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Ryan Chao
6 min read

How to play games using math


I know what you’re thinking–and no, this is not about video games.


In this article, we will address the practical applications of the rich and nuanced field of game theory. We will also offer advice on how you may apply these same concepts to help navigate strategy games, negotiations, and other real-life scenarios.


Even though you may not know it, you are navigating elements of game theory every day.


For example: when checking out of a grocery store, you will want to go to the shortest line to maximize your efficiency. However, other customers will likely do the same, making the “short” line not so short anymore.








At many amusement parks there are "Easy Passes" that you can buy in order to wait for rides in an expedited line. However, imagine 90% of people buy the pass in order to avoid the long lines. In this case, the regular line would actually be much faster.






So, what is a “game” anyway, in the math world?


A game is simply a situation in which there are multiple people, each of whom are trying to maximize their personal gain.






In order to understand how to use the strategies of game theory, as well as model known situations that arise, we have to make some necessary—if seemingly cold—assumptions.


1: Everything has utility or value that can be quantified.

2: Everyone is rational: e.g. everyone is working to maximize their gain of utility


It is risky to mix psychology or human tendencies into game theory because, in this field, we assume everyone acts with perfect strategy.



The simplest example of game theory in action is the prisoner’s dilemma:






This is how it works:

There are two prisoners on death row convicted of committing a crime together. They are given the following ultimatum: if both of them confess to their crime, they will be given a life sentence. If neither of them confess, they will serve 10 years in prison. If one confesses and the other doesn’t, the confessor gets off scot-free, which the other prisoner is executed.


It turns out it is always in the prisoners' best interest to confess, even though if they both confess, they will both be given life sentences. Let's call them prisoner A and prisoner B (very imaginative, I know). If you are prisoner A, you know that prisoner B will either stay silent or confess.


If prisoner B stays silent, prisoner A confessing means prisoner A will get away scot-free. Prisoner A staying silent means that both prisoners will serve 10 years. Clearly the first option is more appealing!


If prisoner B confesses, prisoner A confessing means prisoner A will serve a life sentence. If prisoner A stays silent, they will have been ratted out and be executed. Assuming execution is worse, it is better for prisoner A to confess.


Because prisoner A should confess in both scenarios, it is always best for them to confess. But the same applies to prisoner B! That means that they both would end up with life sentences when they could both remain silent and only serve 10 years. What's going on here? This is a demonstration of an advanced concept in game theory called the Nash Equilibrium. This concept was named after John Nash, a pioneer of the field, originally an electrical engineer and then a now-prominent mathematician who even has a movie made about him: A Beautiful Mind - The story of John Forbes Nash Jr. The essence of the Nash Equilibrium is that sometimes when all individuals do what is best for themselves, the outcome might be suboptimal for all of them.


While the prisoner's dilemma may seem somewhat contrived (which it is) it provides an excellent case-study into games as a whole. This is because, while it may seem complicated, this is one the simplest examples of a game: there are only 4 outcomes!


In fact, a very similar concept has been implemented in the British game show called The Golden Balls, which uses the game split or steal.







The rules of the show are as follows:


There are two players and a pot of a million dollars. They are each given two golden balls: one with the word “split” written inside, and one with the word “steal.” Both players choose which ball to give to the arbiter. If both choose the split ball, they split the money and walk home with 500,000 dollars. If one chooses to steal, and the other splits, the stealer is given all of the money. If both choose to steal, neither gets their share of money.


In an infamous episode – one of the last ever done on the show – one contestant claims that they are going to steal the money, and then give back half afterwards. The other contestant, who had their own plan of trying to manipulate the firsts and steal the money, was forced to split. This is because, if they stole, they knew they would definitely get nothing. If they split, they at least had their opponent's word on live television that they would get half the money, which is at least a chance at $500,000.






What makes the episode particularly funny is that the first player actually does end up choosing the split ball, and they both split the money. This is interesting because while it may seem like a genius play, it was not actually a rational move. If the first opponent were to play the steal ball, as they originally stated, they would not only have the decision to actually share or not but would also not run any risk of being betrayed.


Of course, all of this is assuming that we consider losing your money to betrayal is worse than neither player getting it (in previous terms, the second alternative has a lower utility).

In game theory, you never know your opponent's strategy. They, like you, have multiple strategies to choose from and could deviate at any time from what you might “assume” they are doing.


One thing to be said about game theory is that we assume everyone is rational, which is not often the case in the real world.


Despite its far-reaching impacts, game theory cannot be used for everything. There are some games to which, by their random/luck-based nature, you unfortunately cannot apply the normal tools of game theory.


An example of this is rock-paper-scissors.


Game theory is much more about having your own strategy to maximize personal gain, rather than others’ strategies. For example, a game like rock paper scissors is interesting to study from a psychological point of view but less so for game theory.






Imagine your opponent plays “rock” every single time, 10 times in a row. You might be inclined to believe that they will continue next time, and play paper. However, if they change their strategy suddenly and play “scissor” you will lose.


In game theory you want to maximize your gain on average, meaning your strategy is optimal for dealing with the average of strategies of your opponent. In the case of rock paper scissors, every strategy is the optimal strategy. No matter what you do, you statistically are likely to win 50% of the time, and so is your opponent.


That is why game theory is more applicable to games that involve strategy as well as luck.



Finally, what are some takeaways from this article that you can bring with you into real world situations?


1: A game is really any situation where you apply a strategy to gain the most possible from a situation.

2. When all the players are rational (and sometimes they aren’t), everyone will try to maximize their own gain.

3. These “all or nothing” scenarios, like the ones described, are one of the most ubiquitous examples of games and come up everywhere. Look for them in your own life and decide what is best.

4. Sometimes there are multiple right strategies, and they are all equally effective (think of “rock, paper, scissors”).

5. There are many games such the aforementioned “rock paper scissors” which are highly random. While you may believe that your choice matters, in reality the game does not actually have a true strategy. While this may sound sad, it is really just a reminder to not blame yourself for things you cannot control, even when, from an informed perspective, it seems like you can.



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