| Reference: | Lior Davidovitch and Yakov Ben-Haim |
| Abstract | The modern theory of voting usually regards voters as expected utility maximizers. This implies that voters define subjective probabilities and utilities for different outcomes of the elections. In real life, these probabilities and utilities are often highly uncertain, so a robust choice, immune to erroneous assumptions, may be preferred. We show that a voter aiming to satisfice his expected utility, rather than maximize it, may present a bias for sincere voting, as opposed to strategic voting. This may explain previous results which show that strategic voting is not as prevalent as would be expected if all voters were expected utility maximizers. |
| Keywords | Plurality voting, Info-gap decision theory, Sincere voting, Strategic voting |
| Scores | TUIGF:100% SNHNSNDN:993% GIGO:100% |
Overview
This is a typical info-gap article.
It is typical in the sense that, true to form, it takes no cognizance of some of the important work that has been done in the area of expertise which it discusses. Nor does it acknowledge the role and place of the proposed method in the state of the art of the topic under consideration.
So, as the second author is the Father of info-gap decision theory himself, it is particularly important to call the readers' attention to two points that bear directly on the article's main proposition which is to use info-gap decision theory as a way out of the paradox that the authors discuss.
These two points are:
- The Wald's Maximin connection.
- The Radius of Stability connection.
But, first thing first.
As the authors duly acknowledge, there is a rich literature on the use of formal mathematical models for the analysis of voting systems, processes, policies etc. So, having made this point clear, one would have expected them to refer at least obliquely to the fact that Wald's Maximin model is used extensively as a framework for modeling/analysis/solution of various types of generic voting problems (e.g. Minimax Condorcet).Indeed, it is greatly surprising that Wald's Maximin model is not so much as mentioned in an article discussing a robustness model for voting. Not only because Wald's Maximin model and its many spin-offs figure so prominently as frameworks for modeling/analyzing/solving generic voting problems. But because, as the authors are well aware, the model that they propose, namely info-gap's robustness model, is in fact a simple instance of Wald's Maximin model.
In other words, in spite of the fact that the authors are well aware of the formal proof demonstrating that info-gap's robustness model is a Maximin model , they have no qualms to propose a model that is an instance of this model while keeping mum about the Maximin model itself.
So the obvious question is: Why the secrecy about the Maximin connection?
On this, needless to say, one can only speculate.
What I can say, though, is that this stance is basically in line with the approach adopted in info-gap publications regarding the info-gap/Maximin connection. Thus, while some publications vaguely suggest that info-gap's robustness model has some relevance to Maximin model, others ignore the issue altogether, and others deny the true intimate kinship between the two models. Thus, the article's second author -- the Father of info-gap decision theory -- continues to insist in some of his publications that info-gap's robustness model is not a Maximin model. For instance, in his new book, Ben-Haim (2010, p. 9) claims the following (emphasis is mine):
Info-gap theory is related to robust-control and min-max methods, but nonetheless different from them.So, I take this opportunity to call the readers' attention, not only to the gaping hole in this article, created by the inexplicable absence from it of a requisite discussion of the role of the Maximin model in this context. But also to the fact that the model that is proposed here is in fact a simple instance of the Maximin model -- one that is a staple model in ... control theory.
State of the art
First, for the benefit of readers who are not familiar with the topic of "voting rules," here is a quote from a paper published in the journal Public Choice (emphasis is mine):
First, focusing on three-alternative elections, we compute the coalitional manipulability of some voting rules not considered in the previous investigations. On the whole, eight rules are analyzed: Plurality, Borda, antiplurality, Hare (which reduces to two-stage plurality in three-alternative elections), Coombs (or two-stage antiplurality), Kim-Roush (a variant of Coombs), Copeland and the Min-Max (also referred to as Simpson-Kramer).
DOMINIQUE LEPELLEY and FABRICE VALOGNES (2003, p. 166)
Voting rules, manipulability and social homogeneity
Public Choice, 116, 165-184.But, as is customary in info-gap publications, the paper under review shows total disrespect for the state in the art, in this case in the area of modeling and analysis of voting preferences, behavior, control, systems, etc. The most notable example is the lack of all reference to papers dealing with classical decision theory approaches to voting. And this, take note, in spite of the fact that the paper proposes a classical decision model for this purpose.
It is particularly astounding that not a single reference is made to the seminal article:
Ferejohn, John A. and Fiorina, Morris A. (1974)
The Paradox of Not Voting: A Decision Theoretic Analysis
American Political Science Review, Vol. 68 (2), 525-536.and the numerous discussions on it in this literature.
This omission in fact borders on the incredible because in this seminal paper Ferejohn and Fiorina propose Savage's Minimax Regret model (1951) to explain a paradox associated with an expected utility approach to voting!!! In Ferejohn and Fiorina (1974, p. 528-529) we read (emphasis is mine):
Two questions concern us in this section. First, is it again the case that minimax regret and expected utility maximizing rationality criteria imply significantly different predictions about the voting decision? Second, given an enlarged set of available voting strategies, does a citizen still make his decision between only two of them: abstaining, or voting for his most preferred candidate? We answer the first question positively. The answer to the second question provides a clear test for distinguishing empirically between expected utility maximizers and minimax regretters.And on page 535 we find this:
Rational choice theorists are guilty of equating the notion of rational behavior with the rule of maximizing expected utility. Alternatively, we usually define the first as the second. Yet although the rule of maximizing expected utility is the most widely known, widely used and widely accepted rationality criterion, it is not the only one. Our comparison of expected utility maximization and minimax regret decision criteria shows that the behavior of decision makers using the alternative rules differs considerably. Specifically, expected utility maximizers go to the polls only under the most restrictive conditions, whereas minimax regret decision makers need little incentive to participate. And one should remember that most people do vote.So, in this case we have a double whammy. Not only does Ferejohn and Fiorina's paper deal with the type of problem that is investigated in the article under review. It proposes a Minimax model -- a model that (as attested by their publications) the authors are surely familiar with!Indeed, as indicated above, other Minimax/Maximin models are commonly used to analyze various aspects of voting. For example, consider this (emphasis is mine):
Ambiguity is reflected by the fact that the voter's beliefs are given by a set of probabilities, each of which represents in the voter's mind a different possible scenario. The voter's aversion to ambiguity is reflected by the fact that he evaluates each available option by using the worst possible scenario for that option -- that is, he satisfies the "maxmin expected utility with multiple priors" (MEU) model of Gilboa and Schmeidler (1989).
