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Level k and Cursed Equilibrium

Jörg Oechssler University of Heidelberg November 27, 2018

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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.

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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.

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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)

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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

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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)

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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

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Example: Initial Play Matters

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p-Beauty Contest: The game

Nagel (AER, 1995), a.k.a. guessing game N players choose an integer in xi 2 f0, . . . , 100g Calculate average guess µ = 1

n ∑ xi

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.

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p-Beauty Contest: Nash equilibrium

Unique Nash equilibrium is x = 0

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p-Beauty Contest: Nash equilibrium

Unique Nash equilibrium is x = 0 Suppose µ > 0

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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

3x.

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p-Beauty Contest: Nash equilibrium

In fact, accounting for own guess, can choose y s.t. y

=

2 3 1 ny + n 1 n x

• y

=

3n 3 3n 2x

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p-Beauty Contest: Nash equilibrium

In fact, accounting for own guess, can choose y s.t. y

=

2 3 1 ny + n 1 n x

• y

=

3n 3 3n 2x If everyone chooses y = x, x

=

2 3 1 nx + n 1 n x

• x

=

2 3x

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. y

=

2 3 1 ny + n 1 n x

• y

=

3n 3 3n 2x If everyone chooses y = x, x

=

2 3 1 nx + n 1 n x

• x

=

2 3x unique solution: x = 0.

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p-Beauty Contest: level D

NE can also be found by iterated elimination of (weakly) dominated strategies

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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

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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

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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. D1 : 1 round of elimination, D2 2 rounds etc....

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p-Beauty Contest: Experimental evidence

Many subjects choose values 33 or 22 (and some 0)

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p-Beauty Contest: Interpretations

(At least) two possible interpretations: Iterated dominance (level D) doesn’t …t the peaks at 22 and 33 very well

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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

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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

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Level-k

Type 0 uniformly randomizes over all available pure strategies (some authors assume a di¤erent distribution, e.g., normal)

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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

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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.

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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. Type behavior often converges, i.e., for k > x, type k = type k + 1) Or it cycles (i.e., type k = type k + x c, where c and x are constants), example: matching pennies assuming that type 0 picks one strategy for sure

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Best reply to random play

What is the best reply to uniform randomization over

f0, ..., 100g ?

Jörg Oechssler University of Heidelberg () November 27, 2018 16 / 25

Best reply to random play

What is the best reply to uniform randomization over

f0, ..., 100g ?

1 101 ∑100 x=0 x = 50 and hence br = 2 350 = 33. 3?

Jörg Oechssler University of Heidelberg () November 27, 2018 16 / 25

Best reply to random play

What is the best reply to uniform randomization over

f0, ..., 100g ?

1 101 ∑100 x=0 x = 50 and hence br = 2 350 = 33. 3?

Not true! (Breitmoser, GEB, 2012) Consider 2-player Guessing game, n = 2. What is br against uniform randomization?

Jörg Oechssler University of Heidelberg () November 27, 2018 16 / 25

Best reply to random play

What is the best reply to uniform randomization over

f0, ..., 100g ?

1 101 ∑100 x=0 x = 50 and hence br = 2 350 = 33. 3?

Not true! (Breitmoser, GEB, 2012) Consider 2-player Guessing game, n = 2. What is br against uniform randomization? Unique best reply and weakly dominant strategy: y = 0

Jörg Oechssler University of Heidelberg () November 27, 2018 16 / 25

Best reply to random play

What is the best reply to uniform randomization over

f0, ..., 100g ?

1 101 ∑100 x=0 x = 50 and hence br = 2 350 = 33. 3?

Not true! (Breitmoser, GEB, 2012) Consider 2-player Guessing game, n = 2. What is br against uniform randomization? Unique best reply and weakly dominant strategy: y = 0 For n > 2 not trivial to calculate br

Jörg Oechssler University of Heidelberg () November 27, 2018 16 / 25

Best reply to random play

What is the best reply to uniform randomization over

f0, ..., 100g ?

