Bargaining Under a Deadline: Evidence from the Reverse Ultimatum Game ✩
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Games and Economic Behavior 45 (2003) 347–368 www.elsevier.com/locate/geb Bargaining under a deadline: evidence from the reverse ultimatum game ✩ Uri Gneezy,a,b Ernan Haruvy,c and Alvin E. Roth d,e,∗ a University of Chicago Graduate School of Business, Chicago, IL, USA b Technion, Israel Institute of Technology, Haifa, Israel c School of Management, University of Texas at Dallas, Richardson, TX, USA d Department of Economics, Harvard University, Cambridge, MA, USA e Harvard Business School, Boston, MA, USA Received 8 January 2003 Abstract We study a “reverse” ultimatum game, in which proposers have multiple chances to offer responders a division of some fixed pie. The game ends if the responder accepts an offer, or if, following a rejection, the proposer decides not to make a better offer. The unique subgame perfect equilibrium gives the proposer the minimum possible payoff. Nevertheless, the experimental results are not too different from those of the standard ultimatum game, although proposers generally receive slightly less than half of the surplus. We use the reverse ultimatum game to study deadlines experimentally. With a deadline, the subgame perfect equilibrium prediction is that the proposer gets the entire surplus. Deadlines are used strategically to influence the outcome, and agreements are reached near the deadline. Strategic considerations are evident in the differences in observed behavior between the deadline and no deadline conditions, even though agreements are substantially less extreme than predicted by perfect equilibrium. 2003 Elsevier Inc. All rights reserved. JEL classification: C7; C9 ✩ Accompanying materials to be found at http://www.utdallas.edu/~eharuvy/deadline/. * Corresponding author. E-mail address: [email protected] (A.E. Roth). 0899-8256/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0899-8256(03)00151-9 348 U. Gneezy et al. / Games and Economic Behavior 45 (2003) 347–368 Prologue: In memory of Bob Rosenthal One of Bob’s many contributions was the centipede game (Rosenthal, 1981), which captured the attention of the profession as a model for exploring the kind of perplexing backward induction questions that arise, e.g., in the finitely repeated prisoner’s dilemma. Both of those games, and also the ultimatum game, have received a lot of attention from experimenters interested in the tension between backwards induction and psychological and other strategic considerations. (Bob himself became an experimenter in recent years.) The present paper, dedicated to Bob’s memory, proposes another such game, the “reverse ultimatum game.” It is useful here for studying deadline effects in bargaining because its backward induction equilibrium properties are sensitive to the presence or absence of a deadline. Like many game theory papers, this one would have been better if we could have run it by Bob’s critical eye. We miss him. 1. Introduction Eleventh hour agreements are often reported in negotiation settings. Labor agreements tend to be reached just before contracts expire and court settlements tend to be reached just before or at the deadline. Williams (1983), for example, found that 70% of all civil cases in one sample were settled in the last 30 days before the trial and 13% were settled on the day of the trial (see also Spier, 1992). In experimental settings, the concentration of agreements near the end of the allotted time appears to be far more robust to bargaining parameters than are other aspects of the bargaining outcome (Roth et al., 1988). In some settings, a possible explanation of the deadline effect appears to be straightfor- ward: Consider a finite-horizon negotiation without discounting, and suppose players have the option of delaying. Then the player who moves at the last possible time might hope to capture the lion’s share of the surplus by making an ultimatum offer (Ma and Manove, 1993).1 A one-step backward induction argument reveals that the subgame perfect equilib- rium of an ultimatum game gives all or virtually all the surplus to the proposer, so under this argument, the deadline effect is the result of strategic behavior aimed at creating the conditions to issue an ultimatum. However, when ultimatum bargaining is examined in the laboratory, the large advan- tages to the proposer predicted by subgame perfect equilibrium are most often notable by their absence. A typical experiment of this sort uses the ultimatum game first stud- 1 In other settings, incomplete information is often used to explain delays in bargaining agreements (Fudenberg and Tirole, 1983; Fudenberg et al., 1985; Hart, 1989; and Rubinstein, 1985), where typically an informed bargainer in a “strong” position can signal his position by delaying agreement as the probability that an agreement