Collective actions and expectations

Collective actions and expectations

Belev Sergey, Kalyagin Grigory

Moscow State University

«If the same individuals take part in the repeating game, the tendency to cooperation increases significantly. And there is no doubt that repeating multiplies the cooperative result, involving those, who in single-stage game showed themselves cautious and suspicious» (McCabe, Rassenti, Smith,1996)

Introduction to problemIntroduction to problem

In repeating games individualsIn repeating games individuals gain experience gain experience and according to It form their expectationsand according to It form their expectations concerningconcerning further interactionsfurther interactions..

Individuals learn playing again and againIndividuals learn playing again and again

Possible solutionPossible solution

Macy’s learning modelMacy’s learning model (1990): (1990):

Payoff function of jth individualPayoff function of jth individual : :

The model for simulationThe model for simulation

jJj CN

LRV 1

CCj j – – binary choice (to take part (Cj=1)/not to take part (Cj=0) in the collective action), ,

N N – – group size, ,

J J – – competitiveness of good (J=0 –full competitiveness, J=1 – no competitiveness), ,

R R – – payoff of all group members within 100% level of participation in the collective action, ,

L L – – share of the maximum total payoff (R) received by all group members

Macy’s production function L Macy’s production function L (1990):(1990):

The model for simulationThe model for simulation(2)(2)

MM – – slope parameter (slope parameter (ММ=1 – approaches linearity, =1 – approaches linearity, ММ=10 - s-shaped curve)=10 - s-shaped curve)

p p – – share of volunteers share of volunteers

)5,0(1

1pMe

L

Payoff and participationPayoff and participation

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1 1,2

Дол

я м

акси

мал

ьног

о вы

игры

ша

L

Доля участвующих p

M=1

M=5

M=10

The innovation to MacyThe innovation to Macy (1990): (1990):

IndividualIndividual makes decision to take part or notmakes decision to take part or not, , comparingcomparing expectedexpected benefitsbenefits::


jJ

eej C

N

RLV 1

)5,0(1

1

N

cpM

e

jej

e

L

)5,0(1

1pMe

L jJj C

N

LRV 1

To beTo be, , if expected payoff is more whenif expected payoff is more when he or she takes part he or she takes part than when he or she don’tthan when he or she don’t::

To be or not to beTo be or not to be……… … a volunteera volunteer??

)0()1( jejj

ej CVCV

JpMJN

pM Ne

R

Ne

Re

e

1)5,0(1

)1

5,0( )1(1

)1(

The expected level of The expected level of other members’ other members’ participation participation ::


tti

t

i

ti

e pp t

11

1

1

actual share of volunteers at i period of t-1 periods of interaction. 1t

ip

weight of other members’ actual participation at i period.1ti

11

1

t

ii

random error of individual calculations, uniformly distributed at [-a;a]

shortness of memory (as more, so shortly)

t

)1(1 ti

ti

δ

The example of calculationThe example of calculation

1 2 3 4 5 6 7 8 9 10 11 12 13

1 1,000 0,400 0,211 0,123 0,076 0,048 0,031 0,020 0,013 0,009 0,006 0,004 0,003

2 0,000 0,600 0,316 0,185 0,114 0,072 0,047 0,030 0,020 0,013 0,009 0,006 0,004

3 0,000 0,000 0,474 0,277 0,171 0,108 0,070 0,046 0,030 0,020 0,013 0,009 0,006

4 0,000 0,000 0,000 0,415 0,256 0,162 0,105 0,069 0,045 0,030 0,020 0,013 0,009

5 0,000 0,000 0,000 0,000 0,384 0,244 0,157 0,103 0,068 0,045 0,030 0,020 0,013

6 0,000 0,000 0,000 0,000 0,000 0,365 0,236 0,154 0,101 0,067 0,044 0,029 0,020

7 0,000 0,000 0,000 0,000 0,000 0,000 0,354 0,231 0,152 0,101 0,067 0,044 0,029

8 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,347 0,228 0,151 0,100 0,066 0,044

9 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,342 0,226 0,150 0,100 0,066

10 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,339 0,225 0,149 0,099

11 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,337 0,224 0,149

12 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,336 0,223

13 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,335

14 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000

δ =0,5 ti

= 0,211* +0,316* +0,474* +ep4 1p 2p 3p 4

1.1. The more group size (N) isThe more group size (N) is, , the less expected marginal benefits from participation arethe less expected marginal benefits from participation are

2.2. The less competitiveness The less competitiveness (0→ (0→ J →1) is ORJ →1) is OR the more the maximum payoff (R) isthe more the maximum payoff (R) is,, than the more the maximum size of group, where the than the more the maximum size of group, where the collective action could take place, iscollective action could take place, is

Obvious resultsObvious results

JpMJN

pM Ne

R

Ne

Re

e

1)5,0(1

)1

5,0( )1(1

)1(

Individual takes partIndividual takes part, , if expected level of cooperation is near if expected level of cooperation is near 0,50,5, , and doesn’tand doesn’t, , if he or she expects eitherif he or she expects either full full participation or noparticipation or no participation at allparticipation at all

Obvious resultsObvious results

-1

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

ожидаемое участие pe

)0()1( jejj

ej CVCV

Conditions of simulationsConditions of simulations

N M R δ J a po

20 11 6

With the help of computer simulations we analyze the With the help of computer simulations we analyze the impact of shortness of memory, size of random error impact of shortness of memory, size of random error and initial expected level of cooperationand initial expected level of cooperation. .

