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

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.


Money envelope game.


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, in reality, the contestants both 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 that player more than it would benefit them. 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, each offering a different explanation for this, and no consensus on the topic has yet been reached.[1,2,3,4]

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

  • 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.


Look Forward, Reason Backward: A Primary Principle of Strategic Thinking

One of the basic principles in game theory is that you should look forward and reason back. Essentially, this means that before making a move, you should consider all the possible moves that you and the other players can make, together with the possible outcomes that these moves lead to. Then, consider how desirable each outcome is to each player, and based on this, determine which moves they are likely to make.

In order to think this way, it’s useful to use a game tree (also referred to as a decision tree in cases where there is only one player). In the following example, you will see how this principle works, and how you can implement it in order to make smarter decisions, by taking advantage of game theory.


Example: to Advertise or Not to Advertise?


MegaCorp is currently the only company selling a certain type of high-quality industrial lasers.

Startupo is a new company, which is considering entering the market currently dominated by MegaCorp.

To deter Startupo from taking a part of their market share, MegaCorp can engage in a costly advertising campaign, which would involve significantly reducing their prices.

Since Startupo is a smaller and more flexible company, they can wait and see whether MegaCorp runs their ad campaign before deciding if they should enter the market.

As such, each company has two possible moves: MegaCorp can decide whether or not to run the ads, while Startupo can decide whether or not to enter the market.

Therefore, there are 4 possible outcomes for the game, each ranked differently by the players (1 is the most desirable outcome, while 4 is the least desirable outcome):



  1. No ads, no entry (of Statupo).
  2. Ads, but no entry.
  3. No ads, but entry.
  4. Ads and entry.



  1. No ads, yes entry.
  2. Ads, but no entry.
  3. No ads and no entry.
  4. Ads and entry.


Based on this, we get the following game tree:

Game tree showing the possible moves in the scenario.


When making the decision whether to or not advertise, MegaCorp starts by looking at the possible outcomes, and then asking themselves which moves their competitor will make:

  • If MegaCorp runs the ads, then Startupo will choose not to enter the market (since it gets them their #2 choice, as opposed to their #4 choice). In this case, MegaCorp also get their #2 choice.
  • If MegaCorp doesn’t run the ads, then Startupo will choose to enter the market, since it leads to a better outcome for them than not entering the market (choice #1 versus choice #3). In this case, MegaCorp get their #3 choice.
  • Clearly, MegaCorp should run its ads, to prevent Startupo from entering the market. This way they get their #2 choice (similarly to Startupo), while if they decide not to run the ads, they will get their #3 choice, while Startupo will get their #1 choice.

This rationale is phrased so that it appears we start by considering MegaCorp’s moves. However, we in fact start by considering the outcomes, and then considering which move Startupo will make in each of the two scenarios available to them (ads/no ads). Only then do we consider MegaCorp’s initial move.


Other considerations

This method of backward induction can be used to find the optimal solution of a game when the following conditions apply:

  1. Sequentiality: the game must be sequential, meaning that the players act one after another (as opposed to a simultaneous game, where the players act at the same time).
  2. Finiteness: the game must be finite (have a clear endpoint).
  3. Perfect information: the players must have perfect information regarding the possible moves, outcomes, and the desirability of each outcome.
  4. Assumption of rationality: the players must make the rational choice (i.e. select the option that is best for them).

Of course, in reality, things are more complicated. Perfect information rarely exists, and the potential moves and motives of each player tend to be more complex. However, this method is nonetheless the best way to approach such scenarios, and these extra factors and consideration can be appropriately incorporated into the model. Similarly, the addition of more players to the game doesn’t change the overall strategy either; rather, it just increases the complexity of the game tree.

This all leads to an important caveat: just because the game has an optimal strategy that you should select, doesn’t mean that it’s easy to find it. (Think about chess, for example)


Summary and conclusions

  • When thinking strategically, start by looking forward in order to see the possible moves and outcomes in the scenario.
  • Try to predict how desirable each outcome is for each player.
  • When you have the game tree mapped out, with all the possible moves and consequent outcomes, reason backwards in order to find the best moves for you to make.
  • This allows you to find the optimal solution to a game, given that some conditions are met (sequentiality, finiteness, availability of perfect information, and assumption of rationality).


