The Money Envelope: How an 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 a round 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, 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$.
  • Now, we’re back at the original scenario, where Richard got 2000$. 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.

See the idea here? Basically, 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 wants to trade with him anyway.

 

A note on the math

This setting is based on the two envelopes problem/the exchange paradoxIn practice, 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’). There is a large number of papers on the topic, which address different variations of the paradox, and offer different explanations. So far, no consensus on the topic has been reached.

In general, most formulations of this problem focus on a situation where a single person is given two envelopes, and is allowed to switch between them. However, the current scenario involves two players, engaged in in a zero-sum game (because one contestant’s gain is exactly balanced by the other contestant’s loss).

In practice, it doesn’t matter that the original motivation to exchange may not exist, because the logic behind the players’ decision to keep their envelope, as presented here, still holds. As always, when reading about game-theory models, you should focus on the concept behind it, rather than on the scenario itself.

 

Summary and conclusions

  • The Money Envelope Game involves players trading envelopes filled with different amounts of money if they believe that doing so will help them get more money on average.
  • 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.
  • Use these insights to make better-informed and more strategic decisions.

 

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.

I recommend it over the earlier version of the book (“Thinking Strategically”), because that’s what the authors themselves recommend. However, the difference between the two versions isn’t crucial.