**My Papers:**

Publications:

Learning in a Game of Strategic Experimentation With Three-Armed Exponential Bandits, forthcoming in a volume of the series *Static & Dynamic Game Theory: Foundations & Applications*, edited L.A. Petrosyan, V.V. Mazalov, et N.A. Zenkevich

Abstract: The present article provides some additional results for the two-player game of strategic experimentation with three-armed exponential bandits analyzed in Klein (2013). Players play replica bandits, with one safe arm and two risky arms, which are known to be of opposite types. It is initially unknown, however, which risky arm is good and which is bad. A good risky arm yields lump sums at exponentially distributed times when pulled. A bad risky arm never yields any payoff. In this article, I give a necessary and sufficient condition for the state of the world eventually to be found out with probability 1 in any Markov perfect equilibrium in which at least one player’s value function is continuously differentiable. Furthermore, I provide closed-form expressions for the players’ value function in a symmetric Markov perfect equilibrium for low and intermediate stakes.

Parliament Shapes and Sizes (with Raphaël Godefroy), Economic Inquiry (forthcoming: https://doi.org/10.1111/ecin.12584)

Abstract: This paper proposes a model of Parliamentary institutions in which a Parliament Designer makes three decisions: whether a Parliament should comprise one or two chambers, what the relative bargaining power of each chamber should be if the Parliament is bicameral, and how many legislators should sit in each chamber. We document empirical regularities across countries that are consistent with the predictions of our model.

[« Les Échos » que nous avons donnés à ce papier : Y a-t-il trop de parlementaires en France ?]

Will Truth Out?--An Advisor's Quest To Appear Competent*(with Tymofiy Mylovanov), Journal of Mathematical Economics, 2017, 72, 112-121*

**Abstract: We study a dynamic career-concerns environment with an agent who has incentives to appear competent. It is well known that dynamic career concerns create incentives for an agent to be conservative and to tailor his reports towards a commonly held prior opinion. The existing models, however, have focused on short time horizons. We show that, for long time horizons, there exist countervailing incentives for the agent to report his true opinion. In particular, if the agent is sufficiently patient, the time horizon is sufficiently long given the agent's patience, and the quality of the competent expert is high enough given the time horizon and the discount factor, the beneficial long-term incentives overwhelm any harmful myopic ones, and the incentive problem vanishes.**

**The Importance of Being Honest, **Theoretical Economics, 2016, 11: 773-811

**Abstract:** This paper analyzes the case of a principal who wants to provide an agent with proper incentives to explore a hypothesis that can be either true or false. The agent can shirk, thus never proving the hypothesis, or he can avail himself of a known technology to produce fake successes. This latter option either makes the provision of incentives for honesty impossible or does not distort its costs at all. In the latter case, the principal will optimally commit to rewarding later successes even though he only cares about the first one. Indeed, after an honest success, the agent is more optimistic about his ability to generate further successes. This, in turn, provides incentives for the agent to be honest before a first success.

Strategic Learning in Teams, Games and Economic Behavior, 2013, 82: 636-657

Abstract: This paper analyzes a two-player game of strategic experimentation with three-armed exponential bandits in continuous time. Players play bandits of identical types, with one arm that is safe in that it generates a known payoff, whereas the likelihood of the risky arms' yielding a positive payoff is initially unknown. When the types of the two risky arms are perfectly negatively correlated, the efficient policy is an equilibrium if and only if the stakes are high enough. If the negative correlation is imperfect and stakes are high, there exists an equilibrium that leads to efficiency for optimistic enough prior beliefs.

[An older version, which provides some additional details on certain aspects, is **here.]**

**Negatively Correlated Bandits** *(with Sven Rady), Review of Economic Studies, 2011, 78(2): 693-732.*

**Abstract: **We analyze a two-player game of strategic experimentation with two-armed bandits. Either player has to decide in continuous time whether to use a safe arm with a known payoff or a risky arm whose expected payoff per unit of time is initially unknown. This payoff can be high or low, and is negatively correlated across players. We characterize the set of all Markov perfect equilibria in the benchmark case where the risky arms are known to be of opposite type, and construct equilibria in cutoff strategies for arbitrary negative correlation. All strategies and payoffs are in closed form. In marked contrast to the case where both risky arms are of the same type, there always exists an equilibrium in cutoff strategies, and there always exists an equilibrium exhibiting efficient long-run patterns of learning. These results extend to a three-player game with common knowledge that exactly one risky arm is of the high payoff type.

