Spite and reciprocity in auctions

Games

Volumn 2, Issue 3, 2011, Pages 365-411

  Nishimura, Naoko ^a   Cason, Timothy N ^b   Saijo, Tatsuyoshi ^c,d   Ikeda, Yoshikazu ^e  

^a SHINSHU UNIVERSITY   (Japan)

^b PURDUE UNIVERSITY   (United States)

^c OSAKA UNIVERSITY   (Japan)

^d UNIVERSITY OF CALIFORNIA   (United States)

^e UNIVERSITY OF KITAKYUSHU   (Japan)

Author keywords

Ascending bid auction; Dominant strategy; English auction; Experiment; Reciprocity; Second price auction; Social preferences; Spite

Indexed keywords

EID: 84865059415     PISSN: None     EISSN: 20734336     Source Type: Journal    
DOI: 10.3390/g2030365     Document Type: Article

References (51)

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8. Kagel, J.H.; Harstad, R.M.; Levin, D. Information impact and allocation rules in auctions with affiliated private values: A laboratory study. Econometrica 1987, 55, 1275-1304.
9. Chew, S.H.; Nishimura, N. Revenue Non-Equivalence between the English and the Second-Price Auctions: Experimental Evidence. J. Econ. Behav. Organ. 2003, 51, 443-458.
10. The "Japanese" version of English auction is the type of ascending-bid auction we employed in our theoretical and experimental analyses, where all the bidders are active from the start by pushing their buttons as default and they leave the auction by releasing their buttons. Thus, they are not allowed to leave the auction and reenter at the later stage. It is well known that the value-revealing bidding strategy generates the unique dominant strategy equilibrium and there is no room for any asymmetric equilibrium such as jump-bidding strategy (see Milgrom and Weber [11] for example.)
11. Milgrom, P.; Weber, R. A Theory of Auctions and Competitive Bidding, Econometrica 1982, 50, 1089-1122.
12. Saijo, T.; Nakamura, H. The "spite" dilemma in voluntary contribution mechanism experiments. J. Confl. Resolut. 1995, 39, 535-560.
13. Rabin, M. Incorporating fairness into game theory and economics. Am. Econ. Rev. 1993, 83, 1281-1302.
14. Segal, U.; Sobel, J. Tit for tat: Foundations of preferences for reciprocity in strategic settings. J. Econ. Theor. 2007, 136, 197-216.
15. Dufwenberg, M.; Kirchsteiger, G. A theory of sequential reciprocity. Game. Econ. Behav. 2004, 47, 268-298.
16. Falk, A.; Fischbacher, U. A theory of reciprocity. Game. Econ. Behav. 2006, 54, 293-315.
17. Morgan, J.; Steiglitz, K.; Reis, G. The spite motive and equilibrium behavior in auctions. Contrib. Econ. Anal. Pol. 2003, 2, 5:1-5:27.
18. Cooper, D.; Fang, H. Understanding overbidding in second price auctions: An experimental study. Econ. J. 2008, 118, 1572-1595.
19. Andreoni, J.; Che, Y.-K.; Kim, J. Asymmetric information about rivals' types in standard auctions: An experiment. Game. Econ. Behav. 2007, 59, 240-259.
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21. Arora, A.; Greenwald, A.; Kannan, K.; Krishnan, R. Effects of information-revelation policies under market-structure uncertainty. Manag. Sci. 2007, 53, 1234-1248.
22. Cason, T.N.; Kannan, K.; Siebert, R. An experimental study of information revelation policies in sequential auctions. Manag. Sci. 2011, 57, 667-688.
23. Edelman, B.; Ostrovsky, M.; Schwarz, M. Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords. Am. Econ. Rev., 2007, 97, 242-259.
24. As highlighted in Cason et al. [3], subjects fail to choose dominant strategies when other, non-dominant strategy Nash equilibria exist. This led Saijo et al. [25] to introduce secure implementation, where the goal is achieved by both dominant and Nash equilibria.
25. Saijo, T.; Sjostrom, T.; Yamato, T. Secure implementation. Theor. Econ. 2007, 2, 203-229.
26. Their original paper deals only with the incomplete information environment, which we extend here to the complete information setting.
27. Their model has the simplest form among various reciprocity models proposed, yet general enough to include such models as Rabin [13] and Charness and Rabin [28]. They provide axiomatic foundation for reciprocity models and obtain utility representation theorem which prescribes utility function Ui(σi, σ*j) of the form (3) except that σi is player i's mixed strategy and σ*j is player i's belief about player j's mixed strategy. Also the constant term Rj =0 in the Segal and Sobel's model, whereas Rj =0 in other reciprocity models. Like other reciprocity models, they consider the framework of the psychological game developed by Geanakoplos, Pearce, and Stacchetti [29] in which beliefs play an important role. Segal and Sobel [14] showed that the concept of Nash equilibrium directly applicable in their framework in the sense that each player i considers her best response for a given belief σ*j, and in equilibrium the belief must coincide with the actual strategy choice, i.e., σ*j = σj. As long as we restrict our attentions to the analysis of equilibrium behavior, not off-equilibrium, we can avoid discussions of beliefs. Thus, we conduct our analysis based on a degenerate form of (3) treating beliefs implicitly, although we include an analysis that incorporates beliefs in our working paper Nishimura et al. [30]. For more detail discussions of beliefs, see Sobel [31].
28. Charness, G.; Rabin, M. Understanding social preferences with simple tests. Q. J. Econ. 2002, 117, 817-869.
29. Geanakoplos, J.; Pearce, D.; Stacchetti, E. Psychological games and sequential rationality. Game. Econ. Behav. 1989, 1, 60-79.
30. Nishimura, N.; Cason, T.; Saijo, T.; Ikeda, Y. Spite and Counter-Spite in Auctions; Working paper; Shinshu University: Matsumoto, Nagano, Japan; Purdue University: West Lafayette, IN, USA, 2007.
31. Sobel, J. Interdependent preferences and reciprocity. J. Econ. Lit. 2005, 43, 392-436.
