Reputation effects in bargaining games.

Park, Eun-Soo

1. Introduction

Since the publication of Rubinstein's (1982) alternating-offer bilateral bargaining game with complete information, much of the recent literature on bargaining has been devoted to models with incomplete information. (See Osborne and Rubinstein 1990 for a survey.) One of the most popular approaches in this line of research is to base incomplete information on something tangible about the parties involved, such as uncertainty about the buyer's valuation of the good or the seller's production cost of the good.

In light of the recent experimental bargaining results (e.g., Ochs and Roth 1989), however, it is not surprising to see a growing view that much of the recent literature on alternating-offer bargaining with incomplete information does not directly address what we see in the experimental bargaining literature. For example, Kreps (1990a) states that

These formulations do not directly address what we see in the experimental literature and what we have conjectured will be found in subsequent experiments. There is no "tangible uncertainty" at work in the Ochs and Roth experiments. Rather, what seems to be at work are the beliefs of one player concerning equity and greed and the assessment of the other player about the beliefs of the first. It is not clear a priori that the forms of incomplete information that have been investigated so far will speak adequately to effects driven by these other sorts of considerations. (p. 569)

Kreps (1990b, pp. 123-8) adheres to this view and consequently suggested an alternatingoffer bargaining game that is perturbed with a "crazy" type, in that there is a small probability that one player might irrationally insist on a particular allocation. He conjectured that players of the crazy type may indeed get that particular allocation, regardless who has which discount rate.

Myerson (1991, pp. 399-403) has expressed the same view; he characterized the equilibrium of an alternating-offer bargaining game in which one player might be a crazy type who irrationally insists on a particular allocation. Myerson's model can be summarized as follows. First, consider the following slight variation of Rubinstein's (1982) alternating-offer bargaining game in the context of splitting a dollar, a variation based on Myerson (1991, pp. 394-9). Player 1 makes an offer in every odd-numbered round and player 2 makes an offer in every even-numbered round until some offer is accepted or the game ends in disagreement. In each round, after the offer is made, the other player can either accept or reject it. At any round when player i makes the offer and player j rejects it, there is probability [p.sub.i] that player i will walk out and the bargaining will end in disagreement. In this scenario, both players get nothing. This game has a unique (subgame perfect) Rubinstein equilibrium. In equilibrium, the game will end at round 1 with the following Pareto efficient allocation:

([p.sub.1]/[p.sub.1] + [p.sub.2] - [p.sub.1][p.sub.2], [p.sub.2] - [p.sub.1][p.sub.2]/[p.sub.1] + [p.sub.2] - [p.sub.1][p.sub.2])

Now suppose that player 2 assesses probability q that, before the game begins, player 1 is replaced by a computer that is programmed to play the following r-insistent strategy: Offer r = ([r.sub.1], [r.sub.2]) every time he is able to make an offer in which [r.sub.1] + [r.sub.2] [less than or equal to] 1 and both [r.sub.1] and [r.sub.2] are between 0 and 1, and reject any offer that leaves him with less than [r.sub.0]. In this variation of the alternating-offer bargaining game, Myerson showed that in equilibrium, the bargaining game is sure to end within the first J(q, r) rounds if player 1 follows the r-insistent strategy. Besides the fact that Myerson proved the existence of such an upper bound in an ingenious way, a more striking feature of Myerson equilibrium is that J(q, r) does not depend on [p.sub.1] or [p.sub.2].(1)

This result is in sharp contrast to that of Rubinstein, since Rubinstein's equilibrium allocation crucially depends on [p.sub.1]/[p.sub.2]. Note that in Rubinstein's model described above, [p.sub.i] measures player i's "strength" in the sense that it is the probability that any offer he makes, if rejected, may be final. The Rubinstein model suggests that players would reach an agreement in which their relative shares crucially depend on their relative strength, [p.sub.i]/[p.sub.j], even if the absolute level of strength of each player, [p.sub.i], is very low. (For example, if [p.sub.1] = 0.004 and [p.sub.2] = 0.001, then in Rubinstein equilibrium, player 1 will get approximately 4/5 and player 2 will get approximately 1/5 of the pie to be shared.) This result suggests that player 1 could never convince anyone that there are some offers that would be impossible for him to accept. However, in real bargaining situations (for instance, as when bargaining over automobile or house prices), we often try to convince each other that we would never accept any offer outside a particular range, regardless of one's relative strength. To understand the importance of such tactics, Myerson's model introduced the incomplete information in the form of r-insistent strategy (that is, the crazy type using the jargon of game theory).

