Prior-free cost sharing design Ruben Juarez Department of Economics, University of Hawaii http://www2.hawaii.edu/~rubenj April, 2009 Ruben Juarez Prior-free cost sharing design
Motivation ◮ Traditional mechanisms require the designer to have a lot of information from the agents. ◮ Alternative approach: implement robust mechanisms that work well under different information contexts. ◮ Two structural problems: Little information about the valuations (utilities) of the agents and little information about whether or not the agents can coordinate misreports. ◮ Two solutions: Worst case measures and Group strategyproofness (GSP). Ruben Juarez Prior-free cost sharing design
Related literature ◮ Aumann[1959], Berheim, Peleg and Whinston[1987] ◮ Maskin[1985], Ehlers[2002], Ehlers and Klaus [2003], Papai[2000, 2001] ◮ Roughgarden et al.[2006a, 2006b, 2007], Pal and Tardos [2003] and Immorlica et al.[2005]. ◮ Moulin[1999, 2007], Moulin and Shenker[2003], Juarez[2006, 2007a] ◮ Goldberg and Hartline [2004, 2006], Baliga and Vohra [2003] ◮ Roughgarden et al.[2007], Juarez[2007b] ◮ Segal[2003], Bergemann et al.[2004, 2007], Morris and Bergemann [2006, 2007] Ruben Juarez Prior-free cost sharing design
Cost sharing problems ◮ Group of agents interested in getting a good or service. ◮ v i the valuation of agent i for getting service. ◮ Cost function to produce the service that depends on the players who are served. ◮ Cost-sharing mechanism: elicits bids from the agents, picks winning set of agents S and determines prices for the winners. Ruben Juarez Prior-free cost sharing design
Cost sharing problems ◮ Group of agents interested in getting a good or service. ◮ v i the valuation of agent i for getting service. ◮ Cost function to produce the service that depends on the players who are served. ◮ Cost-sharing mechanism: elicits bids from the agents, picks winning set of agents S and determines prices for the winners. This general model allows applications to several problems, e.g. auctions, network facility location problems, Queuing problems. Ruben Juarez Prior-free cost sharing design
Cost-sharing mechanism: two natural goals Two natural goals: ◮ Economically efficient (maximizes total surplus) ◮ Immune to coordination of the agents (group strategyproofness) Ruben Juarez Prior-free cost sharing design
Cost-sharing mechanism: two natural goals Two natural goals: ◮ Economically efficient (maximizes total surplus) ◮ Immune to coordination of the agents (group strategyproofness) Fact. Two natural goals mutually incompatible in very general cost-sharing settings: ◮ Shummer[2008] ◮ Juarez[2008] Ruben Juarez Prior-free cost sharing design
Trade offs 1. Robust mechanism are very inefficient. ◮ Question: quantifying efficiency loss of robust mechanism Juarez[2008, 2009] Ruben Juarez Prior-free cost sharing design
Trade offs 1. Robust mechanism are very inefficient. ◮ Question: quantifying efficiency loss of robust mechanism Juarez[2008, 2009] 2. Simple mechanisms are efficient but are not robust ◮ Question: quantifying efficiency loss due to lack of robustness: Juarez[2008c] Ruben Juarez Prior-free cost sharing design
Trade offs 1. Robust mechanism are very inefficient. ◮ Question: quantifying efficiency loss of robust mechanism Juarez[2008, 2009] 2. Simple mechanisms are efficient but are not robust ◮ Question: quantifying efficiency loss due to lack of robustness: Juarez[2008c] 3. Mechanism in between: neither fully efficient nor fully robust ◮ Question: Finding simple mechanisms that are ’almost’ robust and ’very’ efficient (Roughgaden et al. 2009, own research in progress) This paper is about point 1. Ruben Juarez Prior-free cost sharing design
Auctioning a single private service/good ◮ N = { 1 , . . . , n } interested to get a unit of good. ◮ Agent i has private monetary valuation u i for getting it. ◮ Seller has reserve price of 1. Ruben Juarez Prior-free cost sharing design
Auctioning a single private service/good ◮ N = { 1 , . . . , n } interested to get a unit of good. ◮ Agent i has private monetary valuation u i for getting it. ◮ Seller has reserve price of 1. Second-price auction: If bid vector is ( b 1 , . . . , b n ) , b 1 ≥ b 2 ≥ · · · ≥ b n , and b 1 > 1 , then agent 1 gets unit at a price max( b 2 , 1) . Ruben Juarez Prior-free cost sharing design
