Exchange Paradox Research Articles

Gain from the two-envelope problem via information asymmetry: on the suboptimality of randomized switching

The two-envelope problem (or exchange problem) is one of maximizing the payoff in choosing between two values, given an observation of only one. This paradigm is of interest in a range of fields from engineering to mathematical finance, as it is now known that the payoff can be increased by exploiting a form of information asymmetry. Here, we consider a version of the ‘two-envelope game’ where the envelopes’ contents are governed by a continuous positive random variable. While the optimal switching strategy is known and deterministic once an envelope has been opened, it is not necessarily optimal when the content's distribution is unknown. A useful alternative in this case may be to use a switching strategy that depends randomly on the observed value in the opened envelope. This approach can lead to a gain when compared with never switching. Here, we quantify the gain owing to such conditional randomized switching when the random variable has a generalized negative exponential distribution, and compare this to the optimal switching strategy. We also show that a randomized strategy may be advantageous when the distribution of the envelope's contents is unknown, since it can always lead to a gain.

Corporate Venturing Deal Syndication and Innovation: the Information Exchange Paradox

Many incumbent corporations make equity investments in young technological start-ups to enhance their innovation effectiveness, and the great majority syndicate at least some of their investments with other incumbents. While syndication is generally held to benefit incumbent corporations, this study demonstrates that it may also be detrimental to corporate innovation, by elaborating the notion of an information exchange paradox – essentially, that information exchanges within CVC networks must, somehow, be both open and closed at the same time. Corporations must try to appropriate the knowledge championed by their investees and fellow-investors, but also protect their own know-how from leaking to competitors. Unlike prior CVC research, we demonstrate that knowledge sharing in open innovation forums may be counterproductive. Using a unique data set of the investment decisions made by 163 corporations over four years we show that, for some, participating in syndicate networks may involve losses that outweigh their gains. Our analysis establishes two key findings. First, corporations need to consider the trade-off between the number of ventures they support and the position they take in their syndication networks. The best strategies appear to be maximizing isolationist (supporting many ventures but staying away from the network centre) or minimizing centralist (supporting few ventures, but occupying a central network position) – the other two options (maximizing centralist and minimizing isolationist strategies) are far less effective in converting CVC investments into corporate innovation. Second, this picture is particularly applicable to highly concentrated industries dominated by several powerful incumbents: in fragmented industries these strategy differences are far less pronounced, so the choice of CVC syndication strategy will depend on other considerations. This supports a contingency view of syndication, implying that ensuring incumbent corporations really benefit from equity investments in start-ups is a not a trivial task for their managers.

Opening Two Envelopes

In the two-envelope problem, one is offered a choice between two envelopes, one containing twice as much money as the other. After seeing the contents of the chosen envelope, the chooser is offered the opportunity to make an exchange for the other envelope. However, it appears to be advantageous to switch, regardless of what is observed in the chosen envelope. This problem has an extensive literature with connections to probability and decision theory. The literature is roughly divided between those that attempt to explain what is flawed in arguments for the advantage of switching and those that attempt to explain when such arguments can be correct if counterintuitive. We observe that arguments in the literature of the two-envelope problem that the problem is paradoxical are not supported by the probability distributions meant to illustrate the paradoxical nature. To correct this, we present a distribution that does support the usual arguments. Aside from questions about the interpretation of variables, algebraic ambiguity, modal confusions and the like, most of the interesting aspects of the two-envelope problem are assumed to require probability distributions on an infinite space. Our next main contribution is to show that the same counterintuitive arguments can be reflected in finite versions of the problem; thus they do not inherently require reasoning about infinite values. A topological representation of the problem is presented that captures both finite and infinite cases, explicating intuitions underlying the arguments both that there is an advantage to switching and that there is not.

THE TWO-ENVELOPE PROBLEM REVISITED

The two-envelope problem has intrigued mathematicians for decades, and is a question of choice between two states in the presence of uncertainty. The problem so far, is considered open and there has been no agreed approach or framework for its analysis. In this paper we outline an elementary approach based on Cover's switching function that, in essence, makes a biased random choice where the bias is conditioned on the observed value of one of the states. We argue that the resulting symmetry breaking introduced by this process results in a gain counter to naive expectation. Finally, we discuss a number of open questions and new lines of enquiry that this discovery opens up.

Conditionals, Probabilities, and Utilities: More on Two Envelopes

Sutton (2010) claims that on our analysis (2007), the problem in the two-envelope paradox is an error in counterfactual reasoning. In fact, we distinguish two formulations of the paradox, only one of which, on our account, involves an error in conditional reasoning. According to Sutton, it is conditional probabilities rather than subjunctive conditionals that are essential to the problem. We argue, however, that his strategy for assigning utilities on the basis of conditional probabilities leads to absurdity. In addition, we show that a crucial presupposition of Sutton’s argument — namely, that one can know that envelope A contains n simply on the basis of a stipulation — is mistaken.

