Abstract
We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE). We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players.
Highlights
A cost-sharing game is an abstract setting that describes interactions of selfish players in environments where the cost of the produced solution needs to be shared among the participants
We propose an alternative, relaxed requirement that we call budget balance in the equilibrium (BBiE)
We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner tree problem
Summary
A cost-sharing game is an abstract setting that describes interactions of selfish players in environments where the cost of the produced solution needs to be shared among the participants. The order of events is as follows; first, the designer specifies the cost-sharing methods, using the product probability distribution over the players’ types, the players interact in the induced Bayesian game, and end up in a Bayesian Nash Equilibrium. A BBiE cost-sharing protocol satisfies budget balance in all equilibria; for any non-equilibrium profile we do not impose this requirement This natural relaxation enlarges the design space but maintains the desired property of balancing the cost in the equilibrium. The use of posted prices, to serve as cost-sharing mechanism, is highly desirable, but not always possible to achieve; a price is posted for each resource and the players behave as price takers, picking up the cheapest possible resources that satisfy their requirements Such a mechanism is desirable because it is extremely easy to implement and induces dominant strategies. We stress that our main results can be implemented by anonymous posted prices
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