Abstract
We study the efficiency of the proportional allocation mechanism that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction proportional to her bids. We quantify the inefficiency of Nash equilibria by studying the Price of Anarchy (PoA) of the induced game under complete and incomplete information. When agents' valuations are concave, we show that the Bayesian Nash equilibria can be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure equilibria Johari and Tsitsiklis (Math. Oper. Res. 29(3), 407---435 2004). Next, we upper bound the PoA over Bayesian equilibria by 2 when agents' valuations are subadditive, generalizing and strengthening previous bounds on lattice submodular valuations. Furthermore, we show that this bound is tight and cannot be improved by any simple or scale-free mechanism. Then we switch to settings with budget constraints, and we show an improved upper bound on the PoA over coarse-correlated equilibria. Finally, we prove that the PoA is exactly 2 for pure equilibria in the polyhedral environment.
Highlights
Allocating network resources, like bandwidth, among agents is a canonical problem in the network optimization literature
We prove that the Price of Anarchy (PoA) over Bayesian Nash equilibria is at most 2
We prove that the PoA of the proportional allocation mechanism for coarse correlated equilibria is at most 1 +φ ≈ 2.618, where φ is the golden ratio
Summary
Allocating network resources, like bandwidth, among agents is a canonical problem in the network optimization literature. Johari and Tsitsiklis [16] relaxed the assumption that the users act as price takers and instead they can anticipate the effects of their actions on the prices of the resources They observed that this strategic bidding in the proportional allocation mechanism leads to inefficient allocations that do not maximize social welfare. They showed that this efficiency loss is bounded when agents’ valuations are concave They proved that the proportional allocation mechanism admits a unique pure Nash equilibrium with Price of Anarchy (PoA) [19] at most 4/3. Non-concave valuation functions were studied by Syrgkanis and Tardos [26] for both complete and incomplete information games They showed that, when agents’ valuations are lattice-submodular, the PoA for coarse correlated and Bayesian Nash equilibria is at most 3.73 by applying their general smoothness framework. Simultaneous first price auctions, and Roughgarden [23] proved general lower bounds for the PoA of all simple auctions by using the corresponding computational or communication lower bounds of the underlying allocation problem
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