## Research

## Research in Economics

**Consumer Surplus in the Dark**, with Jakub Steiner and Colin Stewart

A growing body of evidence suggests that consumers are not fully informed about prices, contrary to a critical assumption of classical welfare analysis. We analyze a model in which consumers can vary in both their preferences and their information about prices. Given an aggregate demand curve and a distribution of prices, we identify the set of rationalizable aggregate consumer surpluses. Each surplus in this set can be rationalized with simple information structures and preferences. We also identify the range of possible changes in surplus associated with a marginal change in the distribution of prices.

**Endogenous Risk Attitudes**, with Nick Netzer, Arthur Robson and Jakub Steiner

In a model inspired by neuroscience, we show that constrained optimal perception encodes lottery rewards using an S-shaped encoding function and over-samples low-probability events. The implications of this perception strategy for behavior depend on the decision-maker's understanding of the risk. The strategy does not distort choice in the limit as perception frictions vanish when the DM fully understands the decision problem. If, however, the DM underrates the complexity of the decision problem, then risk attitudes reflect properties of the perception strategy even for vanishing perception frictions. The model explains adaptive risk attitudes and probability weighting, as in prospect theory and, additionally, predicts that risk attitudes are strengthened by time pressure and attenuated by anticipation of large risks.

**Revealing Private Information in a Patent Race** (Slides)

In this paper I study dynamic and strategic aspects of R&D rivalry. I consider a patent race in which the first firm to make two consecutive breakthroughs wins the prize. A breakthrough arrives with instantaneous probability equal to the firm's R&D effort level, and its arrival is observed privately. A firm varies its effort as it updates its belief about the rival's progress. I find that a firm drops its effort over time until its first breakthrough arrives, in which case the effort jumps up and keeps increasing until one of the players patents. Further, I investigate whether a firm would want to reveal success in order to discourage its rival. I find that a firm never reveals if its rival has, and is first to reveal when a breakthrough is hard to achieve. When breakthroughs arrive quickly the firm prefers secrecy to revelation. For intermediate levels of research difficulty firm's revelation behavior entails randomization or delay. Interestingly, when there are more than two players, equilibrium always entails revelation.

**Multi-Player Discrete All-pay Auctions**

In this paper I study all-pay common-value auctions in which bids are restricted to non-negative integers. I prove that the game has unique symmetric Nash equilibrium in mixed strategies given that there are three or more players. Although players bid on average lower as more players are present, they always randomize on the whole set of bids smaller than the value of the prize, so long as there is sufficiently enough players. Players always receive a positive expected payoff which is bounded by a constant regardless of how large the value of the prize is. Finally, I prove that in limit the equilibrium converges to the equilibrium of a continuous all-pay auction.

**Optimal Stopping in Patent Race Games**

In their study* Optimal Stopping with Private Information*, Kruse and Strack (2013) analyze a single-agent optimal stopping mechanism design problem with transfers. I extend their framework to a general class of problems in which termination can occur exogenously prior to the agent's stopping time. I provide a simple condition under which all cut-off rules are implementable by a posted-price mechanism. As an application, I consider patent race in which each agent has private information about the arrival rate of a discovery, and stopping represents giving up the research efforts.

## Research in Mathematics

P. Kocourek, W. Takahashi and J. C. Yao (2011), **Fixed point theorems and ergodic theorems for nonlinear mappings in Banach spaces**, *Advances in Mathematical Economics*.

W. Takahashi, J.C. Yao, P. Kocourek (2011), **Weak and strong convergence theorems for generalized hybrid nonself-mappings in Hilbert spaces**, *J. Nonlinear Convex Analysis*.

P. Kocourek, W. Takahashi and J. C. Yao (2010), **Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces**, *Taiwanese Journal of Mathematics*.

P. Kocourek (2010), **An elementary new proof of the determination of a convex function by its subdifferential**, *Optimization.*