Auctioning a single private service/good ◮ N = { 1 , . . . , n } interested to get a unit of good. ◮ Agent i has private monetary valuation u i for getting it. ◮ Seller has reserve price of 1. Second-price auction: If bid vector is ( b 1 , . . . , b n ) , b 1 ≥ b 2 ≥ · · · ≥ b n , and b 1 > 1 , then agent 1 gets unit at a price max( b 2 , 1) . Second-price auction is not Group Strategyproof! Ruben Juarez Prior-free cost sharing design
Auctioning a single private service/good ◮ N = { 1 , . . . , n } interested to get a unit of good. ◮ Agent i has private monetary valuation u i for getting it. ◮ Seller has reserve price of 1. Second-price auction: If bid vector is ( b 1 , . . . , b n ) , b 1 ≥ b 2 ≥ · · · ≥ b n , and b 1 > 1 , then agent 1 gets unit at a price max( b 2 , 1) . Second-price auction is not Group Strategyproof! Same problem with other classical auctions (English, Dutch, Sealed-bid). Ruben Juarez Prior-free cost sharing design
Can I prevent coordinated misreports of any coalition? Ruben Juarez Prior-free cost sharing design
Can I prevent coordinated misreports of any coalition? Yes, I can!!! Ruben Juarez Prior-free cost sharing design
Can I prevent coordinated misreports of any coalition? Yes, I can!!! I set a priority on the agents, say 1 , . . . , n . Ruben Juarez Prior-free cost sharing design
Can I prevent coordinated misreports of any coalition? Yes, I can!!! I set a priority on the agents, say 1 , . . . , n . Set arbitrary prices x 1 , x 2 , . . . , x n not less than 1 for each of those agents. Ruben Juarez Prior-free cost sharing design
Can I prevent coordinated misreports of any coalition? Yes, I can!!! I set a priority on the agents, say 1 , . . . , n . Set arbitrary prices x 1 , x 2 , . . . , x n not less than 1 for each of those agents. Finally, from those agents whose offer exceed their price, I choose the one highest ranked. Ruben Juarez Prior-free cost sharing design
Can I prevent coordinated misreports of any coalition? Yes, I can!!! I set a priority on the agents, say 1 , . . . , n . Set arbitrary prices x 1 , x 2 , . . . , x n not less than 1 for each of those agents. Finally, from those agents whose offer exceed their price, I choose the one highest ranked. Proposition. Only the priority mechanisms are immune to coordinated misreports of any group of agents (GSP). Ruben Juarez Prior-free cost sharing design
Can I prevent coordinated misreports of any coalition? Yes, I can!!! I set a priority on the agents, say 1 , . . . , n . Set arbitrary prices x 1 , x 2 , . . . , x n not less than 1 for each of those agents. Finally, from those agents whose offer exceed their price, I choose the one highest ranked. Proposition. Only the priority mechanisms are immune to coordinated misreports of any group of agents (GSP).Unfortunately, they recover only a tiny fraction of the efficient surplus ( 1 n ), where n is the number of agents. Ruben Juarez Prior-free cost sharing design
Single facility location problem ◮ There is a single facility with fixed cost F if opened, and ◮ Agent i has a personalized connection cost c i if connected. Ruben Juarez Prior-free cost sharing design
Efficient average cost mechanism (EAC) ◮ Elicits bids from the agents. ◮ Serve the efficient allocation. Everyone pays the same price. Ruben Juarez Prior-free cost sharing design
Efficient average cost mechanism (EAC) ◮ Elicits bids from the agents. ◮ Serve the efficient allocation. Everyone pays the same price. Fact: EAC is efficient but not immune to coordination. Ruben Juarez Prior-free cost sharing design
Efficient average cost mechanism (EAC) ◮ Elicits bids from the agents. ◮ Serve the efficient allocation. Everyone pays the same price. Fact: EAC is efficient but not immune to coordination. e.g. n = 3 , c 1 = c 2 = c 3 = 0 Ruben Juarez Prior-free cost sharing design
Efficient average cost mechanism (EAC) ◮ Elicits bids from the agents. ◮ Serve the efficient allocation. Everyone pays the same price. Fact: EAC is efficient but not immune to coordination. e.g. n = 3 , c 1 = c 2 = c 3 = 0 u 1 = 2 F 3 + 3 ǫ u 2 = u 3 = F 3 − ǫ. Ruben Juarez Prior-free cost sharing design
Efficient average cost mechanism (EAC) ◮ Elicits bids from the agents. ◮ Serve the efficient allocation. Everyone pays the same price. Fact: EAC is efficient but not immune to coordination. e.g. n = 3 , c 1 = c 2 = c 3 = 0 u 1 = 2 F 3 + 3 ǫ u 2 = u 3 = F 3 − ǫ. k u k = 4 F Since � 3 + ǫ > F , so everyone should get service at price F 3 . Ruben Juarez Prior-free cost sharing design
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