Optimal Mechanism Design with Risk-Loving Agents

Abstract

One of the most celebrated results in mechanism design is Myerson's characterization of the revenue optimal auction for selling a single item. However, this result relies heavily on the assumption that buyers are indifferent to risk. In this paper we investigate the case where the buyers are risk-loving, i.e. they prefer gambling to being rewarded deterministically. We use the standard model for risk from expected utility theory, where risk-loving behavior is represented by a convex utility function. We focus our attention on the special case of exponential utility functions. We characterize the optimal auction and show that randomization can be used to extract more revenue than when buyers are riskneutral. Most importantly, we show that the optimal auction is simple: the optimal revenue can be extracted using a randomized take-it-or-leave-it price for a single buyer and using a loser-pay auction, a variant of the all-pay auction, for multiple buyers. Finally, we show that these results no longer hold for convex utility functions beyond exponential. Related WorkMost work on optimal mechanism design beyond the risk-neutral setting has focused on riskaverse preferences. The classic results of Maskin and Riley [1984] and Matthews [1983] provide a characterization of the optimal mechanism with concave utility functions. A recent result in this area by Dughmi and Peres [2012] is that any mechanism designed for risk-neutral buyers can be adjusted to also align the incentives of risk-averse buyers and obtain similar guarantees. Fu et al.[2013] consider the design of prior-independent mechanisms (that have no access to the buyers' private value distributions) for risk-averse buyers. Finally, Chawla et al. [2018] study the design of robust mechanisms under the cumulative prospect theory model.To the best our knowledge, the only work on mechanism design under risk-loving behavior is by Hinnosaar [2017], who shows that in the absence of regulations, the seller can extract infinite revenue from the buyer with asymptotically risk-loving behavior under both the expected utility theory and prospect theory models.Recently, the duality theory framework has drawn attention in the mechanism design community for understanding optimal mechanisms for selling multiple items. For example, Daskalakis et al. [2017Daskalakis et al. [ , 2013 and Koutsoupias [2014, 2015] discovered the connection between the dual problem and the optimal transport (bipartite matching) problem. Cai et al. [2016] consider a duality framework via linear programming, and identify a connection between the virtual valuations and the dual variables. In our setting, the problem results in a different form of dual problem than in the multi-item setting, hence we seek to establish a new duality framework that diverges from the multi-item setting to different behavior models.