THE MATH BEHIND YOUR DECISIONS

The Mathematician Who Decoded Human Behaviour

The Math Behind Your Decisions

John Forbes Nash Jr. was not a typical mathematician. Born on June 13, 1928, in Bluefield, West Virginia, Nash would go on to reshape how we understand competition, cooperation, and decision-making itself. His work earned him the Nobel Memorial Prize in Economic Sciences in 1994 and the Abel Prize in 2015. He died on May 23, 2015.

Most people know Nash from the film “A Beautiful Mind.” The movie gave audiences a window into his personal struggles with paranoid schizophrenia. What it did not do, at least not with enough precision, is explain why his mathematical contributions matter to you and me.

Nash’s academic body of work spans real algebraic geometry, differential geometry, and partial differential equations. His doctoral dissertation at Princeton, completed at the age of 21, was titled “Non-Cooperative Games.” That 27-page thesis introduced the concept of what we now call the Nash Equilibrium, a framework that fundamentally changed economics, political science, biology, and military strategy.

But this book is not a biography. It is not a math textbook. It is a practical guide to three concepts that emerge from Nash’s work and the broader field of game theory he helped define. These three concepts shape your daily life, whether you recognize them or not:

1. Nash Equilibrium — the state where no one benefits from changing strategy alone

2. Sunk Cost Fallacy — the trap of honouring past investments that cannot be recovered

3. The Prisoner’s Dilemma — the tension between individual gain and collective benefit

Each of these concepts is at work every time you negotiate a salary, choose a restaurant with friends, stay in a failing project, or decide whether to trust a colleague. The math is already running. The question is whether you see it.

Let’s look under the hood.

Part One: Nash Equilibrium

Chapter 1: The Concept

Imagine you and a colleague are both preparing proposals for the same client meeting. You each have a choice: present an aggressive or a conservative pricing strategy. Neither of you knows what the other will present. But you each know this: if you go aggressive and the other goes conservative, you win the deal. If you both go aggressive, the client gets confused, and neither of you wins. If you both go conservative, the deal splits evenly.

A Nash Equilibrium is the point where each person’s strategy is the best response to the other person’s strategy. No one has an incentive to unilaterally change what they’re doing, because doing so would make them worse off.

This is the key insight: it is not about finding the “best” outcome for everyone. It is about finding the stable outcome, the point where no individual gains by deviating alone. The equilibrium might not be optimal. It might not even be good. But it is stable.

Nash proved mathematically that every game with a finite number of players and strategies has at least one such equilibrium point. This was the core of his 1950 dissertation, and it was a seismic shift. Before Nash, economists largely assumed that markets and competition would naturally lead to the best possible outcomes. Nash showed that stable outcomes and optimal outcomes are not always the same thing.

Chapter 2: Equilibrium in Everyday Life

Traffic and Route Selection

Every morning, thousands of commuters choose between the highway and the back roads. If everyone takes the highway, it gets congested and slows down. If everyone takes the back roads, the same thing happens there. Over time, drivers sort themselves into a pattern where switching routes would not improve any individual driver’s commute time. That pattern is a Nash Equilibrium.

You have experienced this. You know the days when you “tried a different route” and ended up arriving at the same time or later. The system had already found its balance. Your deviation didn’t help you, because everyone else was already playing their best strategy.

Pricing Between Competitors

Consider two coffee shops on the same street. If Shop A lowers prices dramatically, it might attract all the customers for a brief period. But Shop B will respond by lowering prices too. Eventually, both shops settle at a price point where neither benefits from going lower (because margins disappear) or higher (because customers leave). That stable price point is an equilibrium.

This is why gas stations next to each other tend to have nearly identical prices. It is not collusion. It is in equilibrium. Neither station benefits from deviating, because the other would simply match.

Choosing a Restaurant with Friends

You are in a group chat. Someone asks where to eat. Nobody wants to be the person who picks a place everyone else dislikes. So people either stay silent or suggest something safe; the chain restaurant everyone tolerates but nobody loves. That safe, mediocre choice is an equilibrium. No individual improves their position by deviating (suggesting something bold and risking rejection), so the group defaults to the stable, but suboptimal, outcome.

