Applying Game Theory to Personal and Professional Relationships

Game theory as we know it today came about in part because of one man’s interest in poker.

This man was not just your average person. John von Neumann was a mathematician, physicist, and computer scientist. His goal was loftier than becoming a better poker player.

According to a Forbes article, he “was only interested in poker because he saw it as a path toward developing a mathematics of life itself.”

He “wanted a general theory – he called it ‘game theory’ – that humans could apply to diplomacy, war, love, evolution or business strategy.”

He moved closer toward that goal when he collaborated with economist Oskar Morgenstern on a book called A Theory of Games and Economic Behavior in 1944.

The Library of Economics and Liberty (Econlib) states that “in their book, von Neumann and Morgenstern asserted that any economic situation could be defined as the outcome of a game between two or more players.”

You know I love math, and success is just math + strategy.

What Is A Game?

What is a game according to game theory?

Yale economics professor Ben Polak notes a game has three basic components:

-Players
-Strategies
-Payoffs

As mentioned, game theory applies to games involving two or more players.

In a game, players share “common knowledge” of the rules, available strategies, and possible payoffs of a game.

However, players do not always have “perfect” knowledge of these elements of a game.

Strategies are the actions that players take in a game. Strategy is at the heart of game theory.

Forbes describes the theory presented in A Theory of Games and Economic Behavior as “the mathematical modeling of a strategic interaction between rational adversaries, where each side’s actions would depend on what the other side would do.”

The concept of strategic interdependence – the actions of one player influencing the actions of the other players -- is a critical aspect of von Neumann’s version of game theory that is still relevant today.

Then, there are payoffs, one source describes as the “outcome of the strategy applied by the player.”

Payoffs could be a wide range of things, depending on the game.

It could be profits, a peace treaty, or getting a great deal on a car.

Limitations

One limitation of Von Neumann’s version of game theory is that it focuses on finding optimal strategies for one type of game called a zero-sum game.

According to Forbes, “one player's loss is the other player's gain” in a zero-sum game.

Another source notes that “players can neither increase nor decrease the available resources” in zero-sum games.

Critics have noted that life is often more complex than a zero-sum game. More complicated game scenarios are possible in the real world.

For example, players can do things like find more resources or form coalitions that increase the gains of several players.

Game theory has evolved to analyze a broader range of games, such as combinatorial and differential games.

Dilemma

A classic example of a game often studied in game theory is “The Prisoner’s Dilemma.”

Different versions of this game are available on the Internet. This version is from the Fundamental Finance website:

There are two prisoners, Jack and Tom, who have just been captured for robbing a bank. The police don't have enough evidence to convict them but know they committed the crime. They put Jack and Tom in separate interrogation rooms and lay out the consequences:

-If both Jack and Tom confess, they will each get ten years in prison.

-If one confesses and the other doesn't, the one who confessed will go free, and the other will spend 20 years in prison.

-If neither person confesses, they will both get five years for a different crime they were wanted for.”

The Prisoner’s Dilemma contains the basic elements of a game.

The two players are Jack and Tom.

There are two strategies available to them: confess or don’t confess.

The game's payoffs range from going free to serving five, ten, or twenty years in prison.

As Fundamental Finance explains, “It is easier to see and compare these outcomes (payoffs) if they are put into a matrix:

Since Tom's strategies are listed in rows or the x-axis, his payoffs are listed first. Jack's payoffs are listed second because his strategies are in columns or on the y-axis. ‘C’ means ‘confess,’ and ‘NC’ means ‘not confess.’

This matrix is called ‘Normal Form’ in game theory. Moves are simultaneous, which means neither player knows the other's decision, and decisions are made simultaneously (in this example, both prisoners are in separate rooms and won't be let out until they have both made their decision).”

One common solution to simultaneous games is the “dominant strategy.”

Fundamental Finance defines it as the “strategy that has the best payoff no matter what the other player chooses.”

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Tom does not know if Jack will confess or not. He takes a look at his options.

