Application of Level-k Model

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Level-k thinking�

• Level-k thinking: Decision-making process which is based on the assumption that players have different degrees of rationality
• It is a non-equilibrium concept that describes how people actually behave

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• Solving games using level-k thinking:
• Level 0: Non-strategic (random, reference point, etc.)
• Level 1: Best respond to level 0
• Level 2: Best respond to level 1
• Level 3: Best respond to level 2

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Limitations of level-k models�

• Hard to define Level 0
• Random
• Reference point

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• Too many parameters (degrees of freedom)
• The percentage of each level is an additional degree of freedom
• Needs more (ad hoc) assumptions in order to constrain degrees of freedom

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• Low predictive power
• Are types fixed across games?
• Georganas et al. (2015) estimate subjects’ types in two different games and find no correlation in estimated types

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Application of Level-k Model

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Application 1: Auctions�

• Auction Hunters:
• https://www.youtube.com/watch?v=7YrkF0oX9Go

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Application 1: Auctions�

• Problem: Economic agents overbid in common-value auctions
• Oil and gas drainage license tend to sell at much higher price when there is significant asymmetric information (Hendricks and Porter 1988)
• Overbidding is also observed in the markets for book publication rights, corporate takeovers, etc.
• In experimental setting overbidding is so intense that participants on average earn negative payoffs (Kagel and Levin 1986)

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• Why overbid?
• People overestimate the value of the prize
• People care about winning itself
• People are bounded rational

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Application 1: Auctions�

• Common-value auction
• The item of common but unknown value is sold at an auction
• Each bidder receives a private signal that is correlated with the actual value of the auctioned item
• After all bidders submit their bids, the bidder with the highest bid wins
• The most common finding is the “winner’s curse” (overbidding)

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• Crawford and Iriberri (2007) provide an explanation based on the level-k model of strategic thinking:
• Level 0 is behaving randomly
• Level 1 is best responding to Level 0, and Level 2 to Level 1
• The analysis show that level-k model rationalizes the existing data and explains overbidding better than other behavioral models

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Application 2: Movies�

• Movie reviews
• Movie studios generally show movies to critics in advance of the release so that critics’ reviews can be published before the movie is shown
• Some movies are deliberately made unavailable until after the initial release, a practice sometimes called “cold opening”

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• Why would a studio choose cold opening?
• If the studio believes the film is of a high quality then it would be beneficial to allow critics to pre-screen the movie so that they can provide positive reviews to attract moviegoers to the theater
• If instead the studio knows that the film is of a low quality then it will expect negative reviews which will only deter moviegoers

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Application 2: Movies�

• Fully rational strategic analysis:
• If moviegoers believe that studios know their movie’s quality then fully rational moviegoers should infer that cold-opened movies are below average in quality
• Anticipating this inference, studios should only cold open the very worst movies
• Consequently, cold-opened movies should be the least profitable

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• Level-k things:
• Level 0 moviegoer thinks that cold-opening decisions are random (they convey no information about quality) and hence infers that the quality of a cold-opened movie is average
• Level 1 studio anticipates that moviegoers think this way and therefore cold opens all below-average movies, and shows all above-average movies to critics
• Consequently, cold-opened movies could generate higher profit

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Application 2: Movies�

• Brown et al. (2012) analyze a dataset of 1414 widely released movies:
• Ratings of critics and moviegoers are highly correlated
• Critics ratings are predictive of box-office revenues
• 11% of the movies are cold opened, and such movies have a much lower rating
• Nevertheless, cold opened movies generate 10%-30% higher revenue, compared to similar-quality movies that are pre-reviewed

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• Conclusion:
• Some consumers are overestimating the quality of movies that are cold opened
• Evidence points out that consumers apply limited (bounded) strategic thinking consistent with level-k model

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Application 2: Movies�

• It appears that studios are aware of moviegoers’ limited rationality and are learning, since cold openings have been increasingly more frequent over the past years

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Application 3: Analysts�

• Problem: Financial analysts bias stock recommendations upward and investors often follow these biased recommendations
• Analysts issue recommendations on individual stocks that range from “strong sell” and “sell” to “hold,” “buy,” and “strong buy”
• Analysts tend to bias their stock recommendations upward when they are affiliated with the underwriter of the stock (Malmendier and Shanthikumar 2007)

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• Why would a financial analyst make a biased recommendation?
• Conflict of interest: (1) reliable recommendations attract customers and enhance the analyst’s reputation; (2) buy recommendations are more likely to generate trading business than sell recommendations
• Pressure from the management: Favorable recommendations are generally viewed as an implicit condition of underwriting contracts

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Application 3: Analysts�

• Fully rational strategic analysis:
• The analyst wants to persuade an investor to buy, which creates an incentive for the analyst to make a favorable recommendation
• The investor does not want to be deceived by the analyst
• Consequently, there should be no correlation between recommendations and investments

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• Level-k things:
• Level 0 analyst always makes a truthful recommendation
• Level 1 investor follows Level 0 analyst’s recommendation
• Level 2 analyst best responds to Level 1 investor by always recommending to buy
• Level 3 investor does not follow Level 2’s recommendation
• Consequently, there should be a positive correlation between recommendations and investments for Level 1 investors and no correlation for Level 3 investors

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Application 3: Analysts�

• Malmendier and Shanthikumar (2007) analyze the New York Stock Exchange trades data:
• Small investors (Level 1), usually novice investors, follow recommendations literally
• Large investors (Level 3), who are usually represented by trade professionals, tend to buy following “strong buy” recommendations, but exhibit no reaction to “buy”, “hold” or “sell” recommendations

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References�

• Dhami, S. (2016). The Foundations of Behavioral Economic Analysis. Oxford University Press.
• Crawford, V.P., & Iriberri, N. (2007). 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.
• Hendricks, K., & Porter, R.H. (1988). An empirical study of an auction with asymmetric information. American Economic Review, 78, 865-883.
• Kagel, J.H., & Levin, D. (1986). The winner's curse and public information in common value auctions. American Economic Review, 76, 894-920.
• Brown, A.L., Camerer, C.F., & Lovallo, D. (2012). To review or not to review? Limited strategic thinking at the movie box office. American Economic Journal: Microeconomics, 4, 1-26.
• Malmendier, U., & Shanthikumar, D. (2007). Are small investors naive about incentives? Journal of Financial Economics, 85, 457-489.

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