Thus, Myerson argues that if one player can create some doubt in another player's mind that he may have some irrational commitment to some allocation, then he can expect to get an agreement that is close to or better than this allocation, no matter what the ratio [p.sub.1]/[p.sub.2] may be. Thus, in both Kreps' and Myerson's models, the key in determining the allocations of bargaining games is the anticipation of what one can expect to get and what one expects must be given, rather than the institutional details of the game (such as the players' discount rates).

The main theme of this article is that the bargaining protocols, such as infinite time horizon or alternating offers, are not the key to the results of Kreps' or Myerson's type of bargaining game with incomplete information. In particular, we consider a model of a simultaneous-offer bilateral bargaining with a finite time horizon in which one player might have some kind of irrational commitment that compels him to insist on some bargaining allocation.(2) In this variation, we show that, as in the work of Kreps and of Myerson, beliefs of one player concerning equity and greed and the assessment of the other player about the beliefs of the first are the key in determining the allocations of bargaining games. Kreps (1990a) shares the same view when he conjectures, "'Institutional details' concerning who has which discount rate and what is the precise protocol of bargaining will be swamped by expectations as to what one can expect to get and what one expects must be given in order to come to agreement" (p. 568).

The bargaining protocols considered in my model are similar in spirit to the experimental bargaining game conducted in Roth and Schoumaker (1983). In Roth and Schoumaker's experiments, the subjects played a 25-round bilateral (simultaneous-move) bargaining game using a computer interface. In order to study the role of expectations and reputation in bargaining, Roth and Schoumaker conditioned their subjects by having them play the bargaining game against computers for the first 15 rounds; the subjects were not aware of this.(3) The computer was programmed to play the 50-50 split for some subjects and to play the 20-80 split for the others. Then conditioned subjects were mixed to play the bargaining game with each other. The results were not surprising in that expectations and reputation played an important role in determining the allocations: When two subjects who had been conditioned in compatible ways were matched, they played the equilibrium they had learned. When two subjects who had been conditioned in incompatible ways were matched, either money was left on the table or disagreement outcomes were reached.(4)

2. The Model and Theorem

Consider a static two-person bargaining game, (F, v), where F [subset or equal to] [R.sup.2] is the set of feasible payoff allocations and v = ([v.sub.1], [v.sub.2]) [element of] [R.sup.2] is the disagreement point.(5) Assume that F is a compact and convex set and that the Pareto frontier of F is strictly downward sloping. Assume further that there is some allocation y = ([y.sub.1], [y.sub.2]) [element of] F that is strictly better for both players than disagreement, that is, [y.sub.1] [greater than] [v.sub.1] and [y.sub.2] [greater than] [v.sub.2]. Let [m.sub.i] denote the maximum payoff that player i can get, that is,

[Mathematical Expression Omitted].

Now consider a T round (where T is sufficiently large, but finite) dynamic version of (F, v) where in each round, two players simultaneously and independently name a demand.(6) Let [x.sub.1t] denote player 1's demand in round t for t = 1, 2, . . ., T. Also let [x.sub.2t] denote player 2's demand in round t. If these two demands are compatible in the sense that ([x.sub.1t], [x.sub.2t]) [element of] F, then ([x.sub.1t], [x.sub.2t]) is implemented and the game ends. If not, with probability [p.sub.1], player 1 walks away and terminates the bargaining process. Similarly, let [p.sub.2] denote the probability that player 2 walks away and terminates the bargaining process if the two demands are not compatible. If the demands are not compatible or the game ends in disagreement in any round, each player gets nothing. If the two demands are incompatible but the game does not end in disagreement in the current round, the game moves to the next round.

