,A„)}. This surface of weights can be used by an arbitrator for evaluation outcomes in different concrete bargains. On the second stage, for a concrete bargain, the arbitrator chooses a vector of weights and an outcome by use of a maximin criterion. We prove that this two-stage game leads to the well-known asymmetric N.b.s.
Keywords: Bargaining powers, Weights of individual utilities, Nash bargaining solution, Imlementation, Egalitarian solution, Utilitarian solution JEL classification C78, D74
1. Introduction
An n-person bargaining problem is defined by a pair (S, d) where S is a convex set in R" (the feasible set) and d G S (the status quo, disagreement point, threat point). Each point in S is interpreted as a bundle of utilities of the players; d is a point of utilities which can be achieved without cooperation, other points in S are available in case of cooperation. It is required to choose a ”good”, in one sense or another, outcome x G S.
Instead of x G S it is convenient to consider differences u = x — d. However, it is often assumed that d = 0 , in such case u = x; this assumption is accepted in the present paper.
(Nash, 1950) considered the bargaining problem from a normative point of view: he formulated a list of properties (axioms) of a good solution and found a unique solution satisfying these axioms. The set of axioms includes Pareto optimality, anonymity (or its weaker version, symmetry), scale invariance and contraction independence (also known as independence of irrelative alternatives). The solution, referred as Nash bargaining solution (hereafter N.b.s.), is the point of maximum
maxU1U2 . ..U".
xES
* This work was supported by the Program of Basic Research of the National Research University Higher School of Economics.
In an asymmetric case, when the players have different bargaining powers, the axiom of symmetry is not true, and the outcome satisfying the remaining three axioms can be found as a point of maximum
max ul1 u2 ... u"n,
xES 12 "
where hi > 0 are bargaining powers of the players, (Harsanyi and Selten, 1972, Kalai, 1977a).
For reviews of results developing the axiomatic approach to the bargaining problem see (Roth, 1979a), (Thomson and Lendsberg, 1989), (Thomson, 1994), (Serrano, 2008).
Nash himself saw a shortage of the axiomatic approach in a lack of an explicit description of a process of negotiations or of dynamics of formation of the outcome. A problem of constructing a non-cooperative strategic game resulting in an outcome satisfying definite axioms is referred now as Nash program. Contributions to the program were provided by (Nash, 1953), (Rubinstein, 1982), (Binmore et al., 1986), (Carlsson, 1991) and many others. Later many authors tried to connect the Nash program with the theory of mechanisms. A term implementation appeared, we understand it here as a rather broad approach collecting games and mechanisms supporting the N.b.s.
The asymmetric N.b.s. finds applications in very different branches of Economics such as wage formation (Grout, 1984 and many others), pricing mechanisms (Bester, 1993), macroeconomic policy (Alesina, 1987), circulation of ideas (Hellman and Perotti, 2006). A recent paper (Eismont, 2010) is devoted to analysis of the European gas market.
The N.b.s. is closely related with the problem of interpersonal comparison of utilities and with the use of weights of individual utilities. Historically, the usage of cardinal utilities of players and sums of individual utilities in Game Theory (in particular in games with transferable utilities) induced a dissatisfaction. (Harsanyi, 1963) and (Shapley, 1969) introduced intrinsically-defined utility weights. Shapley (Shapley, 1969) interpreted them as endogenously determined transfer rates between players’ utilities, i.e. as a kind of prices (analogues of exchange rates, i.e. prices of currencies). Shapley allowed a possibility of a change in weights and considered as solutions both a point of maximum maxxES (^1x1 + ... + jinx",) and a point of equality \1x1 = ... = Xnxn (equivalent to finding maxmin Xixi ). Cor-
xES i
respondingly, he spoke about ’efficiency’ weights and ’equity’ weights. These two criteria correspond two basic concepts of optimality, ’’competing” in economic policies. Later (Yaari, 1981) interpreted these criteria as ’utilitarian’ and ’egalitarian’, correspondingly. A review of utilitarian and egalitarian approaches to moral theory and, in particular, a discussion of papers (Shapley, 1969) and (Yaari, 1981) is provided by Binmore (Binmore, 1998, Binmore, 2005).
In relation with the weights of utilities (Shapley, 1969) advanced the following principle: ”(III) An outcome is acceptable as a ’value of the game” only if there exist scaling factors for the individual (cardinal) utilities under which the outcome is both equitable and efficient”. So, of principle for Shapley was the existence of a vector of weights which could for some outcome serve efficiency and equity simultaneously. For the bargaining problem such solution really exists and it is nothing else but the symmetric N.b.s. The fact can be formulated in the following way.
Theorem 1. Let the set of feasible utilities, S, be restricted by coordinate planes and by a surface g (x1, . .. ,xn) = C , where g is a strictly convex function. Then the following two statements are equivalent.
1.For a point x G S there exists such a vector of weights Ai, .. ., A„ that x is simultaneously (i) a point of maximum of ’utilitarian’ function A1 x1 + . . . + \nxn and (ii) a point where maxmin Aixi is achieved or, what is the same, where ’egalitarian’
xi
condition A1X1 = . .. = Xnxn takes place.
