×

On the beliefs off the path: equilibrium refinement due to quantal response and level-\(k\). (English) Zbl 1294.91022

Summary: The extensive form game we study has multiple perfect equilibria, but it has a unique limiting logit equilibrium (QRE) and a unique level-\(k\) prediction as \(k\) approaches infinity. The convergence paths of QRE and level-\(k\) are different, but they converge to the same limit point. We analyze whether subjects adapt beliefs when gaining experience, and if so whether they take the QRE or the level-\(k\) learning path. We estimate transitions between level-\(k\) and QRE belief rules using Markov-switching rule learning models. The analysis reveals that subjects take the level-\(k\) learning path and that they advance gradually, switching from level 1 to 2, from level 2 to equilibrium, and reverting to level 1 after observing opponents deviating from equilibrium. The steady state therefore contains a mixture of behavioral rules: levels 0, 1, 2, and equilibrium with weights of 2.9%, 16.6%, 37.9%, and 42.6%, respectively.

MSC:

91A18 Games in extensive form
91A26 Rationality and learning in game theory

Software:

NEWUOA; Gambit
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, S.; Goeree, J.; Holt, C., Rent seeking with bounded rationality: an analysis of the all-pay auction, J. Polit. Economy, 106, 4, 828-853 (1998)
[2] Asheim, G., Proper rationalizability in lexicographic beliefs, Int. J. Game Theory, 30, 4, 453-478 (2002) · Zbl 1083.91006
[3] Battalio, R.; Samuelson, L.; Huyck, J., Optimization incentives and coordination failure in laboratory stag hunt games, Econometrica, 69, 3, 749-764 (2001) · Zbl 1032.91546
[4] Battigalli, P., On rationalizability in extensive games, J. Econ. Theory, 74, 1, 40-61 (1997) · Zbl 0887.90185
[5] Bernheim, B., Rationalizable strategic behavior, Econometrica, 52, 4, 1007-1028 (1984) · Zbl 0552.90098
[7] Camerer, C.; Hua Ho, T., Experience-weighted attraction learning in normal form games, Econometrica, 67, 4, 827-874 (1999) · Zbl 1055.91504
[8] Camerer, C.; Ho, T.; Chong, J., A cognitive hierarchy model of games, Quart. J. Econ., 119, 3, 861-898 (2004) · Zbl 1074.91503
[9] Carlsson, H.; van Damme, E., Global games and equilibrium selection, Econometrica, 61, 5, 989-1018 (1993) · Zbl 0794.90083
[10] Cooper, D.; Van Huyck, J., Evidence on the equivalence of the strategic and extensive form representation of games, J. Econ. Theory, 110, 2, 290-308 (2003) · Zbl 1042.91006
[11] Costa-Gomes, M.; Crawford, V., Cognition and behavior in two-person guessing games: an experimental study, Amer. Econ. Rev., 96, 5, 1737-1768 (2006)
[12] Costa-Gomes, M.; Crawford, V.; Broseta, B., Cognition and behavior in normal-form games: an experimental study, Econometrica, 69, 5, 1193-1235 (2001)
[13] Crawford, V.; Iriberri, N., Level-\(k\) auctions: can a nonequilibrium model of strategic thinking explain the winner’s curse and overbidding in private-value auctions?, Econometrica, 75, 6, 1721-1770 (2007) · Zbl 1133.91374
[14] Dixit, A., Clubs with entrapment, Amer. Econ. Rev., 93, 5, 1824-1829 (2003)
[15] Erev, I.; Roth, A., Predicting how people play games: reinforcement learning in experimental games with unique, mixed strategy equilibria, Amer. Econ. Rev., 88, 4, 848-881 (1998)
[16] Frankel, D.; Morris, S.; Pauzner, A., Equilibrium selection in global games with strategic complementarities, J. Econ. Theory, 108, 1, 1-44 (2003) · Zbl 1044.91005
[17] Frühwirth-Schnatter, S., Finite Mixture and Markov Switching Models (2006), Springer Verlag · Zbl 1108.62002
[18] Gaviria, A.; Raphael, S., School-based peer effects and juvenile behavior, Rev. Econ. Statist., 83, 2, 257-268 (2001)
[19] Gneezy, U., Step-level reasoning and bidding in auctions, Manage. Sci., 51, 11, 1633-1642 (2005) · Zbl 1232.91302
[20] Goeree, J.; Holt, C.; Palfrey, T., Quantal response equilibrium and overbidding in private-value auctions, J. Econ. Theory, 104, 1, 247-272 (2002) · Zbl 1015.91028
[21] Haruvy, E.; Stahl, D., Between-game rule learning in dissimilar symmetric normal-form games, Games Econ. Behav., 74, 1, 208-221 (2012) · Zbl 1278.91022
[22] Ho, T.; Camerer, C.; Weigelt, K., Iterated dominance and iterated best response in experimental “\(p\)-beauty contests”, Amer. Econ. Rev., 88, 4, 947-969 (1998)
[23] Hyndman, K.; Ozbay, E.; Schotter, A.; Ehrblatt, W., Convergence: an experimental study of teaching and learning in repeated games, J. Europ. Econ. Assoc., 10, 3, 573-604 (2012)