-If Jack confesses and Tom does not, Tom will get 20 years in prison.

-If Jack and Tom confess, Tom will get only ten years.

-If Jack does not confess and Tom does, Tom will go free.

The best strategy for Tom is to confess because it leads to the best payoffs regardless of Jack’s actions. Confessing will cause Tom to either go free or serve less prison time than if he did not confess.

Jack is in the same situation and has the same options as Tom. As a result, Jack's best strategy is confessing because it leads to the same payoffs that Tom will get.

Investopedia states that a “dominant strategy equilibrium is reached when each player chooses their own dominant strategy.”

Why is the strategy of both not confessing not the best choice?

While this option would give both of them less prison time than if they confessed, it would work only if each could be sure the other would not confess.

It is unknown whether Tom and Jack could work together with that level of cooperation. Additionally, both are unlikely to choose not to confess because it carries a more significant penalty than the one they would face if they confessed.

Confessing also allows each of them to serve no prison time, even less than five years.

The Prisoner’s Dilemma is a good example of how rationality can be problematic in game theory.

University of British Columbia - Vancouver researcher Yamin Htun calls it “one of the most debatable issues in game theory.”

Htun points out that “almost all of the theories are based on the assumption that agents are rational players who strive to maximize their utilities (payoffs).” 

Yet, studies demonstrate that players do not always act rationally and that “the conclusions of rational analysis sometimes fail to conform to reality.”

As we can see from this game, the most rational strategy that would give both players less prison time was not the best choice, while a choice that involves both players doing more prison time was.

The Nash Equilibrium

The Prisoner’s Dilemma also reflects how other game theorists fixed some of the problems with Von Neumann’s version of game theory.

One of them was mathematician John Nash. He found a way to determine optimal strategies in any finite game.

A New Yorker article describes the Nash equilibrium as:

“a particular solution to games—one marked by the fact that each player is making out the best he or she (or it) possibly can, given the strategies being employed by all of the other players.”

When Nash equilibrium is reached in a game, none of the players wants to change to another strategy because doing so will lead to a worse outcome than the current strategy.

In the Prisoner’s Dilemma, the Nash equilibrium is the strategy of both players confessing. There is no other better option for either player to switch to. From this game, we can also see another interesting aspect of the Nash equilibrium.

Mathematician Iztok Hozo points out that “any dominant strategy equilibrium is also a Nash equilibrium.”

He explains that “the Nash equilibrium is an extension of the concepts of dominant strategy equilibrium.”

However, he notes that the Nash equilibrium can be used to solve games without a dominant strategy.

Relationships and Game Theory: Making Sense of the Sums

Our personal and professional relationships often echo these game scenarios.

Think about it:

• It can feel like a zero-sum game in competitive situations, like competing for a promotion against a colleague.

• You're in a positive-sum game when you collaborate with a friend to organize an event and pool resources and ideas.

• Have you ever had a drawn-out argument where both of you felt worse by the end? That’s the negative-sum game in action.

Recognizing these patterns is crucial because relationships are more than just individual interactions. They're a series of ongoing games, and understanding which game you're playing can lead to better outcomes.

Whenever possible, aim to create positive-sum games with work colleagues and especially with romantic partners.

And–at all costs, romantically and professionally–avoid negative-sum games. Winding up in this situation will have dire consequences. You could get demoted, disciplined, or fired. And if it’s a romantic relationship, you could wind up alone.

Consider payoffs for yourself and the best payoffs for all parties involved, and, when possible, play a game in which both parties win.

The BMM Takeaway

I know all of this information may seem overwhelming.

However, it’s critical for your success. You have to understand the basics before you master high-level influence.

When you know how to apply game theory to all your relationships properly, you can increase your value in the marketplace.

You know what to do and what moves to make. Understanding game theory can help you become a master strategist.

Making the correct strategic choices (playing positive-sum games) will allow you to boost your reputation into someone who helps others win.

When you help others win, everyone wants to be around you, hire you, and sleep with you, and you get paid a lot more for it.