Game Theory Explained

Game theory is a really cool subject and it's one of my favourite applications of math.

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 man on the street. He was a mathematician, physicist, and computer scientist named John von Neumann. 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 could be applied 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.

Von Neumann and Morgenstern contended that any economic situation could be defined as the outcome of a game between two or more players.

What is a game according to game theory?

Yale economics professor Ben Polak notes a game has three basic components — players, strategies, and payoffs.

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, it is not always the case that players 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 the game theory.

Forbes describes the theory presented in A Theory of Games and Economic Behavior as “the mathematical modelling 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 one important aspect of von Neumann’s version of game theory that is still relevant today. And then there are payoffs, which 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.

One limitation of Von Neumann’s version of game theory is that it focused on finding optimal strategies for one type of game called a zero-sum game. In a zero-sum game, “one player’s loss is the other player’s gain” according to Forbes.

Another source notes that “players can neither increase nor decrease the available resources” in zero-sum games. Critics have noted that life is often not as simple as a zero-sum game. More complicated game scenarios are possible in the real world. For instance, 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 wider range of games such as combinatorial games and differential games, but we have time to look at only one.

A classic example of a game often studied in game theory is called 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, Rick and Ted, who have just been captured for robbing a bank. The police don’t have enough evidence to convict them but know that they committed the crime. They put Rick and Ted in separate interrogation rooms and lay out the consequences. If both Rick and Ted confess, they will each get 10 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 5 years for a different crime they were wanted for.”

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

The two players are Rick and Ted.

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

The payoffs of the game range from going free to serving 5, 10, or 20 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 Ted’s strategies are listed in rows, or the x-axis, his payoffs are listed first. Rick’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 that neither player knows the other’s decision and decisions are made at the same time (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 known as “dominant strategy.” Fundamental Finance defines it as the “strategy that has the best payoff no matter what the other player chooses.”

Ted does not know if Rick will confess or not. He looks at his options. If Rick confesses and Ted does not, Ted will get 20 years in prison. If both Rick and Ted confess, Ted will get only 10 years. If Rick does not confess and Ted does, Ted will go free.

The best strategy for Ted is to confess because it leads to the best payoffs regardless of Rick’s actions. Confessing will cause Ted to either go free or serve less prison time than if he did not confess. Rick is in the same situation and has the same options as Ted. As a result, the best strategy for Rick is also to confess because it leads to the same best payoffs that Ted will get.

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

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

While this option would give both of them less prison time than if they confessed, it would work only if each of them could be sure the other one would not confess. It is unknown whether Ted and Rick would be able to work together with that level of cooperation. In addition, both are unlikely to choose the strategy of not confessing because it has a greater penalty than they would get if they confessed. Confessing also gives each of them the possibility of serving no prison time, which is even less than 5 years in prison.

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

The 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 Prisoner’s Dilemma also reflects how other game theorists were able to fix 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 this is because “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 that do not have a dominant strategy.

Nash received great praise for the Nash equilibrium and his other work in game theory — but not from John von Neumann.

According to Forbes, “Von Neumann, consumed with envy, dismissed the young Nash’s result as ‘trivial’ — meaning mathematically simple.” Others did not share in Von Neumann’s assessment of Nash’s work. Nash, Reinhard Selten, and John Harsanyi went on to share the 1994 Nobel Memorial Prize in Economic Sciences for their work in game theory.

When Nash died in 2015, one academic news website summed up his accomplishments this way — “Nash’s most fundamental contribution to game theory was in opening the field up to a wider range of applications and different scenarios to be studied. Without his breakthrough, much of what followed in game theory might not have been possible.”

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