Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
1
11/20/98
Chapter 3
THE HARSANYI SOLUTION TO THE BARGAINING PROBLEM
3.1 Introduction
In the present chapter we introduce several additional subjects related to
the formulation and solution of the bargaining problem. Recall, first, that the twoperson bargaining problem discussed in Chapter 2 treated the bargaining parties
b g
“threat (disagreement, conflict) payoffs,” t  t1 , t2 , as given. However, as has
been already recognized by Nash (1953), in many cases, the threats can actually be
decided by the parties themselves. In fact, according to Nash’s construction, the
determination of the threat payoff is the sole reflection of the parties’ strategic
choices (conflict strategies). Hence, the determination of the bargaining parties’
conflict payoffs is considered first.
Recall, next, that Nash’s original formulation of the bargaining problem
dealt exclusively with the two-person bargaining problem. However, more often
than not, the number of bargaining parties exceeds two. This is certainly true in
many real-world political-economies. To deal with the general n -person
b g
bargaining problem n  2 , we adopted Harsanyi’s (1963) formulation and
proposed solution of the generalized n-person bargaining problem. As coalition
formation is always possible, and even likely, when the number of parties exceeds
two, of necessity the bargaining problem is far more complicated in the n -party
b g
case n  2 . Furthermore, both the formulation and the solution concept of the n-
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
2
11/20/98
person bargaining problem reflect a stronger strategic and cooperative
conceptualization.
Finally, as our political-power theoretic approach relies heavily on the
model of political power proposed in Harsanyi (1962a, 1962b), Harsanyi’s model
of the social power relationship is presented in this chapter. While the
presentation is rather brief and concise, it is sufficiently comprehensive to be selfcontained.
3.2 The Two-Person Bargaining Game with Endogenous Disagreement Payoffs
In the bargaining game discussed in Chapter 2 it was assumed that the
b g
disagreement (conflict) payoffs, t  t1 , t2 , were predetermined by the rules of the
game as given fixed values. Such a relationship is not unheard of in real-world
bargaining situations. Nevertheless, quite often the bargaining parties are capable
of influencing the disagreement payoffs, thus affecting the solution payoffs of the
corresponding bargaining game. This suggests extending the “simple” bargaining
game by turning it into the second stage of a more complex game. It is assumed
that in the first stage of this game each player announces the conflict strategies
that will be employed should players fail to agree in the second stage. It is also
assumed that the announced strategies are binding and will be carried out by both
players. Let

i
b g
be the sets of conflict strategies available to player i i  1,2 .
b g
Similarly, let  i i  1,2 be player i ’s disagreement strategy choice. That is,
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
b g
3
11/20/98
d i bi, j  1,2gdenote players i ’s payoff as a function
 i  i i  1,2 . Let U i  i , j
of player i ’s and player j ’s conflict strategies, given player i ’s conflict strategy.
b gbi, j  1,2g.
Hence, ti  U i  i ,   i
In Chapter 2, we defined H as the set of payoff vectors in the payoff space,
P, not dominated, even weakly, by any other payoff vector in P. We also defined
l
q
P*  u  P: ui  ti , u2  t2 and we denoted the upper-right boundary of P* by
b g
H *  H . The bargaining problem in Chapter 2 was then: Given P and i  1,2 ,
what will be the solution, u , that the bargaining parties will eventually reach? In
this chapter, t is made endogenous and the bargaining problem becomes: Given P
and
, what will be the solution u ?
i  1,2g
b
i
Crucially, it is assumed that once
the disagreement payoffs have been determined in the first, noncooperative stage
of the bargaining game, the solution to the second stage is plainly Nash’s
bargaining solution, i.e., the point, u  ( u1, u2 )  H * , satisfying
(u1, t1 )(u2 , t2 )  max (u1, t1 )(u2 , t2 )
(3.1.a)
uP
such that
ui  ti
(3.1.b)
bi  1,2g
Let H ( u1 , u2 )  0 be the Equation of H * and assume that the partial
b g
b g
derivatives, H1 u1 , u2 and H 2 u1 , u2 , are nonzero; then employing the Kuhn-
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
4
11/20/98
b g
b gb g
Tucker condition for maximum u1  t1 u2  t2 with respect to u1 i  1,2 , subject
b g
to H u1 , u2  0 , we get
a1( u1  t1 )  a2 ( u2  t2 ) ,
(3.2.a)
where
c h Hu bu , u g
ai  H i u1,u2 
(3.2.b)
1
bi  1,2g.