This is equilibrium in action: stable does not mean ideal.

Salary Negotiation

When you negotiate a salary, you and the employer are playing a game. You want the highest number possible. They want the lowest. The equilibrium is the number where you would not walk away, and they would not rescind the offer. Both parties are playing their best strategy given what they know about the other.

If you have competing offers, the equilibrium shifts upward; your best alternative changes the game. If the employer has ten other qualified candidates, the equilibrium shifts downward. The math does not care about fairness. It cares about strategy.

Household Chores

Two roommates. One hates dishes. The other hates vacuuming. Over time, they settle into a pattern: one does dishes, the other vacuums. Neither switches because switching means doing the task they hate more. This is a Nash Equilibrium in a domestic game; a stable division of labour where no one benefits from unilateral change.

Chapter 3: Why Equilibrium Matters

The practical takeaway here is not abstract. It is operational. Recognizing equilibrium means recognizing when a situation is stable versus when it can be disrupted. If you are in a negotiation that has reached equilibrium, pushing harder alone will not improve your position; you need to change the game's structure. Add new information. Introduce a new player. Alter the payoffs.

In business, understanding equilibrium tells you when price wars are pointless, when market positions are locked, and when disruption requires not just effort but a fundamentally different game.

In your personal life, it shows why some arguments with your partner never resolve. Both of you are locked in strategies that are individually rational but collectively frustrating. The way out is not to argue harder. It is to change the game; introduce a new variable, reframe the problem, or agree on new rules.

Equilibrium is not destiny. It is a diagnosis. Once you see it, you can decide whether to accept the stable state or redesign the conditions that created it.

Part Two: The Sunk Cost Fallacy

Chapter 4: The Concept

The Sunk Cost Fallacy is not a Nash original, but it is deeply embedded in the game-theoretic landscape his work illuminated. In game theory, rational decision-making requires evaluating future payoffs, not past expenditures. The sunk cost fallacy is the violation of that principle. It is the tendency to continue investing in something because of what you have already put in, rather than what you stand to gain going forward.

The logic feels intuitive: “I’ve already spent $50,000 on this renovation. I can’t stop now.” But that $50,000 is gone. It does not matter whether you continue or quit. The only rational question is: “From this point forward, is the expected return worth the additional investment?”

Nash’s equilibrium framework assumes rational players: those who evaluate strategies based on future outcomes, not on emotional attachment to past outcomes. The sunk cost fallacy occurs when we deviate from rationality. And we do it constantly.

Chapter 5: Sunk Costs in Everyday Life

The Bad Movie

You paid $15 for a movie ticket. Thirty minutes in, the movie is terrible. You think: “I already paid for it, I should stay.” But the $15 is gone whether you stay or leave. Staying costs you 90 more minutes of your time; time you could spend doing something you actually enjoy. The rational move is to leave. The emotional move is to stay because you feel you owe something to the money already spent.

You owe it nothing. It is spent.

The Failing Business Venture

An entrepreneur has invested two years and $200,000 into a startup. The market has shifted. The product is not gaining traction. Every advisor says to pivot or close. But the founder says, “I’ve put too much into this to walk away.”

This is the fallacy at its most dangerous. The two years and $200,000 are sunk. They do not return regardless of the next decision. The only valid question is: “If I had $200,000 and two years today, with no prior history, would I invest them in this exact venture as it stands right now?” If the answer is no, then continuing is not perseverance. It is denial.

Relationships

This one is uncomfortable but necessary. “We’ve been together for seven years. I can’t just throw that away.” The seven years are not currency. They do not buy future happiness. The question is not what you have invested. The question is what the next year looks like, and whether the relationship, as it exists today, warrants continued investment of your time, energy, and emotional capital.