can be reached before the deadline diminishes. Fershtman and Seidmann (1993) showed that in a sequential bargaining model, negotiators delay agreements until the deadline if a player who rejects an offer is committed not to accept any poorer proposal in the future. In general, strategic uses of time can have multiple causes, since a single dimension (time) is being manipulated in response to complicated strategic considerations. For example, see Roth and Ockenfels (2002) for a discussion of the multiple causes of last minute bidding on eBay. (There is also an Organizational Behavior literature that considers the perception of deadlines; see, e.g., Moore, 2000.) U. Gneezy et al. / Games and Economic Behavior 45 (2003) 347–368 349 ied by Güth et al. (1982), in which one player makes a proposal over how to divide a fixed sum, and a second has only the options of accepting the proposal—in which case the proposed division takes effect—or rejecting it, in which case both the proposer and the responder receive zero. If players care only about their own payoffs, the subgame perfect equilibrium prediction for such a game is that the proposer will receive virtually all of the pie to be divided, while the responder will get at most the smallest monetary unit in which proposals can be made. However, the experimental results (see, e.g., the survey in Roth, 1995) are that responders receive much closer to half of the pie. This result is robust (see, e.g., Roth et al., 1991), which seems to involve strong initial pref- erences involving fairness (see, e.g., Ochs and Roth, 1989; Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999), and it is persistent, which seems to involve the different speeds of learning that the game induces in proposers and responders (Roth and Erev, 1995; Slonim and Roth, 1998; Cooper et al., 2003). The goal of the present study was to design and explore an environment in which the subgame perfect equilibrium predictions are as clear, and extreme, as in the ultimatum game, but in which time can be used strategically in a way that will let deadlines become potentially important.2 To this end, we examine the role of deadlines in a novel bargaining environment, which we call the reverse ultimatum game (RUG). In the RUG, the addition of a deadline can drastically shift the subgame perfect equilibrium prediction from one extreme to the other in terms of which bargainer is predicted to gain all but a fraction of the available wealth. In its simplest form, the RUG involves two players. Player 1 proposes a division of a fixed amount (in our case 25 tokens) to player 2 in the form of an integer offer of x tokens. If player 2 accepts the x tokens, then the game ends with this division as the outcome. If player 2 rejects the offer, player 1 is then allowed to make another offer, as long as that offer is strictly higher by a minimum increment (1 token), and as long as both players’ shares remain strictly positive. In addition, player 1 may end the bargaining at any point, in which case both players receive 0 tokens. That is, the game ends either when player 2 accepts a proposal, or when, following a rejection, player 1 declines to make a better offer. The game would also end if player 2 rejects the highest feasible offer player 1 is able to make. We call the game a “reverse” ultimatum game because player 2’s rejection of an offer is a form of “reverse” ultimatum, which may be interpreted as meaning “give me more, or we will each get nothing,” and because the subgame perfect equilibrium division between proposer and responder is the reverse of that in the ultimatum game. The proof that the unique subgame perfect equilibrium gives one token to the proposer and the remainder (N − 1 tokens) to the responder is as follows. At a subgame following 2 This is not to say that imposing a deadline in a conventional ultimatum game could not in some circumstances have an effect on observed behavior, but it does not enter into the strategic calculations that determine the subgame perfect equilibrium, which in the conventional ultimatum game is invariant to the presence of a deadline. See also Güth et al., 2001, which studies deadline effects in an ultimatum game in which the proposer may make (and repeat) a demand over several periods. When the pie size is constant, the ability to issue an ultimatum over several periods does not change the perfect equilibrium prediction, and Güth et al. find that it does not make a big difference in the observed distribution between proposer and responder. 350 U. Gneezy et al. / Games and Economic Behavior 45 (2003) 347–368 a proposal that gives the proposer only a single token, the responder will accept, because the requirement that both players’ shares must be positive implies that the proposer cannot make another offer if this proposal is rejected, so a rejection will lead to both players receiving zero.