Productivity Revelation and the

Dynamics of Group Formation

 

 

 

 

 

Ivanna Ferdinandova

IDEA

 

And

 

Michael Heaney

University of Chicago

 

 

 

 

2001 Graduate Workshop in Computational Economics

Homework Problem (2001) – Solution

 

 

I.                   General Assumptions

 

·       Objective: High grades – Each individual is motivated to belong to the group that will yield her the highest grade.

 

·       Group solutions – Individuals submit solutions as a group; individual grades are determined on the basis of the group product.  Individuals are free to join a group of size one or higher.

 

·       Difficulty – Homework problems do not vary in expected difficulty over the course of the semester.

II.                Theory: What allows groups to earn higher grades?

 

·       Productive members

 

^ Groups with more productive members will earn higher grades than groups with less productive members, ceteris paribus.

 

·       Optimal group size

 

^ Groups that are close to the optimal number of members will earn higher grades than groups than groups with more or less than the optimal number of members, ceteris paribus.

 

·       Stability

 

^ Groups that are able to remain stable over time will earn higher grades than groups that sustain a high degree of turnover, ceteris paribus.

 

III.             Problem: The productivities of individuals are difficult to observe directly.  Therefore, groups will be uncertain as to which individuals will best serve their objectives of grade maximization.

 

 

 

 

IV.           Hypotheses

 

·       H1: Increases in the observability of individual productivities will decrease the time it takes for groups to become stable.

 

·       H2: Increases in the observability of individual productivities will increase the variance of group grades.

 

·       H3: Increases in the observability of individual productivities will increase the average of group grades (i.e., the positive effect of stability overwhelms the negative effect of sorting).

 

V.              Computational Model: Assumptions

 

·       Individuals randomly join groups during the first week.  They lack knowledge of their own productivities or the productivities of their classmates at that time.  Individuals learn their own productivity after the first week.

 

·       Groups form with sizes varying from 1 to 6.

 

·       Grades = average productivity * size * stability

 

·       Average productivity = simple average

 

·       Size =  { 1.0 if the group has one member

{ 1.1 if the group has two members

{ 1.2 if the group has three members

{ 1.3 if the group has four members

{ 1.2 if the group has five members

{ 1.1 if the group has six members

 

·       Stability = Percent turnover in the group.

 

VI.           Computational Model: Algorithm

 

·       Individuals compare their productivities to the group grade.  If their productivities are lower than the group grade, they are satisfied.  If their productivities are higher than the group grade, they are dissatisfied.

 

·       Dissatisfied individuals put up a “flag” indicating that they would like to join another group.

 

·       All groups observe the flagging individual’s productivity with some degree of certainty. 

 

·       Each group calculates the expected value-added by each flagging individual and makes an offer to the flagging individual with the highest expected value-added, assuming this value is positive (if the highest value is negative, then no offer is made).

 

·       Each flagging individual evaluates all her offers and accepts an offer from any group with a grade higher than her individual productivity.  If no group has productivity higher than her individual productivity, she forms her own group (with one member).

 

·       Undertake next homework (if not week 15).

 

·       Repeat for 14 weeks.

 

VII.        Results

 

Observability: High Certainty

 

Week         # of Gps           Avg. Grade      Grd. Var.         # of Gps w/ Turnover

 

1

2

3

…

 

Observability: Moderate Certainty

 

Week         # of Gps           Avg. Grade      Grd. Var.         # of Gps w/ Turnover

 

1

2

3

…

 

Observability: Low Certainty

 

Week         # of Gps           Avg. Grade      Grd. Var.         # of Gps w/ Turnover

 

1

2

3

…

 

VIII.     Statistical Analysis

 

·       H1: # of Groups w/ Turnover=0 will reach 0 first under high certainty, followed by moderate and low certainty.

 

·       H2: Weekly grade variance will be higher under high certainty, followed by moderate and low certainty.

 

·       H3: The average of the average weekly grade will be highest under high certainty, followed by moderate and low certainty.

IX.            Potential Extensions

 

·       Attitudes toward risk – make some of the individuals/groups in the class risk averse/neutral/seeking with regard to grades.  Examine how variations in these attitudes affect grade averages, variance, and group turnover.

 

·       Updating information – What kind of information can be gained from individuals’ decisions to leave/remain in a group?  Can groups draw reliable inferences about member behavior conditioned on grades of the group she is seeking to leave.

 

·       Discounting – to what extent can variations in discount rates affect the willingness of individuals to remain in groups that offer grades lower than their average productivities.

 

 

X.               Standard Social Science Scenarios

 

·       Coalitions – Whenever the benefits are shared evenly (regardless of size}, but individual membership in a coalition may be a liability for the coalition.  E.g., international coalitions when some nations may “misbehave.”

 

·       Vacancy chains – Sorting behavior whenever there are status differences among groups.  E.g., thin labor markets, such as the market for CEOs.

 

·       Mating / Dating behavior – When group size is limited to a maximum of two.