How Can I Use Game Theory to Determine the Most Cost-and-Time-Efficient Method for my School to Go about Offering a More Comprehensive Music Education Program to its Lower and Middle School Students?



by Katie Nichols


Empathy Interview

I conducted an interview with my mom, who agreed on a lot of the issues with my school’s music curriculum I was already thinking about. When I asked her about the impression our middle school concerts gave her, she said there was not one at all, and couldn’t remember them to begin with (an indictment in itself)! Though I was thinking mainly about overworked teachers and poor song selection, she emphasized the connection between these two phenomena; that teachers have no time, or simply are not good enough educators, to devote themselves to fostering a true musical appreciation in their students. She also emphasized the importance of performing as a building block of musical education, something my school does not offer save two or max three times a year. 

Another important point of hers; little kids, for the most part, love to sing, because they don’t recognize or worry about the extent of their musical talent. In middle school, when they begin auditioning for parts in musicals, whether they get a part as an extra or a lead tells them whether they are good, bad, or somewhere in between. But in the years leading up to this development of self-consciousness, children (for the most part) like to sing in that mixture of shouting and whispering to a tune that they do. The school should be taking advantage of this, in getting those performing muscles pumping early on. It helps not just to build foundations in music education, but also with public speaking, which we all must do fairly often as we grow older. Were it not for my interview with my mom, I think I would have missed these crucial pieces of the puzzle, and her thinking on this issue has greatly impacted the project I will now jump into below!

The Players

The players in my game theory model are my school and their clients (including parents and students alike). It is the interests and actions of these two groups of people that are going to determine how the school reforms its music education curriculum. In some cases, they are in conflict with each other, and in others, coincide, which makes for a non-zero-sum game. 

Before this model can be useful at all, the school community must be convinced that music education is not only valuable but essential to a comprehensive general education. I have cited several articles at the bottom of this page that make this point, but for now, I will assume they have already decided to implant reforms. 

The relationship between the two players is unique in that decisions are heavily influenced by impact on their “opponent.” The school may decide to pick a strategy on their own without consulting their clients. Though, if they select one unfit for the clients’ needs, they risk angry parents and disappointed students, which reflects badly on the school (say clients donate less at next year’s annual fundraiser, for example). The clients, further, may decide to implant a strategy that requires numerous resources and thus a lot of money. But if too much money goes into reforming the music program, then other areas of the school’s education lose money, lowering the overall quality of the school (which of course, the clients are using). When a decision is bad, it generally loops back around to hurt the opponent and player who made the choice of strategy, which works to foster cooperation to some extent. The school serves the clients with education, and the clients serve the school by offering the money to sponsor the education. It is a circle of payoffs. Though interests, as I have just demonstrated, may be at odds with each other, and that’s where cooperation grows sticky. Our matrix looks like this:



The Strategies

To understand the reform strategies, one must first understand the breakup of the average lower and middle school music year as it is. Today, as I remember it, ¼ of the class is lessons in basic music theory (learning the notes; length, tune, order) and ¾ is practice with songs for concerts (singing). I would like to keep ¼ of class time allotted to music theory lessons, as nobody can deny that they are essential to a music education. Though I have some ideas about how to best spend the other ¾ of music class time.

There are several ways to improve the music program without spending money or devoting more time to music class than is currently. First, expose students to the music of the greats; Beethoven symphonies, Chopin nocturnes, Schubert impromptus, etc. Currently, there is little to none of this in lower and middle school music classes, which is, frankly, an absolute travesty. A foundation in simply recognizing and appreciating great classical music is perhaps more essential to music education than any other possible reforms I will soon recommend; if students cannot appreciate what they are practicing and performing, how could they possibly hope to further their musical abilities (and really, cultural fluency)? Better and more entertaining songs to sing should be selected: ones that students at least partially pick for themselves. There are many no-cost, little-energy-spent improvements to be made. 

But, with this model, I am going to concentrate on ways to improve the music program that do cost money (and may or may not take up time not already allotted to music class). One might be simply brightening and reconstructing the rooms that the class takes place in; today, they are rather sad places to be learning something as wonderful as music. Funnily enough, we are actually building entirely new musical facilities currently (literally from the ground up), though for the purpose of this project, I am going to pretend as though the school is still deciding where to spend their money. They might also hire more and better teachers who are excited (and get enough sleep) to do their jobs well. Yet another way is creating a mandatory band and orchestra for students to be offered time at an instrument as well as performance with it. Currently, there is an optional orchestra that meets after school hours, though few attend and it is not very good. Ideally, sufficient reform comes from resources spent on a combination of the three (my model is going to focus on how to break up a specific amount of money, rather than what amount of money to allot to reform in the first place). Thus, I will focus on the strategies of the players as a breakup of resources (money, time, and energy) represented by a fraction of ten. So, if 50% of resources go toward reconstruction of class space, 30% go to the new band and orchestra, and 20% to the hiring of new teachers, then I will note that strategy as 5R (R for reconstruction), 3O (O for orchestra), 2H (H for hiring); 5R3O2H.  

