Regret Minimizers and Convergence to Price-Taking

An extended abstract is available here.

This paper studies various regret minimizers in private value sealed bid double auctions; unlike the expected utility maximizers that populate typical market models, these traders do not determine their actions using a single prior. The analysis proves that minimax regret traders will not converge to price-taking as the number of traders in the market increases, contrary to standard economic intuition. However, not all regret-based decision rules fail to respond to market size. Introducing priors over some part of the decision problem to minimize expected maximum regret, or multiple priors to minimize maximum expected regret, have different effects. The robustness of the sealed bid double auction is limited by the need to avoid priors that destroy traders’ incentive to truthfully reveal their redemption values.

The Symmetry Axiom and Strategies Invariant to the Number of Rivals

We consider whether competitive pressures can induce traders to truthfully report their private information under Knightian uncertainty. Traders face Knightian uncertainty if they know the possible outcomes of each available action, but do not know each outcome’s probability. Such uncertainty may motivate use of a decision rule other than expected utility maximization. Two such alternative decision rules are maxmin and minimax regret. Stoye’s (2011) axiomatic characterization of these decision rules reveals that there is one axiom that maxmin and minimax regret share, and that distinguishes them from Bayes rule: the axiom of symmetry.

 We find that if agents use decision rules that accord with the symmetry axiom, then their strategies will be invariant to the number of other players in the game. Consequently, a market populated by traders that follow the symmetry axiom will not converge to efficiency as the market grows.