2.Point x G S is a symmetric N.b.s., i.e. the point of maximum of the product
x1...xn.
A proof will be given for a more general Theorem 3 below.
In the present paper a new approach to the problem of implementation of the Nash bargaining solution (N.b.s.) is proposed. The equivalence described in Theorem 1 (or in Theorem 3 in an asymmetric case) leaves open a positive question of a possible mechanism of weights formation. Can the weights be formed in result of a recurrent procedure of reconciliation of utilitarian and egalitarian interests? Can a set of feasible bundles of weights be a result of a procedure or a game independent on a concrete bargaining situation?
We consider a two-stage game, where on the first stage the players, on base of their bargaining powers, elaborate a set of all admissible bundles of weights A = {(A1,...,An)} which can be used by an arbitrator to assess outcomes in different concrete bargains. On the second stage, for a concrete bargain, the arbitrator chooses a vector of weights and an outcome by use of a Rawlsian-type maximin criterion. We prove that this game leads to the asymmetric N.b.s.
We interpret the arbitrator as a society (or a community) in a framework of which the bargains are fulfilled. In fact, in reality very often bargains take place not in a ’vacuum” but depend on an encirclement and on a choice of moral-ethical assessments.
We suppose that the society confesses a maximin Rawlsian-type principle of justice but attaches weights to individual utilities and is manipulated. The participants, by use of all available means draw up a set A of admissible bundles of weights (reputational moral-ethical assessments); bargaining powers of participants become apparent in this process. (The bargaining powers depend on an access to political power and to media, on image-makers’ abilities, on previous reputations, etc.) In each concrete situation of bargaining the society chooses weights from the set A. The assessments (weights) entering the set A are not unambiguous, and the society chooses those weights which it finds to be fair in a concrete case. Thus, generally, the society is conformist and possesses a whole spectrum of assessments which can be used in case of need. In each concrete bargaining problem there is a correspondence A ^ ParS between admissible bundles of weights A G A and outcomes x G ParS, where ParS is the Pareto border of S.
In Section 2 we consider a relation between ’efficiency’ weights, ’equity’ weights and bargaining powers as well as different characterizations of the N.b.s. The concepts of weights as moral-ethical assessments and of a two-stage game are introduced in Section 3. In Section 4 the same point of view is applied to two other well-known bargaining solutions: Kalai-Smorodinsky and egalitarian (Kalai). In Section 5 the moral-ethical assessments are considered as a mechanism working in a concrete bargaining situation. In Section 6 a relation between the N.b.s. and the Cobb-Douglas production functions is studied. Section 7 concludes.
2. The weights of individual utilities and characterizations of the N.b.s.
The question of existence of a ”good” system of weights is a normative one. However, not less interesting is a positive question of what is the system of weights acting in a concrete bargaining situation. In this relation a question arises about a presence of an iterative process of weights formation similar to a cobweb rule and leading to the N.b.s.
The set S plays a role similar to the production-possibilities set in the theory of trade (see, e.g. (Caves et al., 2006)). For the case of 2 players let the set S be bounded by the coordinate axes and a curve g (x1, x2) = C, where g is differentiable. Following (Shapley, 1969) the weights can be considered as coefficients of conversion,
i.e. prices. ’Efficient’ prices are being formed similarly to commodity prices in the trade model: if the economy is interested in achieving a point x then prices A1, A2 are established that the marginal rate of substitution is equal to the relative price:
dg(x)
dX2 _ dx! M /-I \
dxi [X) dg^l A2 ' W
dx2
Thus, the relative price is a function of the relative utility:
£-*(!)• (2)
On the other hand, given current ’efficient’ prices A1 ,A2, the society tries to redistribute the utility fairly. If the utility is transferable, then an outcome x would be chosen to satisfy the system of equations:
| AA = A2 x2 (3)
| A1x1 + A2'X2 = A1 x1 + A2x2
However, as soon as in reality the utility is not transferable, an outcome is chosen not by use of (3) but using the system:
{A1x1 = A2 x,2 g (x1,x2) = C
Consequently, the new relative utility is connected with the relative price by an inverse relation:
x\_ _ x2 Ai
The point (x1,x2) is used on the next iteration to find new ’efficient’ prices A1, A2 such that
a2 *{-J=rVM
where Lp (k) = tp .
So, an iterative process takes place:
A convergence of this process is of interest.
Theorem 2. A condition of a local stability of the process (4) is inequality \E\ > 1, where E is the elasticity of substitution of function g (x1,x2) at point (x1,x2) of the symmetric N.b.s.
A proof of a more general Theorem 6 is provided below.
EXAMPLE. Let set S be constrained by coordinate axes and by curve
5xP1 + (1 — 6) xp = C, (5)
where 0 < 6 < 1, p > 1, C > 0. Then the condition of stability of the cobweb rule (4) is p < 2.
