The Harsanyi Solution to the Bargaining Problem11/20/98

Chapter 3 (R-Z)1

The Harsanyi Solution to the Bargaining Problem11/20/98

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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 two-person bargaining problem discussed in Chapter 2 treated the bargaining parties “threat (disagreement, conflict) payoffs,” , 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 -person bargaining problem , 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 -party case. Furthermore, both the formulation and the solution concept of the n-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 self-contained.

3.2 The Two-Person Bargaining Game with Endogenous Disagreement Payoffs

In the bargaining game discussed in Chapter 2 it was assumed that the disagreement (conflict) payoffs, , 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 be the sets of conflict strategies available to player . Similarly, let be player ’s disagreement strategy choice. That is, . Let denote players ’s payoff as a function of player ’s and player ’s conflict strategies, given player ’s conflict strategy. Hence, .

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 and we denoted the upper-right boundary of by . The bargaining problem in Chapter 2 was then: Given P and , what will be the solution, , 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 ? 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, , satisfying

(3.1.a)

such that

(3.1.b)

Let be the Equation of and assume that the partial derivatives, and , are nonzero; then employing the Kuhn-Tucker condition for maximum with respect to, subject to , we get

(3.2.a),

where

(3.2.b).

Hence, given , any pair of disagreement payoffs satisfying (3.2.a) will yield as a solution, a situation described in Figure 3.1. Thus, both t and in Figure 3.1 yield as the bargaining solution. Note also that, since , we may select nonnegative ’s where, in general, at least one . In the following, we shall assume that both and . Thus, all t’s yielding the same solution are elements of the line in P. The higher , the more favorable is to player 2.

Let denote the line in P whose Equation is where ; being the partial derivative, at . It can be shown that if is more favorable to player 1 than (i.e., ), then

(3.3),

and the converse also holds. Namely, if is more favorable to the player 2 than (i.e., ), one has

(3.4),

where again

(3.5)

Figure 3.1. Nash solution to the two-person bargaining game in the (ui,,ui) space

Hence, it is in the interest of player 1 to maximize while player 2’s interest is to minimize the same expression. Consequently, the chosen conflict* strategies are given by

(3.6)

and 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 disagreement payoffs, it is natural to assume that in the n-person case 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 will represent a bargaining equilibrium situation between the participants if all other agreements between the players are regarded as given.

He assumed that each player, i, , gets a secure dividend, , from every coalition, S, of which he is a member (i.e., ). The sum of all dividends paid out by S is determined in a bilateral bargaining game between the coalition S and its complementary coalition in N. The final payoff, , of player i from the entire game is the sum of his dividends. That is,

(3.7)for all

It is, of course, required that

(3.8)

It was assumed that each coalition, S, backs the dividends guaranteed its members by announcing a given conflict strategy, . The conflict strategies determine the conflict payoff, , that all members of S and , respectively, get; i.e.,

(3.9)for each

The dividends guaranteed to any member, i, of S by all its subsets, R, should not exceed the conflict payoff, , which is the largest payoff that the cooperative effort of the members of S could secure for i in the event of a conflict between S and . That is,

(3.10)for all

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: for all , j in S and where and when . The disagreement (conflict) payoffs of i and j in the game, , when both i and j belong to S are

(3.11.a)

In particular, the solution payoffs, , and , include the dividends, and , which are determined in the game, where the corresponding disagreement payoffs are

(3.11.b)

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’sdisagreement payoff when he is a member of S is

(3.12.a)

and player j’sdisagreement payoff when he is a member of S is

(3.12.b)

where and are the numbers of members in coalitions S and R, respectively. Moreover, and are similarly expressed in terms of and . Also,

(3.13)

The entire n-person cooperative bargaining game is, in fact, a composite game and the final payoffs, , must satisfy the following conditions,

(3.14.a),

(3.14.b)

where

(3.14.c)

(3.14.d)

(3.14.e)

and

(3.14.f)

subject to

(3.14.g)

In order to obtain conditions that do not assume the existence of the partial derivatives, one can replace (3.14.a) and (3.14.c) by the following conditions:

(3.14.a´)

and

(3.14.c´)

The last two conditions imply that, given that , the bargaining solution, , is the payoff vector maximizing the weighted sum over the payoff space, P. Since P is compact and convex, exists and is a unique element of H, or .

Finally, it is worth noting that, as demonstrated by Harsanyi (1963), the solution payoff, , may also be characterized as the modified Shapley value. That is, is the value

(3.15)

where

(3.16)

and

(3.17)

using rather than the characteristic function, , 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 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 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 when, in the absence of A's intervention, B would perform X with probability only . A offers B the reward, R, should B comply with A’s demand (i.e., B performs X with probability ). A also threatens B with a penalty, T, should B elect to perform X with probability . Suppose the value of the reward R to B is r and the cost of R to A is utility units. Similarly, let the sanction T entail a loss of t units of B’s utility and a cost of 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 utility units. Then, if B complies with A’s demand and performs X with probability, the parties’ expected utilities are:

(3.18.a)

and

(3.18.b).

If B does not comply with A’s demand and a conflict situation arises, then the parties’ expected utilities are:

(3.19.a)

and

(3.19.b).

Maximizing the Nash product with respect to yields the following solution value, :

(3.20).

By the theory of optimal threat strategy presented in Section 3.2, A’s optimal strategy is to select a value of T minimizing and a value of R maximizing .[2] Harsanyi (1962a) then considered Dahl’s measure of the quantity of A’s power over B (Dahl 1957). This dimension of power was defined by Dahl as the probability difference . It follows immediately from Equation (3.20) that

(3.21).

According to Harsanyi (1962a: 77) the numerator of the first term inside the square brackets on the right-hand side of (3.21), ,

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 strength of A’s power over B. Accordingly, the quotient measures the gross relative strength of A’s power over B with respect to action X.

Similarly, the difference “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 performing action .”(Harsanyi 1962a: 78.) Action denotes an action by A of tolerating B’s noncompliance. Hence,

the quotient, , measures the gross relative strength of B’s power over A with respect to . Finally, the difference, , 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, . 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, . 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 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)

The first term on the right side of Equation (27) is a weighted sum of a constant 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 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 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.