Understanding Homelessness with Game Theory

Cities must deal with the problem of homelessness. However, some choose to deal with it by not doing anything at all.

Introduction Video

The most important goal in the UN Sustainable Development Goals for me is “Decent Work and Economic Growth.” This goal directly relates to the problems of unemployment and homelessness, which to me are very important to solve. Unemployment, defined as the state of not being employed, leads many people to poverty and homelessness. According to the Homeless World Cup Foundation, about 150 million people are homeless in the entire world. Many people are left suffering on the streets without the ability to work and make an income. Instead, their lives become a daily battle to find food and shelter, which can take severe tolls on one’s mental and physical state. The problem of homelessness is massive, and although many strides are being made to support people living on the streets, we still have a long way to go to make a significant difference. In this project, I attempt to explain the problem regarding homelessness through a game theoric representation of the decisions facing cities over the matter.

The Problem

One of the most significant problems regarding homelessness is the inactivity on the part of cities when it comes to dealing with supporting homeless populations. Using game theory, we can represent this problem in the form of a matrix game. In the game, the two players are two different cities. Each one has the option to either ignore their homelessness problem and send their homeless population to another city. Or, each city can use extra tax dollars to set up their own homeless support systems such as added shelters and food banks. In game theory, we refer to the level of satisfaction by each player in regards to the outcome of their decision as their “payout”. If both cities ignore the problem of homelessness and do nothing to increase support, they will receive a negative (bad) payout of -2 since they will still have to deal with the issue of homelessness. If only one city sets up support systems, the other city will benefit because they can send their homeless population to that other city, resulting in a positive (good) payout of 1. The other city will still have to deal with the sent-in homeless people and they will have to use up extra tax dollars. Therefore, they will still receive a negative payout of -1. If both cities set up support systems, they will both have succeeded in playing a large role to assuage the problem of homelessness and therefore will receive a positive payout of 2. Although it will cost extra money, it is the best option for both cities as it will help fix the problem of homelessness. In the following matrix, I represent this situation.

Game Theory Matrix Model

In this game, both cities have the choice to either ignore the problem of homelessness or to establish more support in their city. The first number in each pair of numbers represents City #2’s payout for each outcome, and the second number in each pair of numbers represents City #1’s payout for each outcome.

Test Your Thinking

Take the above survey and see if you got the answer to the question right by reading on.

Solving this Game using Game Theory

Using Nash Equilibrium, we can demarcate the outcome of the game in which neither city would have a desire to switch their strategy. In the following model, the models refer to the desired strategy for each city depending on the strategy chosen by the other city. If a vertical arrow and a horizontal arrow point to the same box on the grid, that represents that both cities would have no incentive to change their strategy because if they did, then so would the other city.

Nash Equilibrium solution to the game

Using Pareto Optimality, we can affirm that the Nash Equilibrium is the best strategy to be taken by each city. Since the Pareto Optimal solution to this game is equal to the Nash Equilibrium, there is no reason why both cities would play a strategy different to “Increase Strategy”. The Pareto Optimal solution is the outcome in which no other outcome would result in a higher payout to one player without sacrificing a lesser payout to the opposing player. Therefore, Pareto Optimal outcomes are outcomes in which satisfaction by both players can’t be increased. Sometimes the Nash Equilibrium is different from the Pareto Optimal solution, but not in this game. In the graph below, I graphed the four outcomes with City #1’s payout being the x value and City #2’s outcome being the y value. As you can see, the Pareto Optimal outcome is (2, 2) because no other outcomes have an x value greater than 2 with a y value also greater than 2.

The Pareto Optimal solution to the game is at (2,2)

Call to Action for Cities

The clear answer of this problem according to game theory and according to common sense is for actions to be taken by cities to help address the problem of homelessness. This issue is that more often than not, cities will ship their homeless populations to other cities which have better support for the homeless. Instead of developing support systems in their own city, they will instead send their homeless population to cities which do have such systems in place. This ultimately doesn’t accomplish much since these cities will eventually just need more and more support set up and it is a never-ending cycle. The best way to deal with the problem of homelessness is for all cities to set up their own support systems and to not ignore the problem by sending homeless people in their cities to other cities. Increased numbers of homeless shelters, food banks, or job opportunities for homeless people would go a long way in helping to solve the problem, and cities must realize this fact.

Call to Action for Individuals

While the best way to deal with the problem of homelessness lies in the hands of cities, the second best way to deal with this problem is by doing exactly what this project page attempts to do: raise awareness. Therefore, we all must emphasize that the problem of homelessness is not something to be disregarded in our communities. By standing up for the homeless, we can inspire action by others and the more and more we can do so, the more pressure there will be on cities to take responsibility for fixing the problem at hand. It is time for those of us who care about the issue to advocate for ending the practice of busing homeless populations away from cities to other cities, as that only compounds the problem. Cities must take the initiative to provide better support for their homeless populations, or otherwise the problem will never be solved. Besides raising awareness, serving in a homeless shelter is a great way to get out and help the homeless in a tangible way. I highly recommend visiting this link ( to find the nearest homeless shelter to you. Helping out at homelessness shelters not only benefits homeless people, but it humanizes the situation by putting faces to the problem. In my opinion, there is no better way to help out the homeless than volunteering even a short amount of time at a local homeless shelter.

Link to my Google Doc:

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  1. April 25, 2019 by Hannah Robbins

    Hi! I found your page so interesting because you were able to so clearly articulate why we should help the homeless, and how everyone can benefit. I found it really innovative that you took an economic model like game theory and applied to the argument of homelessness, which can sometimes get lost in idealism and different beliefs about social issues.

  2. April 25, 2019 by Claudia

    I like your use of the actual game theory while also making sure that the graphics were able to keep the audience entertained! Good job!

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