2
i
b g
Hence, given P* , any pair t1 , t2 of disagreement payoffs satisfying
(3.2.a) will yield u as a solution, a situation described in Figure 3.1. Thus, both t
and t' in Figure 3.1 yield u as the bargaining solution. Note also that, since
H1 / H2  0 , we may select nonnegative H i ’s where, in general, at least one
H i  0 . In the following, we shall assume that both H1  0 and H 2  0 . Thus, all
ch
t’s yielding the same solution u * are elements of the line C u * in P. The higher
ch
C u * , the more favorable is u * to player 2.
bg
Let C u denote the line in P whose Equation is
u being the
bgbi  1,2g; H bg
a1t1  a2t2  a1u1  a2u2  constant, where ai  Hi u
i
partial derivative, H u ui at u . It can be shown that if u * is more favorable to
player 1 than u (i.e., u *  u1 ), then
(3.3)
a1u1*  a2u2*  a1u1  a2u2 ,
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
5
11/20/98
and the converse also holds. Namely, if u is more favorable to the player 2 than
*
u * (i.e., u2  u2 ), one has
(3.4)
a1u1*  a2u2*  a1u1  a2u2 ,
where again
(3.5)
bg
ai  H i u
bi  1,2g
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
6
11/20/98
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
7
11/20/98
Figure 3.1. Nash solution to the two-person bargaining game in the (ui ,, ui) space
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
8
11/20/98
b g
b g
Hence, it is in the interest of player 1 to maximize a1U1  1 ,  2  a2U 2  1 ,  2 while
player 2’s interest is to minimize the same expression. Consequently, the chosen
conflict* strategies are given by
(3.6)
c h c h
a1t1  a2t2  a1U1  10 ,  02  a2U 2  10 ,  02
 max min a1U1( 1,  2 )  a2U 2 ( 1,  2 ) .
 1 1  2  2
 10 and  02 are referred to as the mutually optimal conflict strategies. Note again
that the selected conflict strategies affect the ultimate bargaining outcome through
their mutual effects on the disagreement payoffs, t.
3.3 The n-Person Bargaining Game
In contrast to the two-person bargaining game with endogenous
b g
disagreement payoffs, it is natural to assume that in the n-person case n  2
parties will be able to form coalitions. We focus here on Harsanyi’s solution to the
resulting n-person cooperative bargaining game (Harsanyi 1963).
In formulating the relevant model on page 205, Harsanyi stated,
the final payoffs of the game are determined by a whole network of
various agreements among the players and we shall try to define
the equilibrium conditions of mutual consistency and
interdependence that these various agreements have to satisfy.
That is, we shall assume that each such agreement between players
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
9
11/20/98
will represent a bargaining equilibrium situation between the
participants if all other agreements between the players are
regarded as given.
b g
He assumed that each player, i, i  N , gets a secure dividend, uiS , from every
coalition, S, of which he is a member (i.e., i  S , S  N ). The sum of all
dividends paid out by S is determined in a bilateral bargaining game between the
c
h
coalition S and its complementary coalition S in N i.e., S  N  S . The final
payoff, ui , of player i from the entire game is the sum of his dividends. That is,
(3.7)
ui 
w.
S
i
iS
SN
b g
for all i  N
It is, of course, required that
(3.8)
u  ( u1 , u2 ,, un )  H .
It was assumed that each coalition, S, backs the dividends guaranteed its members
by announcing a given conflict strategy,  s . The conflict strategies determine the
conflict payoff, uiS and uSj , that all members of S and S , respectively, get; i.e.,
(3.9)
c h
uiS  Ui  S ,  S
for each S  N and i  S .
The dividends guaranteed to any member, i, of S by all its subsets, R, should not
exceed the conflict payoff, uiS , which is the largest payoff that the cooperative
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
10
11/20/98
effort of the members of S could secure for i in the event of a conflict between S
and S . That is,
uiS 
(3.10)
w
for all S  N .
R
i
iR
R S
Consider next the case where all dividends are not negative. (Although
negative dividends are possible, we shall ignore this case in the following.) The
distribution of dividends between players i and j is determined in a bargaining
game between i and j, when they belong to the same coalition and where the
following payoffs are taken as given: ukS for all k  i , j in S and uiS where j  S
and u Sj when i  S . The disagreement (conflict) payoffs of i and j in the game,
GijS , when both i and j belong to S are
(3.11.a)
tiS 
w
R
i
, and t Sj 
j R
R S
w
R
j
.
iR
R S
In particular, the solution payoffs, ui , and u j , include the dividends, wiN and w Nj ,
which are determined in the game, Gij  GijN where the corresponding
disagreement payoffs are
(3.11.b)
tiN 
w
S
i
j S
SN
and t Nj 
w .
