We offer a new view on the two envelope problem (also called the exchange paradox). We describe it as a zero-sum game of two players, having only partial information. We first explain a standard situation and show that the mean gain-when defined-is really zero. However, there are even more paradoxical situations in which the information obtained by the players supports the exchange of envelopes. We explain that this does not lead to a contradiction and we demonstrate it also by computer simulation. The reason for this paradox is that the mean gain does not exist and that the players have different information, supporting their contradictory decisions.
The two envelope problem (also called the exchange
paradox) is a famous logical puzzle demonstrating a paradox in
logic and probability. We adopt its formulation from [
There are two indistinguishable envelopes, each containing money, one contains twice as much as the other. Player A picks one envelope of his choice; player B receives the second envelope. They can keep the money contained in their envelopes or switch the envelopes (if both agree on it). Should they switch?
Argument 1. The situation is symmetric. Thus there is no reason for (or against) switching.
However, there are other interpretations suggesting something else: Argument 2. The situation is symmetric. Thus the probability of having the envelope with the higher or lower amount is 1/2. If the envelope of player A contains the amount a, then the other envelope contains 2a (and the exchange results in a gain of a) or a/2 (and the exchange results in a loss of a/2). In average, the mean gain is 1 1 a a 2 a − 2 2 = 4 , thus switching is always recommended.
Another point of view is the following: Argument 3. The smaller amount is x, the bigger is 2x. They are assigned randomly (with probabilities 1/2) to players A and B. The mean values for both players are 1 1 2 x − 2 2x = and there is no reason for (or against) switching.
We presented several arguments; each of them seems correct, but their conclusions are contradictory.
Surprisingly, the debate about this paradox is still not
finished (cf. [
“Currently, there is no consensus on a
demonstration, since most people generally reject each
other’s demonstrations." [
The paradox has more variants (cf. [
It is also not specified whether the players know the amounts in their envelopes. This knowledge is useless if the distribution is unknown. On the other hand, knowing the distribution, the amounts bring useful information for the decision. We suppose that the players know the distribution from which x was drawn. We discuss this case in detail. Some of its consequences seem to determine the strategy also without looking inside the envelope, but—as we shall show—this need not correspond to conclusions made in the former case.
We present an explanation of the paradox (based
on [
Exchange Paradox: First Level We introduce a third member of the experiment, the banker C, who controls the game and puts (his) money in the envelopes. We assume that the smaller amount, x, was chosen by a realization of a random variable X . (The larger amount in the second envelope is 2x.) The distribution of random variable X is known to the banker and also to the players. For simplicity, we assume that the distribution is discrete and the amounts are positive. Then we do an independent random experiment (e.g., tossing a coin) with two equally probable results, expressed by a random variable U , whose possible values are 0 and 1 and expectation EU = 1/2. If U = 0, player A receives the smaller amount, x; if U = 1, player A receives the bigger amount, 2x. He does not know x, only the contents of his envelope, specified by realizationa of random variable A, A =
(x A = (1 + U ) X .
if X = x and U = 0 , 2x if X = x and U = 1 , Player B receives the other envelope and knows only its contents, specified by realizationb of random variable B, B =
(x B = (2 − U ) X .
if X = x and U = 1 , 2x if X = x and U = 0 , If the players exchange the envelopes, the gain of A is G = B − A. For player B, G is the loss and −G is the gain.
Random variables X and U are independent. If X has an expectation EX , then
EA = (1 + EU ) EX = EB = (2 − EU ) EX = EG = EB − EA = 0 . for all x. However, this does not apply to conditional probabilities P(U = 0|A = a), P(U = 1|A = a) because U, A are dependent:
P(U = 0|A = a) = P(U = 1|A = a) = = =
P(U = 0, A = a)
P(A = a) P(U = 0, X = a)
P(A = a) P(U = 1, A = a)
P(A = a) P(U = 1, X = a2 )
P(A = a) = =
P(X = a) 2 P(A = a) where
P(A = a) = P(U = 0, A = a) + P(U = 1, A = a) = P(U = 0, X = a) + P(U = 1, X = a2 ) =
Notice that P(U = 0|A = a) is the conditional probability of gain and P(U = 1|A = a) is the conditional probability of loss given A = a. Their ratio is
P(X = a)
P(X = a2 ) .
As there is no uniform distribution on an infinite
countable set (a fact ignored even in [
Given A = a, switching brings a gain with conditional probability distribution
P(G = g|A = a) =
P(G = g, A = a)
P(A = a)
,
where
(after substitution a := 2b). This explains the error in
Argument 2 provided that the expectation of G is defined.
