**Working Papers **

## 1) Learnability and Stochastic Choice

**Abstract :**In this paper, we study a non-parametric approach to prediction in stochastic choice models in economics. We apply techniques from statistical learning theory to study the problem of learning choice probabilities. A model of stochastic choice is said to be learnable if there exist learning rules defined on choice data that are uniformly consistent. We construct learning rules via the procedure of empirical risk minimization, where risk is defined in terms of incentive compatible scoring rules. This approach involves mild distributional assumptions on the model, with the main requirement being a constraint on the complexity of the stochastic choice model. Our main result shows that in the context of stochastic preferences, when the VC dimension of a preference model is finite, then we can recover the distribution over preferences (heterogeneity) in the uniform sense. We provide several applications and obtain general rates of convergence based on sample complexity. Under additional conditions, we establish a converse result for the non-learnability of models. An implication of our non-learnability result is that for any learning rule and sample size, the worst case expected distance between true and estimated choice probabilities is at least half times the maximum possible distance. In binary choice settings, this expected error lower bound is above the error from always predicting that each alter-native will be chosen with equal probability. Lastly, we also address recoverability of cognitive heterogeneity and non-uniform learning.

learnabilitystochastic.pdf |

## 2) Learnability and Models of Decision Making under Uncertainty (*with Federico Echenique) *

**Proceedings of the 19th ACM Conference on Economics and Computation, 2018**

Revise and Resubmit, Theoretical Economics

Revise and Resubmit, Theoretical Economics

**Abstract :**We study whether some of the most important models of decision-making under uncertainty are uniformly learnable. Imagine an analyst who seeks to learn, or estimate, an agent’s preference using data on the agent’s choices. A model is learnable if the analyst can construct a learning rule to learn the agent’s preference, when preferences conforms to the model, with enough data, and uniformly over processes that generate choice problems. We consider the Expected Utility, Choquet Expected Utility and Max-min Expected Utility models: arguably the most important models of decision-making under uncertainty. We show that Expected Utility and Choquet Expected Utility are learnable. Morever, the sample complexity of the former is linear, and of the latter exponential, in the number of states. This means that accurate estimation of Choquet Expected Utility may require very large sample sizes, while Expected Utility requires modest sample sizes. The Max-min Expected Utility model is learnable when there are two states, but not when there are three states or more. Our results exhibit a close relation between learnability and the axioms that characterise the model.

learnabilitypreference.pdf |

## 3) Repeated Coordination with Private Learning (*with Kalyan Chatterjee, Tetsuya Hoshino and Omer Tamuz) *

**Revised and Resubmit, Journal of Economic Theory**We study a repeated game with payoff externalities and observable actions where two players receive information over time about an underlying payoff-relevant state, and strategically coordinate their actions. Players learn about the true state from pri- vate signals, as well as the actions of others. They commonly learn the true state (Cripps et al., 2008), but do not coordinate in every equilibrium. We show that there exist stable equilibria in which players can overcome unfavorable signal realizations and eventually coordinate on the correct action, for any discount factor. For high discount factors, we show that in addition players can also achieve efficient payoffs.

Abstract :Abstract :

repeatedcoordinationgame.pdf |

## 4) On interim rationality, belief formation and learning in decision problems with bounded memory (*with Kalyan Chatterjee*)

**Economics Working Paper No.110, Institute for Advanced Study, School of Social Science, Princeton, NJ***We study the process of decision-making and inference by a single, boundedly rational, economic agent. The agent chooses either a safe or a risky alternative in each period after receiving a signal about the state of the world in that period. The state of the world is changing according to a Markov process with some degree of persistence across time. The agent's decision rule is expressed as a finite-state automaton with a fixed number of memory states. Updating on the basis of the received signal is, for such an agent, making a transition from one state to another. The finiteness of the number of automaton states automatically suggests that beliefs are classified into categories and a signal causes a (possible) change in the category on the basis of which the next action is taken. The problem is one in partially-observed Markov decision processes (POMDP). We characterise the structure of the optimal decision rule in this setting and show how its properties pin down the categories of beliefs and explain some observed, seemingly irrational behaviour. We then specialise to a fixed state of the world, weaken the optimality requirement to admissibility and derive the staircase structure of the admissible automaton. Finally we examine the question of randomisation in the design of an automaton, propose a measure of the extent of such randomisation and show that there exists a minimal degree of randomisation for the set of automata implementing a given strategy. We show that if the number of signals is large, virtually no randomisation is required.***Abstract :** boundedmemory.pdf |

## 5) Dynamic Bayesian Persuasion with a Privately Informed Receiver

*We study a dynamic Bayesian persuasion framework in a finite horizon setting con- sisting of a seller and a buyer. The seller wishes to persuade the buyer to buy a durable good at a given price by providing information about its relevance (match quality). The buyer has private information about his valuation for a good match and we study optimal dynamic information policies employed by the seller in equilibrium. For a fixed horizon, we show that the seller always provides signals which truthfully convey a good match but may garble a bad one. Moreover, if the good is not bought in the first stage, the seller provides information which improves over time. The agents always interact for a fixed amount of time within which a purchase decision is made. The length of this interaction remains fixed even for long horizons and depends only on the prior on the buyer’s valuation. As the horizon goes to infinity, the bulk of the in- formation about match quality is provided in the first period. This allows the buyer to extract a large amount of information from the seller at the beginning of their interaction. Even a slight probability of the buyer being difficult to convince facilitates close to full disclosure immediately.*

**Abstract :** dynbayesperspriv.pdf |

## 6) Ethnic Conflicts with Informed Agents: A Cheap Talk Game with Multiple Audiences and Private Signals (*with Souvik Dutta and Suraj Shekhar*)

*Consider a society where ethnic conflict is imminent due to people's belief about the state of the world. An 'informed agent' is a player who has private information about the state. We analyze whether the informed agent can achieve peace by sending private messages to a fraction of the players. If the informed agent is known to be biased towards her own ethnicity, she is unable to communicate credibly with the other ethnicity. Despite this, we show that peace can be achieved if the informed agent can communicate with many players.*

**Abstract :** ethnicconflicts.pdf |

**Publications :**

Bayesian Updating Rules and AGM Belief Revision

**Journal of Economic Theory,**Volume 179, January 2019