RoomAssignment145

RoomAssignment145 - Room Assignment Ichiro Obara UCLA...

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Room Assignment Ichiro Obara UCLA January 5, 2010 Obara (UCLA) Room Assignment January 5, 2010 1 / 46
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General Introduction General Introduction Obara (UCLA) Room Assignment January 5, 2010 2 / 46
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General Introduction What will we learn in this course? We study the exchange of indivisible “goods”. We focus on market design . Along the way, we learn some useful math. Obara (UCLA) Room Assignment January 5, 2010 3 / 46
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General Introduction Examples? Assignment of dormitory rooms to students Organ (kidney) transplants Matchmaking School admission Medical internship Auctions etc. ... Obara (UCLA) Room Assignment January 5, 2010 4 / 46
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General Introduction Why market design? Free market is just not an option for some goods (ex. organ exchange, school admission etc). On the other hand, it may be still a good thing for people to “trade” them we need to design “market” explicitly to overcome certain constraints and facilitate exchanges. Suppose that you open an auction site. How much money you can make depends on how you design the auction. you need to design the auction right (and auction is a kind of market). Obara (UCLA) Room Assignment January 5, 2010 5 / 46
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Room Assignment Problem Room Assignment Problem Obara (UCLA) Room Assignment January 5, 2010 6 / 46
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Room Assignment Problem Room Assignment Problem Room Assignment Problem Three students 1 , 2 , 3 Three dormitory rooms h 1 , h 2 , h 3 initially assigned to 1 , 2 , 3 respectively. Each student has a (strict) preference. For example, suppose that 1 prefers h 2 most and h 3 least. We denote this by h 2 ± 1 h 1 ± 1 h 3 . How should we (re)allocate the rooms given any preference? Obara (UCLA) Room Assignment January 5, 2010 7 / 46
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Room Assignment Problem Room Assignment Problem Formal Model More generally, a formal model of room assignment is ( N , H , ± ), where N = { 1 , ..., n } . H = { h 1 , ..., h n } ( h j is initially assigned to j ) ± = ( ± 1 , ..., ± n ). Let X be the set of all possible allocations of rooms (Ex. If n = 3, there are 6 allocations. One example of allocation is ( h 2 , h 3 , h 1 ) X , where h n +1 is assigned to student n (modulo 3)). Obara (UCLA) Room Assignment January 5, 2010 8 / 46
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Room Assignment Problem Formal Model Remark We use h s ± j h t as “ h s is weakly preferred to h t ”. Given the assumption of strict preference, h s ± j h t means either h s ² j h t or h s = h t . We may sometimes use more general weak preferences ± that allow students to be indifferent between two different rooms (Ex. n = 3, suppose that j likes h 1 most, but otherwise don’t care. This preference is represented by ± j , which satisfies h 1 ± j h 2 , h 1 ± j h 3 , and h s ± j h t for s , t = 2 , 3). Obara (UCLA)
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RoomAssignment145 - Room Assignment Ichiro Obara UCLA...

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