Remark 1. There is another arrangement suggested in [
P(G = g, A = a) =
P(G = g|A = a) = P(U = 0, X = a) if g = a ,
P(U = 1, X = 2a ) if g = − 2 , a 0 otherwise.
0 = 2211 PP((XX == a2a)) iiff gg == a−,2 ,
a otherwise. P(X = a) P(X = a) + P(X = 2a ) P(X = a2 ) P(X = a) + P(X = 2a ) 0 if g = a ,
a if g = − 2 , otherwise.
The conditional expectation of the gain is always defined and it is
E(G|A = a) = a P(X = a) − 2a P(X = a2 ) .
P(X = a) + P(X = a2 )
P(G = g) = 2121 PP((XX == g−)g) if g < 0 , if g > 0 , 0 otherwise and its expectation is
EG = ∑ g P(G = g) g ∑ g P(X = g) + ∑ g P(X = −g) g>0 g<0 ∑ g P(X = g) − ∑ h P(X = h) g>0 h>0 = 0 (after substitution g := −h), provided that the sum (2) is absolutely convergent. In this case
EG = ∑ P(A = a) E(G|A = a) a (1) (2) (3) (4) a P(X = a) − 2a P(X = 2a ) ∑ a P(X = a) − ∑ b P(X = b) a b = 0 = = 1 2 1 2 = ∑ a 2
1 =
For q = 0.25, it is shown in Fig. 1.
If player A has a = 2s, the conditional probability of his gain by switching is
P(U = 0|A = 2s) =
P(X = a) P(X = a) + P(X = a2 ) =
q q + 1 (5) for all s ∈ {1, 2, . . .} (and 1 for s = 0). As q < 1/2, this probability is less than 1/3. The conditional expectation For q = 0.25, it is shown in Fig. 2. The contributions of conditional expectations E(G|A = a) to the unconditional expected gain EG are E(G|A = a) · P(A = a), see Fig. 3. The conditional expectation is positive only for s = 0 (i.e., a = 1), negative otherwise. This determines the right strategy of switching: Switch only if you hold 1. The two players never both agree on switching the envelopes because at most one of them holds 1.
If player A does not know the contents of his envelope, he may use only its distribution 1 2 P(A = 2s) = 1
qs
1 − q
1
+
qs−1
1 − q
2 1 − q
0
1 qs−1 q + 1
2 1 − q
= 1 1
2 1 − q
0
In Section 2, we presented a standard explanation of the
exchange paradox. It is based on the assumption that the
(7)
amount in the envelopes has an expectation. This can fail
even for some common distributions. (This fact is ignored,
e.g., in [
Let us consider the situation from Section 3 if q > 12 . Then the expectation EX does not exist. (The respective sum of a geometric series with quotient 2q > 1 is +∞.) For q = 0.75, the joint distribution of U and A is shown in Fig. 4.
The expectation of the gain, EG, does not exist because the sum (2) is not absolutely convergent; it is a difference of two infinite sums in (3).
Still formula (6) for the conditional expectation of the gain is valid,
Thus E(G|A = 2s) > 0 for all s ∈ {1, 2, . . .}. For q = 0.75,
see Fig. 5. (Notice that formula (5) for the conditional
probability of gain P(U = 0|A = 2s) for s 6= 0 still holds
and gives a constant value from the interval ( 1 , 12 ).) Player
3
A has a strong argument for switching the envelopes,
independently of the amount in his envelope. (Thus he may
“rationally” decide for switching without looking inside
the envelope.) Such distributions are called paradoxical
in [
The same argument applies to player B. Although he holds a different amount in his envelope, he also prefers switching. We have again a paradox, now supported by a probabilistic analysis. The only thing which does not work as in Section 3 is formula (7) for unconditional gain; the sum is not absolutely convergent. However, the unconditional gain is not needed for decision if the conditional one is always positive. How can we now defend Argument 1?
First of all, we refuse the possibility (considered in [
If the players know only the amount in the envelope they received first, they have different information, and this is the key difference.
Example 1. Suppose that A has 4 and B has 8. Then A knows that B can have 2 or 8, while B knows that A can have 4 or 16. Using Argument 2, A might expect that switching brings him a mean gain of 1. Using the model described in this section (formula (5) remains valid), he knows that his chance of gain is lower, 5
Exchange Paradox: Third Level Another modification of the example of Section 3 withq = 1/2 is also of particular interest. The joint distribution of U and A is shown in Fig. 7. Formula (5) for the conditional q q + 1 ∈ .