Randomized switching in the two-envelope problem

The two-envelope problem is a conundrum in decision theory that is subject to longstanding debate. It is a counterintuitive problem of decidability between two different states, in the presence of uncertainty, where a player’s payoff must be maximized in some fashion. The problem is a significant one as it impacts on our understanding of probability theory, decision theory and optimization. It is timely to revisit this problem, as a number of related two-state switching phenomena are emerging in physics, engineering and economics literature. In this paper, we discuss this wider significance, and offer a new approach to the problem. For the first time, we analyse the problem by adopting Cover’s switching strategy—this is where we randomly switch states with a probability that is a smoothly decreasing function of the observed value of one state. Surprisingly, we show that the player’s payoff can be increased by this strategy. We also extend the problem to show that a deterministic switching strategy, based on a thresholded decision once the amount in an envelope is observed, is also workable.

Taking the Two Envelope Paradox to the Limit

The original version of the two envelope paradox is not all that paradoxical. The fact that (a) one of two sealed envelopes contains twice as much money as the other does not imply that (b) the other envelope is equally likely to contain twice or half as much money as your envelope. And (b) is what is behind the familiar reasoning that the other envelope has greater expected utility than your envelope. However, there are strengthened versions of the two envelope paradox where the familiar reasoning seems to hold, but it still does not make any sense to prefer the other envelope. David Chalmers (1994) and John Norton (1998) have attempted to resolve the paradox by pointing out that the familiar reasoning is flawed in such cases because the expected utility of each envelope is infinite. However, simply noting this fact provides only a partial analysis of the paradox. We should not simply throw up our hands when we are confronted with infinite expectations. In particular, there are two envelope scenarios where the expected utility of each envelope is infinite, but where it does seem rational to prefer one envelope over the other. Thus, we need a way to reason about such scenarios that does not lead us back into a paradox. Peter Vallentyne (2000) has proposed that we reason about infinite lotteries by looking at the behavior of arbitrarily large finite lotteries in the limit. In this paper, I show how the same sort of technique can be used to complete the analysis of the two envelope paradox.

Keep or trade? An experimental study of the exchange paradox

The “exchange paradox”—also referred to in the literature by a variety of other names, notably the “two-envelopes problem”—is notoriously difficult, and experts are not all agreed as to its resolution. Some of the various expressions of the problem are open to more than one interpretation; some are stated in such a way that assumptions are required in order to fill in missing information that is essential to any resolution. In three experiments several versions of the problem were used, in each of which the information given was sufficient to determine an optimal choice strategy when it exists or to justify indifference regarding keeping or trading when such a strategy does not exist. College students who were presented with the various versions of the problem tended to base their choices on simple heuristics and to give little evidence of understanding the probabilistic implications of the differences in the problem statements.

The Puzzle of the Two-Envelope Puzzle

All previous research on the well known two-envelope paradox has focused on how the comparison ought to be made between the possible gain and loss. This paper shows that no sensible comparison can be made at all given all relevant information. Thus the puzzle of the two-envelope puzzle is the ironical fact that making use of more information sometimes reduces the solvability of a problem.

A Tale of Two Envelopes

This paper deals with the two-envelope paradox. Two main formulations of the paradoxical reasoning are distinguished, which differ according to the partition of possibilities employed. We argue that in the first formulation the conditionals required for the utility assignment are problematic; the error is identified as a fallacy of conditional reasoning. We go on to consider the second formulation, where the epistemic status of certain singular propositions becomes relevant; our diagnosis is that the states considered do not exhaust the possibilities. Thus, on our approach to the paradox, the fallacy, in each formulation, is found in the reasoning underlying the relevant utility matrix; in both cases, the paradoxical argument goes astray before one gets to questions of probability or calculations of expected utility.

The exchange paradox: Probabilistic and cognitive analysis of a psychological conundrum

The term “exchange paradox” refers to a situation in which it appears to be advantageous for each of two holders of an envelope containing some amount of money to always exchange his or her envelope for that of the other individual, which they know contains either half or twice their own amount. We review several versions of the problem and show that resolving the paradox depends on the specifics of the situation, which must be disambiguated, and on the player's beliefs. The latter psychological variables are part and parcel of the resolution. Assuming reasonable subjective distributions, exchanging cannot always be advantageous for both players. We suggest several deep-rooted psychological reasons for the considerable difficulty people demonstrably have in dealing with this problem. Implicit widespread and compelling assumptions—that affect judgement in diverse contexts—obstruct the solution. Analysing this paradox underscores the close connection between psychology and probability theory.