S
j
iS
SN
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
11
11/20/98
Given (3.9), (3.11.a), and (3.11.b), the disagreement payoffs of parties i and j,
respectively, may be derived from the coalition conflict payoffs of the players. It
can be shown that player i’s disagreement payoff when he is a member of S is
tiS 
(3.12.a)
 ( 1)
s  r 1
uiR ,
iR
R S
and player j’s disagreement payoff when he is a member of S is
t Sj 
(3.12.b)
 ( 1)
s  r 1
u Rj , 
j R
R S
where s  S and r  R are the numbers of members in coalitions S and R,
respectively. Moreover, tiN and t Nj are similarly expressed in terms of uiS and
b g
usj S  N . Also,
tiN  uiN  wiN and t Nj  uNj  wNj .
(3.13)
The entire n-person cooperative bargaining game is, in fact, a composite game and
b
g
the final payoffs, u  u1 , ..., un , must satisfy the following conditions,
(3.14.a)
H ( u1 , u2 , , un )  0 ,
(3.14.b)
ai (u1  tiN )  a j (u j  t Nj )
i, j  N ,
where
(3.14.c)
(3.14.d)
am  H m ( u1 , u2 , , un ) 
d i
uiS  U i  0S ,  0S
H ( u )
um
m  i, j,
i S, S  N ,
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
tiS 
(3.14.e)
12
11/20/98
 ( 1)
s  r 1
s  1, i  S , S  N ,
U iR
iR
R S
and
(3.14.f)
d ,  i  a U d ,  i
L a U d ,  i a U d ,  iO
max min M
, S, S  N
P



 N
Q
 a u  a u  a U
S
i i
j
iS
j S
S
j
i
iS

S
0
i
S
0
j
S
0
j
j S
S
S
S
0
S
i
S 
S
S
i
j
iS
S
S
j
j S
subject to
(3.14.g)
R
|Sa cu  t h a cu  t h
|T a du  t i  a du  t i
i
S
i
j
S
i
S
j
S
k
k
S
j
m
S
k
S
m
S
m
i, k  S
j, m  S
In order to obtain conditions that do not assume the existence of the partial
derivatives, H1,..., H n , one can replace (3.14.a) and (3.14.c) by the following
conditions:
ai  0
(3.14.a´)
i N
and
(3.14.c´)
a u
i i
iN
 max ai ui
uP
iN
a1 ,, an  constants
The last two conditions imply that, given that a1 ,, an  constants
ba  0 for all i  N g, the bargaining solution, u , is the payoff vector maximizing
i
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
13
11/20/98
the weighted sum aiU i over the payoff space, P. Since P is compact and convex,
bg
u exists and is a unique element of H, or H u  0 .
Finally, it is worth noting that, as demonstrated by Harsanyi (1963), the
solution payoff, ui , may also be characterized as the modified Shapley value.
That is, ui is the value
(3.15)
ui 

SN
iS
s S ,n  N
( s  1)!( n  s )!
 ( S )   ( S )
n!
i N
where
(3.16)
d i
 ( S )  Ui  0S ,  0S
iS
and
(3.17)
d i
 ( S )  U i  0S ,  0S ,
iS
using  ( S ) rather than the characteristic function,  ( S ) , as defined by von
Neumann and Morgenstern (1953).
3.4 Harsanyi’s Bargaining Theoretic Interpretation of Reciprocal Social Power
Relations
Until the early 1960s, the study of social power (including political power)
had been carried out exclusively by social scholars. The latter succeeded to
advance a fairly elaborate and useful theory of social power.1 However, it was
Harsanyi who in 1962 managed to introduce the rigor of mathematical reasoning
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
14
11/20/98
into the theory of social power. This he achieved by applying the analytical tools
and models of game theory to the subject. In addition to certain significant
contributions, Harsanyi provided a clear interpretation which eliminated much of
the ambiguity that had plagued the theory of social power prior to the publication
of Harsanyi’s works on the subject (Harsanyi 1962a, 1962b). In his analyses,
Harsanyi distinguished two principal modes of social influence broadly conceived
as situations where the intervention of one actor, A, causes another actor, B, to
alter his/her behavior in a manner that actor B would not otherwise do. In the first
distinct principal model—the unilateral power relationship—actor A unilaterally
influences B’s behavior by setting the values of certain variables under A’s
control. As B’s reaction as a function of these variables is known to A, A thereby
influences B’s reaction. (Thus, A is a Stackelberg leader, while B is a Stackelberg
follower.) Alternatively, A may irrevocably set certain conditional incentives in
the form of sanctions for noncompliance and rewards for compliant behavior
which leads B to fully comply with A’s demands (ultimatum game). Both the
Stackelberg relation and ultimatum games are unilateral power relations.
