Game Theory & Applications to Operational Research
- Ryan Mancuso
- Apr 26
- 6 min read
Updated: May 20
The other day I was watching A Beautiful Mind, a movie based on the life of John Forbes Nash Jr. Although not the main point of the movie, it briefly touched upon Nash's major contribution to the field of mathematics, his advancements in game theory. This helped inspire me to learn a bit more about this topic and its applications to my interests. This theory aims to mathematically find the best possible outcome for strategic decision makers dubbed "players" in either non-cooperative or cooperative situations. In a non-cooperative situation, the player is aiming to achieve the best possible outcome for itself by choosing the most effective strategy, regardless of its impact on others. However, in a cooperative situation, the players aim to distribute rewards in the most equitable way based on their contributions. Whilst this topic is far too complicated to be fully covered in a single blog post, an easily comprehensible way to understand non-cooperative instances of the application of game theory would be the prisoner's dilemma.
Imagine you committed a robbery with an acquaintance you hold no loyalty towards and you were both caught by the police. The police could not determine if either you, your acquaintance, or both of you committed the crime so the judge offers you a deal: if you both stay quiet, both of you would get one year in prison, you both accuse each other of the crime, you both get two years. However, if either you or your acquaintance betrays but the other one chooses silence, the one who stayed silent would get three years in prison whilst the betrayer would be set free. Game theory would determine that both would betray each other, because the Nash equilibrium recognizes that decision as a dominant strategy. Regardless of what the other player does, you are always better off betraying than staying silent.
As you can see, game theory does not always try to achieve the best possible outcome, rather the best decision possible given certain information. For my football fans out there, have you ever wondered why your team chooses to run the ball as many times as they do despite pass plays being a clearly better tactic? My first introduction to game theory was a youtube video on why the Seahawks choose to run the ball as much as they do when passing the ball generally produces a better result.
Pass plays, on average produce ~9.57 yards/play while run plays only produce ~4.67 yards per play. However, it all comes down to the defense's choice in their tactics; whether they want to try to protect against a pass play, or a run play.
To figure out what would be the most effective choice to make, game theory is used.
Run Defense (yards/play) | Pass Defense (yards/play) | |
Run | 2.8 | 8.41 |
Pass | 12.44 | 5.74 |
You may notice that the average yards when running the ball is lower than passing the ball, however game theory would determine that given this information, as long as both players are acting rationally, the Seahawks should run the ball 54% of the time. While this seem counterintuitive, you must remember the prison dilemma. The best possible outcome for the offense is a pass play paired with a run defense, but you cannot assume that the opposing player will pick that strategy, so in order to make that outcome a possibility, run plays must be used in order to force the other player to choose run defenses to counter a run play, so when the offensive player calls a mix between run and pass plays, they are achieving the best possible equilibrium between the two outcomes to achieve the greatest odds of yards possible. If you are interested in learning more, the video I linked goes into even greater detail and I would highly recommend watching. But how does game theory apply to operational research? Like football, operational research relies on the use of quantitative methods to make the best decisions when the outcomes of those decisions depend on others. There are so many different applications of game theory in this topic, such as network design and routing. The goal of traffic systems are to reduce the average travel time to get from one place to another in the least amount of time possible. However, traffic systems are incredibly hard to optimize given their load dependent nature (ie. how much traffic is traveling on the roads in a given amount of time) and the selfish motivations of drivers. In traffic systems, it is assumed that in the absence of regulation, drivers will operate with selfish interests to shorten their individual transit times as much as possible. Therefore, the drivers will reach a Nash equilibrium based on their own individual preferences, which again referencing the prisoner's dilemma, does not account for societal welfare. Most would assume that adding an additional lane or constructing an additional route would reduce the amount of time to travel, but according to game theory, that is not always the case. This phenomenon is due to Braess's paradox.
The simplest way to describe this paradox is with strings and springs. Picture two springs connected by strings, arranged vertically. The first spring hangs from a fixed point above, with a string hanging below it attached to a heavy weight. The second spring is arranged in the opposite orientation; its string is attached above to the fixed point, and the spring itself sits below, just above the weight. Together, the two springs share the load of the weight between them, each experiencing only half the total force.
