[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
IDEAS home Printed from https://ideas.repec.org/p/uea/wcbess/10-05.html
   My bibliography  Save this paper

On the beliefs off the path: Equilibrium refinement due to quantal response and level-k

Author

Listed:
  • Yves Breitmoser

    (Department of Economics, EUV Frankfurt (Oder))

  • Jonathan H. W. Tan

    (Nottingham University Business School)

  • Daniel John Zizzo

    (School of Economics and CBESS, University of East Anglia)

Abstract
This paper studies the relevance of equilibrium and nonequilibrium explanations of behavior, with respects to equilibrium refinement, as players gain experience. We investigate this experimentally using an incomplete information sequential move game with heterogeneous preferences and multiple perfect equilibria. Only the limit point of quantal response (the limiting logit equilibrium), and alternatively that of level-k reasoning (extensive form rationalizability), restricts beliefs off the equilibrium path. Both concepts converge to the same unique equilibrium, but the predictions differ prior to convergence. We show that with experience of repeated play in relatively constant environments, subjects approach equilibrium via the quantal response learning path. With experience spanning also across relatively novel environments, though, level-k reasoning tends to dominate.

Suggested Citation

  • Yves Breitmoser & Jonathan H. W. Tan & Daniel John Zizzo, 2010. "On the beliefs off the path: Equilibrium refinement due to quantal response and level-k," Working Paper series, University of East Anglia, Centre for Behavioural and Experimental Social Science (CBESS) 10-05, School of Economics, University of East Anglia, Norwich, UK..
    • Handle: RePEc:uea:wcbess:10-05
    as