(As Yet) Unpublished Papers:

NEW VERSION!! NOW WITH BROWNIAN MOTION, TOO!!

Strongly Symmetric Equilibria in Bandit Games *(with Johannes Hörner & Sven Rady)*

Abstract: This paper studies strongly symmetric equilibria (SSE) in continuous-time games of strategic experimentation with bandits. SSE payoffs can be studied via two functional equations similar to the HJB equation used for Markov equilibria. This is valuable for three reasons. First, these equations retain the tractability of Markov equilibrium, while allowing for punishments and rewards: the best and worst equilibrium payoffs are explicitly solved for. Second, they capture behavior of the discrete-time game: as the period length goes to zero in the discretized game, the SSE payoff set converges to their solution. Third, they encompass a large payoff set: there is no perfect Bayesian equilibrium in the discrete-time game with frequent interactions with higher asymptotic efficiency. For Poisson bandits, we give a necessary and sufficient condition for the existence of an asymptotically efficient equilibrium. For Brownian-motion bandits, we show that efficiency can always be achieved in the limit.

NEW VERSION (WITH ADDITIONAL RESULTS)!!

Relational Contracts with Private Information on the Future Value of the Relationship: The Upside of Implicit Downsizing Costs (with Matthias Fahn)

**Abstract: We analyze a relational contracting problem, in which the principal has private information about the future value of the relationship. In order to reduce bonus payments, the principal is tempted to claim that the value of the future relationship is lower than it actually is. To induce truth-telling, the optimal relational contract may introduce distortions after a bad report. For some levels of the discount factor, output is reduced by more than would be sequentially optimal. This distortion is attenuated over time even if prospects remain bad. Our model thus provides an alternative explanation for indirect short-run costs of downsizing.**

Slides for the 6th Workshop on Stochastic Methods in Game Theory

**Strategic Experimentation with**** As****ymmetric**** Players**(with Kaustav Das and Katharina Schmid)

Abstract: We examine a two-player game with two-armed exponential bandits à la [Keller, Rady, Cripps (2005)], where players operate different technologies for exploring the risky option. We characterise the set of Markov perfect equilibria, and show that there always exists an equilibrium in which the player with the inferior technology uses a cutoff strategy. All Markov perfect equilibria imply the same amount of experimentation but differ with respect to the expected speed of the resolution of uncertainty. If and only if the degree of asymmetry between the players is high enough, there exists a Markov perfect equilibrium in which both players use cutoff strategies. Whenever this equilibrium exists, it welfare dominates all other equilibria. This contrasts with the case of symmetric players, where there never exists a Markov perfect equilibrium in cutoff strategies. We also show that the equilibrium where only the player with the inferior technology uses a cutoff strategy is not welfare dominated.

Strategic Investment and Learning with Private Information (with Peter Wagner)

Bandits in the Lab *(with Johannes Hölzemann)*

Abstract: We test Keller, Rady, Cripps’ (2005) game of strategic experimentation with exponential bandits in the laboratory. We find strong support for the prediction of free-riding because of strategic concerns. We also find strong evidence for behavior that is characteristic of Markov perfect equilibrium: non-cutoff behavior, lonely pioneers and frequent switches of action.

[Videos illustrating the operation of the eye-tracking devices can be accessed here.]

Work in Its Earlier Gestational Stages:

*** Competition, Learning and Duplicative Search in a Patent Race (with Kaustav Das)**

*** Generally Correlated Bandits (with Sven Rady)**

** Over-Cautious or Trigger-Happy Advisors---When Best to Stop (with Sidartha Gordon)*

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Figuring stuff out