John Forbes Nash, Jr., renowned mathematician, Nobel Prize winner, Abel Prize

laureate, and subject of the 2001 Academy Award-winning movie A Beautiful Mind, died in a car

accident on Saturday, May 25. In remembrance of his immeasurable contributions to education,

this article plays a non-cooperative game with student-teacher engagement.

Student engagement has been a prominent focus in districts around the nation, yet many

educators believe an equal concentration should be placed on engaging teachers. While

engagement undoubtedly takes many forms, if we operate under the basic assumption that high

engagement is more difficult to supply than low engagement, then we can use “giving high

effort” synonymously with “being highly engaged.” The essential question we as educators then

face is this: How do we cultivate high effort by teachers and students alike?

In light of this question, it may be helpful to think of education, particularly as it relates

to the efforts of teachers and students, as a non-cooperative game. In game theory, a non-

cooperative game is any game in which the players make decisions independent of the decisions

of the other players. Anyone who has even spent one second in a classroom can agree that

education is most certainly non-cooperative; after all, it is possible that the phrase “pulling teeth”

is said more frequently in teachers’ lounges than dental offices.

Furthermore, education is also a nonzero-sum game, in that one player’s gain is not

dependent upon another player’s loss (as opposed to zero-sum games like poker, chess, and

football). So, under these parameters, we can comfortably say that the game being played in

schools all over the country is both non-cooperative and nonzero-sum, that is, the players

(students and teachers) make decisions independent of each other, and the outcomes can be win-

win, lose-lose, or any combination thereof. Encouraged yet?

No? Well that’s why Nash’s mind was more beautiful than our own. He understood that if

we know the rules to the game and make a few logical assumptions, then we can find the ideal

solution concept—the point at which neither player has anything to gain by changing only their

strategy. This solution concept is famously known as the Nash equilibrium, and it still has some

pretty significant implications for modern day education.

If your, um…game, let’s take a look at an example of the Nash equilibrium in work.

We’ll be looking specifically at our essential question from above, focusing our attention to the

effort levels of our player groups: students and teachers. We already know the rules of the game

(non-cooperative and nonzero-sum), so let’s make our assumptions:

1. The greatest reward (win) would be for the high effort of one group to be reciprocated by

the high effort of the other. Every teacher’s dream.

2. The greatest insult (loss) would be for the high effort of one group to be met with low

effort by the other. Every teacher’s nightmare.

3. Low effort is low effort; that is, there are not separate degrees to which one could supply

minimal effort.

The scorecard for these types of games is something called the payoff matrix, and—given

our assumptions—our payoff matrix would look like this,

Table 1.1: Payoff Matrix

The choices for the column group of players (teachers) is denoted by the second value in each cell.

Teachers

High effort Low effort

StudentsHigh effort d, a f, b

Low effort e, c e, b

where a > b > c for teachers and d > e > f for students. If we supply numbers to fit the variables

(e.g. 2 > 1 > 0), the payoff matrix takes the form:

Table 1.2: Payoff Matrix (with numerical values)

Hefty Implications

Table 1.2 shows two clear Nash equilibria, namely high effort by both groups or low

effort by both groups. The implication for looking at engagement in this manner is that one

group’s effort is likely to be predicated by their assumptions of the other group. For example, if

teachers believe that the majority of their students will put forth minimal effort, then the most

rewarding choice for teachers is to put forth minimal effort themselves. Likewise for students, if

they believe that their teacher is “just collecting a paycheck,” then their optimal choice would be

to supply minimal effort.

Of course, the converse is also true, and it’s much more encouraging. If teachers believe

that their students will supply a high level of effort, then the teachers best choice would be to

honor their students’ endeavors with high effort of their own. Similarly, if students think highly

of their teacher’s level of effort, they would be rewarded by providing an increased level of effort

themselves.

Questions Worth Considering

Teachers

High effort Low effort

StudentsHigh effort 2, 2 0, 1

Low effort 1, 0 1, 1

With the scores tallied and results from our game in, it is clear to see that Nash’s work

continues to provide keen insights into the strategies behind the decisions we make as educators

every day. Some questions that may be worthy of additional reflection include:

• How is engagement among our students quantified? We might think we know what “low

effort” looks like, but how do we gauge it formally in our students? (Marzano, Pickering, and

Pollock have developed an Effort and Achievement Rubric to this end).

• What does “low effort” look like in teachers? How can it be quantified? What, if anything,

should be done to hold educators accountable for their own engagement?

• What does “high effort” from students look like? From teachers? What, if anything, can be

done to incentivize high levels of effort from these two groups?

• To what degree are effort assumptions based upon cultural competencies, or a lack thereof?

• To what degree (or percentage) does each group of players assume high effort by the other?

To what degree (or percentage) does each group of players assume low effort by the other?

The last two questions are especially important since, as mentioned above, the game’s

payoff matrix implies that the effort-supply choices of both students and teachers will be wholly

dependent upon their assumptions about the other group’s effort. Depending upon the answers to

the questions listed above, a few final questions may be worth asking:

• How can the assumptions of each group be changed?

• Will a change in assumptions correlate to a change in effort?

• Will a change in effort correlate to a change in achievement?

Conclusion

The answers to these last three questions mentioned above will determine if, and to what

degree, our little thought experiment is worth replicating at your own site or district. If a

transitive relationship exists between assumptions, effort, and achievement, and if assumptions

can indeed be changed, then programs that target the assumptions of teachers are likely to

produce a positive change in achievement, and should therefore be of high importance to

educational leaders. And when these programs are in place and students and teachers are

supplying maximum effort in perfect harmony and test scores go through the roof—we’ll know

whose beautiful mind deserves all of the credit.