    Download full text from publisher

    File URL: https://ueaeco.github.io/working-papers/papers/cbess/UEA-CBESS-10-05.pdf
    File Function: main text
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Ui, Takashi, 2001. "Robust Equilibria of Potential Games," Econometrica, Econometric Society, vol. 69(5), pages 1373-1380, September.
    2. Vincent P. Crawford & Nagore Iriberri, 2007. "Level-k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private-Value Auctions?," Econometrica, Econometric Society, vol. 75(6), pages 1721-1770, November.
    3. Stahl, Dale II & Wilson, Paul W., 1994. "Experimental evidence on players' models of other players," Journal of Economic Behavior & Organization, Elsevier, vol. 25(3), pages 309-327, December.
    4. Frank Schuhmacher, 1999. "Proper rationalizability and backward induction," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 599-615.
    5. Atsushi Kajii & Stephen Morris, 1997. "The Robustness of Equilibria to Incomplete Information," Econometrica, Econometric Society, vol. 65(6), pages 1283-1310, November.
    6. Kreps, David M & Wilson, Robert, 1982. "Sequential Equilibria," Econometrica, Econometric Society, vol. 50(4), pages 863-894, July.
    7. Goeree, Jacob K. & Holt, Charles A. & Palfrey, Thomas R., 2002. "Quantal Response Equilibrium and Overbidding in Private-Value Auctions," Journal of Economic Theory, Elsevier, vol. 104(1), pages 247-272, May.
    8. Ernst Fehr & Klaus M. Schmidt, 1999. "A Theory of Fairness, Competition, and Cooperation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(3), pages 817-868.
    9. Stahl, Dale O., 2000. "Rule Learning in Symmetric Normal-Form Games: Theory and Evidence," Games and Economic Behavior, Elsevier, vol. 32(1), pages 105-138, July.
    10. Erev, Ido & Roth, Alvin E, 1998. "Predicting How People Play Games: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria," American Economic Review, American Economic Association, vol. 88(4), pages 848-881, September.
    11. Stahl Dale O. & Wilson Paul W., 1995. "On Players' Models of Other Players: Theory and Experimental Evidence," Games and Economic Behavior, Elsevier, vol. 10(1), pages 218-254, July.
    12. Vincent P. Crawford & Miguel A. Costa-Gomes & Nagore Iriberri, 2010. "Strategic Thinking," Levine's Working Paper Archive 661465000000001148, David K. Levine.
    13. Costa-Gomes, Miguel & Crawford, Vincent P & Broseta, Bruno, 2001. "Cognition and Behavior in Normal-Form Games: An Experimental Study," Econometrica, Econometric Society, vol. 69(5), pages 1193-1235, September.
    14. Frankel, David M. & Morris, Stephen & Pauzner, Ady, 2003. "Equilibrium selection in global games with strategic complementarities," Journal of Economic Theory, Elsevier, vol. 108(1), pages 1-44, January.
    15. McFadden, Daniel L., 1984. "Econometric analysis of qualitative response models," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 24, pages 1395-1457, Elsevier.
    16. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    17. Carlsson, Hans & van Damme, Eric, 1993. "Global Games and Equilibrium Selection," Econometrica, Econometric Society, vol. 61(5), pages 989-1018, September.
    18. Vincent P. Crawford & Miguel A. Costa-Gomes, 2006. "Cognition and Behavior in Two-Person Guessing Games: An Experimental Study," American Economic Review, American Economic Association, vol. 96(5), pages 1737-1768, December.
    19. Simon P. Anderson & Jacob K. Goeree & Charles A. Holt, 1998. "Rent Seeking with Bounded Rationality: An Analysis of the All-Pay Auction," Springer Books, in: Roger D. Congleton & Arye L. Hillman & Kai A. Konrad (ed.), 40 Years of Research on Rent Seeking 1, pages 225-250, Springer.
    20. Stahl, Dale O. & Haruvy, Ernan, 2008. "Level-n bounded rationality in two-player two-stage games," Journal of Economic Behavior & Organization, Elsevier, vol. 65(1), pages 41-61, January.
    21. Bernheim, B Douglas, 1984. "Rationalizable Strategic Behavior," Econometrica, Econometric Society, vol. 52(4), pages 1007-1028, July.
    22. Rogers, Brian W. & Palfrey, Thomas R. & Camerer, Colin F., 2009. "Heterogeneous quantal response equilibrium and cognitive hierarchies," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1440-1467, July.
    23. Cooper, David J. & Van Huyck, John B., 2003. "Evidence on the equivalence of the strategic and extensive form representation of games," Journal of Economic Theory, Elsevier, vol. 110(2), pages 290-308, June.
    24. Haruvy, Ernan & Stahl, Dale O., 2012. "Between-game rule learning in dissimilar symmetric normal-form games," Games and Economic Behavior, Elsevier, vol. 74(1), pages 208-221.
    25. Avinash Dixit, 2003. "Clubs with Entrapment," American Economic Review, American Economic Association, vol. 93(5), pages 1824-1829, December.
    26. Richard Mckelvey & Thomas Palfrey, 1998. "Quantal Response Equilibria for Extensive Form Games," Experimental Economics, Springer;Economic Science Association, vol. 1(1), pages 9-41, June.
