Level k and Cursed Equilibrium Jörg Oechssler University of Heidelberg November 27, 2018 Jörg Oechssler University of Heidelberg () November 27, 2018 1 / 25
Literature Breitmoser, Y. (2012), “Strategic reasoning in p-beauty contests”, GEB . Camerer, C. F., T.-H. Ho, and J.-K. Chong (2004): “A Cognitive Hierarchy Model of Games,” QJE . Costa-Gomes, M., V. P. Crawford, and B. Broseta (2001): “Cognition and Behavior in Normal-Form Games: An Experimental Study,” Econometrica. Crawford, V. P. and N. Iriberri (2007): “Level-k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner’s Curse and Overbidding in Private-Value Auctions?” Econometrica . Jörg Oechssler University of Heidelberg () November 27, 2018 2 / 25
Literature Ho, T.-H., C. Camerer, and K. Weigelt (1998): “Iterated Dominance and Iterated Best Response in Experimental ‘p-Beauty Contests‘,” AER . Nagel, R. (1995): “Unraveling in Guessing Games: An Experimental Study,” AER . Stahl, D. (1993): “Evolution of smart-n players,” GEB . Eyster, E., & Rabin, M. (2005). “Cursed equilibrium”, Econometrica . Jörg Oechssler University of Heidelberg () November 27, 2018 3 / 25
Why study non-equilibrium models? It is not clear that behavior always converges to some equilibrium Initial play does often not correspond to equilibrium play In some games, play may never converge to an equilibrium depending on how people learn Even if behavior does converge to some equilibrium in the long run, we may still be interested in initial play There are often multiple equilibria. If we know where we start and how people learn, we can predict in which equilibrium we will end up In some applications not only the long-run outcome but also the initial and intermediate outcomes matter (it may take a while to converge to the equilibrium) Jörg Oechssler University of Heidelberg () November 27, 2018 4 / 25
Example: Initial Play Matters Van Huyck, Cook, and Battalio ( JEBO , 1997) run an experiment in which initial play has a substantial e¤ect on the long-run outcome 7 subjects play the same game 15 times Each subject chooses an e¤ort level between 1 and 14 and payo¤s are determined by the subject’s e¤ort as well as the median e¤ort of the group After each one of 15 periods, the median e¤ort is publicly announced Jörg Oechssler University of Heidelberg () November 27, 2018 5 / 25
Example: Initial Play Matters The game has many Nash equilibria but only two of them are symmetric pure-strategy equilibria (median e¤ort 3 and 12) Jörg Oechssler University of Heidelberg () November 27, 2018 6 / 25
Example: Initial Play Matters In the experiment, roughly half of 10 groups happen to have an initial median of 8 or higher These groups typically converged to the equilibrium with a median of 12 Groups that started with a low median tended to converge to the equilibrium with a median of 3 Jörg Oechssler University of Heidelberg () November 27, 2018 7 / 25
Example: Initial Play Matters Jörg Oechssler University of Heidelberg () November 27, 2018 8 / 25
p-Beauty Contest: The game Nagel ( AER , 1995), a.k.a. guessing game N players choose an integer in x i 2 f 0 , . . . , 100 g Calculate average guess µ = 1 n ∑ x i The player whose choice is closest to p times µ wins a …xed price (In Nagel’s exp. p = 2 / 3 ) . The price is shared if there is a tie. The game is called “beauty contest” after a passage in Keynes’ General Theory of Employment, Interest, and Money in which he compares investment to a beauty contest in which competitors pick the most beautiful one out of six faces and those competitors whose guess corresponds to the most frequently chosen face win. Jörg Oechssler University of Heidelberg () November 27, 2018 9 / 25
p-Beauty Contest: Nash equilibrium Unique Nash equilibrium is x = 0 Jörg Oechssler University of Heidelberg () November 27, 2018 10 / 25
p-Beauty Contest: Nash equilibrium Unique Nash equilibrium is x = 0 Suppose µ > 0 Jörg Oechssler University of Heidelberg () November 27, 2018 10 / 25
p-Beauty Contest: Nash equilibrium Unique Nash equilibrium is x = 0 Suppose µ > 0 In equilibrium, everyone must choose same x . If not, the one with highest guess can never win and should lower his guess. But then it would be better to choose 2 3 x . Jörg Oechssler University of Heidelberg () November 27, 2018 10 / 25
p-Beauty Contest: Nash equilibrium In fact, accounting for own guess, can choose y s.t. � 1 � 2 ny + n � 1 y = x 3 n 3 n � 3 y = 3 n � 2 x Jörg Oechssler University of Heidelberg () November 27, 2018 11 / 25