This is not a cold calculation. It is clarity. Sometimes the answer is: yes, the relationship is worth continuing and improving. But that answer should come from forward-looking assessment, not backward-looking guilt.

The Degree You Don’t Want

A college junior realizes they hate their major. They have completed three years of coursework. Switching would add another year. The sunk cost argument says, “Three years wasted if I switch.” The rational argument says: “Three years are done regardless. The question is whether I want to spend the next 40 years in a career built on a foundation I resent.”

One additional year invested in the right direction is worth more than a lifetime spent honouring a decision you made at 18.

The Gym Membership

You bought a year-long gym membership for $600. You stopped going in March. It is now August. You think: “I should go because I already paid.” But the $600 is sunk. The only question is: Do you want to go to the gym today? If yes, go. If not, do something else that improves your health. The membership fee should not be the deciding factor.

Technology Projects

A company has spent 18 months building a custom internal tool. Halfway through deployment, a commercial product launches that does the same thing at a fraction of the cost. The CTO argues: “We’ve invested too much to switch.” This is sunk cost reasoning applied to engineering decisions. The 18 months are gone. The question is: over the next five years, will maintaining and extending the custom tool cost more or less than licensing the commercial product?

The code you already wrote does not care about your feelings.

Chapter 6: How to Escape the Trap

Escaping the sunk cost fallacy requires a specific mental discipline. It is a practice, not a one-time insight.

The Clean Slate Test: Before making any continuation decision, ask yourself: “If I were starting from zero today, with no history, no prior investment, no emotional baggage, would I choose this path?” If the answer is no, you have your signal.

Separate Identity from Investment: We often continue because quitting feels like admitting failure. But quitting a losing strategy is not failure. It is an adaptation. The most disciplined operators in any field know when to cut losses. They do not confuse identity with allocation.

Track Forward-Looking Metrics: In business and personal life, build the habit of measuring projected returns rather than historical investment. What has this cost so far is an accounting question. What this will cost going forward is a strategic question. Live in the second one.

The sunk cost fallacy is not a character flaw. It is a cognitive pattern that evolution wired into us; losing feels roughly twice as painful as winning feels good, a phenomenon psychologists call loss aversion. Knowing this does not eliminate the feeling. But it gives you a framework to override it with logic when the stakes are high.

Part Three: The Prisoner’s Dilemma

Chapter 7: The Concept

The Prisoner’s Dilemma is the most famous game-theoretic construct. It was formalized by Merrill Flood and Melvin Dresher in 1950 at the RAND Corporation, and Albert Tucker gave it the narrative framing that made it famous. Nash’s equilibrium concept provides the mathematical backbone for analyzing it.

Here is the classic setup. Two suspects are arrested and placed in separate rooms. The police offer each one the same deal:

• If you confess and your partner stays silent, you go free, and your partner gets 10 years.

• If you both stay silent, you each get 1 year on a lesser charge.

• If you both confess, you each get 5 years.

• If you stay silent and your partner confesses, you get 10 years, and your partner goes free.

The rational move for each individual player, analyzed in isolation, is to confess. Here is why: regardless of what the other person does, confessing gives you a better personal outcome. If they stay silent, confessing gets you freedom instead of 1 year. If they confess, you get 5 years instead of 10. Confessing is the dominant strategy.

The Nash Equilibrium of this game is mutual confession; both players confess, and both get 5 years. And here is the uncomfortable part: this equilibrium is worse for both players than if they had both stayed silent and each received only 1 year.

This is the dilemma. Individual rationality leads to collective failure. The best strategy for me, pursued independently by both of us, produces an outcome that is worse for both of us than if we had cooperated.

Chapter 8: The Dilemma in Everyday Life

The Arms Race at Work

Two departments in the same company are competing for the budget. Each could cooperate, share resources, present a unified case to leadership, and both would get adequate funding. Instead, each department inflates its requests, sandbagging the other, trying to grab a bigger share. The result: leadership sees dysfunction, cuts both budgets, and hires a consultant to “fix the culture.” Both departments end up with less than they would have received through cooperation.