Now suppose that before the game begins, player 2 assesses probability [Epsilon] that player 1 always names the demand of [r.sub.1] whenever he is able to name a demand. Call this player l's strategy [r.sub.1]-insistent strategy. Let [r.sub.2] denote the maximum payoff that player 2 can get when player 1 gets [r.sub.1]. That is,

[r.sub.2] = max {[y.sub.2][where]([r.sub.1], [y.sub.2]) [element of] F}.

Note that [m.sub.2] [greater than or equal to] [r.sub.2]. To simplify the analysis, suppose [v.sub.1] = [v.sub.2] = 0.

THEOREM. In any sequential equilibrium of the game described above, if player 1 follows the [r.sub.1]-insistent strategy, then there exists a constant M([Epsilon],][m.sub.2], [r.sub.2]) depending on [Epsilon], [m.sub.2], and [r.sub.2] but independent of T, [p.sub.1], or [p.sub.2], such that the game is sure to end within the first M([Epsilon], [m.sub.2], [r.sub.2]) rounds. Furthermore, M([Epsilon], [m.sub.2], [r.sub.2]) is the smallest integer n such that

[Epsilon] [greater than or equal to] [([m.sub.2]/[m.sub.2] + [r.sub.2]).sup.n-1.

So the expected payoff to player 1 in equilibrium cannot be less than [r.sub.1][(1 - max{[p.sub.1], [p.sub.2]}).sup.M]. If [p.sub.1] and [p.sub.2] are small, then this lower bound is close to [r.sub.1].

REMARKS. Some remarks are in order. First, note that in the establishment of such an upperbound M, we need to look at equilibria in which player 1 is not observed deviating from the [r.sub.1]-insistent strategy throughout the game, since any deviation from the [r.sub.1]-insistent strategy immediately reveals that player 1 is not a machine. An important point is that if player 1, a rational type, tries to mimic the crazy type, the relative strength of the two players is of little importance. Second, note that Myerson's proof for the alternating-offer model cannot be extended to the current simultaneous-offer model. The reason is that [p.sub.i] is defined slightly differently because of the difference between the simultaneous- and alternating-offer formats. In Myerson's model, at any round when player i makes an offer ([x.sub.1], [x.sub.2]) and player j rejects it, there is a probability [p.sub.i] that player i walks out and the bargaining will end in disagreement. On the other hand, in the current model, in any round, each player names his own demand [x.sub.i], and if ([x.sub.1], [x.sub.2]) [not an element of] F, where F is the set of feasible allocations, then there is a probability [p.sub.i] that player i walks away.

PROOF. Consider the first k rounds and count the time backward from the kth round. So t = k, k - 1, . . ., 1. Denote by [[Mu].sub.t], the probability assessed by player 2 at the beginning of round t that player 1 is an [r.sub.1]-insistent machine, given that player 1 has always played [r.sub.1] previously.

We claim that if

[[Mu].sub.t] [greater than or equal to] f(t) [equivalent to] [([m.sub.2]/[m.sub.2] + [r.sub.2]).sup.t-1],

then player 2 cannot be willing to withstand the [r.sub.1]-insistent strategy for the next t rounds (including this round) by demanding more than [r.sub.2] during these rounds, if the game does not end in disagreement.

To prove this claim, I will use mathematical induction. For t = 1, it is clear that the claim is true. Now fix the integer t and suppose the claim is true for t - 1. For notational simplicity, let [Mu] = [[Mu].sub.t].

Suppose player 1, as a rational player, plans to follow the [r.sub.1]-insistent strategy throughout these t rounds with some positive probability. In particular, let [[Pi].sub.t] be the probability that player 1 plays the [r.sub.1]-insistent strategy as a rational player in this round. Now suppose that player 2 is following a strategy under which she withstands the [r.sub.1]-insistent strategy throughout these t rounds by demanding more than [r.sub.2]. Call this strategy for player 2 [c.sub.2].