According to the second distinct principal mode—the reciprocal power
relationship—both parties possess some power over each other. As demonstrated
by Harsanyi, this mode of power relationship is best modeled as a bargaining
game. To simplify the exposition and render the relevant underlying principles
more visible, Harsanyi focused first on the two-person bargaining game as a
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
15
11/20/98
model of a reciprocal bilateral power relationship. Surveying the then existing
literature, Harsanyi argued that the accepted set of dimensions characterizing the
power relationship was inadequate and that two additional dimensions are
required for a meaningful theoretical model of the social power relationship. In
particular, modelers should always include the “cost and strength of A’s power
over B” in their models. The “cost dimension” refers to the cost borne by A in
influencing B’s behavior. The “cost of power,” whether a direct cost or the cost
of missed opportunities (opportunity cost), comprises the cost to A of rewarding B
for compliance or the cost to A of penalizing B when the latter ignores A’s
demands. The other needed dimension is the “strength of A’s power over B.”
This term refers to B’s utility loss when not complying with A’s demands. It is
the sum of B’s utility losses due to unrealized rewards and the subjective cost
caused by A’s sanctions under noncompliance.
Harsanyi offered the following expository illustration: Suppose A wants B
to perform X with probability p2 when, in the absence of A's intervention, B
b
g
would perform X with probability p1 only p1  p2 . A offers B the reward, R,
should B comply with A’s demand (i.e., B performs X with probability p2 ). A
also threatens B with a penalty, T, should B elect to perform X with probability
p1 . Suppose the value of the reward R to B is r and the cost of R to A is r * utility
units. Similarly, let the sanction T entail a loss of t units of B’s utility and a cost
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
16
11/20/98
of t * utility units to A. Suppose, further, that if B performs X, a subjective loss of
x utility units is suffered by B, while A enjoys a gain of x* utility units. Then, if
B complies with A’s demand and performs X with probability p2 , the parties’
expected utilities are:
uA  u0*  r *  p2 x*
(3.18.a)
and
uB  u0  r  p2 x .
(3.18.b)
If B does not comply with A’s demand and a conflict situation arises, then
the parties’ expected utilities are:
u~A  u0*  t *  p1x*
(3.19.a)
and
u~B  u0  t  p1x .
(3.19.b)
Maximizing the Nash product with respect to p2 yields the following solution
value, p20 :
p20  p1 
(3.20)
L
M
N
O
P
Q
1 r*  t* t  r
.

2 x*
x
By the theory of optimal threat strategy presented in Section 3.2, A’s
b gc hand a value
optimal strategy is to select a value of T minimizing t / x  t * / x *
of R maximizing
 r / x g.
cr / x hb
*
*
2
Harsanyi (1962a) then considered Dahl’s
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
17
11/20/98
measure of the quantity of A’s power over B (Dahl 1957). This dimension of
power was defined by Dahl as the probability difference p  p20  p1 . It follows
immediately from Equation (3.20) that
L
M
N
O
P
Q
1 ( t  r ) t *  r *
.
p  p  p1 

2
x
x*
0
2
(3.21)
0
2
According to Harsanyi (1962a: 77) the numerator of the first term inside
b g
the square brackets on the right-hand side of (3.21),  t  r ,
measures the difference it would make for B to have A as his enemy
instead of having him as his friend. It represents the total opportunity
costs to B of choosing noncompliance instead of choosing
compliance.... In our terminology, it measures the (gross) absolute
b g
strength of A’s power over B. Accordingly, the quotient r  t / x
measures the gross relative strength of A’s power over B with respect
to action X.
c h
Similarly, the difference t *  r * “measures the difference it would make for
A to have B as an enemy instead of having him as a friend. It represents the
net opportunity costs to A of choosing the conflict situation rather than
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
18
11/20/98
performing action X * .” (Harsanyi 1962a: 78.) Action X * denotes an action
by A of tolerating B’s noncompliance. Hence,
c h
the quotient, t *  r * / x * , measures the gross relative strength of B’s
power over A with respect to X * . Finally, the difference,
b g c h
 r  t / x  t *  r * / x * , is the difference between the gross relative
strength of A’s power over B with respect to action X and the gross
relative strength of B’s power over A with respect to the
complementary action, X * . It may be called the net strength of A’s
power over B with respect to action X. (Harsanyi 1962a: 78.)
