While the classic Vickrey-Clarke-Groves mechanism ensures incentive compatibility for a static one-shot game, it does not appear to be feasible to construct a dominant truth-telling mechanism for agents that are stochastic dynamic systems. The contribution of this paper is to show that for a set of LQG agents a mechanism consisting of a sequence of layered payments over time decouples the intertemporal effect of current bids on future payoffs and ensures truth-telling of dynamic states by their agents, if system parameters are known and agents are rational. Additionally, it is shown that there is a "Scaled" VCG mechanism that simultaneously satisfies incentive compatibility, social efficiency, budget balance as well as individual rationality under a certain "market power balance" condition where no agent is too negligible or too dominant. A further desirable property is that the SVCG payments converge to the Lagrange payments, the payments corresponding to the true price in the absence of strategic considerations, as the number of agents in the market increases. For LQ but non-Gaussian agents the optimal social welfare over the class of linear control laws is achieved.
The problem of how to conduct bidding to achieve social welfare optimality in stochastic dynamic systems is examined in @cite_0 . It assumes that all agents are price-takers.
@cite_0
2482076478 2326228838 1489333206 2949060750
In a combinatorial auction with item bidding, agents participate in multiple single-item second-price auctions at once. As some items might be substitutes, agents need to strategize in order to maximize their utilities. A number of results indicate that high welfare can be achieved this way, giving bounds on the welfare at equilibrium. Recently, however, criticism has been raised that equilibria are hard to compute and therefore unlikely to be attained. In this paper, we take a different perspective. We study simple best-response dynamics. That is, agents are activated one after the other and each activated agent updates his strategy myopically to a best response against the other agents' current strategies. Often these dynamics may take exponentially long before they converge or they may not converge at all. However, as we show, convergence is not even necessary for good welfare guarantees. Given that agents' bid updates are aggressive enough but not too aggressive, the game will remain in states of good welfare after each agent has updated his bid at least once. In more detail, we show that if agents have fractionally subadditive valuations, natural dynamics reach and remain in a state that provides a @math approximation to the optimal welfare after each agent has updated his bid at least once. For subadditive valuations, we can guarantee an @math approximation in case of @math items that applies after each agent has updated his bid at least once and at any point after that. The latter bound is complemented by a negative result, showing that no kind of best-response dynamics can guarantee more than an @math fraction of the optimal social welfare. We study the following simple Bayesian auction setting: m items are sold to n selfish bidders in m independent second-price auctions. Each bidder has a private valuation function that specifies his or her complex preferences over all subsets of items. Bidders only have beliefs about the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular or, more generally, fractionally subadditive (aka XOS) valuation functions, every Bayes-Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game, a pure Nash always exists and can be found in time that is polynomial in both m and n. We study the following Bayesian setting: mitems are sold to nselfish bidders in mindependent second-price auctions. Each bidder has a privatevaluation function that expresses complex preferences over allsubsets of items. Bidders only have beliefsabout the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both mand n. We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders are submodular, in many interesting settings (e.g., constant number of bidders, budget additive bidders) computing an equilibrium with good welfare is essentially as easy as computing, completely ignoring incentives issues, an allocation with good welfare. On the other hand, for subadditive valuations, we show that computing an equilibrium requires exponential communication. Finally, for XOS (a.k.a. fractionally subadditive) valuations, we show that if there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from our current ones.