1 101 ∑100 x=0 x = 50 and hence br = 2 350 = 33. 3?

Not true! (Breitmoser, GEB, 2012) Consider 2-player Guessing game, n = 2. What is br against uniform randomization? Unique best reply and weakly dominant strategy: y = 0 For n > 2 not trivial to calculate br Extreme example: n = 3, p = 0.9. y = 0.5217 > 1

2! (see Breitmoser, 2012, p. 560).

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Some further problems with level-k models

Types are not identi…ed unless many pure strategies available. In that case: poor …t unless error term included More than one observation per subject combined with the assumption that types remain stable Not de…ned what happens when a type’s beliefs are falsi…ed in an extensive form game Many degrees of freedom (speci…cation of type 0, speci…cation

• f what types do when their assumptions are falsi…ed,

speci…cation of error term if included in the model) Estimation of the type-distribution: Nonparametric if identi…ed

• r assume a speci…c distribution (e.g., Poisson)

Distribution of types is not constant across games

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Poisson Cognitive Hierarchy

Is it reasonable to assume that type k is only aware of type k 1?

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Poisson Cognitive Hierarchy

Is it reasonable to assume that type k is only aware of type k 1? Higher types are probably aware of all lower types, not just k 1 Are lower types really unaware of the fact that there might be higher types? Cognitive Hierarchy models (Camerer, Ho, Chong, QJE, 2004) Assume that type k is aware of the entire distribution of types lower than k However, type k thinks he is the only type k and that there are no types higher than k.

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Poisson Cognitive Hierarchy

Typically, they assume that the distribution of types follows a Poisson distribution (P(X = xjτ) = eττx

x!

, x = 0, 1, . . . ; 0 < τ < ∞) Also, assume that there are no types higher than K. The probability that a player is of type k is then given by

P(X =kjτ) ∑K

i=0 P(X =ijτ)

Can estimate τ using maximum likelihood.

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Cursed equilibrium

Well know fact: winners’ curse in common value auctions Bidders don’t take into account that they only win the auction because others bid low and they do so since they have bad info Very robust phenomenon (Bazerman and Samuelson, 1983, Journal of Con‡ict Resolution; Kagel and Levin, 1986, AER).

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Winners’ curse: An example

Firm has book value v U[0, 1]. Firm knows book value, Raider does not.

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Winners’ curse: An example

Firm has book value v U[0, 1]. Firm knows book value, Raider does not. Firm values at v and Raider at 3

2v

Raider makes o¤er b to Firm.

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Winners’ curse: An example

Firm has book value v U[0, 1]. Firm knows book value, Raider does not. Firm values at v and Raider at 3

2v

Raider makes o¤er b to Firm. PBE: Firm sells i¤ v < b ! given b is accepted, average v is b

2

Jörg Oechssler University of Heidelberg () November 27, 2018 21 / 25

Winners’ curse: An example

Firm has book value v U[0, 1]. Firm knows book value, Raider does not. Firm values at v and Raider at 3

2v

Raider makes o¤er b to Firm. PBE: Firm sells i¤ v < b ! given b is accepted, average v is b

2

average value to raider: 3

2 b 2 = 3 4b < b

Jörg Oechssler University of Heidelberg () November 27, 2018 21 / 25

Winners’ curse: An example

Firm has book value v U[0, 1]. Firm knows book value, Raider does not. Firm values at v and Raider at 3

2v

Raider makes o¤er b to Firm. PBE: Firm sells i¤ v < b ! given b is accepted, average v is b

2

average value to raider: 3

2 b 2 = 3 4b < b

Optimal bid: b = 0, no trade.

Jörg Oechssler University of Heidelberg () November 27, 2018 21 / 25

Winners’ curse: An example

Firm has book value v U[0, 1]. Firm knows book value, Raider does not. Firm values at v and Raider at 3

2v

Raider makes o¤er b to Firm. PBE: Firm sells i¤ v < b ! given b is accepted, average v is b

2

average value to raider: 3

2 b 2 = 3 4b < b

Optimal bid: b = 0, no trade. Experimental Evidence by Bazerman and Samuelson (1985): 59% of subjects bid in [0.5, 0.75], and 92% bid more than 0. Subjects lose money on average.