I’m going to assume that reconstruction requires at least 50% of resources and at most 70% (high because constructing/redesigning a building, paying those who are doing the building, and managing the construction, is all very time-and-energy-consuming as well as expensive). Then, I will assume that 20% to 40% will be allotted to the orchestra (instruments and maintaining them and teaching children how to use them are also expensive and time-and-energy-consuming, though not as much so as an entire building). Finally, 10% to 20% will go to hiring new teachers (the least resource-eating of the three). I cannot possibly include all the combinations of resource allocation, even given my own constraints; so, I have selected what I think are diverse options and thus will produce very different outcomes in this hypothetical new music education program. Strategies are as follows, and are the same for each player; 5R4O1H, 6R2O2H, 7R2O1H. Our matrix now looks like this:



Utilities and Payoffs

Now we move onto outcomes. Essentially, what occurs after the clients and the school have picked a method of resource allocation moving forward? Well, I have decided that when the clients and school pick the same strategy, that strategy will be implemented. When they pick different strategies, however, the average of the two is taken, and that is the strategy that will be implemented. So, for example, if the clients choose 6R2O2H and the school chooses 5R4O1H, then the strategy that will be implemented is 5.5R3O1.5H (6+5=11/2=5.5 so 55% of resources go to reconstruction “5.5R,” etc.). Our matrix then looks like this: 



Now for a very important question: what are the school and their clients feeling about these outcomes? On a scale of 1 to 10, how much do they like each one? That is the question I am asking when I assign “utilities;” relative preferences of outcomes for each player. 

Well, the clients are thinking in the very short-term. What is going to most benefit my child right now? They are not thinking (for the most part) about the children going through the music program as their child is having children of their own. So, they value the new orchestra and band and the hiring of new teachers to support it more than they do construction; it can take ages, not just to build, but to plan for the build (not to mention permits and such). They prefer 5R to 7R, and care more about not wasting money on R to spending on H and O. Now, a band and orchestra is nothing without good teachers to conduct it; parents would much rather listen to their child practice well at a cheaper cello than horrendously at an expensive one as they tune into their next Zoom meeting. So, they prefer 2H to 1H, and H to O in general (though if they can get it, 4O to 2O as well). That leaves us with 7R2O1H < 5R4O1H < 6R2O2H. The latest prioritizes hiring new teachers (and not overspending on orchestra and band) to reconstruction, where it does not spend too many resources: a happy medium. Our matrix now looks like this, for 7R2O1H = 1, 5R4O1H = 3, and 6R2O2H = 5 in the eyes of the clients. For conflicting choices of strategies, I took the average of the utilities for each (i.e. if 6R2O2H is 5 and 7R2O1H is 1, then 6.5R2O1.5H is 3).



On the other hand, the school is thinking long-term. Though they do care about their students today (they must if the education is to be good), they are thinking equally about those students to come. Not only their happiness once they get to the school, but also attracting them to it in the first place. Shining new buildings look good on promotional pamphlets. The school is thinking most about reconstruction (as their clients are), though for a different reason: they want to spend a higher percentage of resources on it. Additionally, they could do with pushing hiring new teachers down the line, which takes (on average) more time and energy of the administrators’ time than simply putting in an order for several bucketloads of instruments. However, there is little distinction between hiring and the orchestra otherwise; they both equally and positively impact the school long-term. So, it will be spending money on R to O and, with very slight distinction, to H. The utilities, therefore, are 5R4O1H < 6R2O2H < 7R2O1H. Again, I believe this order is an honest reflection of preferences; the school cares a whole lot more about spending on reconstruction than they do on where resources go to the other two possible reform areas, founding the orchestra and hiring new teachers. 

Our final matrix, then (I have done the same here in calculating conflicting strategy payoffs as I have in calculating the clients’), for 5R4O1H = 1,  6R2O2H = 3, and 7R2O1H = 5, looks like the following. These payoffs may seem confusing; how have I now allotted 5 to 7R2O1H when I just gave it 1 in assessing the clients’ payoffs? Well, that is exactly it; these are the school’s payoffs, and the above matrix the clients’. As I have just explained, the players feel differently about the outcomes. So, though I have used the same numbers here as I did previously, do not be confused; what these numbers reflect are very different utilities.




And now, without further ado, we answer my research question! How can I use game theory to determine the most cost and time efficient method for my school to go about offering a more comprehensive music education program to its lower and middle school students? Well, let’s find out!


