Prior-free cost sharing design Ruben Juarez Department of - - PowerPoint PPT Presentation

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. ◮ vi 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. ◮ vi 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 ui 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 ui for getting it. ◮ Seller has reserve price of 1.

Second-price auction: If bid vector is (b1, . . . , bn), b1 ≥ b2 ≥ · · · ≥ bn, and b1 > 1, then agent 1 gets unit at a price max(b2, 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 ui for getting it. ◮ Seller has reserve price of 1.

Second-price auction: If bid vector is (b1, . . . , bn), b1 ≥ b2 ≥ · · · ≥ bn, and b1 > 1, then agent 1 gets unit at a price max(b2, 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 ui for getting it. ◮ Seller has reserve price of 1.

Second-price auction: If bid vector is (b1, . . . , bn), b1 ≥ b2 ≥ · · · ≥ bn, and b1 > 1, then agent 1 gets unit at a price max(b2, 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 x1, x2, . . . , xn 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 x1, x2, . . . , xn 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 x1, x2, . . . , xn 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 x1, x2, . . . , xn 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 ci 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, c1 = c2 = c3 = 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, c1 = c2 = c3 = 0 u1 = 2F

3 + 3ǫ u2 = u3 = 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, c1 = c2 = c3 = 0 u1 = 2F

3 + 3ǫ u2 = u3 = F 3 − ǫ.

Since

k uk = 4F 3 + ǫ > F, so everyone should get service at price F 3 .

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, c1 = c2 = c3 = 0 u1 = 2F

3 + 3ǫ u2 = u3 = F 3 − ǫ.

Since

k uk = 4F 3 + ǫ > F, so everyone should get service at price F 3 .

However, agents 2 and 3 are not happy to pay those prices! If they coordinate bids, say b2 = b3 = 0, then the facility is not

• pened and they are better off.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)

◮ Elicits bids from the agents. ◮ All agents are offered to get connected at prices given by

pN = (F

n + c1, . . . , F n + cn).

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)

◮ Elicits bids from the agents. ◮ All agents are offered to get connected at prices given by

pN = (F

n + c1, . . . , F n + cn). ◮ If only agents in S accepted prices in previous step, then only

them are re-offered at prices given by pS

i = F |S| + ci for i ∈ S.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)

◮ Elicits bids from the agents. ◮ All agents are offered to get connected at prices given by

pN = (F

n + c1, . . . , F n + cn). ◮ If only agents in S accepted prices in previous step, then only

them are re-offered at prices given by pS

i = F |S| + ci for i ∈ S. ◮ We iterate this until all agents who are getting the offer

accepts it, or all agents refused an offer.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)

◮ Elicits bids from the agents. ◮ All agents are offered to get connected at prices given by

pN = (F

n + c1, . . . , F n + cn). ◮ If only agents in S accepted prices in previous step, then only

them are re-offered at prices given by pS

i = F |S| + ci for i ∈ S. ◮ We iterate this until all agents who are getting the offer

accepts it, or all agents refused an offer. Fact: EFC is immune to coordination of the agents.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.Indeed, consider the utility profile: ˜ u = (c1 + F n − ǫ, c2 + F n − 1 − ǫ, c3 + F n − 2 − ǫ, . . . , cn + F − ǫ) for ǫ > 0 small.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.Indeed, consider the utility profile: ˜ u = (c1 + F n − ǫ, c2 + F n − 1 − ǫ, c3 + F n − 2 − ǫ, . . . , cn + F − ǫ) for ǫ > 0 small. At ˜ u the mechanism does not serve any of the agents.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.Indeed, consider the utility profile: ˜ u = (c1 + F n − ǫ, c2 + F n − 1 − ǫ, c3 + F n − 2 − ǫ, . . . , cn + F − ǫ) for ǫ > 0 small. At ˜ u the mechanism does not serve any of the agents. Indeed:

◮ Since ˜

u1 < c1 + F

n , then agent 1 declines the offer in the first

iteration.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.Indeed, consider the utility profile: ˜ u = (c1 + F n − ǫ, c2 + F n − 1 − ǫ, c3 + F n − 2 − ǫ, . . . , cn + F − ǫ) for ǫ > 0 small. At ˜ u the mechanism does not serve any of the agents. Indeed:

◮ Since ˜

u1 < c1 + F

n , then agent 1 declines the offer in the first

iteration.

◮ Since ˜

u2 < c2 +

F n−1, then agent 2 declines it in the second

iteration,

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.Indeed, consider the utility profile: ˜ u = (c1 + F n − ǫ, c2 + F n − 1 − ǫ, c3 + F n − 2 − ǫ, . . . , cn + F − ǫ) for ǫ > 0 small. At ˜ u the mechanism does not serve any of the agents. Indeed:

◮ Since ˜

u1 < c1 + F

n , then agent 1 declines the offer in the first

iteration.

