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The Competitive Ultimatum Game The Competitive Ultimatum Game When Competition Matters, Although It Should Not, and Backward Induction Appears Through the Backdoor Klaus Abbinkb, Ron Darzivj, Zohar Gilulaj, Harel Gorenj, Bernd Irlenbusche, Arnon Kerenj, Bettina Rockenbache, Abdolkarim Sadrieht, Reinhard Seltenb, and Shmuel Zamirj University of Bonnb, Hebrew University Jerusalemj, University of Erfurte, Tilburg Universityt Preliminary version, April 2000. Comments are welcome. Abstract A game is introduced and experimented that allows for a direct comparison of the two predominant equilibrium concepts of economic theory: the competitive versus the strategic equilibrium. Neither of the two extreme and completely disjoint equilibrium predictions fully explains the data. Observed outcomes, however, are closer to the competitive than the subgame perfect (sequential) equilibrium predictions. As subjects gain experience in repeated plays of the game, outcomes move even closer to the competitive equilibrium. But, within play behavior frequently exhibits patterns that qualitatively correspond to the game theoretic predictions. Thus, although the competitive equilibrium concept clearly is the better predictor of behavior in the given environment, the strategic equilibrium concept also retains some of its behavioral relevance. Keywords Competition, backward induction, equilibrium concepts, experimental economics JEL Classification Codes C78, C91, C92, D82 Acknowledg ements The authors thank the teams of the RatioLab at the Hebrew University of Jerusalem and the Laboratorium für experimentelle Wirtschaftsforschung at the University of Bonn for their aid in collecting the data. Many thanks also to the seminar participants at the ENDEAR Bari workshop and summer school (1999) as well as at the IAES meeting in Munich (2000) for many helpful comments. Support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 303, by the German-Israeli Foundation ! GIF !, by the European Union through the TMR program ENDEAR (FMRX-CT98-0238), and by the Land Nordrhein-Westfalen is gratefully acknowledged. Correspondence Karim Sadrieh, Dept. of Economics, Tilburg University, POBox 90153, 5000 LE Tilburg, Netherlands - [email protected] 1 1. Introduction Ever since game theory introduced a new equilibrium concept to economic theory, the game-theoretic strategic equilibrium has co-existed with the traditional competitive equilibrium. The co-existence has been “friendly” in the sense that the followers of neither concept have seriously attempted to extinguish the application of the concept they do not fancy. One reason for this persistent duality lies in the fact that the theoretic delineation of the concepts has proven to be intricate. Not only do both concepts (in principle) allow for multiplicity of equilibria with a given set of assumptions, but both also allow an application to models with a broad range of possible assumptions. Perhaps it is due to these difficulties, that general results concerning the relationship of the strategic and the competitive equilibrium have not been derived so far. Generally, it seems possible to devise economic institutions in which the two equilibrium concepts produce any constellation of conclusions: equilibrium outcome sets that coincide, equilibrium outcome sets that are disjoint, or equilibrium outcome sets of which one is partially or wholly contained in the other. Clearly, the exciting case for research is the case in which equilibrium predictions do not coincide. In such cases, when the equilibrium concepts are in competition with one another, the problem of comparing and evaluating the conflicting equilibrium concepts emerges. This is the issue that is at the heart of the present paper. Instead of taking an axiomatic approach to tackle the issue of equilibrium concept evaluation, an empirical approach is attempted with this research. Using the method of experimentation, we study a game in which the set of outcomes predicted by the competitive equilibrium and the set of outcomes predicted by the strategic equilibrium are disjoint. The goal is to evaluate the two equilibrium predictions in terms of their behavioral relevance. Obviously, if one of the equilibrium outcomes predominates our experimental observations, conclusions concerning the behavioral relevance of the concepts in our setting will be apparent. Our design, however, also will allow us to draw more intricate conclusions from the comparison of observed decision paths. Finally, the robustness of the results with respect to changes in the informational setting will be explored by comparing the results of experimental sessions with a perfect to those of sessions with an imperfect information version of the underlying game. To our knowledge, this is the first systematic experimental study concerned with the evaluation of the competing equilibrium concepts. However, an interesting benchmark for our work is supplied by PRASNIKAR and ROTH (???). Interested in tests of an adaptive learning model, they conducted experi- ments with a game in which the predictions of the two equilibrium concepts coincide at an extreme outcome. They find that subjects’ behavior rapidly converges to this jointly strategic and competitive equilibrium of the game. In a way, we find this result comforting, since it confirms the behavioral validity of at least one of the two equilibrium concepts. But, since the predicted outcomes fully overlap, it is not clear whether observed behavior is actually driven by competition or by the backward induction logic of the sub-game perfect equilibrium. 