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 both contestants agree, the exchange occurs.

The game

Before we start, note that this example involves some simple math. You don’t have to follow the specific numbers too closely; what matters is the concept behind it all.

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 that he stands to gain more from making the exchange that way, because 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 (and 0.5*1000+0.5*4000=2500).

However, Bob is thinking the exact same thing, regardless of whether he got $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 want to make the switch. However, what happens is that both contestants end up choosing 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. Rather, 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 the highest amount of money possible ($8000) won’t ever agree to an exchange, but neither will someone who got the second-highest amount of money ($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 shouldn’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.

Practical implications

The main takeaway from the Money Envelop game is this: in certain situations, your opponent’s moves or their willingness to conduct certain moves can tell you a lot about their position, and you can use this information in order to make optimal decisions for yourself.

For example, in a negotiation situation, if your opponent is surprisingly willing to agree to your demands, it’s possible that your position, relative to theirs, is stronger than expected. Accordingly, this means that you could leverage your position, and increase your demands, in order to benefit more from the negotiation.

Similarly, based on various background factors, such as the type of negotiation that you’re conducting and the other person’s reputation, you might reach a different conclusion based on their willingness to agree to your demands.

For example, if you’re looking to buy a second-hand car from someone, and they are surprisingly willing to reduce their initial asking price, it’s possible that there is an issue with the car that they’re not disclosing, which means that you should be much more cautious about the trade. This represents the well-known saying that if a deal looks too good to be true, then it probably is.

Overall, the important thing to remember is to be aware of your opponent’s moves, and of their willingness to conduct certain moves, so that you can get clues about their position. When implementing this principle, it’s important to try and notice any deception on your opponent’s part, since people can sometimes bluff or disguise their moves and intentions, in order to make it more difficult for you to discern their position.

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. Many papers which address different variations of the paradox have been published, and each offers a different explanation for the math behind it. However, 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.

Each contestant is allowed to look only 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.

You can implement this principle in practice, by looking at what your opponent says and does, and then using this information in order to make optimal decisions. For example, in a negotiation situation, if your opponent is surprisingly willing to agree to your demands, this could indicate that your position is stronger than you thought, or that there is an unexpected catch that your opponent is not telling you about.

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.