Indeed, (5) can be written as
(■Sx\ + (1 — p) xf)p = C.
The LHS, the well-known CES function, has elasticity of substitution a = .
The stability condition \a\ > 1 reduces to p < 2 . So, the stability takes place under 1 <p < 2.
The natural generalization of Theorem 3 is the following statement.
Theorem 3. Let S and g satisfy conditions of Theorem 1, and let h1,...,hn be
positive bargaining powers. Then the following two statements are equivalent.
1. For a point x there exists such a vector of weights Ai,...,A„ ; that x is simultaneously
(i) a point of maximum of ’utilitarian’ function ^1x1 + ... + Vnxn with weights
Vi = hiAi (i = 1,...,n) (6)
and
(ii) a point where maxmin Aixi is achieved or, what is the same, where ’egalitarian’
xES i
condition
Aixi = ... = Xnxn (7)
takes place.
2. Point x is the asymmetric N.b.s. with bargaining powers b 1, . .. ,bn, i.e. the point of maximum of function
xbi xbn
1n
Proof. For the problem of maximization of the ’utilitarian’ function a necessary and sufficient condition of optimality is the equation
Vi f. ■ N f0N
^ !£(*)
For the problem of maximization of function x^1 ... x" a necessary and sufficient condition of optimality is the equation
bjXj _ §j:(x)
few
^ = (9)
dxj v■L)
Equations (6), (7) and (8) imply (9), i.e. Statement 1 implies Statement 2.
Let Statement 2 be true. Then (9) takes place. For point x weights Ai,..., A„ can be found for which equations (7) are true. Equations (6), (7) and (9) imply (8). Thus, Statement 1 follows from Statement 2.
Binmore (Binmore, 2009) writes that ”A small school of psychologists who work on ’’modern equity theory” come closest to my own findings. They find experimental support for Aristotle’s ancient contention that ”what is fair is what is proportional”. More precisely, they argue that an outcome is regarded as fair when each person’s gain over the status quo is proportional to that person’s ’social index”. This conclusion is consistent, for example, with the widespread concern about preserving differentials between the wages paid to different trades (such as electricians or carpenters) when there is a general wage increase.”
According to (7), in role of a ’social index’ of player i the value reverse to her ’egalitarian’ weight appears.
The relation (6) between ’utilitarian’ weights /ii and ’egalitarian’ weights Ai can be interpreted in the following way. The weights Ai (i = 1,...,n) are constructed in a framework of the ’egalitarian’ criterion and are not suitable for the ’utilitarian’ function because they too strongly reflect personalities of the players: players with higher bargaining powers hi have understated weights Ai while for players with lower bargaining powers the weights are overstated. The products /ii = hiAi contain ’’correcting” coefficients and provide objective, in some sense, estimates of utilities of players for inclusion into the ’utilitarian’ function.
Theorem 4. The asymmetric N.b.s. can be characterized as a point in the Pareto border, where the pair-wise elasticity of the border is equal to the corresponding ratio of the bargaining powers.
Proof. From (9)
dxi ^X ^
xi b,
— (i,j = l,...,n).
b>
Geometrically the point of the N.b.s., x, is characterized as follows. Consider a (hyper)plane tangent to surface g{xi,..., xn) = C at point x . Let X* (i = 1,..., n) be points of intersection of the plane with coordinate axes.
Theorem 5.
x = f3\X\ + P2X2 + ... + /3nXn, where f3i = bl + bi+b is the relative bargaining power of player i (i = 1 ,...,n),
n
EA = 1.
i=1
Proof. Let n\x\ +... + = /xixi +... + VnXn = C be an equation of the tangent
plane at point x. The point X* of intersection of the plane with the *-th coordinate
axis has the i-th coordinate
C /^1— Vn —
— = —xi + ... H------xn,
№i №i №i
and other coordinates are zero. By Theorem 3, the N.b.s. satisfies equations
x
Hence,
From here
bj Vi — ( • \
3 - 7-------Xi (j = l,...,n).
bi Vj
C b\ + ... + bn _
Vi
bi
-Xi.
C
C
b 1X1 + ... + bnXn — bi —,..., bn— — (6i + ... + bn) x.
Vi
Vn
Vice versa, an arbitrary point x G S with positive coordinates satisfies equation
__ _ n _
x = (3iXi + ... + /3nXn with some coefficients (3i > 0, Pi = 1; where Xt is
i=l
the point of intersection of the tangent plane at point x with i-th coordinate axis. These coordinates define point x uniquely. According to Theorem 4, point x is the asymmetric N.b.s. with bargaining powers (3i > 0 (i = 1,...,n) (or any proportional bargaining powers).
In particular, in case of two players with equal bargaining powers, the N.b.s. x is characterized by the property that the segment of the straight line tangent to curve g{xi,X2) = C at point x and contained between coordinate axes is divided in halves by point x. In case of three players with equal bargaining powers, x is the point of intersection of medians of the triangle formed by intersections of the coordinate planes with the plane tangent to surface y>(xi, x2, X3) =0 at point x.