 I am going to employ two methods of solving zero-sum games; Nash’s and Pareto’s. Below is how I uncovered the Nash equilibrium for the game, (3, 4), for clients select 6R2O2H and the school selects 7R2O1H…


Change this Subheading


And below is how I uncovered the Pareto optimality for the game (by graphing each point): any solution along the bolded line Y = -1/2 X + 11/2 , or (1, 5), (3, 4), and (5, 3). As sometimes occurs, the solutions are conflicting (though Pareto optimality does acknowledge the Nash equilibrium as one of its solutions). 



Conclusion and Where We Go From Here

After thinking carefully about the solutions, I have determined that (3, 4) is the optimal solution to the game (though Pareto’s method resulted in more than just one). This way, the school and the clients pick their highest-ranked method of resource allocation, and the resulting reforms to the lower and middle school music programs reflect both player’s interests equally (as the strategy adopted is the average of the two preferences). Here, 65% of resources goes to redesigning the music wing, 20% goes to the foundation of a mandatory orchestra and band, and 15% goes to hiring new and better teachers. 

Now, it is up to the school and their clients to decide what these percentages are truly of; in other words, the amount of resources they want to devote to reforming the music education program in the first place. Is the more comprehensive music education of students worth 100,000 dollars? 10,000? 10? My game cannot give my community the answer to this question; though I hope, once we find one, that it may help to organize reform along the way.

And, finally, for some of the articles I mentioned long ago in my introduction paragraphs! It is apparent to me now, after having done some research, that my school is not the only one falling victim to a decreasing quality of their music education program (particularly in the lower and middle schools). Currently, across the country, those at the top, making the important decisions about our schools’ curriculum, and those at the bottom, parents at public schools, do not value music education as they do, say, math or history. 

Due to it being increasingly undervalued, music education programs in the U.S. have grown increasingly worse for almost three decades: and that’s just what’s on the books. The New York Times took notice in 1993. The Music For All Foundation recorded its decline from 1999 to 2004. The School Band and Orchestra Magazine discussed it in 2004. Give A Note Foundation reported on the national state of it in 2017. Why are we not making a collective effort to improve upon this broken system? We need to be; if we continue down this steep hill of music education decline, it is actually really bad, for everyone. As reported by the American Psychological Association, music students score better in math, science, and english than their nonmusical peers. According to Healthline, music connects us. It can improve learning and memory, while also helping to treat mental illness, like anxiety and depression. It can contribute not just to a healthy mind but a healthy body as well, boosting heart health and exercise performance while decreasing fatigue and managing pain. The bottom line: music is important. Music education is important. And we should all care about the betterment of it

Thanks for reading through my 2021 Catalyst Conference Project! I cannot wait to check out yours. Your visit to my page is greatly appreciated, and I hope you not only enjoyed but learned something new as well. If you have any further thoughts on the topic of music education across the U.S. or on my specific project, please comment them down below! I would love to hear your feedback.

Works Cited

Chira, Susan. “As Schools Trim Budgets, The Arts Lose Their Place.” The New York Times, 3 February 1993.

“The Sound of Silence: The Unprecedented Decline of Music Education in California Public Schools, A Statistical Review.” The Music For All Foundation, September 2004.

Guhn, Martin, Emerson, Scott D., and Gouzouasis, Peter. “A Population-Level Analysis of Associations Between School Music Participation and Academic Achievement.” Journal of Educational Psychology, The University of British Columbia, 20 June 2019.

Gouzouasis, Peter. “Music Students Score Better in Math, Science, English Than Nonmusical Peers.” American Psychological Association, 24 June 2019.

Stanborough, Rebecca Joy. “The Benefits of Listening to Music.” Healthline, 1 April 2020.




  1. Katie!! This project is fabulous! So well thought out, and such an important issue. I cannot express firmly enough how impressed I am with your research and the realistic and detailed math. @GOAlearning #GOAndchangeit

    1. Hi Bess! Thank you so much for the wonderful feedback!! I really appreciate it :))

  2. Hi! Your project definitely focuses on a very important issue and I can deeply relate to that. I appreciate how you made a clear presentation and how you used various approaches to solve your model. Thank you for you work!

    1. Hi, Kevin! Thanks for visiting my page – so glad you enjoyed it!

  3. You’ve done really well to take a unique and personal topic and apply it to the curriculum of your course. I really appreciate this novel approach. I wonder who would be your next group of people to share this with to refine it? Teachers? fellow students? What constituency might enhance the project and help you identify some things you’ve missed?

    1. Thank you for your feedback, Lucas! I am actually starting to really think about sharing it with my school community, as you have suggested. If I were to do so, I think I would need to throw in a little more evidence and research to back up some of my opening arguments, both applying across the country and to just my school. Thanks again for visiting my page!

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