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Imagine a committee of people in charge of hiring a CEO. A committee member can vote "No hire" (+0), "Maybe"(+1), "Hire"(+2) for a potential CEO candidate. Each CEO is scored based on the votes, and the CEO with the highest score is hired.

Now, imagine a situation where two candidates are tied... both of them have a score of 5. However, candidate1 had 2Hires and 1Maybe (ie, 2+2+1), whereas candidate2 had 5Maybes (ie, 1+1+1+1+1).

Here is my question:

  • Intuitively, I think we can say that we are more confident about candidate1 even though the scores are tied. What specific domain of statistics am I dealing with here (so that I can investigate on my own)
  • In statistics, how does one go about resolving such situations? And I am not just merely talking about resolving a tie... but rather, how to resolve this in a generic manner?
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  • $\begingroup$ Why does candidate 1 have only 3 votes? Does that mean he also has 2 "no hire"? In that case one may choose number 2!!! $\endgroup$ – nico Jan 2 '11 at 21:23
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    $\begingroup$ Or were two members of the committee out to lunch while candidate 1 was being interviewed? In that case one might treat their votes as 'missing data'. $\endgroup$ – onestop Jan 2 '11 at 22:40
  • $\begingroup$ These kinds of examples are also better understood if you list all of the ballots cast.In the example you gave, the ballots are based on a point system, so a board member could theoretically give +2 for several candidates, while another board member might only give one candidate a +2. $\endgroup$ – David J. Jan 3 '11 at 2:10
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To give some context, I don't view this as a "statistical" question as much of a "group preference" question. Economists and policy wonks do a lot of thinking about questions of how to convert individual preferences into a "will of the people." You will find lots of interesting reading if you search the web for "political economy" and "voting system" together.

Reasonable people will disagree on which candidate "wins" for the example you gave. There is no objectively correct voting system. Baked into the idea of a voting system are assumptions about fairness and representation. Each voting system has advantageous and disadvantageous properties. (Digging into the pros/cons of various voting systems is super interesting but would take many pages. I recommend by starting with Wikipedia's entry on "Voting Systems". Also be sure to read about Arrow's Impossibility Theorem.)

Don't expect to find a universally accepted "best" voting system. Rather, pick a voting system that seems most reasonable for your domain (i.e. the upsides of the system outweigh the downsides). Make sure that your constituency buys-in to the voting system if you want the results to be taken seriously.

Resolving this described situation cannot be done with statistics alone. You will need to choose a voting system -- that choice will drive the ballot choice and how the ballots are converted into results (e.g. one or more winners, some sort of ranking, or some sort of scoring).

This is a big question. I think you'll enjoy digging into these topics.

Finally, for what it is worth, I have strong reservations about the described ballot (0 for no hire, +1 for maybe, +2 for hire) as a way for a board to pick a CEO.

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  • $\begingroup$ Good info Dave. Why do you have strong reservations about my proposed "preference ranking" based voting system? Note that the CEO bit was just an example... my domain is actually different. $\endgroup$ – varuman Jan 3 '11 at 11:19
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    $\begingroup$ @varuman - I tend to prefer voting systems when you have relatively large constituencies that are unlikely, unwilling, or unable to engage in dialog and deliberation as part of the selection process. On the other hand, small groups that already have open lines of communication would benefit strongly by attempting to select the CEO by a deliberative and consensus-oriented approach. If the group does end up needing to vote numerically, I would be agnostic about which voting system to use until I know more details about the problem space. $\endgroup$ – David J. Jan 3 '11 at 15:16
  • $\begingroup$ @DavidJames, just to nitpick I really think this is still a statistical question because we -humans- are probabilistic models that try to maximize/minimize the expected value of some optimization objective. $\endgroup$ – caveman Feb 27 '16 at 21:17
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You're asking an intriguing question. I agree with the comments that are showing some apprehension at the "one-man-one-vote" system. I also agree that knowing the basic statistics (like standard deviation and mean) will not give you an insight into the will of the voters.

I would like to play off of David James's answer, keying in on stakeholders. Instead of a vote, perhaps you could give stakeholders a virtual account, which they must "spend" on the candidates.

If they had $100 each, then perhaps one stakeholder would show a strong preference for candidate A by spending all $100 on him or her. Another stakeholder might like candidate B slightly more than candidate A and spend $60/$40. A third stakeholder might find all three candidates equally (un)appealing and spend $33/$33/$34.

A variation would be to give different stakeholders accounts of different sizes. For example, perhaps the exiting CEO gets $200 and a worker's representative gets $150.

You could even ask for an open vote, where each stakeholder explains his reasoning.

Highest earner wins the position. Or maybe the top two get the most careful look and a runoff.

This betting technique is an adaptation of what is done in Blind Man's Bluff: The Untold Story of American Submarine Espionage (1998, S. Sontag and C.Drew) A B-52 bomber collided with an air tanker, and they lost an H-bomb.

Craven asked a group of submarine and salvage experts to place Las-Vegas-style bets on the probability of each of the different scenarios that might describe the bomb's loss.... Each scenario left the weapon in a different location.... He was relying on Bayes' theorem of subjective probability. (pp. 58-59)

Whatever you choose, please make sure that the rules are clear before you start voting on candidates. A perception that the rules changed will not help the transition.

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    $\begingroup$ rajah9's suggestion to have the stakeholders divvy up $100 is a variation of a range voting system (with a rating ballot). $\endgroup$ – David J. Jan 3 '11 at 15:11
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A little OT, but one of my favourite nuggets of science is Arrow's theorem, so in case you're not familiar here's the wikipedia page:

http://en.wikipedia.org/wiki/Arrow's_impossibility_theorem

And all from a PhD thesis, too. Quite inspiring really. Mine was rubbish.

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