Similarly to the previous one, consider an iterative ” cobweb” process of weights formation. Let vi, V2 be ’efficient’ prices under which the society tries to redistribute utilities x1,x2. In case of transferable utility, an outcome (X1,X2) would be chosen to satisfy the system of equations
Aixi = A2 x<2 Vixi + V2x2 = Vixi + V2x2:
(10)
where Ai = vi/bi , and bi is the bargaining power of player i (i = 1, 2). As soon as utility is not in fact transferable, the outcome (xi,x2) has to satisfy the system of equations
{Aixi = A2 x,2 g (xi,x2) = C
Hence, the relative utility is linked with the relative price by equations:
(11)
xi
x2
^2
Ai
h V2_ &2 Ml
On the next iteration the point (xi, x2) is used to define new ’efficient’ prices (Vi, V2) such that
* dg(x) / - \ /7, '
- = iSk = *p(-)=*p( y-
A*2 VX2/ \02V1,
So, the following iterative process takes place:
i
te)
t+i
2
(12)
where cp(k) = if, .
Theorem 6. A condition of a local stability of the process (12) is the inequality \E \ > 1 , where E is the elasticity of substitution of function g(xi,x2) at point x of the asymmetric N.b.s.
Proof. A condition of stability is inequality
M2
< 1 at the steady state. The
weights Mi, m2 are responded by utilities xi,x2 such that
Mi -
— Xl bi
M2 _ b2
The following is true:
1 El't (El
x2 ) b2\ Mi
dg(x)
dxi
dg(x)
dx2
_ dg(x) X1 d(x2)
xM2
dg(x)
dxi
d
2
where E is the elasticity of substitution of function g(xi,x2) at point x. Thus, the stability condition is the inequality \E\ > 1.
3. Weights as reputational moral-ethical assessments. Formation of weights
Many real bargains and negotiations are characterized by two common features. Firstly, it is a presence of a community in role of an arbitrator in a framework of which the bargains are fulfilled. For example, in case of international trade and international relations it is a so called ’’international community” including governments of countries, and different international organizations. In case of labor relations, it is a ’collective” of a firm (or, in the West, a union). In other cases it can be a local community, a ’scientific community”, an ’artistic community”, and so on.
Secondly, bargains between a fixed set of participants are often not ’one-shot’ but represent a routine repeated process, and, along with bargains, a ’’public opinion” of the community is being formed. The community acts as an arbitrator realizing a control for bargains in such way that unfair, from the point of view of the community, bargains are not possible, at least as routine ones. An outcome of an unfair bargain can be revised, if not formally, than in result of a conflict. Such conflicts often arise, both in local communities and organizations (conflicts between separate members of community or between a worker and administration), and on a national level (conflicts between social groups) and in international relations (conflicts between countries).
The process of formation of the public opinion goes uninterruptedly, but often one does not know what bargains will take place in the future, so the public opinion
is elaborated by parties not in conformity to a concrete bargain but in relation to all admissible situations of bargaining. Moral-ethical assessments formed by use of propaganda emphasize an insufficiency of utilities received by some participants and an excessiveness of utilities of others. In other words, the essence of the moral-ethical assessments is that they act as decreasing or increasing coefficients (weights) applied to utilities of the participants.
As an example we can compare judgments concerning the annual income of President D. Medvedev on site Compromat.ru of media Gazeta.ru (December 2008) and in Izvestia newspaper (April 2009). The site emphasizes that D. Medvedev has a largest, in comparison with other members of the government, apartment. Izvestia, on the contrary, stresses that D. Medvedev, by the World measure, has one of the lowest wages among state managers of such level.
It is important to emphasize that moral-ethical assessments are usually not univalent but allow a variance: the public opinion practically always stresses both positive and negative features of participants.
Formation of public opinion is performed by means of propaganda and by use of any available tools (e.g. media, Internet, political meetings, creation of gossips). Possibilities of formation of public opinion are limited by an access to media and by professional abilities of ” image-makers” as well as by a presence of adaptive or rational expectations of the audience: it is rather difficult to instill an opinion which does not correspond to expectations.
Concrete weights can differ in different bargains depending on circumstances. Thus, the community deals not with a unique vector of assessments but with a curve (in case of two bargainers) or with a surface of possible assessments. It can be supposed that the community in its approval or disapproval of an outcome of a bargain acts in accordance with a Rawlsian-type maximin principle, paying attention to the most infringed participant and taking into account the whole spectrum of admissible weights of utilities formed in result of propaganda. In result a parity is reached in the model: weighted utilities of the participants coincide.
Curve of weights in case of two participants. Let us consider a two-stage game. On the first stage the participants form a curve of weights, i.e. a spectrum of all possible assessments of utilities. When the curve of bundles of weights, A = {(Ai, A2)}, is formed it is passed to the arbitrator. On the second stage, for a concrete bargain, by use of the system of assessments, A, the arbitrator finds the maximin
max min{Aixi ,A2x2}.
x £ S,\£A
What is the same, the problem
max {v : v = Aixi = A2x2)
x£S,\£A
is solved.
