A decision-making puzzle that shows how probability and intuition can clash
The Two Envelopes Paradox is a famous thought experiment about decision-making under uncertainty. You are given two sealed envelopes, each containing some amount of money. You know that one envelope contains twice as much money as the other, but you do not know which is which. You pick one envelope and then are offered the chance to switch. A certain line of reasoning seems to say that switching is always better, no matter which envelope you first chose. This strange conclusion is what makes it a paradox.
BASIC SETUP OF THE PARADOX
The situation can be described like this:
There are two envelopes. One contains a smaller amount of money, call it A. The other contains a larger amount, 2A.
You do not know which is which, and from your point of view, each envelope is equally likely to contain A or 2A
You choose one envelope and call the amount inside it X (you have not opened it yet, you just use X as a symbol for “whatever is inside”).
Before opening the envelope, you are offered the option to switch to the other one.
The question is: should you switch?
At first glance, it seems there is no reason to prefer one envelope over the other. If everything is perfectly symmetric, then keeping your first choice or switching should be equally good. But the paradox arises because there is a tempting argument that appears to show that switching is always better.
WHY IT LOOKS LIKE YOU SHOULD ALWAYS SWITCH
Here is the usual “paradoxical” argument:
Let X be the amount in your chosen envelope.
If your envelope currently contains the smaller amount A, then the other envelope contains 2A = 2X.
If your envelope currently contains the larger amount 2A, then the other envelope contains A = X/2.
From your point of view, it seems that each of these cases (having A or having 2A) has probability 1/2.
So you might calculate the expected value of switching like this:
With probability 1/2, switching gives you 2X.
With probability 1/2, switching gives you X/2.
Expected value of switching:
(1/2) · 2X + (1/2) · (X/2)
= X + X/4
= 5X/4.
This is bigger than X, the amount in your current envelope. So this argument seems to say:
No matter what X is,
The expected gain from switching is 5X/4,
Therefore you should always switch.
But this creates a paradox: if the reasoning is correct, then you would always want to switch, even if you had originally chosen the “better” envelope. And if you apply the same reasoning after switching, it again tells you to switch back. You would end up wanting to switch forever, which clearly makes no sense.
So there must be something wrong with the argument.
WHERE THE LOGIC GOES WRONG
The key problem is hidden in how we use the symbol X and how we assign probabilities.
When we write “the envelope may contain X, and the other envelope may contain X/2 or 2X”, we are silently changing the meaning of X:
In the case where your envelope contains the smaller amount A, X = A and the other envelope contains 2X = 2A.
In the case where your envelope contains the larger amount 2A, X = 2A and the other envelope contains X/2 = A.
So X is not a fixed, known value; it stands for “whichever amount happens to be in your chosen envelope”. The probabilities “1/2 for X/2 and 1/2 for 2X” are not correctly defined for a single fixed X. The calculation mixes two different situations as if they were the same.
A more careful analysis needs to start with a description of how the amounts in the envelopes are chosen in the first place. For example:
Is A chosen from a limited range (for instance, always between 1 and 1 000)?
Is A chosen from some probability distribution (for example, all powers of 2 with certain probabilities)?
Can A be arbitrarily large, with no upper bound?
Depending on these assumptions, the probabilities that the other envelope contains half or double your amount are not necessarily 1/2 and 1/2 once you condition on the amount you see.
When you use a consistent probability model for how A is chosen, you find that:
Either switching and not switching have exactly the same expected value, or
The “always switch” rule does not hold for all possible amounts.
In other words, the basic “5X/4” argument is based on a misuse of the symbol X and unrealistic assumptions about the probabilities.
COMMON WAYS TO RESOLVE THE PARADOX
Different explanations focus on different aspects of the mistake. Here are some of the most common ones.
Bounded or realistic amounts
In the real world, money is not infinite. There is always some practical upper limit to how much could be inside an envelope. When you include such limits in your model, the expected value of switching is no longer always greater than staying. For very large amounts, it may even be more reasonable not to switch, because it is unlikely that there is an even larger amount in the other envelope.
The role of the prior distribution
To talk about probabilities and expected values, we need to specify how the initial amounts are chosen. This starting assumption is called the prior distribution. If the prior distribution is chosen badly (for example, in a way that does not make sense for very large amounts), then the paradox appears. With a sensible prior, the expected value of switching is not strictly better than staying for every possible X.
Confusing conditional probabilities
The paradoxical argument treats “probability of larger amount” and “probability of smaller amount” as if they were always 1/2. But once you see or assume a particular amount X, these probabilities can change. It may become more likely that X is the larger amount, or more likely that it is the smaller one, depending on your model. Ignoring this fact leads to wrong conclusions.
Ignoring strategy and focusing only on one step
Another way to look at it is to consider complete strategies, not just a single switch. If you always use the “switch” argument in a symmetric situation, you end up with a strategy that tells you to keep switching back and forth without gaining anything. This shows that the reasoning cannot be right, because a good decision rule should not lead to endless indecision.
WHY THIS PARADOX MATTERS
The Two Envelopes Paradox is more than just a curious puzzle. It illustrates several important ideas:
Our intuition about probability and expectation can be unreliable when information is incomplete.
It is easy to write down formulas that look correct but secretly use inconsistent assumptions.
Realistic decision-making requires careful thinking about how the situation is set up, not only about algebraic manipulations.
The paradox is related to topics in probability theory, decision theory, and even philosophy. It is often used to show how “infinite” or unbounded models can cause strange results, and how crucial it is to specify exactly how probabilities are defined.
CONCLUSION
The Two Envelopes Paradox looks simple: two envelopes, one with twice as much money as the other, and a free choice to switch. A quick calculation seems to show that switching always increases your expected gain. But when we look more closely, we see that this calculation misuses the symbol X and ignores how the amounts in the envelopes are actually chosen.
Once we correct these assumptions, the paradox disappears. Switching is not always better; in many reasonable models, the expected value of switching and not switching is the same. The paradox is a useful reminder that careful thinking about assumptions is just as important as doing the math itself.
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