Ascending auctions and Walrasian equilibrium

We present a family of submodular valuation classes that generalizes gross substitute. We show that Walrasian equilibrium always exist for one class in this family, and there is a natural ascending auction which finds it. We prove some new structural properties on gross-substitute auctions which, in turn, show that the known ascending auctions for this class (Gul-Stacchetti and Ausbel) are, in fact, identical. We generalize these two auctions, and provide a simple proof that they terminate in a Walrasian equilibrium

Time bounds for iterative auctions : a unified approach by discrete convex analysis

We investigate an auction model where there are many different goods, each good has multiple units and bidders have gross substitutes valuations over the goods. We analyze the number of iterations in iterative auction algo- rithms for the model based on the theory of discrete convex analysis. By making use of L♮-convexity of the Lyapunov function we derive exact bounds on the number of iterations in terms of the ℓ1-distance between the initial price vector and the found equilibrium. Our results extend and unify the price adjustment algorithms for the multi-unit auction model and for the unit-demand auction model, offering computational complexity results for these algorithms, and reinforcing the connection between auction theory and discrete convex analysis

On the Economic Efficiency of the Combinatorial Clock Auction

Since the 1990s spectrum auctions have been implemented world-wide. This has provided for a practical examination of an assortment of auction mechanisms and, amongst these, two simultaneous ascending price auctions have proved to be extremely successful. These are the simultaneous multiround ascending auction (SMRA) and the combinatorial clock auction (CCA). It has long been known that, for certain classes of valuation functions, the SMRA provides good theoretical guarantees on social welfare. However, no such guarantees were known for the CCA. In this paper, we show that CCA does provide strong guarantees on social welfare provided the price increment and stopping rule are well-chosen. This is very surprising in that the choice of price increment has been used primarily to adjust auction duration and the stopping rule has attracted little attention. The main result is a polylogarithmic approximation guarantee for social welfare when the maximum number of items demanded $\mathcal{C}$ by a bidder is fixed. Specifically, we show that either the revenue of the CCA is at least an $\Omega\Big(\frac{1}{\mathcal{C}^{2}\log n\log^2m}\Big)$-fraction of the optimal welfare or the welfare of the CCA is at least an $\Omega\Big(\frac{1}{\log n}\Big)$-fraction of the optimal welfare, where $n$ is the number of bidders and $m$ is the number of items. As a corollary, the welfare ratio -- the worst case ratio between the social welfare of the optimum allocation and the social welfare of the CCA allocation -- is at most $O(\mathcal{C}^2 \cdot \log n \cdot \log^2 m)$. We emphasize that this latter result requires no assumption on bidders valuation functions. Finally, we prove that such a dependence on $\mathcal{C}$ is necessary. In particular, we show that the welfare ratio of the CCA is at least $\Omega \Big(\mathcal{C} \cdot \frac{\log m}{\log \log m}\Big)$

Dynamic strategic interactions : analysis and mechanism design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 225-232).Modern systems, such as engineering systems with autonomous entities, markets, and financial networks, consist of self-interested agents with potentially conflicting objectives. These agents interact in a dynamic manner, modifying their strategies over time to improve their payoffs. The presence of self-interested agents in such systems, necessitates the analysis of the impact of multi-agent decision making on the overall system, and the design of new systems with improved performance guarantees. Motivated by this observation, in the first part of this thesis we focus on fundamental structural properties of games, and exploit them to provide a new framework for analyzing the limiting behavior of strategy update rules in various game-theoretic settings. In the second part, we investigate the design problem of an auctioneer who uses iterative multi-- item auctions for efficient allocation of resources. More specifically, in the first part of the thesis we focus on potential games, a special class of games with desirable equilibrium and dynamic properties, and analyze their preference structure. Exploiting this structure we obtain a decomposition of arbitrary games into three components, which we refer to as the potential, harmonic, and nonstrategic components. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents conflicts between the interests of the players. We make this intuition precise by studying the properties of these two components, and establish that indeed they have quite distinct and remarkable characteristics. The decomposition also allows us to approximate a given game with a potential game. We show that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of a potential game that approximates it. The decomposition provides a valuable tool for the analysis of dynamics in games. Earlier literature established that many natural strategy update rules converge to a Nash equilibrium in potential games. We show that games that are close to a potential game exhibit similar properties. In particular, we focus on three commonly studied discrete-time update rules (better/best response, logit response, and discrete-time fictitious play dynamics), and establish that in near-potential games, the limiting behavior of these update rules can be characterized by an approximate equilibrium set, size of which is proportional to the distance of the original game from a potential game. Since a close potential game to a given game can be systematically found via decomposition, our results suggest a systematic framework for studying the limiting behavior of adaptive dynamics in arbitrary finite strategic form games: the limiting behavior of dynamics in a given game can be characterized by first approximating this game with a potential game, and then analyzing the limiting behavior of dynamics in the potential game. In the second part of the thesis, we change our focus to implementing efficient outcomes in multi-agent settings by using simple mechanisms. In particular, we develop novel efficient iterative auction formats for multi-item environments, where items exhibit value complementarities/substitutabilities. We obtain our results by focusing on a special class of value functions, which we refer to as graphical valuations. These valuations are not fully general, but importantly they capture value complementarity/substitutability in important practical settings, while allowing for a compact representation of the value functions. We start our analysis by first analyzing how the special structure of graphical valuations can be exploited to design simple iterative auction formats. We show that in settings where the underlying value graph is a tree (and satisfies an additional technical condition), a Walrasian equilibrium always exists (even in the presence of value complementarities). Using this result we provide a linear programming formulation of the efficient allocation problem for this class of valuations. Additionally, we demonstrate that a Walrasian equilibrium may not exist, when the underlying value graph is more general. However, we also establish that in this case a more general pricing equilibrium always exists, and provide a stronger linear programming formulation that can be used to identify the efficient allocation for general graphical valuations. We then consider solutions of these linear programming formulations using iterative algorithms. Complementing these iterative algorithms with appropriate payment rules, we obtain iterative auction formats that implement the efficient outcome at an (ex-post perfect) equilibrium. The auction formats we obtain rely on simple pricing rules that, in the most general case, require offering a bidder-specific price for each item, and bidder-specific discounts/markups for pairs of items. Our results in this part of the thesis suggest that when value functions of bidders exhibit some special structure, it is possible to systematically exploit this structure in order to develop simple efficient iterative auction formats.by Utku Ozan Candogan.Ph.D

