1) Learnability and Models of Decision Making under Uncertainty (with Federico Echenique)
Proceedings of the 19th ACM Conference on Economics and Computation, 2018
Abstract : We study whether some of the most important models of decision-making under uncertainty are uniformly learnable, in the sense of PAC (probably approximately correct) learnability. Many studies in economics rely on Savage's model of (subjective) expected utility. The expected utility model is known to predict behavior that runs counter to how many agents actually make decisions (the contradiction usually takes the form of agents' choices in the Ellsberg paradox). As a consequence, economists have developed models of choice under uncertainty that seek to generalize the basic expected utility model. The resulting models are more general and therefore more flexible, and more prone to overfitting. The purpose of our paper is to understand this added flexibility better. We focus on the classical expected utility (EU) model, and its two most important generalizations: Choquet expected utility (CEU) and Max-min Expected Utility (MEU). Our setting involves an analyst whose task is to estimate or learn an agent's preference based on data available on the agent's choices. A model of preferences is PAC learnable if the analyst can construct a learning rule to precisely learn the agent's preference with enough data. When a model is not learnable we interpret it as the model being susceptible to overfitting. PAC learnability is known to be characterized by the model's VC dimension: thus our paper takes the form of a study of the VC dimension of economic models of choice under uncertainty. We show that EU and CEU have finite VC dimension, and are consequently learnable. Morever, the sample complexity of the former is linear, and of the latter is exponential, in the number of states of uncertainty. The MEU model is learnable when there are two states but is not learnable when there are at least three states, in which case the VC dimension is infinite. Our results also exhibit a close relationship between learnability and the underlying axioms which characterise the model.
2) Repeated Coordination with Private Learning (with Kalyan Chatterjee, Tetsuya Hoshino and Omer Tamuz)
Abstract : 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 private 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 it is possible to construct equilibria in which players 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.
3) 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
Abstract : 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.
4) Bayesian Updating Rules and AGM Belief Revision
Revise and Resubmit, Journal of Economic Theory
Abstract : We interpret the problem of updating beliefs as a choice problem (selecting a posterior from a set of admissible posteriors) with a reference point (prior). We use AGM belief revision to define the support of admissible posteriors after observing zero probability events and investigate two classes of updating rules for probabilities : 1) "minimum distance" updating rules which select the posterior closest to the prior by some metric. 2) "lexicographic" updating rules where posteriors are given by a lexicographic probability system. For the former, we show bayesian updating as a special case and for specific AGM belief revisions, provide necessary and sufficient conditions for a minimum distance representation. For the latter, we show that an updating rule is lexicographic if and only if it is bayesian, AGM-consistent and satisfies a weak form of path independence. Lastly, we study a sub-class of lexicographic up- dating rules, which we call ”support-dependent” rules. We show that such updating rules have a minimum distance representation.
5) Dynamic Bayesian Persuasion with a Privately Informed Receiver
Abstract : 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.
6) Ethnic Conflicts with Informed Agents: A Cheap Talk Game with Multiple Audiences and Private Signals (with Souvik Dutta and Suraj Shekhar)
Abstract : 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.