Paolo Ghirardato and Jonathan Katz (2006, p. 381-2)
Indecision theory: weight of evidence and voting behavior
Journal of Public Economic Theory, 8 (3), 379-399.But none of this is so much as mentioned in the article under review. And this to reiterate, when the authors are fully aware of the info-gap/minimax/maximin connection.
Perhaps even more inexcusable is the fact that there is no reference to the classic book
Downs, Anthony (1957)
An Economic Theory of Democracy
New York, Harper and Rowthat popularized the paradox associated with voting models based on expected utility theory.
To appreciate more fully why the author's lack of reference to the Maximin's role in the area that they investigate is so objectionable, it is important that readers be aware of the info-gap/maximin connection.
The Wald's Maximin connection
As I discuss in detail the whole question of the info-gap/Maximin connection on other pages on this site (see for instance FAQs about info-gap decision theory), all I need to do here is remind the readers of the following.
A formal rigorous proof demonstrating that info-gap's robustness model is a simple instance of Maximin model has been available to the public since the end of 2006, and in peer reviewed publications since 2007.
Yet , Ben-Haim continues to obfuscate on this point, vacillating between admission that the proof is mathematically correct to claims in a recent publication maintaining that info-gap's robustness model is not a Maximin model.
I have no explanation for this.
All I can do is again, give the reader easy access to the theorem and its proof. And all you have to do is click on the show/hide for the theorem and its proof.
I also call the readers' attention to the fact that the info-gap robustness model that is presented in this article -- as a framework for handling the voting paradox in question -- is not only a simple instance of the Maximin model, it is in fact a simple instance that is known universally as the Radius of Stability model.
The Radius of Stability connection
The reference to robust-control in Ben-Haim's (2010, p. 9) new book prompted me to remind the Father of info-gap decision theory and his followers that the most popular model of local robustness in control theory is the Radius of Stability model (circa 1960).
Recall that
The Radius of Stability of a system is the radius of the largest ball around a given nominal value of the parameter of interest all of whose elements satisfy pre-determined stability requirements.The info-gap robustness of a system is the size of the largest ball around a given estimate of the parameter of interest all of whose elements satisfy a single pre-determined performance requirement of the "≤" or "≥" type.
Obviously, info-gap's robustness model is a very simple instance of the Radius of stability model, namely the instance where the stability requirement is specified by a single "≤" or "≥" performance constraint. The picture is this:
So clearly:
Find the differences Radius of Stability (circa 1960) Info-gap decision theory (circa 2000) max {α ≥ 0: p∈P(q),∀p∈B(α,p*)} max {α ≥ 0: r* ≤ r(q,p),∀p∈B(α,p*)} The rectangle represents the parameter space, P. The shaded area represents the set P(q) that consists of the values of the parameter p for which the system satisfies given stability requirements. The center of the circles, p*, represents a given nominal value of the parameter p. B(α,p*) denotes a ball of radius α around p* The rectangle represents the parameter space, P. The shaded area represents the values of the parameter p for which the system satisfies given a given performance requirement, namely r* ≤ r(q,p). The center of the circles, p*, represents a give estimate of the parameter p. B(α,p*) denotes a ball of radius α around p*. Theorem
Info-gap's robustness model is a simple instance of the radius of stability model, that is the instance specified by P(q) = {p∈P: r* ≤ r(q,p)}.Proof.
Substituting P(q) = {p∈P: r* ≤ r(q,p)} in the expression defining the radius of stability model, we obtain info-gap's robustness model.
For the record, a trivial formal proof is provided in my recent paper entitled "A bird's view of info-gap decision theory" (Journal of Risk Finance, 11(3), 263-268, 2010).
And to see how elementary the formal proof is, simply click here to hide/show it.
For the benefit of readers who encounter this theorem for the first time I want to reiterate its implications for info-gap decision theory.
The Radius of stability model is a model of local robustness. This means that it functions as a tool for the modeling/analysis/determination of small perturbations in a given nominal value of the parameter of interest.The implication is that using the Radius of stability as a robustness model for the treatment of severe uncertainty of the type considered by info-gap decision theory -- where the estimate is poor, the uncertainty space is vast and the uncertainty model is likelihood free -- amounts to a misapplication of this model.
This misapplication renders info-gap decision theory a voodoo decision theory par excellence.
What Next?
It will be interesting to see how long will the Father of info-gap decision theory and his followers continue to propound the myth that info-gap decision theory is a distinct, novel theory that is radically different from all current theories of decisions under uncertainty.
And it will be even more interesting to see how long will peer reviewed journals continue to accept publications that conceal from the public the Maximin and Radius of stability connections.