“Follow the Follower”: a Lesson in Strategy from Sailboat Racing

Sailboat racing offers the chance to observe an interesting reversal of a “follow the leader” strategy… The leader imitates the follower even when the follower is clearly pursuing a poor strategy. Why? Because in sailboat racing (unlike ballroom dancing) close doesn’t count; only winning matters. If you have the lead, the surest way to stay ahead is to play monkey see, monkey do.

The Art of Strategy: A Game Theorist’s Guide to Success in Business and Life

America’s Cup is a prestigious sailboat race, and one of the world’s oldest international sports competitions. In 1983, the American boat Liberty was leading 3-1 against the Australian Australia II, in a best-of-seven competition. Since they needed only one more victory in order to win the cup, it appeared that Liberty was ready to extend the US’s 131 years long winning streak.

Right at the start of the race, Liberty took the lead when Australia II was penalized for crossing the starting line early. The Australian skipper then attempted to catch up by sailing to the left side of the course, in hopes of catching good winds. The American skipper decided to keep his ship on the right side of the course, believing that it would have more favorable winds.

Soon after this, the wind shifted in favor of the left side of the course, leading Australia II to win the race. Following this victory, Australia II went on to win two more consecutive wins, thus winning the cup and breaking the long-standing American winning streak.


Picture of a boat sailing.


What should have happened

In this situation, the speed of each ship depended on the wind, and each ship’s skipper can only make an educated guess regarding which course is the best to take.

Since the Liberty was already in the lead, if it had simply imitated the strategy of the runner-up, Australia II, it would have sailed at the same rate as her, thus maintaining the initial advantage, and winning the race. Regardless of how sureLiberty‘s skipper was that his course was the better one, the smarter strategy in this case would have been to imitate his runner-up.


Recognizing when the strategy is applicable

“Follow the follower” is by no means a strategy that always works. In the above scenario, there are only two ‘players’, and the only thing that matters in the race is whether you win or lose. However, if these conditions were different, the strategy may have been ineffective. For example, if the race had more than two ships, and changing course was not an immediate action, so that the leading ship couldn’t always adjust to match the runner-up, then the strategy wouldn’t necessarily work. This is because each follower can take a difference course, while the leader can only commit to one of those courses.

In addition, the original scenario discussed here is a relatively clean and simple view of reality. There could have been other considerations that affected the American skipper’s decision:

  • Maybe it’s considered more prestigious to win the race by a bigger gap, and imitating the loser’s strategy can be construed as a lack of confidence.
  • Perhaps there is a high cost or risk in changing course, which could have caused the ship to lose its advantage.
  • We also don’t know how confident the American skipper was in his choice of course; it’s possible that his calculation showed a very high probability that his original course was significantly better.

While these reasons don’t negate the fact that imitating the runner up was the correct choice from a purely strategic perspective, they offer some possible explanations as to why the American skipper made the choice to maintain his course. If, for example, the benefits (in terms of prestige) that come from winning the race by a large gap were significant enough to be worth a small chance of losing, then his choice may have been smart after all. Of course, it’s also entirely possible that the choice of strategy was driven by ego, and not from careful calculation.

This illustrates an important lesson regarding the applications of game theory in real life: reality is messy. There is a reason why simplified models are preferred in game theory; the more factors you add in, the more complicated the game becomes.


Summary and conclusions

  • In certain cases, the best strategy for the leader is to imitate his runner-up.
  • By doing the exact same thing as his follower, the leader can win a ‘race’ by maintaining his original lead. This is true even in cases where the follower select a non-optimal course.
  • Be selective in using this strategy, as it’s only applicable in certain cases. For example, it may not be relevant when there are more than two ‘players’.
  • Once you are familiar with the strategy, the important thing is learning to recognize situations where you can implement it.
  • Ego may lead people to avoid using this strategy. Make sure to overcome this issue in yourself, and to take advantage of other people’s failure to do the same.


The sailboat example and the rationale behind the strategy come 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 too crucial.