Working Hard As an Evolutionary Phenomenon

 

 

Jung-Kyoo Choi

Alexander Peterhansl

July 18, 2001

 

 

1.  Introduction

 

Some of the characteristics of the “Homework Problem” are analogous to the public goods game.  We can assume, for example, that the class as a whole is interested in maximizing effort in completing the assignments.  Each individual, however, would like to expend as little effort as possible.  This is the prototypical public goods scenario where the social and individual objectives conflict.  Our approach will investigate the “Homework Problem” in light of this analogy.  In particular, our approach focuses on the students’ formation of their strategies in the total effort expended.

 

 

2.  The Model

 

The repeated prisoner’s dilemma game provides a good starting point to investigate individual versus collective objectives.  It is the main framework for our analysis.  Students are paired up randomly and play the game repeatedly.  The payoff matrix is as follows:

 

 

H

S

H

10,10

-5, 15

S

15, -5

0, 0

 

There are three strategies:  work hard (H), slack off (S), conditionally work hard (TFT).  A student playing strategy H, will work hard in every period.  A student playing strategy S, will slack off in every period.  Strategy TFT (corresponding to Tit-for-Tat in the usual setup) will work hard if his partner played hard in the previous period.

 

There are 20 students playing the game over the course of 15 homework assignments (one per week).  Each homework assignment consists of 5 sub problems.  A game is played for each sub problem of the assignment so that there are a total of 75 rounds of the game.

 

At the beginning of the game, students are assigned an initial seed of one of three strategies.  They are paired up and play for the first assignment (5 rounds).  Unlike strategies H and S, strategy TFT has an additional complication.  In our setup, TFT players start with working hard and remember their partner’s moves throughout the assignment.  If their partner slacks off in the preceding sub problem, the TFT player will punish him by slacking off for the rest of the assignment.  A TFT player’s memory is lost after submitting the homework.

 

For the next assignment, the strategies are updated in the following fashion:  The strategies of the players of the lowest payoff quintile are replaced with the strategies of the players of the highest payoff quintile.  This updating occurs after each assignment (after every 5 games).

 

Our simulation starts with a fixed initial seed of strategies.  There are 6 unconditional hard workers, 6 slackers, and 8 conditional hard workers.

 

The purpose of this rudimentary model is to provide a first attempt at showing the possibility of evolving strategies.  How does strategy formation affect the effort of the students over the course of the 15 assignments?  The role of the TFT strategy is very important here.  It plays the role of enforcing the objective of the collective over the objective of the individual.  Our model is designed to shed more light on the role of TFT in sustaining hard workers. 

 

 

3.  Results

 

The following graphs show three sample outcomes of evolving strategies with fixed initial seeds.    Each graph plots the changing strategy frequencies over the 15 weeks.  There are three types of outcomes that can be reached in this model.  In outcome 1, figure 1, TFT quickly evolves to be the unique strategy within 9 weeks.  In outcome 2, figure 2, the slacking off strategy quickly gains a foothold (within 5 weeks) at the expense of the other strategies.  In outcome 3, figure 3, TFT players support a high proportion of unconditional hard workers.

 

<Figure 1> Sample Outcome I (TFT dominant equilibrium)

 

<Figure 2> Sample Outcome II (Slacker dominant equilibrium)

 

<Figure 3> Sample Outcome III (TFT and Hard Worker

 

 

4.  Conclusion

 

The rudimentary model yields some promising outcomes for investigating the emergence of strategies in heterogeneous populations.  The logical next step would be to see how robust these results are in face of varying initial seeds, varying number of sub problems, and varying number of weeks.

 

Future directions might be looking at different types of sanctioning mechanisms, such as second-order punishment.  More generally, the investigation of evolving strategies with genetic algorithms might also be an interesting approach.