Working Papers

Stochastic Equilibria: Noise in Actions or Beliefs?

Much is known about the empirical content of quantal response equilibrium (QRE) which relaxes the rationality requirement of Nash equilibrium by allowing for “noise in actions” while maintaining that beliefs are correct. By contrast, little is understood of the testable restrictions of equilibrium models which allow for “noise in beliefs” while maintaining best response. We introduce noisy belief equilibrium (NBE) for normal form games in which axioms restrict belief distributions to be unbiased with respect to and responsive to changes in the opponents’ behavior. The axioms impose testable restrictions both within and across games, and we compare these restrictions to those of regular QRE (Goeree et al. 2005) in which axioms are placed on the quantal response function as the primitive. We show that NBE generates similar predictions as QRE such as the “own payoff effect”, and yet is more consistent with the empirically documented effects of changes in payoff magnitude. Unlike QRE, NBE is a refinement of rationalizability and invariant to affine transformations of payoffs.

We endogenize the precision parameter of logit quantal response equilibrium (LQRE) (McKelvey and Palfrey, 1995). In the first stage of an endogenous quantal response equilibrium (EQRE), each player chooses precision optimally subject to costs, given correct beliefs over other players’ (second-stage) actions. In the second stage, players’ actions form a heterogeneous LQRE given the first-stage choices of precision. EQRE satisfies a modified version of the regularity axioms (Goeree et al., 2005), nests LQRE as a limiting case for a sequence of cost functions, and admits analogues of classic results for LQRE such as those for equilibrium selection. For generalized matching pennies, the sets of EQRE and LQRE (i.e. indexed by their respective parameters) are curves in the unit square that cross at finite points that we give explicitly, and hence the models’ predictions are generically distinct. We apply EQRE to experimental data.

We study an equilibrium model in which players make stochastic choices given their beliefs

Several behavioral theories suggest that, when choosing between multi-attribute goods, choices are systematically affected by the range of values in each attribute. Two theories provide such predictions explicitly in terms of attribute ranges. According to the theory of Focusing (Koszegi and Szeidl, 2013), attributes with larger ranges receive more attention. On the other hand, Relative thinking (Bushong, Rabin, and Schwartzstein, 2017) posits that fixed differences look smaller when the range is large. It is as if attributes with larger ranges are over- and under-weighted, respectively. Since the two theories make opposing predictions, it is important to understand what features of the environment affect their relative prevalence. We conduct an experiment designed to test for both of these opposing range effects in different environments. Using choice under risk, we use a two-by-two design defined by high or low stakes and high or low dimensionality (as measured by the number of attributes). In the aggregate, we find evidence of focusing in low-dimensional treatments. Classifying subjects into focusers and relative thinkers, we find that focusers are associated with quicker response times and that types are more stable when the stakes are high.

The Cost of Capital of the Financial Sector, with Tobias Adrian and Tyler Muir

Standard factor pricing models do not capture the common time series or cross sectional variation in average returns of financial stocks well. We propose a five factor asset pricing model that complements the standard Fama and French (1993) three factor model with a financial sector ROE factor (FROE) and the spread between the financial sector and the market return (SPREAD). This five factor model helps to alleviate the pricing anomalies for financial sector stocks and also performs well for nonfinancial sector stocks when compared to the Fama and French (2014) five factor or the Hou, Xue, and Zhang (2014) four factor models. We find the aggregate expected return to financial sector equities to correlate negatively with aggregate financial sector ROE, which is puzzling, as ROE is commonly used as a measure of the cost of capital in the financial sector.

*Revise and resubmit,***AEJ: Micro**Much is known about the empirical content of quantal response equilibrium (QRE) which relaxes the rationality requirement of Nash equilibrium by allowing for “noise in actions” while maintaining that beliefs are correct. By contrast, little is understood of the testable restrictions of equilibrium models which allow for “noise in beliefs” while maintaining best response. We introduce noisy belief equilibrium (NBE) for normal form games in which axioms restrict belief distributions to be unbiased with respect to and responsive to changes in the opponents’ behavior. The axioms impose testable restrictions both within and across games, and we compare these restrictions to those of regular QRE (Goeree et al. 2005) in which axioms are placed on the quantal response function as the primitive. We show that NBE generates similar predictions as QRE such as the “own payoff effect”, and yet is more consistent with the empirically documented effects of changes in payoff magnitude. Unlike QRE, NBE is a refinement of rationalizability and invariant to affine transformations of payoffs.