If player 2 follows [c.sub.2], then her expected payoff cannot be greater than (1 - [Mu])[[Pi].sub.t](1 - [p.sub.1])(1 - [p.sub.2])[m.sub.2] + (1 - [Mu])(1 - [[Pi].sub.t])[m.sub.2], since the largest probability that bargaining will not end in disagreement at the end of this round is [[Pi].sub.t](1 - [p.sub.1])(1 - [p.sub.2]) and (1 - [Mu])(1 - [[Pi].sub.t])[m.sub.2] is the maximum expected payoff to player 2 when player 1 is not a machine and demands other than [r.sub.1]. On the other hand, if player 2 plays a strategy other than [c.sub.2] against player 1's proposed strategy, then player 2's expected payoff is at least [[Mu].sub.t][r.sub.2] + (1 - [Mu]) [[Pi].sub.t][r.sub.2].

Thus, in order for [c.sub.2] to be optimal, we must have

[Mathematical Expression Omitted],

or equivalently,

[Mathematical Expression Omitted]. (1)

On the other hand, by the induction hypothesis and Bayes' rule, the optimality of [c.sub.2] requires

[Mathematical Expression Omitted],

or equivalently,

[Mathematical Expression Omitted]. (2)

Combining Equation 1 and Equation 2, we have

[MU](1 - f(t - 1))/f(t - 1) [greater than] (1 - [Mu])[m.sub.2] - [Mu][r.sub.2] / [r.sub.2] + [m.sub.2] - [m.sub.2] (1 - [p.sub.1])(1 - [p.sub.2]),

or equivalently,

[Mu] [less than] f(t - 1). [m.sub.2] / [m.sub.2] + [r.sub.2] - [m.sub.2] (1 - [p.sub.2])(1 - f(t - 1)).

Now define g(t) as

[Mathematical Expression Omitted].

So if [Mu] [greater than or equal to] f(t - 1)g(t), then [c.sub.2] cannot be optimal.

Now it can be shown that

g(t) [greater than] [m.sub.2]/[m.sub.2] + [r.sub.2]

Therefore, if

[Mu] [greater than or equal to] f(t - 1). [m.sub.2]/[m.sub.2] + [r.sub.2] = f(t),

then player 2 cannot be willing to withstand the [r.sub.1]-insistent strategy throughout these t rounds. This completes the proof of the claim.

Now consider when t = k. Since [Epsilon] is the prior, if player 1 follows the [r.sub.1]-insistent strategy and if

[Epsilon] [greater than or equal to] [([m.sub.2]/[m.sub.2] + [r.sub.2]).sup.k-1],

then player 2 cannot be willing to withstand the [r.sub.1]-insistent strategy for the first k rounds if the game does not end in disagreement. Let M([Epsilon], [m.sup.2], [r.sub.2]) be the smallest integer k such that the above inequality holds. QED.

Note that as in Myerson's equilibrium discussed earlier, M([Epsilon], [m.sub.2], [r.sub.2]) does not depend on [p.sub.1] or [p.sub.2], and thus player 1 can expect to get an agreement that is close to his desired allocation, no matter what the ratio of the power of commitment of the players.

As a concrete example, consider a T-round (T sufficiently large, but finite) split-a-dollar game with the same dynamic structure as above, where in each round, two players simultaneously and independently name a demand in [0, 1]. For this game,

F = {([x.sub.1], [x.sub.2]) [Epsilon] [[0, 1].sup.2] [where] [x.sub.1] + [x.sub.2] 1}.

