2' A bargaining set 91 is called symmetric if

(n 1,n 2) (ri2, rij) e 9C

A generalized bargaining game (^,11^,0) is a symmetric game if and only if SK is symmetric, Ilf = f l2 and that 0 t = 0 2.

It is obvious that if a generalized bargaining game is symmetric, then the players will receive equal payoffs at the solution of the game. The argument is as follows: The game is symmetric implies that for any permutation r on players we have

( %n ', 0 ) = ( r % r U d,rQ). Therefore, p ( % U d , S) = p ( r % r U d ,rQ). Since, the solution function p is symmetric, we must have p ( r % r U d, T0) = TJX (SR, IT*, 0 ). That is, (px, p 2)= r(pl, p 2) = (p2, pl). Therefore, we have/ij = p 2.

We know from theorem 6.2 that a solution to the bargaining problem satisfies the axioms 1-3 if and only if it maximizes the "asymmetric" Nash product. The

Corollary 6.1 shows that the solution also satisfies the axiom of symmetry. This result holds as long as the bargaining powers are assigned to the players, not to the axes. Since Roth has shown that satisfaction of axiom 1-3 implies satisfaction of the axiom of efficiency, it follows that the solution to any arbitrary bargaining game, for given distribution of bargaining powers, satisfies all the five axioms as satisfied by the

original Nash's solution if and only if it maximizes the corresponding asymmetric Nash product over the bargaining set. The only difference is that the original Nash solution applies in the case when all players have equal bargaining power. In the presence of uneven bargaining power it loses its symmetric property. The solution that follows from theorem 6.2, however, applies to any arbitrary distribution of bargaining power

provided that bargaining power of each player is strictly positive.12

For this reason, in what follows the so-called ‘asymmetric’ Nash solution to any bargaining game with arbitrary distribution of bargaining power will be called the generalized Nash solution. The product ( H - n f 'p ( n 2 - n 2)ÖJ will be called the generalized Nash product.

6 3 The Bargaining Process

Since the bargaining problem in the tariff game is essentially the same as any abstract bargaining game studied by game theorists, we can obtain insights into the underlying bargaining process from their studies as well. The explanation of the bargaining process most frequently referred to is that of Zuthen (1930). He assumed that in a bargaining process players offer proposals to each other, and postulated that each party will make concessions to his opponent once he finds that his opponent's determination is firmer than his own. Harsanyi (1956) has shown that this process of negotiation is mathematically equivalent to that of Nash's solution.13

In recent times, game theorists have started to study the bargaining process by specifying every move of the players (for example, Rubinstein, 1982). This is often called the sequential strategic approach. Rubinstein showed the existence of a unique

12

13

Otherwise the solution will not be unique. See Binmore (1987b).

perfect Nash equilibrium in any bargaining game, if players have sufficiently high time- preference rates and/or every player has to bear a fixed cost of bargaining for each period.

Binmore, Rubinstein and, Wolinsky (1986) not only studied bargaining problems using the sequential strategic approach but also studied the relationship between the solutions obtained from this approach and that of the axiomatic approach or static representation of a bargaining problem. Moreover, they examined the two motives behind the bargaining process that may induce the players to agree rather than to insist indefinitely on incompatible demands.

One of the motives studied by Binmore, Rubinstein and, Wolinsky (hereafter BRW) was the player's "impatience to enjoy the bruits of an agreement", which is concerned with the relative time preference of the players, and the other motive was the player's relative fear of disagreement.

If the players do not have a high enough time-preference rate, then they may keep on insisting on incompatible demands and no agreements may be reached. Making use of Rubinstein's seminal study (Rubinstein, 1982) Binmore (1987b) showed that the relative difference in the time preference rate of the players can be a source of unequal bargaining power of the players in the static representation of the game. Player with relatively higher time preference rate will have lower bargaining power. This was also obtained by BRW.

In their comprehensive study of the strategic models of bargaining BRW also studied a game in which players faced an exogenous risk of breakdown of negotiations. They also found the existence of a unique perfect equilibrium in this game. They showed that if the players differ in their beliefs concerning the likelihood of a breakdown of the negotiation, then the unique perfect equilibrium of the game approaches to the ‘asymmetric’ Nash bargaining solution to the static version of the game. The correspondence is that if a player’s estimate of the probability of breakdown is higher relative to his opponent, his bargaining power will be correspondingly

lowered.

These results have two important implications. First, if players differ in their time-preference rate and/or in the probability of exogenous breakdown of negotiation, then the solution to the bargaining problem is given by the generalized Nash bargaining solution, whereas if the players do not differ in their beliefs (hold equal probability of breakdown, and have identical time preference rates), then the original Nash solution is applicable. Second, if each player assigns a constant probability to the breakdown of the negotiation, then the fear of breakdown consistent with this probability is captured by

the asymmetric bargaining powers of the players. The player who is relatively more fearful will have lower bargaining power.

63.1 Fear of Ruin: Another Fear of Disagreement

To probe the fear of disagreement further, let us assume that the players hold identical beliefs about the external environments, have identical time preferences, etc., such that the players end with having equal bargaining powers in BRW’s sense. Now let us consider a situation as described in figure 6.3.

Fear of ruin Figure 6.3

Assume that RF is the rent transformation frontier (the Pareto efficient boundary of the bargaining set in an arbitrary bargaining game) and FI0 is a distribution of rents at any mutually agreeable14 relative price P°. Without loss of generality, let us take the case of player 1. Suppose further that for a small change of [SPX in Px the resulting distribution of rent in market equilibrium is given by the point FI0 + ATI. This means that player 1 gains by AF^ and player 2 looses by AFI2 (ATI2 < 0) if the price increase actually takes place. Therefore, player 1 may insist on such a price increase and player 2 is likely to oppose (or reject) it. Would player 1 insist on a price increase?

If player 1 insists on an increase in Px, player 2 has two options: accept it, or quit the negotiation table and play a noncooperative game. Under this circumstance,

14 To each player we can always take an offer o f the other party as mutually agreeable price. Because, it will be agreed upon if the player in question accepts it.

insisting on such a price increase implies a gamble on the part of player 1. If it is accepted by player 2, player 1 will get a gain of A F^; if it is rejected by player 2, and player 2 quits the table (player 1 is ruined), player 1 will lose the entire gain of (111 - n f ), and end up with the disagreement payoff of I l f .

Suppose that at FT, player 1 believes that if he insists on a small increase in price, then the probability that player 2 quits the negotiation and the disagreement results is (ft (FT). Let (ft ( n j \ AF^) be such that player 1 is indifferent between insisting

on a price increase for a contingent gain of AF^ with probability (1 - qx) and accepting

FI" with certainty. That is, let

Then, qx provides the threshold probability at FI0 such that qx > qx implies that player 1 will insist on a price increase; qx < (ft implies that player 1 will prefer the certain outcome n j ’ and will not insist on a higher price for his commodities. This means that (ft measures the ‘boldness’ of player 1 in an environment defined by (FT, FT*).

Certainly, player 1, as can be seen from equation (6.3), will not risk the same amount of gain (FI, - n f ) for smaller amounts of contingent gains with identical probability distribution. To induce him to risk (FI" - n f ) for a gain that is smaller than

Arij and remain indifferent ex ante, the probability of ‘ruin’ - that player 2 quits and disagreement results - has to be smaller than that corresponds to AFIj. Therefore, for given (FI" - H x), the threshold probability t ATlj) declines as Aft; gets smaller and smaller, and needs to be standardized to make a measure o f player l's boldness.

Symmetric arguments can be made for player 2. Hence the following definition due to Aumann and Kurz (1977a).