Numerical Solution of Asymmetric Auctions

We propose the backward indifference derivation (BID) algorithm, a new method to numerically approximate the pure strategy Nash equilibrium (PSNE) bidding functions in asymmetric first-price auctions. The BID algorithm constructs a sequence of finite-action PSNE that converges to the continuum-action PSNE by finding where bidders are indifferent between actions. Consequently, our approach differs from prevailing numerical methods that consider a system of poorly behaved differential equations. After proving convergence (conditional on knowing the maximum bid), we evaluate the numerical performance of the BID algorithm on four examples, two of which have not been previously addressed.

Shapley-Folkman-Lyapunov theorem and Asymmetric First price auctions

In this paper non-convexity in economics has been revisited. Shapley-Folkman-Lyapunov theorem has been tested with the asymmetric auctions where bidders follow log-concave probability distributions (non-convex preferences). Ten standard statistical distributions have been used to describe the bidders’ behavior. In principle what is been tested is that equilibrium price can be achieved where the sum of large number non-convex sets is convex (approximately), so that optimization is possible. Convexity is thus very important in economics.