Under such mechanism of partition, utility xi gained by player i is negatively connected with her own weight Ai and is positively connected with Aj, j = i. That is why on the first stage each participant is interested in diminishing the weight of her own utility and in increasing the weight of the others’ utility. However, in negotiations on the system of weights the player i would agree to a decrease in the other’s weight Aj in some part of the curve A at the expense of an increase of her
own weight Ai as soon as her partner similarly temporizes in another part of the curve A.
So far as the system of weights is essential only to within a multiplier, the participants may start bargaining from an arbitrary vector of weights.
To what increase in her own weight (under a decrease in the other’s weight) will a player agree? Bargaining powers become apparent here. We suppose that a constancy of bargaining powers of participants means a constancy of elasticities of Ai with respect to Aj. In other words, player j for 1% decrease in the other’s utility will agree to an increase in her own utility only to bj/bi per cent, where bi, bj are bargaining powers of the players. The more the relative bargaining power of player
i is, the fewer the increase in her utility is, when the opponent attacks. Thus, the following differential equation is fulfilled:
dAi Aj bj
377-r = = const' (13)
dAj Ai bi
By solving this differential equation we receive:
dAi dA2
-T--Ol —-------7--0 2,
Ai A2
ln Ai1 = — ln A22 + const,
Ai1 a22 = C.
The curve A is defined.
Differential game of the weights curve formation. The process of the formation of A can be described more explicitly as a differential game. We assume that the absolute values of the growth rates of weights, \gi\ and \g2\, are constants chosen by the players. Evidently, gi < 0 when player i attacks, and gi > 0 when she defends. The growth rates gi will be considered as control variables of the players. It is assumed that when player i defends, she solves the following problem:
min gi
h_i bi'
s-t. №>1^1/-
The meaning of the coefficients in the RHS is following: the opponent’s bargaining power, bj, sharpens the constraint and leads to increasing the weight Ai, while the own bargaining power, bi relaxes the constraint and thus prevents increasing Ai. Thus the differential game is described by a pair of problems:
min gi
and
s-t. gi > \q2| t~-bi
min g2
bi
s-t. 92 > \gi| t—• b2
A continuum of Nash equilibria exists, each of them satisfying equation
9i_ _ _bj_
9j h
which is equivalent to (13).
Properties of the curve of weights. Let us see, how does the curve of weights depend on the relative bargaining power. Let the players start formation of the curve of weights from a point (Ai, A2). The equation of the curve of weights is:
\&1 \ b2 _____ ri ______ Abl \&2
Ai A2 = C = Ai Ai .
Hence,
A2
A,= ^j A,. (14)
When player 2 diminishes her weight (attacks) and player 1 prevents increasing her weights (defends), under A2 < A2, an increase in the relative bargaining power of player 1 would lead, according to (14), to a decrease in her weight. In other words, player 1, defending, reaches the higher success (i.e. the lower weight) the higher her relative bargaining power is.
Under Ai < Ai , if player 1 attacks, to see the role of her relative bargaining power we have to look at the weight of player 2:
A2 = 2 A2. (15)
With an increase in the relative bargaining power of player 1, the weight of player
2 increases, i.e., while attacking, player 1 achieves also a higher (i. e. worse) weight of her opponent.
If b2 ^ 0, then, according to (14), Ai ^ Ai. It means that if player 1 possesses a very high bargaining power then, when player 2 attacks, the weight of player 1 increases negligibly. When player 1 attacks, as (15) shows, with an increase in Ai/Ai, the weight of player 2 increases significantly.
Utility of outcome and transformation of scale. Let us consider the case of two participants. The curve A is described as above. Then to any outcome x G ParS a unique vector of weights A G A corresponds such that Aixi = A2x2 . Namely, if Ai1 A22 = then
„ __1__ f Xn\ 61+62 , __1__ ( Xl \ 6l+62
Ai = c6i+62 I — J , A2 = c6i+62 I — J . xi x2
The value v(x) = Ai xi = A2x2 will be called a value of outcome x. Thus,
j fcl h2 v(x) = Cbl+62 X^1+b2 X21+b2 ■
Evidently, maxx£S v(x) is reached at the point x of the asymmetric N.b.s.
Let us see, how the value of outcome changes under a change in scales of utilities, when each possible outcome x = (xi, x2) is transformed into x = (aixi, a2x2), where
b
2
b
1
b
ai, a2 are constant. Let A, A G A be vectors of weights before and after the change in scales:
Thus, under a change in scales of utilities with coefficients ai,a2, the value of
(x = argmaxx£s v(x)) to stay invariant under changes in scales of utilities.
Proof. Let A be a vector of weights satisfying v(x) = Aixi = A2x2 and p G A an
arbitrary vector of weights, p = A . Let p^ < Ai,pj > Aj . Then pixi < Aixi = Ajxj
and, hence, min pixi < v(x) = min Aixi. i=i,2 i=i,2
The problem of the arbitrator is to find
max max min Ai xi.
x£S \£
Theorem 7. For each outcome x = (xi ,x2) G S
v(x) = Aixi = A2x2, v(x) = Aiaixi = \2a2x2.
