# How to find a flat using game theory?

I had the idea that maybe probability/game theory knowledge helps finding a flat more systematically. I assume that I have some online offers with number parameters:

• prize
• size (square meters)
• distance (to work in minutes)

I have some limits for all of them, but there are still many offers remaining. Unfortunately, I have to decide immediately if I take a flat or not since otherwise it will be gone.

My main idea is to estimate how much more I can contraint these three parameters numerically and still find a flat within some reasonable time. Basically I define a stronger "soft limit" rather than the neccessary "hard limit". If I filter out too much, there will be too few offers and the search would take too long. From watching the offers I can estimate the distribution of these three parameters.

Of course there are more non-numerical parameters so just being within range doesn't qualify the flat completely. It still might be dismissed, due to other factors.

Any suggestions? (I know about the related "Secretary problem", but it wouldn't give me new limit numbers. And here, I wish a cardinal approach, rather than an ordinal approach)

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I don't understand why the game-theory tag is present. what's the game? Who are the agents? – blubb Mar 13 '12 at 19:58
It's me against nature. I heard that's a common saying in min-max problems similar to game theory. But admins should feel free to relabel it. – Gerenuk Mar 13 '12 at 22:54
game theory fits very well. The game is to find a flat. – Ronald Apr 8 '12 at 1:32
I was confused for the longest time when I read the word "prize", "cost" makes it much more clear. – picakhu Aug 16 '12 at 13:43
Well, "price" also would make it more clear. – celtschk Aug 16 '12 at 13:50

This question made me immediately think of the Secretary Problem!

But then, you mentioned it :) Still, I think your problem is really the same as the Secretary Problem.

Perhaps having three variables makes things too complicated. Rather than thinking about the three variables individually, it would be better to think about your 'utility' as a single variable. Consider how much you value each variable in comparison to the others - and then set a single utility value for each flat (higher utility value = a better flat for your needs). Note that your utility calculation need not be linear/additive - if a flat becomes too expensive for you, its overall utility likely drops down to 0 quickly.

When looking for a flat, I would consider other variables as part of my personal utility: noise levels, energy efficiency, damp, bathroom state, neighbours, etc.

As long as you have a single variable 'utility', the problem simplifies direcly to the secretary problem.

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This is not really the secretary problem. In the secretary problem you would try to maximize the probability of getting the best flat, but in this problem you have to weight the utility of a flat against the disutility of further search and the expected utility of obtaining something better when searching. These problems are in general quite different. – Michael Greinecker May 9 '12 at 16:05
Hum, the secretary problem is about finding the optimal stopping point. It's my understanding that this is exactly what you described. I'm not clear why these are generally different? – Ronald May 9 '12 at 20:55
If the cost of search is high enough, you might settle for something you are sure is very far from optimal. – Michael Greinecker May 9 '12 at 21:07
In economics, there is a rather huge literature on consumer search and on job search, in which agents maximize expected utility. There are some pointers on the Wikipedia entry on Search theory. – Michael Greinecker May 9 '12 at 22:19
@Ronald: Scratch that: the secretary problem is about finding the best and only the best candidate, and considers selecting the second best and the worst candidate as equal outcomes. In the real world, second best is much better than worst. – Hurkyl Jun 10 '12 at 4:23