p-Beauty Contest: Nash equilibrium In fact, accounting for own guess, can choose y s.t. � 1 � 2 ny + n � 1 y = x 3 n 3 n � 3 y = 3 n � 2 x If everyone chooses y = x , � 1 � nx + n � 1 2 x = x 3 n 2 x = 3 x Jörg Oechssler University of Heidelberg () November 27, 2018 11 / 25
p-Beauty Contest: Nash equilibrium In fact, accounting for own guess, can choose y s.t. � 1 � 2 ny + n � 1 y = x 3 n 3 n � 3 y = 3 n � 2 x If everyone chooses y = x , � 1 � nx + n � 1 2 x = x 3 n 2 x = 3 x unique solution: x = 0. Jörg Oechssler University of Heidelberg () November 27, 2018 11 / 25
p-Beauty Contest: level D NE can also be found by iterated elimination of (weakly) dominated strategies Jörg Oechssler University of Heidelberg () November 27, 2018 12 / 25
p-Beauty Contest: level D NE can also be found by iterated elimination of (weakly) dominated strategies 2/3 of the average is always lower than 67 Therefore, choices higher than 66 are dominated Jörg Oechssler University of Heidelberg () November 27, 2018 12 / 25
p-Beauty Contest: level D NE can also be found by iterated elimination of (weakly) dominated strategies 2/3 of the average is always lower than 67 Therefore, choices higher than 66 are dominated If all players are rational, nobody will choose a number above 66 Therefore, 2/3 of the average has to be lower than 2 / 3 � 66 = 44 . 4 Jörg Oechssler University of Heidelberg () November 27, 2018 12 / 25
p-Beauty Contest: level D NE can also be found by iterated elimination of (weakly) dominated strategies 2/3 of the average is always lower than 67 Therefore, choices higher than 66 are dominated If all players are rational, nobody will choose a number above 66 Therefore, 2/3 of the average has to be lower than 2 / 3 � 66 = 44 . 4 2 / 3 � 44 = 29 . 3 2 / 3 � 29 . 3 = 19 . 533 etc. etc. Jörg Oechssler University of Heidelberg () November 27, 2018 12 / 25
p-Beauty Contest: level D NE can also be found by iterated elimination of (weakly) dominated strategies 2/3 of the average is always lower than 67 Therefore, choices higher than 66 are dominated If all players are rational, nobody will choose a number above 66 Therefore, 2/3 of the average has to be lower than 2 / 3 � 66 = 44 . 4 2 / 3 � 44 = 29 . 3 2 / 3 � 29 . 3 = 19 . 533 etc. etc. D 1 : 1 round of elimination, D 2 2 rounds etc.... Jörg Oechssler University of Heidelberg () November 27, 2018 12 / 25
p-Beauty Contest: Experimental evidence Many subjects choose values 33 or 22 (and some 0) Jörg Oechssler University of Heidelberg () November 27, 2018 13 / 25
p-Beauty Contest: Interpretations (At least) two possible interpretations: Iterated dominance ( level D ) doesn’t …t the peaks at 22 and 33 very well Jörg Oechssler University of Heidelberg () November 27, 2018 14 / 25
p-Beauty Contest: Interpretations (At least) two possible interpretations: Iterated dominance ( level D ) doesn’t …t the peaks at 22 and 33 very well Iterated Best responses: “level k -thinking” Assume everybody else chooses randomly ! best respond against that Jörg Oechssler University of Heidelberg () November 27, 2018 14 / 25
p-Beauty Contest: Interpretations (At least) two possible interpretations: Iterated dominance ( level D ) doesn’t …t the peaks at 22 and 33 very well Iterated Best responses: “level k -thinking” Assume everybody else chooses randomly ! best respond against that If they try to get as close as possible to 2 / 3 � 50, they would choose 33 Many subjects seem to anticipate that behavior and best respond to it by choosing 2 / 3 � 33 = 22 Few pick the NE and very few go through more than 2 iterations Jörg Oechssler University of Heidelberg () November 27, 2018 14 / 25
Level-k Type 0 uniformly randomizes over all available pure strategies (some authors assume a di¤erent distribution, e.g., normal) Jörg Oechssler University of Heidelberg () November 27, 2018 15 / 25
Level-k Type 0 uniformly randomizes over all available pure strategies (some authors assume a di¤erent distribution, e.g., normal) Type 1 assumes all other players are type 0 and strictly best responds Jörg Oechssler University of Heidelberg () November 27, 2018 15 / 25
Level-k Type 0 uniformly randomizes over all available pure strategies (some authors assume a di¤erent distribution, e.g., normal) Type 1 assumes all other players are type 0 and strictly best responds Type 2 assumes all other players are type 1 and strictly best responds, etc. Jörg Oechssler University of Heidelberg () November 27, 2018 15 / 25
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