    27. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    28. Turocy, Theodore L., 2005. "A dynamic homotopy interpretation of the logistic quantal response equilibrium correspondence," Games and Economic Behavior, Elsevier, vol. 51(2), pages 243-263, May.
    29. McKelvey Richard D. & Palfrey Thomas R., 1995. "Quantal Response Equilibria for Normal Form Games," Games and Economic Behavior, Elsevier, vol. 10(1), pages 6-38, July.
    30. Battigalli, Pierpaolo, 1997. "On Rationalizability in Extensive Games," Journal of Economic Theory, Elsevier, vol. 74(1), pages 40-61, May.
    31. Selten, Reinhard & Stoecker, Rolf, 1986. "End behavior in sequences of finite Prisoner's Dilemma supergames A learning theory approach," Journal of Economic Behavior & Organization, Elsevier, vol. 7(1), pages 47-70, March.
    32. Colin F. Camerer & Teck-Hua Ho & Juin-Kuan Chong, 2004. "A Cognitive Hierarchy Model of Games," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 119(3), pages 861-898.
    33. Battalio, Raymond & Samuelson, Larry & Van Huyck, John, 2001. "Optimization Incentives and Coordination Failure in Laboratory Stag Hunt Games," Econometrica, Econometric Society, vol. 69(3), pages 749-764, May.
    34. Alejandro Gaviria & Steven Raphael, 2001. "School-Based Peer Effects And Juvenile Behavior," The Review of Economics and Statistics, MIT Press, vol. 83(2), pages 257-268, May.
    35. Kuran, Timur, 1998. "Ethnic Norms and Their Transformation through Reputational Cascades," The Journal of Legal Studies, University of Chicago Press, vol. 27(2), pages 623-659, June.
    36. Reinhard Selten, 1998. "Axiomatic Characterization of the Quadratic Scoring Rule," Experimental Economics, Springer;Economic Science Association, vol. 1(1), pages 43-61, June.
    37. Colin Camerer & Teck-Hua Ho, 1999. "Experience-weighted Attraction Learning in Normal Form Games," Econometrica, Econometric Society, vol. 67(4), pages 827-874, July.
    38. Goeree, Jacob K. & Holt, Charles A., 2004. "A model of noisy introspection," Games and Economic Behavior, Elsevier, vol. 46(2), pages 365-382, February.
    39. Ho, Teck-Hua & Camerer, Colin & Weigelt, Keith, 1998. "Iterated Dominance and Iterated Best Response in Experimental "p-Beauty Contests."," American Economic Review, American Economic Association, vol. 88(4), pages 947-969, September.
    40. Dorothea Kübler & Georg Weizsäcker, 2004. "Limited Depth of Reasoning and Failure of Cascade Formation in the Laboratory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 71(2), pages 425-441.
    41. Johnson, Eric J. & Camerer, Colin & Sen, Sankar & Rymon, Talia, 2002. "Detecting Failures of Backward Induction: Monitoring Information Search in Sequential Bargaining," Journal of Economic Theory, Elsevier, vol. 104(1), pages 16-47, May.
    42. Daniel L. McFadden, 1976. "Quantal Choice Analysis: A Survey," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 5, number 4, pages 363-390, National Bureau of Economic Research, Inc.
    43. Stahl, Dale O., 1998. "Is step-j thinking an arbitrary modelling restriction or a fact of human nature?," Journal of Economic Behavior & Organization, Elsevier, vol. 37(1), pages 33-51, September.
    44. Uri Gneezy, 2005. "Step-Level Reasoning and Bidding in Auctions," Management Science, INFORMS, vol. 51(11), pages 1633-1642, November.
    45. Nagel, Rosemarie, 1995. "Unraveling in Guessing Games: An Experimental Study," American Economic Review, American Economic Association, vol. 85(5), pages 1313-1326, December.
    46. Stahl, Dale O., 1996. "Boundedly Rational Rule Learning in a Guessing Game," Games and Economic Behavior, Elsevier, vol. 16(2), pages 303-330, October.
    47. Geir B. Asheim, 2002. "Proper rationalizability in lexicographic beliefs," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(4), pages 453-478.
    48. Kawagoe, Toshiji & Takizawa, Hirokazu, 2009. "Equilibrium refinement vs. level-k analysis: An experimental study of cheap-talk games with private information," Games and Economic Behavior, Elsevier, vol. 66(1), pages 238-255, May.
    49. Kyle Hyndman & Erkut Y. Ozbay & Andrew Schotter & Wolf Ze’ev Ehrblatt, 2012. "Convergence: An Experimental Study Of Teaching And Learning In Repeated Games," Journal of the European Economic Association, European Economic Association, vol. 10(3), pages 573-604, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Klaus Abbink & David Masclet & Daniel Mirza, 2018. "Inequality and inter-group conflicts: experimental evidence," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 50(3), pages 387-423, March.
    2. Anna Conte & M. Vittoria Levati & Natalia Montinari, 2019. "Experience in public goods experiments," Theory and Decision, Springer, vol. 86(1), pages 65-93, February.
    3. Wright, James R. & Leyton-Brown, Kevin, 2017. "Predicting human behavior in unrepeated, simultaneous-move games," Games and Economic Behavior, Elsevier, vol. 106(C), pages 16-37.
    4. Rapoport, Amnon & Qi, Hang & Mak, Vincent & Gisches, Eyran J., 2019. "When a few undermine the whole: A class of social dilemmas in ridesharing," Journal of Economic Behavior & Organization, Elsevier, vol. 166(C), pages 125-137.