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Bayesian Nash equilibrium

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Cursed equilibrium

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Cursed equilibrium

χ = 0 ! standard Bayesian eq. χ = 1 ! fully cursed eq.: player assumes that opponents play according to their average action independent of their types

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Back to the example

Consider fully cursed case χ = 1 Firm accepts i¤ b > v ! ¯ σF (accept) = R b

0 1dt + R 1 b 0dt = b

Raider thinks all …rms sell with prob. b ! value cond. on sale: E[v] = 1/2 Raider’s preceived payo¤: b(3

2E[v] b) = 3 4b b2

Maximized at b = 3

8 = .375

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Back to the example

Consider fully cursed case χ = 1 Firm accepts i¤ b > v ! ¯ σF (accept) = R b

0 1dt + R 1 b 0dt = b

Raider thinks all …rms sell with prob. b ! value cond. on sale: E[v] = 1/2 Raider’s preceived payo¤: b(3

2E[v] b) = 3 4b b2

Maximized at b = 3

8 = .375

Still too low to explain experiments Winners’ curse because true payo¤ from bidding b = 3

8 is

3 8(3 4 3 8 3 8) = 9 256

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Another example: Lemon’s market

Simple variant of Akerlof’s (1970) lemons model Buyer B might purchase a car from a seller at price of $1,000

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Another example: Lemon’s market

Simple variant of Akerlof’s (1970) lemons model Buyer B might purchase a car from a seller at price of $1,000 Seller S knows whether the car is a lemon, worth $0 to both or a peach, worth $3,000 to B and $2,000 to S.

Jörg Oechssler University of Heidelberg () November 27, 2018 25 / 25

Another example: Lemon’s market

Simple variant of Akerlof’s (1970) lemons model Buyer B might purchase a car from a seller at price of $1,000 Seller S knows whether the car is a lemon, worth $0 to both or a peach, worth $3,000 to B and $2,000 to S. B believes each occurs with probability 1/2 Car is sold if and only if both say yes

Jörg Oechssler University of Heidelberg () November 27, 2018 25 / 25

Another example: Lemon’s market

Simple variant of Akerlof’s (1970) lemons model Buyer B might purchase a car from a seller at price of $1,000 Seller S knows whether the car is a lemon, worth $0 to both or a peach, worth $3,000 to B and $2,000 to S. B believes each occurs with probability 1/2 Car is sold if and only if both say yes Fully rational B would realizes that S wishes to trade if and only if the car is a lemon, and hence refuses to buy

Jörg Oechssler University of Heidelberg () November 27, 2018 25 / 25

Another example: Lemon’s market

Simple variant of Akerlof’s (1970) lemons model Buyer B might purchase a car from a seller at price of $1,000 Seller S knows whether the car is a lemon, worth $0 to both or a peach, worth $3,000 to B and $2,000 to S. B believes each occurs with probability 1/2 Car is sold if and only if both say yes Fully rational B would realizes that S wishes to trade if and only if the car is a lemon, and hence refuses to buy A χcursed B believes that with probability χ, S sells with probability 1/2 irrespective of type of car.

Jörg Oechssler University of Heidelberg () November 27, 2018 25 / 25

Another example: Lemon’s market

Simple variant of Akerlof’s (1970) lemons model Buyer B might purchase a car from a seller at price of $1,000 Seller S knows whether the car is a lemon, worth $0 to both or a peach, worth $3,000 to B and $2,000 to S. B believes each occurs with probability 1/2 Car is sold if and only if both say yes Fully rational B would realizes that S wishes to trade if and only if the car is a lemon, and hence refuses to buy A χcursed B believes that with probability χ, S sells with probability 1/2 irrespective of type of car. So that the car being sold is a peach with probability

(1 χ)0 + χ 1

2 = χ/2

and therefore worth χ

2 3000 = χ1500 > 1000 if χ > 2/3

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