Combinatorial auctions for electronic business

Combinatorial auctions (CAs) have recently generated significant interest as an automated mechanism for buying and selling bundles of goods. They are proving to be extremely useful in numerous e-business applications such as e-selling, e-procurement, e-logistics, and B2B exchanges. In this article, we introduce combinatorial auctions and bring out important issues in the design of combinatorial auctions. We also highlight important contributions in current research in this area. This survey emphasizes combinatorial auctions as applied to electronic business situations

Combinatorial Optimization (hybrid meeting)

Combinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years

Distributed optimisation techniques for wireless networks

Alongside the ever increasing traffic demand, the fifth generation (5G) cellular network architecture is being proposed to provide better quality of service, increased data rate, decreased latency, and increased capacity. Without any doubt, the 5G cellular network will comprise of ultra-dense networks and multiple input multiple output technologies. This will make the current centralised solutions impractical due to increased complexity. Moreover, the amount of coordination information that needs to be transported over the backhaul links will be increased. Distributed or decentralised solutions are promising to provide better alternatives. This thesis proposes new distributed algorithms for wireless networks which aim to reduce the amount of system overheads in the backhaul links and the system complexity. The analysis of conflicts amongst transmitters, and resource allocation are conducted via the use of game theory, convex optimisation, and auction theory. Firstly, game-theoretic model is used to analyse a mixed quality of service (QoS) strategic non-cooperative game (SNG), for a two-user multiple-input single-output (MISO) interference channel. The players are considered to have different objectives. Following this, the mixed QoS SNG is extended to a multicell multiuser network in terms of signal-to-interference-and-noise ratio (SINR) requirement. In the multicell multiuser setting, each transmitter is assumed to be serving real time users (RTUs) and non-real time users (NRTUs), simultaneously. A novel mixed QoS SNG algorithm is proposed, with its operating point identified as the Nash equilibrium-mixed QoS (NE-mixed QoS). Nash, Kalai-Smorodinsky, and Egalitarian bargain solutions are then proposed to improve the performance of the NE-mixed QoS. The performance of the bargain solutions are observed to be comparable to the centralised solutions. Secondly, user offloading and user association problems are addressed for small cells using auction theory. The main base station wishes to offload some of its users to privately owned small cell access points. A novel bid-wait-auction (BWA) algorithm, which allows single-item bidding at each auction round, is designed to decompose the combinatorial mathematical nature of the problem. An analysis on the existence and uniqueness of the dominant strategy equilibrium is conducted. The BWA is then used to form the forward BWA (FBWA) and the backward BWA (BBWA). It is observed that the BBWA allows more users to be admitted as compared to the FBWA. Finally, simultaneous multiple-round ascending auction (SMRA), altered SMRA (ASMRA), sequential combinatorial auction with item bidding (SCAIB), and repetitive combinatorial auction with item bidding (RCAIB) algorithms are proposed to perform user offloading and user association for small cells. These algorithms are able to allow bundle bidding. It is then proven that, truthful bidding is individually rational and leads to Walrasian equilibrium. The performance of the proposed auction based algorithms is evaluated. It is observed that the proposed algorithms match the performance of the centralised solutions when the guest users have low target rates. The SCAIB algorithm is shown to be the most preferred as it provides high admission rate and competitive revenue to the bidders