On Similarities between Inference in Game Theory and Machine Learning

In this paper, we elucidate the equivalence between inference in game theory and machine learning. Our aim in so doing is to establish an equivalent vocabulary between the two domains so as to facilitate developments at the intersection of both fields, and as proof of the usefulness of this approach, we use recent developments in each field to make useful improvements to the other. More specifically, we consider the analogies between smooth best responses in fictitious play and Bayesian inference methods. Initially, we use these insights to develop and demonstrate an improved algorithm for learning in games based on probabilistic moderation. That is, by integrating over the distribution of opponent strategies (a Bayesian approach within machine learning) rather than taking a simple empirical average (the approach used in standard fictitious play) we derive a novel moderated fictitious play algorithm and show that it is more likely than standard fictitious play to converge to a payoff-dominant but risk-dominated Nash equilibrium in a simple coordination game. Furthermore we consider the converse case, and show how insights from game theory can be used to derive two improved mean field variational learning algorithms. We first show that the standard update rule of mean field variational learning is analogous to a Cournot adjustment within game theory. By analogy with fictitious play, we then suggest an improved update rule, and show that this results in fictitious variational play, an improved mean field variational learning algorithm that exhibits better convergence in highly or strongly connected graphical models. Second, we use a recent advance in fictitious play, namely dynamic fictitious play, to derive a derivative action variational learning algorithm, that exhibits superior convergence properties on a canonical machine learning problem (clustering a mixture distribution)

Cournot Competition and Endogenous Firm Size

Barr and Saraceno (JEDC, forthcoming) model the firm as a type of artificial neural network (ANN) which plays a repeated Cournot game. Each period, the network/firm must estimate the relationship between environmental conditions and optimal output. Among other results, the paper develops the notion of a Network Size Equilibrium (NSE): which is an optimal network size for each of the players. The concept of NSE allows us to map environmental complexity to a type of industrial structure, i.e., the average network size in equilibrium. This paper builds on the previous work by exploring the dynamic adjustment process of networks. That is to say, we explore how the network (firm) evolves over time in reaction to the environmental complexity and the behavior of its rival. We model how firms endogenously "grow" over time in the adjustment process toward a network size equilibrium by exploring different adjustment algorithms, which may involve different costs. Further we explore the stability and the types of equilibria that can emerge, given different environmental scenarios.Cournot Competition, Neural Networks, Adjustment Dynamics

Imitation - Theory and Experimental Evidence

We introduce a generalized theoretical approach to study imitation and subject it to rigorous experimental testing. In our theoretical analysis we find that the different predictions of previous imitation models are due to different informational assumptions, not to different behavioral rules. It is more important whom one imitates rather than how. In a laboratory experiment we test the different theories by systematically varying information conditions. We find significant effects of seemingly innocent changes in information. Moreover, the generalized imitation model predicts the differences between treatments well. The data provide support for imitation on the individual level, both in terms of choice and in terms of perception. But imitation is not unconditional. Rather individuals' propensity to imitate more successful actions is increasing in payoff differences

Using problem-based learning for introducing producer theory and market structure in intermediate microeconomics

This paper shows how instructors can use the problem-based learning method to introduce producer theory and market structure in intermediate microeconomics courses. The paper proposes a framework where different decision problems are presented to students, who are asked to imagine that they are the managers of a firm who need to solve a problem in a particular business setting. In this setting, the instructors’ role is to provide both guidance to facilitate student learning and content knowledge on a just-in-time basis.

Imitation - Theory and Experimental Evidence

We introduce a generalized theoretical approach to study imitation and subject it to rigorous experimental testing. In our theoretical analysis we find that the different predictions of previous imitation models are due to different informational assumptions, not to different behavioral rules. It is more important whom one imitates rather than how. In a laboratory experiment we test the different theories by systematically varying information conditions. We find significant effects of seemingly innocent changes in information. Moreover, the generalized imitation model predicts the differences between treatments well. The data provide support for imitation on the individual level, both in terms of choice and in terms of perception. But imitation is not unconditional. Rather individuals' propensity to imitate more successful actions is increasing in payoff differences.Evolutionary game theory; Stochastic stability; Imitation; Cournot markets; Information; Experiments; Simulations

Imitation - Theory and Experimental Evidence

We introduce a generalized theoretical approach to study imitation models and subject themodels to rigorous experimental testing. In our theoretical analysis we find that the differentpredictions of previous imitation models are due to different informational assumptions, notto different behavioral rules. It is more important whom one imitates rather than how. In alaboratory experiment we test the different theories by systematically varying informationconditions. We find that the generalized imitation model predicts the differences betweentreatments well. The data also provide support for imitation on the individual level, both interms of choice and in terms of perception. But imitation is not unconditional. Ratherindividuals’ propensity to imitate more successful actions is increasing in payoff differences.evolutionary game theory, stochastic stability, imitation, Cournot markets,experiments

Dynamic Price Competition with Price Adjustment Costs and Product Differentiation

We study a discrete time dynamic game of price competition with spatially differentiated products and price adjustment costs. We characterise the Markov perfect and the open-loop equilibrium of our game. We find that in the steady state Markov perfect equilibrium, given the presence of adjustment costs, equilibrium prices are always higher than prices at the repeated static Nash solution, even though, adjustment costs are not paid in steady state. This is due to intertemporal strategic complementarity in the strategies of the firms and from the fact that the cost of adjusting prices adds credibility to high price equilibrium strategies. On the other hand, the stationary open-loop equilibrium coincides always with the static solution. Furthermore, in contrast to continuous time games, we show that the stationary Markov perfect equilibrium converges to the static Nash equilibrium when adjustment costs tend to zero. Moreover, we obtain the same convergence result when adjustment costs tend to infinity.Price adjustment costs, Difference game, Markov perfect equilibrium, Open-loop equilibrium

Imitation - Theory and Experimental Evidence

We introduce a generalized theoretical approach to study imitation models and subject the models to rigorous experimental testing. In our theoretical analysis we find that the different predictions of previous imitation models are due to different informational assumptions, not to different behavioral rules. It is more important whom one imitates rather than how. In a laboratory experiment we test the different theories by systematically varying information conditions. We find that the generalized imitation model predicts the differences between treatments well. The data also provide support for imitation on the individual level, both in terms of choice and in terms of perception. But imitation is not unconditional. Rather individuals' propensity to imitate more successful actions is increasing in payoff differences.Evolutionary game theory; Stochastic stability; Imitation; Cournot markets; Experiments