Fully mixed strategies in auction games

Auctions are very old. Herodotus reports that auctions have been used 2500 years ago in Babylonia. Since this time, a lot of new auctions have been invented and methods have been refined. But the basic idea remained the same. This thesis tries to show that auctions are more than a simple tool to maximize seller's profit. In the introductory first chapter we give an intuition why it is so important to detect a class of auction games. We discuss some problems that can be solved much easier by applying auction instruments. Chapters 2 - 4 examine three special auction games. Chapter two adds the problem of cost uncertainty to the literature of Bertrand Edgweworth competition. We consider a Bertrand-Edgeworth duopoly where cost are private information. The chapter links two branches of the literature. On the one hand we have the literature about Bertrand-Edgeworth competition. On the other hand there is the literature about the variable quantity auctions. This chapter brings together the full information Bertrand-Edgeworth model of Kovenock and Deneckere (1996) and the Bertrand under uncertainty model of Spulber (1995). Firms set prices in a game where demand and capacities are common knowledge. The setting corresponds to a first price variable quantity auction with capacity constraints. In equilibrium firms charge supracompetitive prices. Prices for a given cost can be lower than in the simple Bertrand under uncertainty model. The equilibria typically are not differentiable and sometimes even not continuous. The third chapter analyzes varieties and similarities of the simple Chicken Game and the War of Attrition Game. We extend the Chicken Game to a continuous environment and find a simple parameterization that links the two models. The War of Attrition Game typically has many equilibria. By contrast, the continuous Chicken Game does not have any Nash Equilibrium - neither in pure nor in mixed strategies. In an extension we mix the two models. We emphasize that for continuous games the definition of a Nash Equilibrium is not weak enough to guarantee existence of equilibrium. In chapter four we examine a multistage model. We consider a capacity precommitment game where costs are unknown. We extend the model of Kreps and Scheinkman (1983) to the uncertainty case. At the capacity setting stage (first stage) both firms do not know the marginal cost they will have at the price setting stage (third stage). The setup seems plausible as in reality firms do not know marginal costs exactly at the capacity stage. In the second stage nature draws costs. Firms observe their cost. In the final stage firms simultaneously set prices. We can show that equilibria are asymmetric. Equilibrium quantities are large. In an extension we show that if units costs differ, results change substantially.