## Optimal Allocation of a Divisible Good to Strategic Buyers (2004)

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Venue: | Proceedings of the 43d IEEE conference on Decision and Control |

Citations: | 37 - 2 self |

### BibTeX

@INPROCEEDINGS{Sanghavi04optimalallocation,

author = {Sujay Sanghavi and Bruce Hajek},

title = {Optimal Allocation of a Divisible Good to Strategic Buyers},

booktitle = {Proceedings of the 43d IEEE conference on Decision and Control},

year = {2004},

pages = {2748--2753}

}

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### Abstract

We address the problem of allocating a divisible resource to buyers who value the quantity they receive, but strategize to maximize their net payoff (value minus payment). An allocation mechanism is used to allocate the resource based on bids declared by the buyers. The bids are equal to the payments, and the buyers are assumed to be in Nash equilibrium. For two buyers such an allocation mechanism is found that guarantees that the aggregate value is always greater than of the maximum possible, and it is shown that no other mechanism achieves a larger ratio. For a general finite number of buyers an allocation mechanism is given and an expression is given for its worst case efficiency. For three buyers the expression evaluates to 0.8737, for four buyers to 0.8735 and numerical computations suggest that the numerical value does not decrease when the number of buyers is increased beyond four. A potential application of this work is the allocation of communication bandwidth on a single link.

### Citations

1498 |
Rate control in communication networks: shadow prices, proportional fairness and stability
- Kelly, Maulloo, et al.
- 1998
(Show Context)
Citation Context ...sumption is reasonable to make for distributed markets like that of connections vying for Internet bandwidth, where buyers do not have access to sophisticated market information. Kelly and colleagues =-=[2, 3]-=- investigated using a uniform-price mechanism where each buyer submits a bid that is equal to the payment it is willing to make. They showed that if the buyers do not anticipate the effect of their ow... |

741 | Charging and rate control for elastic traffic
- Kelly
- 1997
(Show Context)
Citation Context ...f divisible goods have also received much attention. Besides [1] mentioned above, Maheswaran and Bas¸ar [7] and Gopalkrishnan and Hajek [2] also deal with allocation of a single divisible good. Kelly=-= [6]-=- shows that if the buyers are not strategic but pricetaking, allocation of a divisible good can be made in a socially optimum way. In the economics literature this model appears in Back and Zender [8]... |

547 | How bad is selfish routing
- Roughgarden, Tardos
- 2002
(Show Context)
Citation Context ...Recent research contains many examples of efforts to quantify the inefficiency of Nash equilibria in games related to resource allocation. Koutsoupias and Papadimitriou [5] and Roughgarden and Tardos =-=[4]-=- quantify worst-case inefficiencies of Nash equilibria in routing games. Johari and Tsitsiklis [1] quantify the inefficiency in allocating a divisible good with a uniform price. All of these papers us... |

275 |
Existence and uniqueness of Equilibrium points for concave N-Person Games
- Rosen
- 1965
(Show Context)
Citation Context ...uyers are forced to bid at least ɛ. Also, the payments would never exceed the value obtained from having the entire unit of the resource, so we can assume that wi ≤ Ui(1). Then, by Theorems1 in Ros=-=en [3] there exist-=-s an NEP �wɛ for the ɛgame. A sequence ɛn → 0 will have a sequence of corresponding equilibria �wɛn. Consider a convergent subsequence in the sequence of equilibria, and let �w be its limi... |

267 |
Characterizations of satisfactory mechanisms for the revelation of preferences for public goods
- Green, Laffont
- 1977
(Show Context)
Citation Context ...n “where it is needed.” Finally, it may be the case that bids have to be public knowledge, and buyers may not want to reveal their private information as part of an auction. Nevertheless, as shown in =-=[1]-=-, VCG mechanisms are the only ones that ensure efficient allocations as dominant-strategy outcomes. Any mechanism that allocates based only on real-valued payments must forgo dominant-strategy as a so... |