◮ Since ˜

u2 < c2 +

F n−1, then agent 2 declines it in the second

iteration,

◮ Similarly, agent i declines the offer in the i-th iteration.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.Indeed, consider the utility profile: ˜ u = (c1 + F n − ǫ, c2 + F n − 1 − ǫ, c3 + F n − 2 − ǫ, . . . , cn + F − ǫ) for ǫ > 0 small. At ˜ u the mechanism does not serve any of the agents. Indeed:

◮ Since ˜

u1 < c1 + F

n , then agent 1 declines the offer in the first

iteration.

◮ Since ˜

u2 < c2 +

F n−1, then agent 2 declines it in the second

iteration,

◮ Similarly, agent i declines the offer in the i-th iteration.

That is clearly inefficient.

Ruben Juarez Prior-free cost sharing design

Equal fixed cost mechanism (EFC)...Continued

However, EFC is inefficient.Indeed, consider the utility profile: ˜ u = (c1 + F n − ǫ, c2 + F n − 1 − ǫ, c3 + F n − 2 − ǫ, . . . , cn + F − ǫ) for ǫ > 0 small. At ˜ u the mechanism does not serve any of the agents. Indeed:

◮ Since ˜

u1 < c1 + F

n , then agent 1 declines the offer in the first

iteration.

◮ Since ˜

u2 < c2 +

F n−1, then agent 2 declines it in the second

iteration,

◮ Similarly, agent i declines the offer in the i-th iteration.

That is clearly inefficient. The efficient allocations serve all the agents and give a surplus: n

i=1 F i − F − nǫ ≈ (ln(n) − 1)F > 0.

Ruben Juarez Prior-free cost sharing design

Main result

• Proposition. [Moulin and Shenker(2001), Juarez(2008b)] The

EFC mechanism is GSP and has a worst absolute surplus loss equal to:

n

• i=1

F i − F ≈ (ln(n) − 1)F. Moreover, it has the smallest worst absolute surplus loss among all feasible GSP mechanisms for the single facility location problem.

Ruben Juarez Prior-free cost sharing design

More general results

◮ We characterize GSP wal-optimal mechanism for two

• rthogonal classes of cost functions.

◮ EFC type mechanisms optimal when the cost function has

decreasing average cost.

◮ Priority type mechanisms is optimal when the cost function

has increasing average cost.

Ruben Juarez Prior-free cost sharing design

What’s Next?

◮ Optimal mechanisms for alternative shapes of cost functions

(Roughgarden, Tardos[2008]).

Ruben Juarez Prior-free cost sharing design

What’s Next?

◮ Optimal mechanisms for alternative shapes of cost functions

(Roughgarden, Tardos[2008]).

◮ What is the optimal GSP mechanism when we know the

network:

◮ If all agents are connected, then only the very inefficient GSP

mechanisms presented above are GSP.

◮ If no agents are connected, then well known class of VCG

mechanisms (Green/Laffont, Roberts 80’s).

◮ How about mechanism for different networks. Ruben Juarez Prior-free cost sharing design

What’s Next?

◮ Optimal mechanisms for alternative shapes of cost functions

(Roughgarden, Tardos[2008]).

◮ What is the optimal GSP mechanism when we know the

network:

◮ If all agents are connected, then only the very inefficient GSP

mechanisms presented above are GSP.

◮ If no agents are connected, then well known class of VCG

mechanisms (Green/Laffont, Roberts 80’s).

◮ How about mechanism for different networks.

◮ When in a repeated network, can we learn the network of

agents?

Ruben Juarez Prior-free cost sharing design

What’s Next?

◮ Optimal mechanisms for alternative shapes of cost functions

(Roughgarden, Tardos[2008]).

◮ What is the optimal GSP mechanism when we know the

network:

◮ If all agents are connected, then only the very inefficient GSP

mechanisms presented above are GSP.

◮ If no agents are connected, then well known class of VCG

mechanisms (Green/Laffont, Roberts 80’s).

◮ How about mechanism for different networks.

◮ When in a repeated network, can we learn the network of

agents?

◮ Can we relax GSP and improve the efficiency?

◮ Roughgarden et al.[2007]: A very clever construction of GSP

mechanisms when indifferences are not important. Efficiency can exponentially increase.

◮ Juarez[2007]: The full characterization of the GSP

mechanisms when indifferences are not important.

Ruben Juarez Prior-free cost sharing design

What’s next? (cont’d)

Seller owns the cost function (i.e. maximizes profit):

◮ Given a cost function, what are the optimal GSP mechanisms

for the seller? (Worst-case measures don’t work well, have to assume stochastic distribution of types)

◮ Auction of a single good/service: priority auctions are optimal

(Work in progress).

◮ Profit-loss comparison between GSP, SP and mechanisms in

between.

◮ GSP when there are multiple sellers of the same good: Cannot

do much, only fixed cost mechanisms (Work in progress).

Ruben Juarez Prior-free cost sharing design