[24] Johnson, E.; Camerer, C.; Sen, S.; Rymon, T., Detecting failures of backward induction: monitoring information search in sequential bargaining, J. Econ. Theory, 104, 1, 16-47 (2002) · Zbl 1015.91016
[25] Kajii, A.; Morris, S., The robustness of equilibria to incomplete information, Econometrica, 65, 6, 1283-1309 (1997) · Zbl 0887.90186
[26] Kawagoe, T.; Takizawa, H., Equilibrium refinement vs. level-\(k\) analysis: an experimental study of cheap-talk games with private information, Games Econ. Behav., 66, 1, 238-255 (2009) · Zbl 1161.91339
[27] Kreps, D.; Wilson, R., Sequential equilibria, Econometrica, 50, 4, 863-894 (1982) · Zbl 0483.90092
[28] Krolzig, H., Markov-Switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis (1997), Springer: Springer Berlin · Zbl 0879.62107
[29] Kübler, D.; Weizsäcker, G., Limited depth of reasoning and failure of cascade formation in the laboratory, Rev. Econ. Stud., 71, 2, 425-441 (2004) · Zbl 1096.91008
[30] Kuran, T., Ethnic norms and their transformation through reputational cascades, J. Legal Stud., 27, S2, 623-659 (1998)
[31] McFadden, D., Quantal choice analysis: a survey, Ann. Econ. Soc. Meas., 5, 4, 363-390 (1976)
[32] McFadden, D., Econometric analysis of qualitative response models, (Handbook of Econometrics, vol. 2 (1984)), 1395-1457
[33] McKelvey, R.; Palfrey, T., Quantal response equilibria for normal form games, Games Econ. Behav., 10, 1, 6-38 (1995) · Zbl 0832.90126
[34] McKelvey, R.; Palfrey, T., Quantal response equilibria for extensive form games, Exper. Econ., 1, 1, 9-41 (1998) · Zbl 0920.90141
[35] McKelvey, R. D.; McLennan, A. M.; Turocy, T. L., Gambit: software tools for game theory, version 0.2007.01.30 (2007)
[36] McLachlan, G.; Peel, D., Finite Mixture Models (2000), Wiley-Interscience, vol. 299 · Zbl 0963.62061
[37] Morris, S.; Rob, R.; Shin, H., \(p\)-dominance and belief potential, Econometrica, 63, 1, 145-157 (1995) · Zbl 0827.90138
[38] Myerson, R., Refinements of the Nash equilibrium concept, Int. J. Game Theory, 7, 2, 73-80 (1978) · Zbl 0392.90093
[39] Nagel, R., Unraveling in guessing games: an experimental study, Amer. Econ. Rev., 85, 5, 1313-1326 (1995)
[40] Pearce, D., Rationalizable strategic behavior and the problem of perfection, Econometrica, 52, 4, 1029-1050 (1984) · Zbl 0552.90097
[41] Powell, M., Developments of NEWUOA for minimization without derivatives, IMA J. Numer. Anal., 28, 4, 649 (2008) · Zbl 1154.65049
[42] Rogers, B.; Palfrey, T.; Camerer, C., Heterogeneous quantal response equilibrium and cognitive hierarchies, J. Econ. Theory, 144, 4, 1440-1467 (2009) · Zbl 1166.91310
[43] Schelling, T. C., The Strategy of Conflict (1960), Harvard University Press: Harvard University Press Cambridge, MA
[44] Schuhmacher, F., Proper rationalizability and backward induction, Int. J. Game Theory, 28, 4, 599-615 (1999) · Zbl 0940.91016
[45] Selten, R., Reexamination of the perfectness concept for equilibrium points in extensive games, Int. J. Game Theory, 4, 1, 25-55 (1975) · Zbl 0312.90072
[46] Selten, R.; Stoecker, R., End behavior in sequence of finite prisoner’s dilemma supergames: a learning theory approach, J. Econ. Behav. Organ., 7, 1, 47-70 (1986)
[47] Stahl, D., Boundedly rational rule learning in a guessing game, Games Econ. Behav., 16, 2, 303-330 (1996) · Zbl 0863.90140
[48] Stahl, D., Is step-\(j\) thinking an arbitrary modelling restriction or a fact of human nature?, J. Econ. Behav. Organ., 37, 1, 33-51 (1998)
[49] Stahl, D., Rule learning in symmetric normal-form games: theory and evidence, Games Econ. Behav., 32, 1, 105-138 (2000) · Zbl 0956.91013
[50] Stahl, D.; Haruvy, E., Level-\(n\) bounded rationality in two-player two-stage games, J. Econ. Behav. Organ., 65, 1, 41-61 (2008)
[51] Stahl, D.; Wilson, P., Experimental evidence on players’ models of other players, J. Econ. Behav. Organ., 25, 3, 309-327 (1994)
[52] Stahl, D.; Wilson, P., On players’ models of other players: theory and experimental evidence, Games Econ. Behav., 10, 1, 218-254 (1995) · Zbl 0831.90135
[53] Turocy, T., A dynamic homotopy interpretation of the logistic quantal response equilibrium correspondence, Games Econ. Behav., 51, 2, 243-263 (2005) · Zbl 1099.91005
[54] Ui, T., Robust equilibria of potential games, Econometrica, 69, 5, 1373-1380 (2001) · Zbl 1041.91006
[55] Vuong, Q., Likelihood ratio tests for model selection and non-nested hypotheses, Econometrica, 57, 2, 307-333 (1989) · Zbl 0701.62106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.