The Money Envelope: How Your Opponent’s Choices Reveal Their Position

Money envelope game.


In game theory, a smart player accounts for the fact that the choices that other players make reveal a lot about their position. This article will show you how this concept works and how you can take advantage of it, by looking at the example of the Money Envelope game.


The setting

Richard and Bob are the final contestants in a game show called “The Money Envelope”. As the winners of tonight’s episode, the host gives each of them one of five possible envelopes. They know the following things:

  • Each envelope contains a check with either 500$, 1000$, 2000$, 4000$, or 8000$ (so that there is only one envelope with 500$, one with 1000$, etc.)
  • One of the contestants is getting an envelope with twice as much money as the other.
  • After each of them sees how much money he got (in private), he can ask the other person to exchange envelopes (without knowing how much money the other person got). If they both agree, the exchange occurs.


The game

Note: this example involves some simple math. You don’t have to follow the specific numbers too closely, what matters is the concept behind it.

In the current game, Richard opened his envelope and found 2000$. Based on the rules, he knows that Bob got either 1000$ or 4000$, with equal probability.

Richard calculates that he should make the exchange, since he stands to gain more that way; on average, he will earn 2500$ dollars after the exchange, compared to the 2000$ that he’s getting now. This is because there’s a 50% chance he will get 1000$, but there’s also a 50% chance that he will get 4000$ (0.5*1000+0.5*4000=2500).

However, Bob is thinking the exact same thing, regardless of whether he found 1000$ or 4000$ in his envelope:

  • If he got 1000$ and he makes the switch, he will get 1250$ on average. (0.5*500+0.5*2000=1250)
  • If he got 4000$ and he makes the switch, he will get 5000$ on average. (0.5*2000+0.5*8000=5000)

Based on this, we would assume that both players would want to make the switch. However, what happens is that both contestants choose to keep their original envelope. How come?


Game-theory analysis

The issue here is that if both Richard and Bob are perfectly rational, and both know that the other person is also perfectly rational, then an exchange is never going to take place. We can see why by considering the situation step by step, starting from a slightly different angle:

  • Let’s say that Richard opens his envelope and finds 8000$. Since he knows that he already has the envelope with the most money, he won’t agree to an exchange.
  • In this scenario, Bob has to get 4000$ in his envelope (since Richard got the maximal amount). However, Bob doesn’t know whether Richard got 8000$ or 2000$. What he does know is that Richard won’t agree to an exchange if he got 8000$. Instead, the only way Richard will agree to an exchange, is if he got 2000$. Therefore, Bob can conclude that he himself should not agree to an exchange, since he will lose out on money if he does.
  • Based on this, we know that a player who gets 8000$ won’t ever agree to an exchange, but neither will someone who got 4000$.
  • Now, we’re back to the original scenario, where Richard got 2000$. If Bob has 4000$, then he’s not going to agree to an exchange, as we saw above. Therefore, if Bob is interested in an exchange, Richard can conclude that Bob got only 1000$, in which case Richard will be the one that doesn’t agree to an exchange.
  • Furthermore, if Bob has only 1000$, he knows that the only way Richard will agree to an exchange is if he has 500$ (the minimal amount), in which case Bob shouldn’t want an exchange in the first place.

Essentially, before making a decision, each player looks at the other player’s behavior. If the other player wants to make an exchange, the original player can conclude that it would benefit the other player more than it would benefit him. Eventually, the only person willing to trade is the guy who got the minimal amount, but no one is willing to trade with him.


A note on the math

This setting is based on the two envelopes problem, which usually focuses on a situation where a single person is given two envelopes, and is allowed to switch between them. However, the scenario in this article involves two players, who are engaged in a zero-sum game, since one contestant’s gain is exactly balanced by the other contestant’s loss.

Based on studies that analyze the two envelopes problem, it is possible that the assumption that each player stands to gain from making the switch may be inherently flawed, which is why it was originally termed the exchange paradox. Essentially, there is a large number of papers on the topic, which address different variations of the paradox, and offer different explanations for the math behind it. So far, no consensus on the topic has been reached.

Nevertheless, the exact math behind this problem isn’t crucial, since the logic behind the players’ decision to keep their envelope, as presented here, still holds.


Summary and conclusions

  • The Money Envelope Game is a game where two players each get a certain amount of money in an envelope. The amount of money each player gets is one of a certain set of values (e.g. 500$, 1000$, 2000$…), and both players know that one of them is going to get twice as much money as the other person.
  • Each contestant is allowed to look inside their own envelope. Then, they can ask the other contestant to exchange envelopes, without knowing how much money the other person got. If both contestants agree, then the exchange occurs.
  • By seeing the choice that the other player makes regarding the exchange, each player can decide whether or not to make the trade.
  • This provides an important lesson about game theory: in many situations, the decisions that your opponents make can give you insights regarding their position.


The basis for this strategy and example comes from “The Art of Strategy: A Game Theorist’s Guide to Success in Business and Life”. It’s a good read for someone looking to understand basic game theory and how it applies to real-life situations.


Make Your Password Manager More Secure by Changing the Number of Password Iterations

If you store all of your passwords in a password manager, you might be worried about someone trying to hack it by brute-forcing their way in.

The best way to avoid this is by using a strong, random password, which will be impossible for a hacker to brute force. However, with the constant advances in computing power and hacking techniques, the bar for how strong your password should be is constantly increasing.

Luckily, there is a way to significantly improve the security of your password manager, without increasing the complexity of your password, and without much effort on your part.

All you have to do is change a single setting in your password manager: the number of password iterations (also known as key transformations). Basically, when you input the master password into your password manager in order to access your database, the software transforms this password multiple times before checking its validity (a process known as key stretching).