    5. Tan, Jonathan H.W. & Breitmoser, Yves & Bolle, Friedel, 2015. "Voluntary contributions by consent or dissent," Games and Economic Behavior, Elsevier, vol. 92(C), pages 106-121.
    6. Xefteris, Dimitrios, 2023. "Coordinated democracy," Games and Economic Behavior, Elsevier, vol. 142(C), pages 33-45.
    7. Benjamin Patrick Evans & Mikhail Prokopenko, 2021. "Bounded rationality for relaxing best response and mutual consistency: The Quantal Hierarchy model of decision-making," Papers 2106.15844, arXiv.org, revised Mar 2023.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Breitmoser, Yves, 2010. "Hierarchical Reasoning versus Iterated Reasoning in p-Beauty Contest Guessing Games," MPRA Paper 19893, University Library of Munich, Germany.
    2. Breitmoser, Yves, 2012. "Strategic reasoning in p-beauty contests," Games and Economic Behavior, Elsevier, vol. 75(2), pages 555-569.
    3. Choo, Lawrence C.Y & Kaplan, Todd R., 2014. "Explaining Behavior in the "11-20" Game," MPRA Paper 52808, University Library of Munich, Germany.
    4. Nagel, Rosemarie & Bühren, Christoph & Frank, Björn, 2017. "Inspired and inspiring: Hervé Moulin and the discovery of the beauty contest game," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 191-207.
    5. Georganas, Sotiris & Healy, Paul J. & Weber, Roberto A., 2015. "On the persistence of strategic sophistication," Journal of Economic Theory, Elsevier, vol. 159(PA), pages 369-400.
    6. Wright, James R. & Leyton-Brown, Kevin, 2017. "Predicting human behavior in unrepeated, simultaneous-move games," Games and Economic Behavior, Elsevier, vol. 106(C), pages 16-37.
    7. Vincent P. Crawford & Miguel A. Costa-Gomes & Nagore Iriberri, 2010. "Strategic Thinking," Levine's Working Paper Archive 661465000000001148, David K. Levine.
    8. Jacob K. Goeree & Charles A. Holt, 2001. "Ten Little Treasures of Game Theory and Ten Intuitive Contradictions," American Economic Review, American Economic Association, vol. 91(5), pages 1402-1422, December.
    9. Breitmoser, Yves, 2019. "Knowing me, imagining you: Projection and overbidding in auctions," Games and Economic Behavior, Elsevier, vol. 113(C), pages 423-447.
    10. Breitmoser, Yves, 2017. "Knowing Me, Imagining You:," Rationality and Competition Discussion Paper Series 36, CRC TRR 190 Rationality and Competition.
    11. Kawagoe, Toshiji & Takizawa, Hirokazu, 2012. "Level-k analysis of experimental centipede games," Journal of Economic Behavior & Organization, Elsevier, vol. 82(2), pages 548-566.
    12. Camerer, Colin F. & Ho, Teck-Hua, 2015. "Behavioral Game Theory Experiments and Modeling," Handbook of Game Theory with Economic Applications,, Elsevier.
    13. Teck-Hua Ho & So-Eun Park & Xuanming Su, 2021. "A Bayesian Level- k Model in n -Person Games," Management Science, INFORMS, vol. 67(3), pages 1622-1638, March.
    14. Strzalecki, Tomasz, 2014. "Depth of reasoning and higher order beliefs," Journal of Economic Behavior & Organization, Elsevier, vol. 108(C), pages 108-122.
    15. David Danz & Dietmar Fehr & Dorothea Kübler, 2012. "Information and beliefs in a repeated normal-form game," Experimental Economics, Springer;Economic Science Association, vol. 15(4), pages 622-640, December.
    16. Vincent P. Crawford & Miguel A. Costa-Gomes, 2006. "Cognition and Behavior in Two-Person Guessing Games: An Experimental Study," American Economic Review, American Economic Association, vol. 96(5), pages 1737-1768, December.
    17. García-Pola, Bernardo & Iriberri, Nagore & Kovářík, Jaromír, 2020. "Non-equilibrium play in centipede games," Games and Economic Behavior, Elsevier, vol. 120(C), pages 391-433.
    18. Teck-Hua Ho & Xuanming Su, 2013. "A Dynamic Level-k Model in Sequential Games," Management Science, INFORMS, vol. 59(2), pages 452-469, March.
    19. Feng, Jun & Qin, Xiangdong & Wang, Xiaoyuan, 2021. "A Bayesian cognitive hierarchy model with fixed reasoning levels," Journal of Economic Behavior & Organization, Elsevier, vol. 192(C), pages 704-723.
    20. Miguel A Costa-Gomes & Vincent P Crawford & Nagore Iriberri, 2008. "Comparing Models of Strategic Thinking in Van Huyck, Battalio, and Beil’s Coordination Games," Levine's Working Paper Archive 122247000000002346, David K. Levine.

    More about this item

    Keywords

    incomplete information; equilibrium refinement; logit equilibrium; rationalizability; quantal response; level-k; inequity aversion; experiment;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C91 - Mathematical and Quantitative Methods - - Design of Experiments - - - Laboratory, Individual Behavior
    • D62 - Microeconomics - - Welfare Economics - - - Externalities

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:uea:wcbess:10-05. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Cara Liggins (email available below). General contact details of provider: https://edirc.repec.org/data/esueauk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.