We vary only the size of J instead of varying M, N, R We vary only the size of J instead of varying M, N, R within Jwithin J..

NN=20=20 is less than maximum group size for certain M, R is less than maximum group size for certain M, R andand JJ..

What are the model conditions when the actual What are the model conditions when the actual share of volunteers is dynamically stableshare of volunteers is dynamically stable and and equalsequals 0? 0?

AssumptionAssumption 1: 1: if initial expectations (pif initial expectations (p00) are near) are near 0 0

oror 1 1

AssumptionAssumption 2: 2: if size of error (a) is smallif size of error (a) is small

AssumptionAssumption 3: 3: if the memory is short if the memory is short ((δ is bigδ is big))

SimulationsSimulations..

Assumption Assumption 1. 1. According to inputted dataAccording to inputted data individual takes part individual takes part if and only if pif and only if pe € e € [0,334; 0,616] . p0 ≤ 0,3 (≤ 6 from 20).

SimulationsSimulations(2).(2).

N M R δ J a po20 11 6 0,5 0,25 1/6 0,3

Expected participation

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0 10 20 30 40 50 60 70 80 90 100

Actual participation

0,00

0,20

0,40

0,60

0,80

1,00

1,20

0 10 20 30 40 50 60 70 80 90 100

N M R δ J a po20 11 6 0,5 0,25 1/6 0,25


0,00

0,05

0,10

0,15

0,20

0,25

0,30

0 10 20 30 40 50 60 70 80 90 100


-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0 10 20 30 40 50 60 70 80 90 100

Assumption Assumption 1. 1. According to inputted dataAccording to inputted data individual takes part individual takes part if and only if pif and only if pe € e € [0,334; 0,616] . p0≥0,65 (≥13 из 20).

Simulations Simulations (3).(3).

N M R δ J a po

20 11 6 0,5 0,25 1/6 ≥0,65

N M R δ J a po20 11 6 0,5 0,25 1/6 1


0,000,100,200,300,400,500,600,700,800,901,00

0 10 20 30 40 50 60 70 80 90 100


0,00

0,20

0,40

0,60

0,80

1,00

1,20

0 10 20 30 40 50 60 70 80 90 100

New AssumptionNew Assumption. . If expected level atIf expected level at the thirdthe third iteration is less than critical value iteration is less than critical value (0,334), than we could get at low expectation’s trap. If J=0,15 the interval is [0,395; 0,556]


N M R δ J a po20 11 6 0,5 0,15 1/6 1

N M R δ J a po20 11 6 0,5 0,15 1/6 1


0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

0 10 20 30 40 50 60 70 80 90 100


-0,10

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0 10 20 30 40 50 60 70 80 90 100

Assumption 2. The impact of the error sizeAssumption 2. The impact of the error size. False expectations near critical values could change the destiny of collective action.


N M R δ J a po20 11 6 0,5 0,25 0,25 0,25

N M R δ J a po20 11 6 0,5 0,25 1/6 0,25


0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0 10 20 30 40 50 60 70 80 90 100


0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

0 10 20 30 40 50 60 70 80 90 100

AssumptionAssumption 3 3 short memoryshort memory. The longer memory is, the less possible the high expectation’s trap is


N M R δ J a po20 11 6 0,05 0,15 1/6 1

N M R δ J a po20 11 6 0,5 0,15 1/6 1

expected participation

0,000,100,200,300,400,500,600,700,800,901,00

0 10 20 30 40 50 60 70 80 90 100


0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

0 10 20 30 40 50 60 70 80 90 100

actual participation

-0,10

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0 10 20 30 40 50 60 70 80 90 100


-0,10

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0 10 20 30 40 50 60 70 80 90 100

• The decision to take part or not dependsThe decision to take part or not depends onon expected expected level of other group member’s cooperation and could be level of other group member’s cooperation and could be illustrated as U-inverted curveillustrated as U-inverted curve: : take part close to take part close to 0,50,5 and and notnot close to close to 1 1 oror 0 0

• The impact of initial expectations is crucialThe impact of initial expectations is crucial: : group group members could face the low expectation’s trapmembers could face the low expectation’s trap

• The high expectation’s trap is possible only if the The high expectation’s trap is possible only if the average expected share of participation at the second average expected share of participation at the second iteration is less than crucial value iteration is less than crucial value

• Random errors could help collective action to take place Random errors could help collective action to take place (or not) if expected participation is close to critical value(or not) if expected participation is close to critical value

ConclusionsConclusions

Thank you !!!Thank you !!!

Collective actions and expectations

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