__Endogenous Quantal Response Equilibrium__*Revise and resubmit,***Games and Economic Behavior**We endogenize the precision parameter of logit quantal response equilibrium (LQRE) (McKelvey and Palfrey, 1995). In the first stage of an endogenous quantal response equilibrium (EQRE), each player chooses precision optimally subject to costs, given correct beliefs over other players’ (second-stage) actions. In the second stage, players’ actions form a heterogeneous LQRE given the first-stage choices of precision. EQRE satisfies a modified version of the regularity axioms (Goeree et al., 2005), nests LQRE as a limiting case for a sequence of cost functions, and admits analogues of classic results for LQRE such as those for equilibrium selection. For generalized matching pennies, the sets of EQRE and LQRE (i.e. indexed by their respective parameters) are curves in the unit square that cross at finite points that we give explicitly, and hence the models’ predictions are generically distinct. We apply EQRE to experimental data.

__Stochastic Choice and Noisy Beliefs in Games: an Experiment__, with Jeremy WardWe study an equilibrium model in which players make stochastic choices given their beliefs

*and*there is noise in the beliefs themselves. The model primitives are an action-map, which determines a distribution of actions given beliefs, and a belief-map, which determines a distribution of beliefs given opponents' behavior. These are restricted to satisfy axioms that are stochastic generalizations of “best response” and “correct beliefs”, respectively. In our laboratory experiment, we collect actions data and elicit beliefs for each game within a family of asymmetric 2-player games. These games have systematically varied payoffs, allowing us to “trace out” both the action- and belief-maps. We find that, while both “noise in actions” and “noise in beliefs” are important in explaining observed behaviors, there are systematic violations of the axioms. In particular, although all subjects observe and play the same games, subjects in different roles have qualitatively different belief biases. To explain this, we argue that the player role itself induces a higher degree of strategic sophistication in the player who faces more asymmetric payoffs. This is confirmed by structural estimates.__Range Effects in Multi-Attribute Choice: an Experiment__, with Tommaso Bondi and Dániel CsabaSeveral behavioral theories suggest that, when choosing between multi-attribute goods, choices are systematically affected by the range of values in each attribute. Two theories provide such predictions explicitly in terms of attribute ranges. According to the theory of Focusing (Koszegi and Szeidl, 2013), attributes with larger ranges receive more attention. On the other hand, Relative thinking (Bushong, Rabin, and Schwartzstein, 2017) posits that fixed differences look smaller when the range is large. It is as if attributes with larger ranges are over- and under-weighted, respectively. Since the two theories make opposing predictions, it is important to understand what features of the environment affect their relative prevalence. We conduct an experiment designed to test for both of these opposing range effects in different environments. Using choice under risk, we use a two-by-two design defined by high or low stakes and high or low dimensionality (as measured by the number of attributes). In the aggregate, we find evidence of focusing in low-dimensional treatments. Classifying subjects into focusers and relative thinkers, we find that focusers are associated with quicker response times and that types are more stable when the stakes are high.

The Cost of Capital of the Financial Sector, with Tobias Adrian and Tyler Muir

Standard factor pricing models do not capture the common time series or cross sectional variation in average returns of financial stocks well. We propose a five factor asset pricing model that complements the standard Fama and French (1993) three factor model with a financial sector ROE factor (FROE) and the spread between the financial sector and the market return (SPREAD). This five factor model helps to alleviate the pricing anomalies for financial sector stocks and also performs well for nonfinancial sector stocks when compared to the Fama and French (2014) five factor or the Hou, Xue, and Zhang (2014) four factor models. We find the aggregate expected return to financial sector equities to correlate negatively with aggregate financial sector ROE, which is puzzling, as ROE is commonly used as a measure of the cost of capital in the financial sector.

Works in Progress

__Incomplete Preferences and Preference for Flexibility__, with Marina Agranov, Mark Dean, Han Huynh, and Pietro Ortoleva

__Mediating Conflict in the Lab__, with Alessandra Casella and Manuel Perez Archila

__Learning to Ignore Non-rationalizable Actions__, with Jeremy Ward and Dilip Ravindran