This is a Prisoner’s Dilemma playing out in a conference room.

Price Wars

Two airlines fly the same route. If both maintain reasonable fares, both profit. If one slashes prices, it captures the market temporarily, but the other follows suit. Now both are selling tickets at a loss. Neither can raise prices first because the other would capture all the customers. They are trapped in a mutually destructive equilibrium.

This pattern has played out in airlines, ridesharing, food delivery, and countless other industries. The Prisoner’s Dilemma does not care about your revenue targets.

Climate Change and Shared Resources

Every country benefits from reducing global carbon emissions. But each country bears the cost of reducing its own emissions while potentially watching competitors gain economic advantage. The rational move for any single country is to defect; keep polluting while hoping others cut back. When every country reasons this way, nobody cuts back, and the collective outcome is catastrophic.

This is the Prisoner’s Dilemma scaled to the level of civilization.

Splitting the Bill

Five friends agree to split a dinner bill evenly. Each person now has an incentive to order slightly more expensively than they normally would, because the cost is distributed across five people. If everyone does this, the bill is significantly higher than if everyone had ordered what they actually wanted. Each person’s individual “rational” choice to upgrade their order results in everyone paying more.

Group Projects

Any student who has survived a group project knows this one. The grade is shared. Each individual benefits from letting others do the work while still receiving the group grade. If everyone reasons this way, nobody does the work, and everyone fails. The Nash Equilibrium of the unmonitored group project is mutual free-riding, and it is worse for everyone than if all members had contributed.

Neighbours and Noise

You live in an apartment building. Your neighbour plays loud music. You could ask them to turn it down (cooperate), or you could retaliate by being equally loud (defect). If both of you cooperate, the building is quiet. If both defects are present, the building becomes unlivable. But here is the tension: if you cooperate and they defect, you suffer in silence. The temptation to defect, to match their behaviour or escalate, is the Prisoner’s Dilemma in residential form.

Chapter 9: Escaping the Dilemma

The Prisoner’s Dilemma seems hopeless in a single round. But life is not a single round. Most of our interactions are repeated games; we deal with the same coworkers, neighbours, competitors, and partners over and over. This changes everything.

In the 1980s, political scientist Robert Axelrod ran a tournament where experts submitted strategies for a repeated Prisoner’s Dilemma. The winning strategy was remarkably simple. It was called Tit-for-Tat, submitted by mathematician Anatol Rapoport.

The rules of Tit-for-Tat:

1. Start by cooperating. Give the other party the benefit of the doubt.

2. Mirror what the other party did last round. If they cooperated, cooperate. If they defected, defect.

3. Forgive quickly. If they return to cooperation, return to cooperation immediately.

This strategy worked because it was clear, predictable, and retaliatory without being vindictive. It communicated: I will cooperate as long as you do. If you betray me, I will respond proportionally. But I am always willing to reset if you are.

The implications for everyday life are direct. In long-term business relationships, reputation matters more than short-term advantage. In marriages and partnerships, the willingness to cooperate and forgive, without being a pushover, creates stability that benefits both parties. In team dynamics, holding people accountable while remaining open to renewed trust is more productive than permanent grudges or unconditional tolerance.

The Prisoner’s Dilemma teaches us that cooperation is not naive. It is strategic, but only when combined with accountability.

Part Four: Where the Concepts Converge

Chapter 10: Seeing the Game

These three concepts, equilibrium, sunk costs, and the Prisoner’s Dilemma, are not isolated ideas. They are lenses. Once you have them, you begin to see the structure underneath situations that previously felt chaotic or emotional.

Consider a common workplace scenario. You have been leading a project for six months. The project is behind schedule and over budget. Your team is frustrated. A competitor has launched a similar product. Leadership is asking tough questions.

Through the lens of sunk costs: the six months are gone. The question is not what you have invested but whether the project’s projected returns justify further investment from this point forward.

