a herding perspective on global games and multiplicity - SSRN papers

Working Paper 03-29 Economics Series 08

Departamento de Economía Universidad Carlos III de Madrid

May 2003

Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624 98 75

A HERDING PERSPECTIVE ON GLOBAL GAMES AND MULTIPLICITY * James S. Costain1 Abstract Recently, it has been claimed that full-information multiple equilibria in games with strategic complementarities are not robust, because generalizing to allow slightly heterogeneous information implies uniqueness. This paper argues that this "global games" uniqueness result is itself not robust. If we generalize by allowing most agents to observe a few previous actions before choosing, instead of forcing players to move exactly simultaneously, then multiplicity of outcomes is restored.

Only a small sample of observations is needed to make our herding

equilibrium behave like a full-information sunspot equilibrium instead of a global games equilibrium.

Keywords: Herding, information cascades, multiplicity, uniqueness, heterogeneity, strategic complementarities, global games, adaptive learning, stochastic recursive algorithms

JEL Classification: C62, C72, C73, E00

1

Departamento de Economía, Universidad Carlos III de Madrid. E.mail: [email protected]

* Thanks to Antonio Ciccone for encouraging me to study some related questions. Thanks also for helpful comments to Fabrizio Germano, Joel Shapiro, Stephen Morris, Hyun Song Shin, Frank Heinemann, Dan Friedman, Berthold Herrendorf, Marcel Jansen, Juan Ruiz, and Antonio Cabrales. This paper was first presented under the title "On Multiplicity and Heterogeneity in a Herding Model", at CEMFI, where I thank Guillermo Caruana and other seminar participants. Financial support from the Spanish Ministry of Science and Technology (MCyT grant SEC2002-01601) is gratefully acknowledged. Errors are the responsibility of the author.


 



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T mapping: Z=0 dash, Z=1 solid. weightSTC=0.5 1

0.8

0.8

probability of choosing x=1

T mapping: Z=0 dash, Z=1 solid. weightSTC=0.3 1

0.6

0.4

0.2

0

probability of choosing x=1



0

0.6

0.4

0.2

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0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

T mapping: Z=0 dash, Z=1 solid. weightSTC=0.7 1

T mapping: Z=0 dash, Z=1 solid. weightSTC=0.9 1

0.8

0.8

probability of choosing x=1

probability of choosing x=1



0.6

0.4

0.2

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0.6

0.4

0.2

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0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

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CDF over S , i=300; weightSTC = 0.3

CDF over S , i=300; weightSTC = 0.5 probability of S (Z=0 dash, Z=1 solid)

0.5 0.4

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i

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0.2

0.4

0.6

0.8

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0.2

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0.6

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Si

Si

CDF over Si, i=300; weightSTC = 0.7

CDF over Si, i=300; weightSTC = 0.9

0.5 0.4 0.3

1

0.5 0.4 0.3

i

probability of Si (Z=0 dash, Z=1 solid)

i

probability of S (Z=0 dash, Z=1 solid)

probability of Si (Z=0 dash, Z=1 solid)

i

0.2 0.1 0

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0.2

0.4

0.6

0.8

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Sample of 200 HISTORIES of Si: Agg state Z=1, weightSTC = 0.5 fraction playing x=1 so far

1 0.8 0.6 0.4 0.2 0

0

50

100

150

200

250 300 350 400 count i Sample of 200 HISTORIES of S : Agg state Z=1, weightSTC = 0.7

450

500

100

450

500

i

fraction playing x=1 so far

1 0.8 0.6 0.4 0.2 0

0

50

150

200

250 count i

300

120

140

100

120

80 60 40 20 0

400

Frequency of Si: weightSTC=0.7

Number of histories

Number of histories

Frequency of Si: weightSTC=0.5

350

100 80 60 40 20

0

0.2

0.4

0.6

0.8

0

1

Si

0

0.2

0.4

0.6 Si

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T mappings: Z by steps of 0.02. weightSTC=0.3 1 0.9

probability of choosing x=1

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 fraction of predecessors choosing x=1

0.9

1

0.9

1

T mappings: Z by steps of 0.02. weightSTC=0.6 1 0.9

probability of choosing x=1

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 fraction of predecessors choosing x=1