For q = 0.75, this chance is 3/7 = 0.43, still sufficient to give a positive conditional mean gain of Player B may apply the same arguments, leading to twice higher estimates of his gain.
Example 2. Suppose now that A has 4 and B has 2. Then A knows that B can have 2 or 8, while B knows that A can have 1 or 4. From the point of view of player A, the situation is the same as in Ex. 1. Player B may apply the same arguments, leading to twice lower estimates of his gain, still supporting the decision to switch.
In Exs. 1 and 2, we saw that probabilistic analysis suggests switching to both players. This apparently brings a gain to only one of them, but their arguments overestimate their chances. This explains why they may have contradictory views on the effect of the switching of envelopes (both thinking that the other envelope is “better”).
To understand this paradox better, imagine the reverse
game: Suppose that the players see the contents of the
other player’s envelope (and not of their own). Then the
same reasoning (based on the information received) would
support keeping the envelopes (and no switching). This
shows that it is the different incomplete information which
supports their paradoxical behavior. (The role of
incomplete information and other arrangements of the
experiment are discussed in [
This situation is not so counterintuitive. Imagine for instance a poker game where two players hold a poker in their hands. They both evaluate their chances of winning as very high, although it is clear that only one is in the winning position. In the reverse game, where they see the cards of the opponent (and not their own), each player would estimate the chances of his opponent as very high, and he would surrender.
The two envelope problem in this setting possesses the same feature: the partial information given to players is overly optimistic. Thus looking inside the envelopes is not so helpful as it seems. Therefore, if rational players do not look inside any of the envelopes, the latter argument makes their choice ambivalent, and they would accept Argument 1. Even if they look in the envelopes, they will not accept Argument 2, knowing (from the above analysis of the model) that its prediction is biased and too optimistic. probability of gain gives P(U = 0|A = 2s) = 1/3 if s 6= 0. The loss is twice more probable but twice smaller. Thus the conditional expectation of the gain simplifies to
It seems that player A (as well as B) may only gain by switching (if he holds 1), in all other cases the risk of loss is compensated by the same expected gain. In the sum in (7), only the first summand is nonzero, and it is positive. So the sum exists and evaluates to 1
1 2 1 − q
The arguments from Section 4 are applicable. Moreover, switching is supported by a computation which results in a positive unconditional gain (of a player not looking in his envelope). However, this argument is wrong. As in Section 4, the expectation of the gain, EG, does not exist because the sum (2), as a difference of two infinite sums in (3), is not absolutely convergent. Formula (7) uses only one possible arrangement of the summands, leading to an invalid conclusion. If player B applies the same argument, he uses another arrangement of the summands and gets a positive expected gain for himself, loss for A. Thus the sum (2) does not exist and the discussion from Section 4 fully applies, despite the seemingly trivial (wrong) sum (7). 6
Simulations To verify the results, we also used computer simulations. We computed the average gain from 1000 samples and repeated this 5000 times. The results were displayed as histograms of the averages, see Figs. 10, 11, 12 for quotients q = 0.25, 0.5, 0.75, respectively. For q = 0.75, the linear scale could not be used for the horizontal axis. The semilogarithmic scale would not allow negative values. Therefore, we used the 31st root as a compromise which combines non-linearity similar to the logarithm and possibility of displaying negative values.
As expected, the histograms show relatively frequent occurrences of averages with high absolute value in cases of q = 0.5, 0.75, where the expectation does not exist. The values are distributed approximately symmetrical with respect to zero, verifying that no envelope appears “better” and Argument 2 does not apply in practice, as predicted by the theoretical analysis in previous sections. We explained the two envelope paradox in its classical form, as well as in two advanced instances in which the players find rather convincing (and still insufficient) probabilistic arguments for switching the envelopes. The latter is our novel contribution to the discussion of the paradox. We confirmed the following conclusion “a perfectly rational player would simply recognize that his subjective probabilities provide a
misleading account using Bayesian decision
theory and would therefore ignore those results” [
This topic has consequences in psychology, but it is also important in economics because it explains behavior at a market which is seemingly well-motivated, but in fact wrong. Besides, this paradox can be a good example and motivation for the study of statistics and information theory.
Acknowledgments. The first author was supported by the Ministry of Education of the Czech Republic under Project RVO13000.