Through the lens of equilibrium: your team, leadership, and competitors have all settled into a pattern. Nobody is unilaterally changing strategy. To break the pattern, you need to change the game, not just work harder within the existing structure.

Through the lens of the Prisoner’s Dilemma: your relationship with leadership is a repeated game. If you hide problems (defects), you may survive this quarter but lose trust for the next five years. If you are transparent (cooperative) and they respond constructively, you build a foundation for long-term credibility. The short-term pain of honesty serves the long-term payoff of trust.

This is what game theory gives you. Not answers. Frameworks. The ability to decompose messy situations into their structural components and make decisions based on logic rather than emotion, habit, or ego.

Chapter 11: A Real-World Composite

Let me walk through a scenario that ties all three concepts together.

You are a small business owner. You and your main competitor have been in an unspoken price war for months. You both keep lowering prices to attract the same customers. Your margins are razor-thin. You have also invested $40,000 in a marketing campaign that is not performing.

Sunk cost analysis: The $40,000 is gone. Whether you continue the campaign or kill it, that money does not return. The question is whether the next $10,000 of marketing spend will generate returns that justify it. If the data says no, stop. Kill the campaign. Redirect the budget.

Equilibrium analysis: You and your competitor are stuck in a pricing equilibrium that benefits neither of you. Lowering prices further will not help; they will match you. Raising prices alone will cost you customers. The way out is to change the game: differentiate your product, target a different customer segment, or signal to the competitor that a price floor benefits both parties.

Prisoner’s Dilemma analysis: You and your competitor are in a repeated game. If you continue to defect (race to the bottom on price), both of you lose. If one of you cooperates (raises prices) and the other defects (keeps prices low), the cooperator loses. The solution lies in signalling and reciprocity; perhaps by gradually raising prices and observing whether the competitor follows, establishing a pattern of mutual cooperation.

No single concept solves the problem. Together, they give you a map.

Conclusion: Decisions Are Not Feelings

John Nash did not give us a theory about how people should feel about their decisions. He gave us a theory about the structure of decisions themselves, the mathematical relationships between choices, strategies, and outcomes.

Equilibrium tells you where systems settle and why they resist change. Sunk cost awareness tells you to evaluate decisions based on future returns, not past investments. The Prisoner’s Dilemma tells you that individual optimization can destroy collective value, and that sustained cooperation requires both goodwill and accountability.

None of this requires a mathematics degree. It requires a willingness to step outside the emotion of a moment and ask a structural question: What is the game? Who are the players? What are the incentives? And is my current strategy rational, or am I operating on autopilot?

Nash spent much of his life wrestling with his own mind. His contributions to mathematics endure because they describe something true about how strategic interactions work, not in theory, but in practice. In your commute. In your career. In your relationships. In your business.

The math is already running.

Now you can see it.

A Note on Further Reading

The concepts in this paper are introductions, not exhaustive treatments. If you want to go deeper, the following areas of study are worth your time:

Game Theory Foundations: Nash’s original 1950 paper “Non-Cooperative Games” is dense but readable for anyone with a quantitative background. The mathematical core of equilibrium theory is laid out in its clearest form there.

Behavioural Economics: The work of Daniel Kahneman and Amos Tversky on prospect theory and cognitive biases provides the psychological counterpart to game-theoretic models. Their research explains why humans consistently deviate from rational decision-making, including through sunk cost reasoning.

Repeated Games and Strategy: Robert Axelrod’s “The Evolution of Cooperation” remains one of the most accessible treatments of how cooperation emerges over time in Prisoner’s Dilemma scenarios.

Nash’s Broader Work: Beyond game theory, Nash made significant contributions to the theory of partial differential equations (the Nash–De Giorgi–Moser theorem) and algebraic geometry (the Nash embedding theorem). These are technical subjects, but they demonstrate the range and depth of a mind that reshaped multiple fields.

The goal of this paper was not to make you a game theorist. It was to hand you three tools that sharpen your ability to analyze decisions: your own and others’. Use them. The clearer you see the game, the better you play it.