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T mapping: Z=0 dash, Z=1 solid. precision=0.1 1 probability of choosing x=1

probability of choosing x=1

T mapping: Z=0 dash, Z=1 solid. precision=0.03 1

0.8

0.6

0.4

0.2

0

0

probability of choosing x=1

probability of choosing x=1

0.6

0.4

0.2

probability of choosing x=1

probability of choosing x=1

0.2

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

0.8

0.6

0.4

0.2

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

T mapping: Z=0 dash, Z=1 solid. precision=10 1

0.8

0.6

0.4

0.2

0

0.4

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

T mapping: Z=0 dash, Z=1 solid. precision=3 1

0

0.6

T mapping: Z=0 dash, Z=1 solid. precision=1 1

0.8

0

0.8

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

T mapping: Z=0 dash, Z=1 solid. precision=0.3 1

0



0.8

0.6

0.4

0.2

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

¶kµ

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1









  








%
 # '    


 probability of Si (Z=0 dash, Z=1 solid)

0.5 0.4 0.3 0.2 0.1 0

probability of Si (Z=0 dash, Z=1 solid)

CDF over Si, i=300; precision = 0.1

0

0.2

0.4

0.6

0.8

0.3 0.2 0.1

0

0.2

0.4

CDF over Si, i=300; precision = 1

0.2 0.1

0.2

0.4

0.6

0.8

0.4 0.3 0.2 0.1 0

1

1

0.5

0

0.2

0.4

Si

0.6

0.8

1

Si

CDF over Si, i=300; precision = 3

CDF over Si, i=300; precision = 10 0.6 probability of Si (Z=0 dash, Z=1 solid)

0.6 probability of Si (Z=0 dash, Z=1 solid)

0.8

CDF over Si, i=300; precision = 0.3

0.3

0.5 0.4 0.3 0.2 0.1 0

0.6 Si

0.4

0

0.4

Si

0.5

0

0.5

0

1

probability of Si (Z=0 dash, Z=1 solid)

probability of Si (Z=0 dash, Z=1 solid)

CDF over Si, i=300; precision = 0.03



0

0.2

0.4

0.6 Si

0.8

0.5 0.4 0.3 0.2 0.1 0

1

¶q¾

0

0.2

0.4

0.6 Si

0.8

1






 
   




 

 

T mapping: Z=0 dash, Z=1 solid. N=2

probability of choosing x=1

probability of choosing x=1

1

0.8

0.6

0.4

0.2

0

0.8

0.6

0.4

0.2

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

0

T mapping: Z=0 dash, Z=1 solid. N=10

probability of choosing x=1

probability of choosing x=1

1

0.8

0.6

0.4

0.2

0

0.8

0.6

0.4

0.2

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

0

T mapping: Z=0 dash, Z=1 solid. N=40 1 probability of choosing x=1

probability of choosing x=1

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1 T mapping: Z=0 dash, Z=1 solid. N=80

1

0.8

0.6

0.4

0.2

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1 T mapping: Z=0 dash, Z=1 solid. N=20

1

0



T mapping: Z=0 dash, Z=1 solid. N=5

1

0


 
 

0

0.8

0.6

0.4

0.2

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1

¶q¿

0

0.2 0.4 0.6 0.8 1 fraction of predecessors choosing x=1









  










 

probability of Si (Z=0 dash, Z=1 solid)

probability of Si (Z=0 dash, Z=1 solid)

0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

0.4 0.3 0.2 0.1

0

0.2

0.4

CDF over Si, i=300; N = 10

CDF over Si, i=300; N = 20

0.2 0.1

0.2

0.4

0.6

0.8

0.4 0.3 0.2 0.1 0

1

1

0.5

0

0.2

0.4

Si

0.6

0.8

1

Si

CDF over Si, i=300; N = 40

CDF over Si, i=300; N = 80 0.6 probability of Si (Z=0 dash, Z=1 solid)

0.6 probability of Si (Z=0 dash, Z=1 solid)

0.8

Si

0.3

0.5 0.4 0.3 0.2 0.1 0

0.6

Si

0.4

0

0.5

0

1

0.5

0



CDF over Si, i=300; N = 5

probability of Si (Z=0 dash, Z=1 solid)

probability of Si (Z=0 dash, Z=1 solid)

CDF over Si, i=300; N = 2


 
 

0

0.2

0.4

0.6

0.8

0.5 0.4 0.3 0.2 0.1 0

1

Si



·

0

0.2

0.4

0.6 Si

0.8

1