32. We set reference losing payoff to be zero for both buyers. The reciprocity model prescribes utility as a sum of own monetary payoff and psychological payoff. Note that buyer i's psychological payoff in the event of winning is zero and omitted in (3), since both her rival's payoff and reference losing payoff in that event are zero.
33. To be precise, the negative weight means spite motivation.
34. It is common in reciprocity models to normalize the deviation from some reference by the payoff range defined by the maximum and the minimum payoff available under the relevant consideration. Here, we add ccto the denominator to be consistent with the case of ascending-bid auction, without loss of generality. Such need arises due to the requirement of a discrete bid strategy space.
35. In order to identify B, we first solve for a real number xe [0,6] that satisfies i.e. equating buyer I s winning and losing payoff when fej =b2 = x. We then find the nearest 0, e B satisfying xe It is easy to check that exists and A For more detail, see Appendix A-1.
36. Based on Lemma 1, we can derive the best response correspondences for both buyers. See Appendix A-2.
37. The position of buyer's threshold bids depend on Ri The conclusions of our qualitative analyses summarized at the end of subsection 2.1A, 2.1B, and 2.2, however, are unaffected by the size of the reference payoffs.
38. This is not the only case where allowing for the possibility of reciprocity leads to eliminating some of Nash equilibria generated by the standard model. The well-known example is an ultimatum game which has multiple Nash equilibria under the standard model. Allowing reciprocity eliminates the proposer's strategy of choosing the most self-advantageous offer expecting the receiver to accept it that is sub-game perfect Nash equilibrium under the standard model, but keeps other Nash equilibria that would have failed the sub-game perfect refinement.
39. A change in the weight j iiiias r increases can lead to a different bid decision. As shown in Proposition 2 below and related discussion to follow, each equilibrium bid decision at the every decision point reached in the ascending-bid auction is also equilibrium in the previous decision point (not vice versa), and all the way back to the very start of auction. In this sense our model maintains consistency. The related issue can be found in the context of sequential decision making through partial resolution of risk particularly via non-expected utility preferences. If we want to maintain the dynamic consistency in preferences, it is known that preferences must satisfy consequentialism (Hammond [40]). It seems, however, inappropriate to impose consequentialism on preferences especially when we deal with players' intentions explicitly.
40. Hammond, P. Consequentialist foundations of expected utility. Theor. Decis. 1988, 25, 25-78.
41. In the same manner as described in [35], p-Lr is derived from the solution i.e. equating buyer l's winning and losing payoff when Then, find the smallest number e Br satisfying It can be shown that. For more details, see Appendix A-4.
42. Though we omit the argument regarding beliefs in our analysis, it may be worthwhile to point out the following; First, it is clear from Definition 3 that every equilibrium bidding strategy profile (&, b'2) belongs to interim equilibrium Er for all >-Second, since £f has the lowest upper bound, there would not arise any need to revise a belief that is consistent with equilibrium bid on the equilibrium path. With these two properties, our Definition 3 is consistent with the definition of Sequential Reciprocity Equilibrium by Dufwenberg and Kirschteiger [15].
43. The psychological dynamics are different in nature from the comparison of Nash equilibrium versus subgame perfect Nash equilibrium The subgame perfect equilibrium set in the conventional self-interested preferences ascending-bid auction is equivalent to the Nash equilibrium set in the second price auction without its inefficient elements. However, with the spite motivation the maximum equilibrium price in the ascending bid auction is f. Thus, the ascending-bid auction price predicted for buyers of spite-with-reciprocity is bounded by the maximum price predicted in the conventional model.
44. The corresponding tie bid xr defined in [41] is 748.3 for r = 720, 746.3 for r = 730, and 744.2 for r = 740. It follows that t= 740.
45. It is well known that the symmetric equilibrium bidding strategy must be strictly increasing in the buyer type (value in this model) in the auction where the highest bidder wins; for example, see Milgrom [46].
46. Milgrom, P. Putting Auction Theory to Work; Cambridge University Press: Cambridge, UK, 2004.
47. Kagel, J. Auctions: A survey of experimental research. In The Handbook of Experimental Economics; Kagel, J., Roth, A., Eds.; Princeton University Press: Princeton, NJ, USA, 1995.
48. These figures pool the data from the treatments with fixed groups and randomly-reformed groups of bidders. The subsequent analysis controls for different matching rules in the parametric regression models.
49. The censoring problem is much greater for the high-value bidder, since in the ascending-bid auction this bidder wins in 278 of the relevant 309 auctions. Therefore, we do not report a bid distribution for the high-value bidder for this auction institution, nor do we use such bids in any of the statistical tests that follow.
50. Cameron, C.; Trivedi, P. Microeconometrics: Methods and Applications; Cambridge University Press: Cambridge, UK, 2005.
51. As in Table 1, we only employ the sealed bid auction data in these regressions because the high-value bidders' bids are rarely observed in the ascending price auction.