Meiburg's paradox

In mathematical game theory and optimal stopping theory, Meiburg's paradox is a variant on the secretary problem for highly risk-averse individuals, which appears to lead to them acting reliably asymmetrically in a symmetric situation; a form of irrationality.

Formulation

The treatment most analogous to the secretary problem follows an employer seeking to fill a position with one of two candidates. Based on prior information (e.g. resumes), the employer's expected utility from either single candidate follows a PDF of

$P(x)={\begin{cases}0,&{\text{if }}x>0,\\{\frac {1}{(x-1)^{2}}},&{\text{if }}x\leq 0.\end{cases}}$

Note that this distribution has an expectation value of $-\infty$, or rather not a defined expectation value at all. The employer will interview the first candidate, determining the exact quality of that candidate, so that the utility is known. The employer must then either accept or reject the candidate. Upon acceptance, the second candidate will not be interviewed. Upon rejection, the second, unknown candidate must be hired.

Each candidate has, without any further information, an infinitely negative expected utility. After the employee has completed the first interview, however, the first candidate will merely have a finite negative utility. Therefore, if the employer is a rational agent, they will choose at this point to hire the first candidate. In turn, since the employer knows ahead of time that they will always hire the first candidate over the second, they can make this decision before the interview. That is, before the employer knows about any differences between the employees, they can confidently hire the one they were planning to interview first. This violates the apparent principle of symmetry that the two candidates should be treated on an equal footing.

Variations

The problem is sometimes discussed with other similar distributions, such as a one-sided Cauchy distribution, so that the values are only negative. Often there is a reversal of signs: two options both have infinitely positive expected value. Now the first candidate has merely positive finite expected value compared with the second candidate, so the employer knows ahead of time that they will always accept the second candidate.

To emphasize how dramatic uninformed decision is, sometimes the paradox is presented with a different distribution for each candidate. For instance, the candidates' qualities $q_{1}$ and $q_{2}$ can have two distinct PDFs $f_{1}$ and $f_{2}$:

$f_{1}(x)={\begin{cases}0,&{\text{if }}x>0\\{\frac {1}{(x-1)^{2}}},&{\text{if }}x\leq 0\end{cases}}$

$f_{2}(x)={\begin{cases}0,&{\text{if }}x>0\\{\frac {0.1}{(x-0.1)^{2}}},&{\text{if }}x\leq 0\end{cases}}$

Now candidate 2 dominates candidate 1 in sense of the CDF: For any given cutoff quality $\alpha <0$, we have that $P(q_{1}>\alpha )<P(q_{2}>\alpha )$. In this sense, the second candidate should be the "better" choice by many reasonable descriptions, but following the argument above the employer would still choose to employ the first candidate – since the expectation of the second candidate is still infinitely negative.

Interpretations and resolutions

Although it is tempting to claim the paradox only occurs because the expectation value is not well defined, the analysis can be similarly done without divergent quantities. This places it on similar grounding as the St. Petersburg paradox. If the first candidate is observed after interviewing as having finite quality $\alpha$, then there is a cutoff $\beta <\alpha$ such that the expectation of the second candidate conditioned on being at least $\beta$ is less than $\alpha$; the remainder of the possibilities are all worse than $\beta$, and hence $\alpha$ as well. This partitions the space of possibilities into two parts, each of which is concretely shown to be worse than the observed first candidate $\alpha$. The original treatment, completed in the context of stochastic extended real numbers, can also be made rigorous.

There is the objection that situations with undefined expected utility are unphysical, and thus the entire premise of the problem is not well-founded. The problem can be in a weak sense "approximated" by bounded-expectation distributions, by truncating each distribution at a finite negative value. As the point of truncation is taken to more and more negative values, the likelihood that the employer selects the first candidate goes to 1. It can then be stated that this is reasonable behavior, and the truly asymmetric aspect of the problem – the employer making a decision before receiving information – is banished.

Applications

The employer's prior distributions represent an extremely risk-averse agent. Risk aversion is not generally interpreted as a form of irrational behavior, but rather an aspect of a rational agent's utility function. In this case, the risk aversion leads (in a limit) to behavior that would otherwise be considered irrational. In reality, many humans demonstrate behavior that is even more risk-averse that the employer in the problem, with certain outcomes that they will mentally completely rule out. This leads to the subject of deontological behavior, as opposed to utilitarian – in the sense that the agent follows a set of prioritizes rules, rather than weighting each with real numbers. In modeling applications where the agents are likely prone to such rulesets – but the model is formulated in terms of utility functions – such divergent utility functions as illustrated in the paradox can help describe the behavior of the system.

References

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