[v.sub.1] = [v.sub.2] = 0, and [m.sub.1] = [m.sub.2] = 1. Let [r.sub.1] [Epsilon] (0, 1), [r.sub.2] = 1 - [r.sub.1], and r = ([r.sub.1], [r.sub.2]). For this game, we have the following corollary:

COROLLARY. In any sequential equilibrium of the dynamic split-a-dollar game described above, if player 1 follows the [r.sub.1]-insistent strategy, then there exists a constant M([Epsilon], r) depending on e and r but independent of T, [p.sub.1], or [p.sub.2], such that the game is sure to end within the first M([Epsilon], r) rounds. Furthermore, M([Epsilon], r) is the smallest integer n such that

[Epsilon] [greater than or equal to] [(1/1+ [r.sup.2]).sup.n-1].

For example, when [Epsilon] = 0.1 and player 1 follows the 0.9-insistent strategy, the game will end within the first 25 rounds. Note that M([Epsilon], r) does not depend on [p.sub.1] and [p.sub.2]. Also note the natural monotonicity of the upper bound: M([Epsilon], r) increases in [r.sub.1] and decreases in [Epsilon].

3. Conclusion

Before closing, I discuss a potential experimental test for the results presented in this paper. As noted earlier, Roth and Schoumaker's (1983) experiment results are similar in spirit to the current results in that expectations are an important factor in reaching agreement. However, in Roth and Schoumaker's experiment, they conditioned their subjects by having them play the bargaining game against computers that were programmed to play a fixed split for the first 15 rounds, though subjects were not aware of this. Thus, in their experimental design, the reputation-building behavior of the player who enjoys the uncertainty about his true type cannot be explained.

In this respect, experiments conducted by Camerer and Weigelt (1988) and Neral and Ochs (1992) provide useful insights. In Camerer and Weigelt's experiment, they looked at the reputation-building behavior for a game with one-sided uncertainty. By changing the payoff parameters and the probability of the crazy type, they found that reputation-building behavior played a key role, as the theory predicted. On the other hand, Neral and Ochs pointed out that Camerer and Weigelt changed the two parameters (payoffs and the probability of the crazy type) simultaneously, so the effect of the change in payoff could not be estimated independently of the effect of a change in the probability. Thus, they examined the reputation-building behavior by changing the payoff parameters while holding the probability of the crazy type fixed. In their experiment, they found that while there is a systematic reputation-building behavior in response to changes in payoff parameters, the behavior is not in the direction predicted by the theory.

For a potential experimental test of the results in this paper (or in Myerson 1991), Roth and Schoumaker's experiment could be modified using the insights provided by Camerer-Weigelt's and Neral-Ochs' experiments. For example, we can develop an experiment in which subjects play a two-person (finite-horizon) dynamic bargaining game over a pie of fixed size (say, $10) and one subject (say, player 1) is designated as the one who may be of, say, $7-insistent type with some known probability, [Epsilon]. Also, in order to represent the probability [p.sub.i] that player i walks out after the disagreement, we can specify the rate at which the pie is shrinking for each player by providing the payoff table for each round. In this design, we can control [Epsilon] and [p.sub.i]. The equilibrium prediction from the model in this paper and Myerson's is that for a larger value of [Epsilon], ceteris paribus, the normal type of player 1 will play the insistent strategy with a higher probability in earlier rounds and player 2 will concede earlier. Also, the round by which player 2 concedes would not be sensitive to the changes in the payoff structure, that is, to the changes in the rate at which the pie is shrinking, ceteris paribus. It would be interesting to test these hypotheses in a well-designed experiment.

1 Drawing upon the results from the repeated game theory, one might suspect that this dynamic (not repeated) bargaining game will end within the first several rounds; however, it needs a lot of work to prove the existence of an upper bound that is independent of players' discount rates.

2 See Dekel (1990) for some features of a simultaneous-offer bargaining game that the model of alternating offer fails to capture.

3 See Roth and Schoumaker (1983) for the detailed rules of their bargaining game.

4 Roth and Schoumaker did not report when agreements were reached. The current paper offers the maximum periods that will elapse before agreement is reached. This result cannot be tested with Roth-Schoumaker's experimental results.

5 See, for example, Myerson (1991, chap. 8) for a rigorous introduction to this subject.

6 The same result holds for the infinite horizon.

References

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