(16)
Ai a2A2
On the other hand, Ai1 A22 = Ai1 A22 = c and, therefore,
(17)
It follows from (16) and (17) that
Hence,
v(x)
v(x) .
bl b2
outcome is multiplied by ai1+b2 a^1+b2. It is a fundamental property of the N.b.s.
Proposition 1. v(x) = max min Aixi for any x G S.
AeA i=i,2
v(x) = max min Aixi = Axi1 x22,
where A = const, and fSi,^2 are relative bargaining powers.
See below a proof for the case of n players (Theorem 8).
In such way the arbitrator’s problem reduces to finding
max Axi1 x22.
x£S i 2
The solution is the asymmetric N.b.s.
Surface of weights in case of n players. Similarly to the case of two players, negotiations of n players apropos the structure of the surface of weights (assessments) of utilities start from an arbitrary bundle A and process in such way that player j for a 1 per cent decrease in utility xi agrees to an increase in her own utility xj to bj/bi per cent only, where bi,bj are bargaining powers. This leads to the following system of differential equations:
= ~TT = const = 1> •••>«,
i j i
Solving this system, as in the case of two players, we receive
Ah Ab = const
Multiplying out these equations, we come to equation
\b1 \bn fi
Ai ...An = C,
which defines the surface of weights A .
The problem of the arbitrator is to find
maxmaxmin {Aixi,..., Anxn\,
x£S AeA
where x = (xi,..., xn) are admissible outcomes.
Theorem 8. For each vector of utilities x = (xi, ..., xn) the following equality takes place:
maxmin {Aixi,..., Anxn} = Ax^1...xn, where A is a constant independent on x and jii, ..., fin are relative bargaining powers.
Proof. Let us fix an arbitrary utility vector x > 0 . There is a unique vector A G A proportional to the vector of inverse elements of vector x:
A ..., An) c(x- , ..., xn ).
Hence, From here
cb1 + ---+bn x — b1 x-bn
where & = , , bl, , are relative bargaining powers of players i = 1, ...,n .
b1 + ••• +
Evidently, min {Aixi, ...,Anxn} = c. Let us show that if A G A, A = A then min {Aixi, ...,Anxn} < c. Assume the opposite: min {Aixi, ...,Anxn} > c. This implies Ai > cx,nx = Ai (i = 1,..., n) . But, on the other hand, for any vector A G A such that A = A there exists an element Ak , such that Ak < Ak . We come to a contradiction. Hence,
maxmin{Aixi,..., Anxn} = c = Ax^1...xn
______1
where A = C bi + ---+i>n = const .
In such way the arbitrator’s problem reduces to finding
max Axi1 ...xtn, xes i n
and we come to the asymmetric N.b.s.
The maximin criterion, used here, reminds the Rawlsian criterion of fairness. However, in our case the society (the arbitrator) looking in each point x for maximum in A plays in favor of a wealthier player by making worse (i.e. increasing) the weight of a more restrained player.
Similarly to the proof of Theorem 8, it can be proved that
and, thus, openly acting in favor of the most wealthy player by improving (i.e. decreasing) her weight.
In such way the model demonstrates that societies with different political mechanisms may negligibly differ in economic and other objective characteristics of their activities.
Now let us show that the same surface of weights A provides the N.b.s. as a utilitarian solution.
Theorem 9. Solution to the problem
Proof. For problem min E Vixi (with a fixed x) we construct a Lagrangian
and, by use of Theorem 3, the solution of problem (18) is the asymmetric N.b.s.
minmax{Aixi,..., Anxn} = c = Ax^1...xn.
It means that a ’’Pharisee just” society for which the criterion is
maxmaxmin {Aixi,..., Anxn\
xeS AeA
does not differ in its accepted outcome from a society searching for
max min max! Aixi,..., Anxn\
xeS AeA
n
(18)
i=i
where Vi = biAi (i = 1,..., n), is
x,A= (x- i,....,Xn1),
where x is the asymmetric N.b.s.
n
n
i=i
and derive the first order optimality conditions
bixi - vbiAbi i Abkk =0 (i =1,...,n).
It is interesting to compare, from the point of view of our two-stage game, the N.b.s. with two other well-known solutions of the bargaining problem: the Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) and the Kalai egalitarian solution (Kalai, 1977b). In both of them, instead of a surface of weights, a fixed bundle of weights Ai,..., An is used. The arbitrator chooses an outcome by solving the problem
maxmin {Aixi,..., Anxn}. (19)
xeS
Proposition 2. Problem (19) has a unique solutionn. This is vector x € ParS
proportional to ( J— ) .
\A1 An J
Proof. The vector x satisfies
Aixi = ... = Axn = c.
Let us show that min{Aixi,..., Anxn} < c for any x € S,x = x. If the opposite is true then Aixi > c for all i = 1, ...,n. On the other hand, there exists i such that xi > xi and, hence, c/Ai > xi . The contradiction implies that
maxmin{Aixi,..., Anxnj = c,
xeS
and solution is reached at the unique point x .