The more times the password is transformed, the longer it takes to check whether a certain password is correct, and the longer it will take someone to brute force their way into your database.

KeePass, for example, currently specifies a default value of 6,000 transformations. However, on most modern computers you could easily change that number to several million transformations, without experiencing a noticeable increase in the time it takes the software to load your database when you input the correct password.

This means that if you increase the number of transformation to 6,000,000, a hacker will now take 1,000 times as long to crack your password, while the program will still load almost instantaneously when you input the correct password.

In fact, KeePass lets you set the number of transformation so that it takes the computer approximately 1 second to check the password. On modern computers, these values can often far exceed the 6 million transformations.


The KeePass database setting for changing the number of iterations/transformation the master password goes though.


Keep in mind that the loading time will vary for different devices; this will be especially noticeable if you access the password manager on mobile, so make sure you’re not setting the value too high for that.

Likewise, the optimal value might be different if you use an online service, such as LastPass, which currently recommends not exceeding 10,000 password iterations for client-side encryption (compared to the default 5,000), though they do allow users to go as high as 200,000.

The setting itself is generally easy to find on all platforms. In KeePass, for example, you will go to File > Database Settings > Security. In LastPass you go to Account Settings > General > Show Advanced Settings > Password Iterations.


LastPass setting for controlling the number of password iterations.


One more thing worth noting: while this post focused on password managers, this advice is also applicable to other types of encryption software. VeraCrypt, for example, allows you to set the number of iterations used for encrypting volumes, using their Personal Iterations Multiplier.


Summary and Conclusions

  • The master password to your password manager undergoes multiple transformations/iterations before being verified; this number scales linearly with the time required to check whether a certain password is correct.
  • By increasing the number of transformations, you can easily improve the security of your password database, without increasing the complexity of the password.
  • This will lead to a negligible increase in software loading time for you, but will significantly increase the time it takes to brute force the password.
  • When setting the number of transformations, make sure to account for any differences in the processing power of the devices you will use to log in, especially if you use mobile.
  • Instructions on how to do this are generally easy to find. If unsure, search for ‘software name + key transformations’ or ‘software name + password iterations.


Why You Should Stop Sleeping with Your Phone Next to You (and What to Do if You Decide to Keep It)

Literally a picture of a phone laying on a bed.


Most of us go to sleep with our phone within reach. This can be a problem, because keeping your phone next to you in bed often leads to a variety of sleep, health, and productivity issues.

In the following article, you will learn why you are better off putting your phone out of reach when you sleep, and what you can do to reduce its negative effects if you decide to leave it near you anyway.


Why sleeping with your phone next to you is problematic

There are a few main reasons why keeping your phone within reach when you’re in bed is bad for you:

  • You’re more likely to browse your phone, which makes it harder for you to fall asleep, and reduces your sleep quality. The negative impact of electronic-media usage before bedtime is well-documented, and occurs due to a multitude of reasons. Because of this, avoiding using your phone and other electronic devices before you go to sleep is considered to be an important aspect of sleep hygiene, which improves your sleep quality.
  • It increases the temptation to use the phone if you wake up throughout the night. Normally, if you wake up in the middle of the night you would want to just go back to sleep. However, if you have your phone nearby, you now have the temptation to check messages or perform some other unnecessary activity on your phone, which is obviously not good for your sleep.
  • It makes it more likely that your phone will be the first thing you see in the morning. This isn’t inherently a bad thing, but you should consider whether this is what you want for yourself. Basically, think about how starting your day by immediately checking your phone is affecting the way you feel, in terms of factors such as anxiety and productivity.

Because of these issues, it’s better if you keep your phone out of reach when you go to bed. The further it is, the less temptation you will have to look at it before or during your sleep. You can also take into consideration how its placement will affect your overall morning routine, and put it somewhere where it won’t be one of the first things you look at in the morning.


What to do if you decide to keep your phone next to you

First, if the main reason why you need your phone next to you is so that you can use the alarm to wake up, you should consider getting a separate alarm clock instead.

Even if that’s not a viable solution, many people will still decide to keep their phone next to them, despite the issues that doing this presents. If you choose to keep it, there are some things you can do which will help you avoid some of these issues:

  • Use an app to filter out blue light from the phone screen. While exposure to light before bedtime is bad in general, exposure to blue light leads to the most issues in terms of sleep quality. By filtering it out, you solve a big part of the problem, without much effort on your part.
  • Turn off the sound on your phone. This prevents late-night messages from interrupting your sleep or waking you up. In addition, you should also turn off any vibrations, which will have a similar effect. Note that if you’re using your phone as an alarm, there will usually be a setting which allows you to disable general sounds on your phone, while leaving the alarm sound enabled.
  • Turn on airplane mode and turn off the WiFi. This can also help reduce the amount of sound coming from your device (with the potential exception of some notifications). More importantly, this removes the temptation to check messages or to look at things online.
  • Use an app that blocks other apps during set hours. This is especially helpful if you have a specific site or app that you know is problematic for you (e.g. something that you browse for hours in bed instead of going to sleep).


Summary and conclusions

  • Keeping your phone next to you in bed can lead to problems falling asleep and reduce sleep quality.
  • Keeping your phone next to you can also cause anxiety and productivity issues if it’s the first thing that you look at in the morning.
  • If your main reason for keeping your phone within reach while you sleep is using it as an alarm, consider getting a separate alarm clock instead.
  • If you decide to keep your phone next to you, there are some things you can do to reduce its negative effects, such as using apps that filter out blue light, or turning on flight mode.