Week 8: Fair Division Part 2
Week 8: Fair Division Part 2 MATH 1014
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This 2 page Class Notes was uploaded by Amy Brogan on Sunday March 6, 2016. The Class Notes belongs to MATH 1014 at University of Cincinnati taught by Mary Koshar in Spring 2016. Since its upload, it has received 12 views. For similar materials see Mathematics of Social Choice in Mathematics (M) at University of Cincinnati.
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Date Created: 03/06/16
Koshar Amy Brogan February 29, March 2 & 4, 2016 Week 8 Fair Division Part 2 Taking Turns, and Divide and Choose Taking Turns Used in division involving 2 parties only A decision must be made as to which party will go first Each party takes turns selecting the available items until all are distributed Problems How to choose who goes first? o Flip a coin or by bidding? Choosing first can be a huge advantage o Do we compensate the second party? Based on the knowledge of the other’s player’s preference, are there any situations a player could follow that would be in their best interests? Strategy 2 players divide items (goods, issues) Neither knows preferences Example 1: Bob and Carol are splitting their assets and they rank them from most want to least want. Bob Carol Pension House house investments Investments Pensions vehicles vehicles If Bob gets to choose first: Bob Carol Pension House investments vehicles And in Carol chooses first: Bob Carol Pension House vehicles investments Is there a strategy that either could use to profit from knowing the other’s preferences? Bottom-Up Strategy: working backwards from the items of least want to most want to gain an advantage over the other party We use this chart: Player 1: __(6)____ ____(4)____ _____(2)_____ Player 2: ____(5)____ ____(3)____ _____(1)_____ As player 2 is selecting second from the beginning, moving backwards they are the first to choose. In this strategy, they choose the item that is at the bottom of the other person’s list. Then person 1 chooses the item from the bottom of 2’s list. After each selection, the item is marked off the board on both sides since it cannot be picked again. The chart can also alternate so that person 1 goes second and gets the st rd th 1 , 3 , and 5 choices, and person 2 gets the rest. With Bob and Carol, it could work as follows: Bob: pensions vehicles Carol: house investments Bob: house pension Carol: investments vehicles It all depends on who has the knowledge of the other person’s preferences as the choices are made at separate times. A mediator, who would know both, could also use this process to come to the most envy-free outcome that satisfies both parties. Divide and Choose Cake-cutting method: the first person cuts, and the second person chooses o Works best with two people o Three or more creates complexities that can’t be fully generalized Example 2: Pizza cutting Suppose you and a friend were going to cut a pizza, using the cake-cutting method, would you rather be the cuter or the chooser? As the cutter, you could slice down the middle evenly, or into an even number of slices so your friend could choose theirs with their preferable toppings. As the chooser, you would have first pick of whatever your friend cut. Now suppose there are three of you. Who gets to cut, and who gets to choose? Do you each take turns cutting and then choosing, or does one person cut and then the others choose? It gets complicated at this point. Now image having to do this with four or more people. Allocation Schemes: o Schemes that work in context of 2 or 3 often do not generalize easily to context 4 or more o Procedures that yield envy-free allocations are better to obtain than procedures that yield proportional allocations Vickrey Auctions o Bidders independently submit sealed bids for the object being sold o Winner is highest bidder, but they only pay as much as the second-highest bid Strategy/Theorem o A bidder can never do better than that achieved by a bid of exactly what the object is worth to that bidder
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