2 In the competitive ultimatum game, which is introduced in this paper, the competitive and the strategic equilibrium predictions are separated at the two extreme ends of the range of possible outcomes. In this game, three proposers, one after the other, can make an offer to a single responder. If the responder accepts the first offer for the division of the cake, the game ends. Otherwise, the second proposers makes an offer, and so forth. The third stage of the game, when only one proposer is left, corresponds to a common ultimatum game. Backward induction implies that the strategic bargaining power of the responder is very limited in this game. Anticipating that the responder will accept any non-negative amount that is proposed in the last stage, the last proposer will offer no more than the smallest money unit. Anticipating this, the one to last proposer will offer no more than two times the smallest money unit. Finally, the backward induction logic predicts that the responder accepts the first proposer’s offer that is no more than three times the smallest money unit. Thus, in subgame perfect (or sequential) equilibrium, the responder receives a tiny piece of the cake, while the first proposer receives almost the entire cake. From a competitive equilibrium perspective, however, the responder has an extremely strong monopolis- tic position, while the three proposers face sharp competition. Only one of the proposers can succeed. In fact, supposing that the three proposers are three sellers and the responder is a buyer and looking at the corresponding demand and supply curves, we find that the competitive equilibrium price is zero, since the marginal cost of all sellers is zero. Thus, in competitive equilibrium, the responder, who is predicted to receive almost nothing in the strategic equilibrium, receives almost the entire cake, leaving the pro- poser with close to nothing. The experiments show that the competitive pressure on the proposers is clearly greater than the bargain- ing power that the game theoretic reasoning attributes to them. The competition amongst proposers leads to high offers: On average more than half of the cake is offered to the responder. Furthermore, as the subjects gain experience with the game, the offers to the responder tend to rise. The stiff competition pushes first proposer offers from slightly above 50% in the first round to well over 60% in round 35. Since an upward trend in the offers can be clearly seen in 22 of 24 independent subject groups, the conclusion seems justified that observed outcomes develop in the direction predicted by the competitive equilibrium concept, although the competitive equilibrium outcome is not reached. However, a second strong regularity appears in the data. In general, the observed offers in a game decline as the game enters later stages, going from one proposer to the next. In the last stage, most observed offers are in the range that is typically also observed in ultimatum bargaining experiments. Thus, as the responder’s threat of rejection loses its credibility, because the end of the game comes closer, the proposers' offers drop. It is in this effect that the backward induction reasoning strongly reappears in our data. 3 The rest of the paper is organized as follows. First, in section 2, the competitive ultimatum game is defined precisely and analyzed game-theoretically. Then, after the experimental design is described in section 3, the results are reported in section 4. 2. The Competitive Ultimatum Game The competitive ultimatum game is an extension of the 'classical' ultimatum game (GÜTH, SCHMITTBERGER, and SCHWARZE 1982) with three (potential) proposers P1, P2, and P3 and one responder R. The three proposers sequentially propose an allocation of the cake C to the responder. First, P1 proposes an allocation a1=(x1,C-x1) of C to R, where x1 denotes the proposed payoff for the responder R and C-x1 denotes the own payoff desired by P1. The game ends if R accepts the proposal of P1. Otherwise, if P1's proposal is rejected by R, P2 proposes an allocation a2=(x2,C-x2) of C to R. If R accepts P2's proposal, the game ends. Otherwise, if R also turns down P2's proposal, it is P3's turn to propose an allocation a3=(x3,C-x3) of C to R. If R accepts the proposal of proposer P3, R receives x3, P3 receives C-x3, and each of both other proposers receives 0. If R rejects all three proposals all four players receive 0. Suppose that each proposer Pi is completely informed about the proposal(s) of the proposer(s) who have decided before Pi.
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