It follows that Dahl’s quantity-of-power measure in a reciprocal bilateral power
relationship is equal to one half of the net strength of A’s power over B with
respect to the action, X—the net strength being defined as the difference between
the gross relative strength of A’s power over B with respect to action X and the
gross relative power of B over A with respect to the complementary action, X * .
The foregoing definitions and derivations reflect the reciprocal power relationship
inherent in the bargaining model of social power.
In a subsequent paper (Harsanyi 1962b), the measurement of social power
in a n-person reciprocal power situation is addressed using Harsanyi’s own
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
19
11/20/98
formulation of the n-person cooperative bargaining game as a model (Harsanyi
1963). Participants in the bargaining process are assumed to aim at maximizing
the probability that their preferred policy be selected by the set, N, of all
participants. We shall refrain from a full presentation of Harsanyi’s arguments in
this chapter, focusing instead on some of its highlights. Recall that Harsanyi’s
proposed solution payoffs of the n-person cooperative bargaining game could be
expressed as “modified Shapley values” (see Section 3.3). Many of Harsanyi’s
principal arguments, in fact, are based on the particular modified Shapley values
derived by Harsanyi from his specific assumption concerning the participants’
objective functions presented above. The modified Shapley values thus obtained
by Harsanyi comprise several components which may be interpreted in power
theoretic terms. We include Harsanyi’s Equation (27) only for completeness
(1962b, 88). The terms are interpreted in words following the equation.
Equation (27)
L
M
M
M
N
O
P
P
P
Q
L
M
M
M
N
O
P
P
P
Q
z 1
( s  1)!( n  s )!( ps  ps )
z 1
r
( s  1)!( n  s )! s
~
pi  i

 i
(R  i )  
(T  T s )
xii n iS
n!
xii n
zi iS
n!
SN
SN
The first term on the right side of Equation (27) is a weighted sum of a
constant 1 / n and the amounts of generic power that individual i and his
coalition partners would possess in various possible conflict situations,
less a weighted sum of the amounts of generic power that members of the
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
20
11/20/98
opposing coalitions would possess.... This term can be regarded as a
measure of the net strength of individual i’s (and of his potential allies)
independent power, i.e., of his ability to implement his policy preferences
without the consent, or even against the resistance, of various individuals
who may oppose him (and of his ability to prevent the latter from
implementing their policy preferences without his consent).
On the other hand, the second term is a weighted sum of net rewards
that all other individuals would obtain in case of full agreement, and of the
net penalties that they would suffer in various possible conflict situations
if they opposed individual i, less a weighted sum of the net rewards that
individual i would obtain if full agreement were reached, and of net
penalties that he and his coalition partners would suffer in conflict
situations. This term can be regarded as a measure of the net strength of
individual i’s incentive power, i.e., of his ability to provide incentives (or
to use incentives provided by nature or by outside agencies) to induce
other participants to consent to implementing his own policy preferences.
Finally, the sum of these two terms, i.e., the whole expression (of
the modified Shapley value), which we have called the strength of i’s
power can be taken as a measure of the full strength of i’s bargaining
position, if his power to get his policy preferences satisfied within the
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
21
11/20/98
group, based on his independent power and on his incentive power.
(Harsanyi 1962b: 89.)
In concluding the argument, Harsanyi also emphasizes that his modified Shapley
value as a measure of the “strength of power” can also “take account of the effects
of alliances and party alignments among participants.” It “can also take account
of improvements in all participants’ power positions when suitable compromise
policies are discovered, which may increase the chances for all participants at the
same time, of reasonable satisfactory outcome” (Harsanyi 1962b: 91).
3.5 Conclusion
In Chapters 2 and 3 we have laid the foundation of our political power
theory of endogenous policy formation. The Nash-Zeuthen-Harsanyi bargaining
theory provided the needed theoretical framework for the political power theory.
Essentially it was Harsanyi’s imaginative model of social power as a bargaining
relation that provided the basic idea. Chapter 4 will be dedicated to the detailed
description and formulation of the political process in terms of Harsanyi’s
underlying framework. In the rest of this book we shall outline the many
ramifications of the basic idea, thus providing a rather broad view of the
impressive tree that grew out of the seed planted by Harsanyi in the early 1960’s.
Chapter 3 (R-Z)
The Harsanyi Solution to the Bargaining Problem
D:\533566470.doc
22
11/20/98
FOOTNOTES
1
See Chapter 1 for more detail on the theory of social power and its evolution.
2
Recall that (by assumption) by selecting T, A uniquely determines the values of
bg bg
r bg
R .
bg
t T and t * T ; and by selecting R, A uniquely determines the values of r R and
*