176 research outputs found
Computing Bayes Nash Equilibrium Strategies in Auction Games via Simultaneous Online Dual Averaging
Auctions are modeled as Bayesian games with continuous type and action
spaces. Computing equilibria in auction games is computationally hard in
general and no exact solution theory is known. We introduce algorithms
computing distributional strategies on a discretized version of the game via
online convex optimization. One advantage of distributional strategies is that
we do not have to make any assumptions on the shape of the bid function.
Besides, the expected utility of agents is linear in the strategies. It follows
that if our regularized optimization algorithms converge to a pure strategy,
then they converge to an approximate equilibrium of the discretized game with
high precision. Importantly, we show that the equilibrium of the discretized
game approximates an equilibrium in the continuous game. In a wide variety of
auction games, we provide empirical evidence that the method approximates the
analytical (pure) Bayes Nash equilibrium closely. This speed and precision is
remarkable, because in many finite games learning dynamics do not converge or
are even chaotic. In standard models where agents are symmetric, we find
equilibrium in seconds. The method allows for interdependent valuations and
different types of utility functions and provides a foundation for broadly
applicable equilibrium solvers that can push the boundaries of equilibrium
analysis in auction markets and beyond
On the Inefficiency of the Uniform Price Auction
We present our results on Uniform Price Auctions, one of the standard
sealed-bid multi-unit auction formats, for selling multiple identical units of
a single good to multi-demand bidders. Contrary to the truthful and
economically efficient multi-unit Vickrey auction, the Uniform Price Auction
encourages strategic bidding and is socially inefficient in general. The
uniform pricing rule is, however, widely popular by its appeal to the natural
anticipation, that identical items should be identically priced. In this work
we study equilibria of the Uniform Price Auction for bidders with (symmetric)
submodular valuation functions, over the number of units that they win. We
investigate pure Nash equilibria of the auction in undominated strategies; we
produce a characterization of these equilibria that allows us to prove that a
fraction 1-1/e of the optimum social welfare is always recovered in undominated
pure Nash equilibrium -- and this bound is essentially tight. Subsequently, we
study the auction under the incomplete information setting and prove a bound of
4-2/k on the economic inefficiency of (mixed) Bayes Nash equilibria that are
supported by undominated strategies.Comment: Additions and Improvements upon SAGT 2012 results (and minor
corrections on the previous version
On the Convergence of Learning Algorithms in Bayesian Auction Games
Equilibrium problems in Bayesian auction games can be described as systems of
differential equations. Depending on the model assumptions, these equations
might be such that we do not have a rigorous mathematical solution theory. The
lack of analytical or numerical techniques with guaranteed convergence for the
equilibrium problem has plagued the field and limited equilibrium analysis to
rather simple auction models such as single-object auctions. Recent advances in
equilibrium learning led to algorithms that find equilibrium under a wide
variety of model assumptions. We analyze first- and second-price auctions where
simple learning algorithms converge to an equilibrium. The equilibrium problem
in auctions is equivalent to solving an infinite-dimensional variational
inequality (VI). Monotonicity and the Minty condition are the central
sufficient conditions for learning algorithms to converge to an equilibrium in
such VIs. We show that neither monotonicity nor pseudo- or quasi-monotonicity
holds for the respective VIs. The second-price auction's equilibrium is a
Minty-type solution, but the first-price auction is not. However, the
Bayes--Nash equilibrium is the unique solution to the VI within the class of
uniformly increasing bid functions, which ensures that gradient-based
algorithms attain the {equilibrium} in case of convergence, as also observed in
numerical experiments
Computing Perfect Bayesian Equilibria in Sequential Auctions
We present a best-response based algorithm for computing verifiable
-perfect Bayesian equilibria for sequential auctions with
combinatorial bidding spaces and incomplete information. Previous work has
focused only on computing Bayes-Nash equilibria for static single-round
auctions, which our work captures as a special case. Additionally, we prove an
upper bound on the utility loss of our approximate equilibria and
present an algorithm to efficiently compute based on the
immediate loss at each subgame. We evaluate the performance of our algorithm by
reproducing known results from several auctions previously introduced in the
literature, including a model of combinatorial split-award auctions used in
procurement.Comment: 12 pages, 8 figure
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