Contrary to intuition, when a shorter third string is added connecting the two springs directly to each other, the weight falls rather than rises. Because the stretched length of a spring is a linear function of the force applied to it, connecting the springs with the third string forces each spring to bear the full weight of the load individually, rather than sharing it; causing both springs to stretch further and the weight to drop. Without the connecting string, even though the springs are shorter, the weight sits higher because each spring is only under half the force. The system, left alone, rests at Nash equilibrium.
In traffic circumstances where the flow from two points in a road network is linear and one path is more desirable for an individual driver and the drivers shift their routes to achieve Nash equilibrium, the paradox states that an extension of the road network could increase individual travel times.
In the example road network, the travel time of segment start to A and B to end is the number of travelers (T) divided by 100, and the travel time of segment start to B and A to end is a constant 45 minutes. There are 4000 drivers trying to travel from the start to the endpoint as quickly as possible. If you calculate the time it would take an individual to travel from the start to end without the road from A to B, it would take a driver 65 minutes to travel either route. However, if a road from point A to point B was added, assuming the road has a travel time of 0, it will take a driver 80 minutes to travel from start to end if they are operating with selfish interests. The drivers would all take the start to A segment because, at worst, the travel time of that individual segment will be 40 minutes (⁴⁰⁰⁰⁄₁₀₀), and then the drivers would switch routes and take B to end for the same reasoning. However, without that road, the number of cars traveling each route would be half as much as when they were operating under Nash equilibrium, so therefore the time it would take to travel from start to A would be only 20 minutes (²⁰⁰⁰⁄₁₀₀). The most famous real-world example of Braess's paradox is the Cheonggyecheon Expressway in Seoul, South Korea. Built in 1968, the elevated highway was a symbol of the city's rapid modernization. It functioned well until the 1980s, when proliferation in car ownership caused it to become permanently congested; precisely because it was the shortest route between so many destinations, every driver selfishly chose it, and no one willingly took the longer alternatives.
In 2005, the mayor of Seoul initiated a project to demolish the expressway and restore the Cheonggyecheon stream that had been buried beneath it. The proposal was met with widespread skepticism, but however, once the expressway was gone, drivers redistributed across the road network, and travel times across the city improved. Contributing factors included increased government investment in public transit, but the redistribution of traffic away from a single dominant route was central to the improvement.
Works Cited:
Davis, Morton D, and Steven J Brams. “Game Theory.” Encyclopædia Britannica, 4 Oct. 2018, www.britannica.com/science/game-theory.
Hey, Jono. “The Prisoner’s Delemma,” Sketchplantations, 2 Sept. 2017, sketchplanations.com/the-prisoners-dilemma.
Michael MacKelvie. “Passing Is Better…So, Why Run?” YouTube, 30 Oct. 2025, www.youtube.com/watch?v=pPXZlK9puqw. Accessed 5 Nov. 2025.
Reinecke, David. “Braess’ Paradox Proves That Sometimes Less Is More.” David Reinecke, 10 Nov. 2019, davidreinecke.com/blogs/articles/braess-paradox-proves-that-sometimes-less-is-more-1. Accessed 21 May 2026.
Roughgarden, Tim, and Éva Tardos. “How Bad Is Selfish Routing?” Journal of the ACM (JACM), vol. 49, no. 2, Mar. 2002, pp. 236–259, https://doi.org/10.1145/506147.506153.
“Seoul and Braess’ Paradox : Networks Course Blog for INFO 2040/CS 2850/Econ 2040/SOC 2090.” Cornell.edu, 2016, blogs.cornell.edu/info2040/2016/09/16/seoul-and-braess-paradox/. Accessed 21 May 2026.
Wikipedia Contributors. “File:Braess Paradox Road Example.svg.” Wikipedia, Wikimedia Foundation, 24 Jan. 2012,
---. “Game Theory.” Wikipedia, Wikimedia Foundation, 18 Mar. 2019, en.wikipedia.org/wiki/Game_theory.
Perfect
How is the socioeconomical impact of game changing players such as Jaxon ‘Emerald City Route Artist’ Smith-Njigba changing the optimization in game theory assumptions?
very cool
Wow this was a really interesting take