Group Preference Discovery: a parable of choosing movies

This is a parable from which we'll see elements of game theory and information content to forward an understanding about communication in a group.

A group of friends is deciding what movie to go watch.  To keep things simple, let's say the group is just of two people, Patrick and Sally.

Well, every time they go see a movie, Patrick asks Sally what movie she'd like to see.  Almost inevitably, Sally always defers back to Patrick to go see whatever he wants instead.  No matter how much Patrick tries to get her to pick or even make some of her own suggestions, she pretty well never does.  Once in a while, she may veto a movie Patrick chose or say that, after seeing the movie, that next time they should see something else for whatever reason (bad actor, terrible director, etc.).

Patrick is mostly fine with choosing, but does Sally realize that her not revealing her preferences for what she wants to see, it makes it next to impossible for Patrick to get to know her via getting to know what she likes, desires, and prefers absent other people's choices.

Patrick doesn't get to know her because by only ever vetoing or not-vetoing choices, she provides just one bit of information regarding any movie he decides on:  i.e. veto vs no-veto, true vs false, 1 vs 0. It's literally one bit of data.  Worse, Patrick can't even tell if a no-veto is due to her really liking a choice he made or if it's just barely passing by her very lowest minimum standards of movie watching.

If when she's asked what she wants to see, she would throw out a few suggestions (e.g. Batman, Man of Steal, and Spiderman), then Patrick could at least start clustering her suggestions and draw a boundary around them to see what other movies she didn't mention might belong to the same cluster. Over time, with more suggestions thrown out freely from her and with more context, Patrick could essentially map and re-map those suggestions on different and more dimensional axis to figure out what her true preferences are like.

Information wise, it's also much more than one bit of information that Patrick would be getting if she would throw out a few suggestions.  If there are M number of movies that she might reasonably consider, by offering 3 suggested movies to see, Patrick actually gets M bits of information: a yay or nay for each of the M restaurants she might reasonably be considering.  That is, out of 2^M (two to the power of M) number of possible combinations to choose from, she offered one.  Whereas in the veto-or-not case, she offers one choice out of two only.  That's a drastically different amount of information she'd be giving up about her personal taste by simply throwing out a few suggestions for movies to watch, which helps others get to know her!

For example, initially Patrick might graph her suggestions as points on a fiction vs non-fiction one-dimensional line and see that Batman, Man of Steal, and Spiderman are all fictional.  Later, with more context and suggestions from her, maybe he can re-map those points onto a 2-D plane by adding a vertical "book-based" axis, then see that Batman, Man of Steal, and Spiderman are all on the higher side of the book-based axis. From that graph, maybe Patrick could infer that Iron Man is likely a good suggestion to counteroffer, and that hypothesis can be tested by seeing if she agrees or veto the counteroffer suggestion.

By hypothesis testing based on those prior observations from those previous iterations of this repeated game (of figure out which movie to watch), over time Patrick could start forming a theory of how she thinks, what she likes, what her preferences are for various things, etc.

Of course, the same can be said in terms of how the repeated game can offer more and more information and clues as to her preferences even in the case of her only giving out one bit of information (i.e. veto, no-veto) in each iteration of the game, but it will take much more iterations to know her.  That's obvious because the amount of information collected in each iteration of the game is, well, one bit.  Plus we can never be quite sure if her no-veto is really just because a choice was barely making it over her lowest minimum standards or if it was at the top of her list of preferred movies to watch.

Now of course, it's also possible that her suggestion of Batman, Man of Steal, and Spiderman are also a (biased) sampling scrapping at the bottom near her minimum standards. But if she goes through the trouble of volunteering so much information, we may also assume she's being charitable by offering suggestions that are in fact reflective of what she wants (maybe slightly biased towards what she thinks the rest of the group, i.e. Patrick, would accept).  Whereas if all Patrick gets from her is a veto-or-not for what is really his decision, he cannot make the same assumption of her being charitable, because one bit of information is literally only one bit more than not communicating with Patrick at all!

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