The logic of the Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) is that ’social indexes’ of players are defined by their maximal possible gains. Then in problem (18) such weights Ai,An are used that the elements of vector (J-,J-) are
A1 An
proportional (with a common coefficient) to maximal values xi,..., xn of coordinates in set S. The solution is a vector x € ParS proportional to x = (xi,..., xn)
The egalitarian solution (Kalai, 1977b) starts from assumption that all players have equal ’social indexes’. So, the arbitrator chooses a point solving the problem
maxmin{xi, ...,xn}.
xeS
It follows from Proposition 2 that the solution is a point x* with equal elements:
* _ _____ *
xi ... xn.
In case of weighted egalitarian solutions (Roth, 1979b) the surface of weights also consists of a single point.
5. Moral-ethical assessments as a mechanism of bargaining
Earlier we supposed that on the first stage of the game the surface (or curve) of weights A is constructed, and on the second stage the arbitrator (society) in a concrete bargaining situation takes a maximin solution. Now let us consider another version of the model: in a concrete bargain participants change weights gradually. Each player tries to diminish the weight of her own utility (or not to allow it to increase) and to increase the weights of the others’ utilities. The arbitrator controls the bargaining process and does not allow the value of outcome, v(x), to diminish.
The following scheme can serve as a model. In a two-player game, if for a current relative weight, Ai/Aj, the value of outcome decreases in Ai then player i attacks by decreasing her weight, Ai , and increasing the other’s weight, Aj. The arbitrator allows this as long as the value of outcome, v(x), increases. In these circumstances a counter-attack of player j is impossible because it would lead to a decrease in v(x) what is not allowed by the arbitrator. Player j in this situation can only defend herself using her bargaining power to restrict the decrease in weight Ai and the increase in weight Aj. The attack of player i can continue only as long as the value of outcome, v(x), increases. With continuous time, this process stops as soon as the value of outcome v(x) reaches its maximum at point x of the asymmetric N.b.s.
The process can be modeled, for example, by a differential game reducing to the following differential equation which describes a path in the curve A as well as a corresponding path in the Pareto frontier of the set S:
Ai _ AT "
—ft, if Aixi + Aixi > 0 [and A2x2 + A2x2 < 0 ft, if Aixi + Aixi < 0 ^and ^2x2 + A2x2 > 0 0, if Aixi + Aixi = 0 ^and ^2x2 + A2x2 = 0 This game leads to a point x described by equations
A i xi + Ai x i = 0,
A2 x2 + A2 x 2 = 0.
(20)
Hence,
dAi A2 dA2 Ai
dxi(x) x2
dx2 xi
The LHS is equal to —b2/bi. According to Theorem 4, x is the asymmetric N.b.s.
A more straightforward proof follows. Let the Pareto frontier of set S be defined by curve x2 = g(xi). The curve of weights, A, is Ai1 A22 = 1. To each vector A € A an outcome x € ParS corresponds satisfying equation Aixi = A2x2. It follows that
Ai
'J'2,
{ g(xi)A61+62
Transforming, we come to the following form of equation (20):
b2g'(xi )xi + big(xi) = 0.
(21)
Equation (21) defines the symmetric N.b.s., as soon as maxxes xi1 x22 is reached under (21).
6. N.b.s. and Cobb-Douglas production function
One more possible area of application of the N.b.s. is the theory of economic growth. Remind that, according to the neoclassical theory, the Cobb-Douglas production function Y = AKaL1-a (where Y is output, K is capital, and L is labor) reflects not only production but also distribution of product: under conditions of perfect competition, parameters a and 1 — a are not only elasticities of output with respect
to production factors but are also shares of the owners of production factors. In recent time a series of research appeared demonstrating a difference in labor and capital shares in countries of the World (in average, in developing countries the capital share is higher than in industrial countries) and changes in the factor shares in time (the capital share increases in all groups of countries). It is possible to use the N.b.s. concept to explain the differences between countries and the tendency of changes in the shares. A constancy of bargaining powers can explain a constancy of factor shares (a validity of the corresponding Kaldor’s stylized fact) in some countries on a definite stage of their development.
Let us consider a model with two social groups: workers and owners of capital (entrepreneurs) who possess bargaining powers bL and bK, correspondingly, and bargain apropos partition of the national product. The N.b.s. is realized by use of the mechanism of propaganda and moral-ethical assessments described above and leads to a distribution of the product in shares bL and bK. To ensure a possibility of such distribution the social groups choose those production technologies (and fields of specialization of the economy) for which factor elasticities 1 — a = b^b and a = b^b are used in the Cobb-Douglas production function. The choice of technologies can take place both on stage of R&D and on stage of production (e.g., workers can violate technologies to receive in result a demanded share of output. This can explain, for example, a tolerance of society to small and big plundering in manufacturing in the USSR).