Trying to Resolve the Two-Envelope Problem

After explaining the well-known two-envelope ‘paradox’ by indicating the fallacy involved, we consider the two-envelope ‘problem’ of evaluating the ‘factual’ information provided to us in the form of the value contained by the envelope chosen first. We try to provide a synthesis of contributions from economy, psychology, logic, probability theory (in the form of Bayesian statistics), mathematical statistics (in the form of a decision-theoretic approach) and game theory. We conclude that the two-envelope problem does not allow a satisfactory solution. An interpretation is made for statistical science at large.

The Two-Envelope Paradox: An Axiomatic Approach

In this paper, we present a simple axiomatic justification for indifference before opening, avoiding any expectation reasoning, which is often considered problematic in infinite cases. Although the two-envelope paradox assumes an expectation-maximizing agent, we show that analogous paradoxes arise for agents using difierent decision principles such as maximin and maximax, and that our justification for indifierence before opening applies here too.

The Classical and Maximin Versions of the Two-Envelope Paradox

The Two-Envelope Paradox is classically presented as a problem in decision theory that turns on the use of probabilities in calculating expected utilities. I formulate a Maximin Version of the paradox, one that is decision-theoretic but omits considerations of probability. I investigate the source of the error in this new argument, and apply the insights thereby gained to the analysis of the classical version.

Two envelope problems and the roles of ignorance

Four variations on Two Envelope Paradox are stated and compared. The variations are employed to provide a diagnosis and an explanation of what has gone awry in the paradoxical modeling of the decision problem that the paradox poses. The canonical formulation of the paradox underdescribes the ways in which one envelope can have twice the amount that is in the other. Some ways one envelope can have twice the amount that is in the other make it rational to prefer the envelope that was originally rejected. Some do not, and it is a mistake to treat them alike. The nature of the mistake is diagnosed by the different roles that rigid designators and definite descriptions play in unproblematic and in untoward formulations of decision tables that are employed in setting out the decision problem that gives rise to the paradox. The decision maker’s knowledge or ignorance of how one envelope came to have twice the amount that is in the other determines which of the different ways of modeling his decision problem is correct. Under this diagnosis, the paradoxical modeling of the Two Envelope problem is incoherent.

Clark and Shackel on the Two-Envelope Paradox

Clark and Shackel (2000) have recently argued that previous attempts to resolve the two-envelope paradox fail, and that we must look to symmetries of the relevant expected-value calculations for a solution. Clark and Shackel also argue for a novel solution to the peeking case, a variant of the twoenvelope scenario in which you are allowed to look in your envelope before deciding whether or not to swap. Whatever the merits of these solutions, they go beyond accepted decision theory, even contradicting it in the peeking case. Thus if we are to take their solutions seriously, we must understand Clark and Shackel to be proposing a revision of standard decision theory. Understood as such, we will argue, their proposal is both implausible and unnecessary. In the version of the two envelope paradox they give, you pick one of two sealed envelopes — let A be the one you pick, and B be the other — each containing some natural power of 2 in dollars. You know that one of the envelopes has twice as much money in it as the other, and you further know that the probability of each possibility is: p0 = p(A = 2 , B = 2) = p(A = 2, B = 2) = 1/12 and pn+1 = p(A = 2 , B = 2) = p(A = 2, B = 2) = /2(pn + (1/4) ), n ≥ 1. You are then offered the opportunity to swap envelopes. Should you? Standard decision theory tells us to maximize expected utility (EU ), where the expected utility of an act is the sum of the products of the probability and utility for all possible outcomes. If there are an infinite number of possibilities over which to sum, the EU is the value to which the series

Decision Theory, Symmetry and Causal Structure: Reply to Meacham and Weisberg

1Department of Philosophy, University of Nottingham, University Park, Nottingham NG7 2RD, UK. michael.clark@nottingham.ac.uk2Department of Philosophy, University of Nottingham, University Park, Nottingham NG7 2RD, UK. n.shackel@ntlworld.com

The non-probabilistic two envelope paradox

Journal Article The non-probabilistic two envelope paradox Get access James Chase James Chase Victoria University of WellingtonWellington, New Zealandjchase@matai.vuw.ac.nz Search for other works by this author on: Oxford Academic Google Scholar Analysis, Volume 62, Issue 2, April 2002, Pages 157–160, https://doi.org/10.1093/analys/62.2.157 Published: 01 April 2002

The non-probabilistic two envelope paradox

The non-probabilistic two envelope paradox

The St. Petersburg two-envelope paradox

Journal Article The St. Petersburg two-envelope paradox Get access David J. Chalmers David J. Chalmers University of ArizonaTucson, AZ 85721, USAchalmers@arizona.edu Search for other works by this author on: Oxford Academic Google Scholar Analysis, Volume 62, Issue 2, April 2002, Pages 155–157, https://doi.org/10.1093/analys/62.2.155 Published: 01 April 2002