157 | Efficiency loss in a network resource allocation game
- Johari, Tsitsiklis
(Show Context)
Citation Context ...in games related to resource allocation. Koutsoupias and Papadimitriou [5] and Roughgarden and Tardos [4] quantify worst-case inefficiencies of Nash equilibria in routing games. Johari and Tsitsiklis =-=[1]-=- quantify the inefficiency in allocating a divisible good with a uniform price. All of these papers use the ratio of the welfare at Nash equilibrium to the welfare at the social optimum, thus evaluati... |

115 |
Auctions of divisible goods: on the rationale for the Treasury experiment
- Back, Zender
- 1993
(Show Context)
Citation Context ...[6] shows that if the buyers are not strategic but pricetaking, allocation of a divisible good can be made in a socially optimum way. In the economics literature this model appears in Back and Zender =-=[8]-=- who advocate using discriminatory pricing for the sale of treasury bonds. For two buyers this paper finds the mechanism that has the lowest worst-case fractional inefficiency for allocating a divisib... |

74 |
Inefficiency of Nash equilibria
- Dubey
- 1986
(Show Context)
Citation Context ...regate value are inefficient. Nash equilibria are fundamental in the study of games of this nature. That inefficient allocations may occur at Nash equilibria is well known in the economics literature =-=[9]-=-. The question addressed in this paper is: how to design an allocation mechanism that results in the worst-case inefficiency of a Nash equilibrium being as low as possible ? Recent research contains m... |

33 | Nash equilibrium and decentralized negotiation in auctioning divisible resources
- Maheswaran, Basar
- 2003
(Show Context)
Citation Context ...ilibrium to the welfare at the social optimum, thus evaluating fractional efficiency. Auctions of divisible goods have also received much attention. Besides [1] mentioned above, Maheswaran and Bas¸ar=-= [7]-=- and Gopalkrishnan and Hajek [2] also deal with allocation of a single divisible good. Kelly [6] shows that if the buyers are not strategic but pricetaking, allocation of a divisible good can be made ... |

29 | Charging and rate control for elastic traffic,” European Trans. on Telecom - Kelly - 1997 |

26 |
Worst case equilibria
- Koutsoupias, Papadimitriou
- 1999
(Show Context)
Citation Context ...ium being as low as possible ? Recent research contains many examples of efforts to quantify the inefficiency of Nash equilibria in games related to resource allocation. Koutsoupias and Papadimitriou =-=[5]-=- and Roughgarden and Tardos [4] quantify worst-case inefficiencies of Nash equilibria in routing games. Johari and Tsitsiklis [1] quantify the inefficiency in allocating a divisible good with a unifor... |

22 |
Do greedy autonomous systems make for a sensible Internet
- Hajek, Gopalakrishnan
(Show Context)
Citation Context ...n allocation is done according to τ ∗ . � The above technique shows the uniqueness of Nash equilibria by formulating an equivalent optimization problem. Similar methodologies have been used befor=-=e in [2]-=- and also in [1]. V. n BUYERS The optimal valid mechanism for two buyers has a natural extension to the case when there are n buyers. Specifically, we saw that the optimal twobuyer mechanism was the o... |

17 | Social Welfare of Selfish Agents: Motivating Efficiency of Divisible Resources
- Maheswaran, Basar
- 2004
(Show Context)
Citation Context ...h the allocation and payment vectors as functions of the bid vector. Non-uniform pricing in this more general scenario has been shown to achieve full efficiency, independently by Maheswaran and Basar =-=[4]-=- and Yang and Hajek [5]. Section 2 of this paper formulates the mechanism design problem. Section 3 builds on a lemma in [6] to show that when designing for increased fractional efficiency in the wors... |

15 | Strategic Buyers in a Sum Bid Game for Flat Networks. Manuscript, http: //tesla.csl.uiuc.edu/∼hajek/Papers/HajekYang.pdf
- Hajek, Yang
- 2004
(Show Context)
Citation Context ...yment vectors as functions of the bid vector. Non-uniform pricing in this more general scenario has been shown to achieve full efficiency, independently by Maheswaran and Basar [4] and Yang and Hajek =-=[5]-=-. Section 2 of this paper formulates the mechanism design problem. Section 3 builds on a lemma in [6] to show that when designing for increased fractional efficiency in the worst-case, it is sufficien... |