The process can be modeled by the two-stage game described above. On the first stage two players (entrepreneurs and workers) form a curve A = (AK,AL) of admissible moral-ethical assessments (weights). On the second stage an arbitrator (society) chooses an admissible pair of weights and divides the product Y among the players (Y = YK + YL) to achieve the maximin
maxmin{AKYK, ALYL}.
The curve of weights formed on the first stage of the game is
AbK AbL = C.
For the second stage of the game, Theorem 7 implies the following statement. Theorem 10. For each outcome (Yk,Yl) (where Yk + Yl = const):
hK bL
hK+hL VhK+hL
maxmin{AKYK, ALYL} = AYKK L YL
where A = const.
Thus, the arbitrator’s problem reduces to the asymmetric N.b.s. The solution, as can be easily seen, implies that the players receive shares proportional to their bargaining powers:
YK = h_K Yl bL '
For such distribution to be possible, the players will choose those technologies (and those fields of specialization of the economy) under which factor elasticities used in a Cobb-Douglas production function are proportional to the bargaining powers.
In relation with globalization, there is a question of stability of a product partition between participants of the global production process. The basic argument of anti-globalists is a reduction in the labor share in advanced countries which is explained by extension of production in developing countries.
Let us consider a model with three participants: an entrepreneur (a transnational company) acting in two countries and workers (unions) of these two countries. Capital is mobile, in contrast to labor. Will an equilibrium reached inside the countries on base of ’inner’ bargaining powers of the participants coincide with an equilibrium which would arise if the three participants meet in ’global’ negotiations? (In case of a non-coincidence, one of the social groups would be interested in a revision of the ”World order”, and the equilibrium would not be stable). We will find bargaining powers under which such coincidence takes place.
Let inner bargaining powers of the entrepreneur and the workers be ai , 1 — ai in country 1 and a2, 1 — a2 in country 2. As a result of a choice of technologies corresponding to the desired partition of the product, the production functions are AK1 Li~a1 and A2K2“2L^-™2. These functions can also be written as K^1 Li-a1
and K^2 L2~a2 where Li = A\~a Li is the effective labor in country i (i = 1,2). The entrepreneur will place the total capital K in a way to equalize the marginal products of capital in the countries:
a iK?1- iLi -a1 = a^2-iL2-a2.
Denote a, ft, y bargaining powers of participants in a global bargain, a + ft + 7 = 1. A coincidence of the equilibria achieved in the inner and in the global bargains means holding equations:
“ = a‘iTT« + "TTTvV=(1 “ ai)iTTiv7 =(1 “ “2)JTTvv
Thus, in equilibrium, the workers of country i possess in the global bargain a bargaining power equal to their inner bargaining power, 1 — ai with a discounting weight Xi = y!+y2 eclual to the country’s share in the global output. The global bargaining power of the entrepreneur is equal to the convex combination of her inner bargaining powers ai with the weights xi (i = 1, 2).
Let us consider a case when inner bargaining powers in both countries are the same: a and 1 — a . Then
Kf-i L i -a = K%-i Li -a.
From here,
= KU
L i + L2 L i + L 2
Thus, the entrepreneur places the capital proportionally to the effective labor. In the global bargaining,
a = a,
ft = {1 — a) ———ry = (1 — ot) 1
Yi + Y2 L i + L 2
L 2
7 = (1 — a)
L i + L 2
Therefore, for holding the equilibrium, the bargaining power of the entrepreneur in the global bargaining has to be the same as in the inner bargaining, and the bargaining powers of the workers have to be proportional to their effective labor. Each of the two groups of workers has in the global bargaining a bargaining power less than in the inner bargaining. If the bargaining powers of the two groups of workers are summed up, the ” united labor” would have the same bargaining power as the workers have in inner bargains in each of the countries. In general, this does not contradict an intuitive idea of a global bargaining and a partition of the world gross product.
7. Conclusion
The Nash bargaining solution (N.b.s.) takes a central place in the theory of bargaining. Research in this paper differs essentially from known approaches to the N.b.s. which were based, as a rule, either on a choice of axioms of a fair distribution when participants play a purely passive role, or on a choice of a stopping rule in negotiations with reference to a concrete bargain. In the present model negotiations relate to a system of weights, A. In the negotiations participants exhibit their bargaining powers. The set A can be used not in a single bargain but in a set of bargains of the same participants. A special role is plaid by an arbitrator interpreted here as a community (society).
A relation between bargaining powers, ’utilitarian’ weights and ’egalitarian weights is studied. In particular, a cobweb procedure leading to the N.b.s. is considered.
We studied a two-stage game, where on the first stage the players form a set A of all possible bundles of weights, and on the second stage, for a concrete bargain, the arbitrator finds a maximin solution. We showed that this game leads to the asymmetric N.b.s. Also it is shown that two other well-known bargaining solutions, Kalai-Smorodinsky and egalitarian, can be received as special cases with A consisting of a single bundle of weights.
As an example of application of the ideas elaborated in the paper we propose a model where the asymmetric N.b.s. leads to the Cobb-Douglas production function.
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