Multiple equilibria and selection by learning in an applied setting

doi:10.1016/j.econlet.2009.03.004

Economics Letters

Volume 104, Issue 1, July 2009, Pages 13-16

https://doi.org/10.1016/j.econlet.2009.03.004 Get rights and content

Abstract

We explore two complementary approaches to counterfactual analysis in an empirical ATM network example with multiple equilibria. First we simply enumerate and compare the possible equilibria. Second, we examine how different learning algorithms select among them.

Introduction

One reason for obtaining detailed empirical estimates of the primitives underlying market outcomes is to enable a realistic analysis of the likely impact of policy and environmental changes. The analysis of these changes, or “counterfactuals,” when there are multiple possible equilibria poses a problem to applied researchers: the model will not generate a unique counterfactual prediction. If the change in primitives moves the market to a state not observed in the past, the available data cannot provide an answer to the choice among possible equilibria. Indeed, even if the counterfactual moves the market to a state that has been observed and assumptions which allow us to use the data to identify which of the possible equilibria were played in the past are viewed as acceptable, we can not rule out the possibility that the change in primitives will change the equilibrium selection mechanism. Moreover the appropriate selection mechanism is likely to depend on historical events the researcher does not have information on.¹

This paper explores two potentially complementary approaches to addressing the problem of multiple equilibria in counterfactual analysis. We do so in the context of a particular application; a network example drawn from Ishii (2005). The example is a simultaneous move game where competing banks within a local market choose the number of ATM machines to install, with payoffs to the game generated from Ishii's estimated structural model. We propose a merger of two banks which is accompanied by an unexpected cost shock of running an ATM, and analyze the reallocation of ATMs among the remaining banks. There are multiple potential equilibria of this game.

We begin by simply enumerating all the potential equilibria. Although there are nearly 200,000 possible allocations, we find that at most three of them constitute equilibria for the specifications of primitives we investigate. The equilibria generated by a given specification are very similar to one another in that the number of ATMs operated by each bank differ across equilibria by at most one. Further the “comparative static” relationships between the sets of equilibria when we vary the specification make economic sense — e.g., if an allocation becomes a new equilibrium when costs of running an ATM are lower and another allocation no longer is an equilibrium, the new equilibrium always has a larger total number of ATMs than the other. This indicates that, when feasible, enumeration may lead to useful bounds on possible counterfactual results.

Next we explore the implications of using different learning algorithms to “select out” equilibria. That is, we model the process by which agents adjust over time to changes in their environment, and then follow those adjustments until no agents wishes to deviate from their chosen strategies. Though much theory has been written on learning models and their properties (c.f. Fudenberg and Levine, 1998, Young, 2004, and the literature cited therein), there has been little if any experience with applying them to choose among equilibria. We examine both a best-response dynamic where agents choose the best reaction to the actions of its competitors in the previous period, and a fictitious play variant where agents instead play a best response to the distribution of previous play by competitors. We find that the variance in the cost shocks can cause a distribution of rest points from a given initial condition, and that distribution has a notable dependence on both the cost specification and on the learning process. Thus if we are to use learning algorithms to improve on the bounds obtained from enumeration, it would be helpful to obtain some evidence on which learning processes are more relevant in which environments.²

Section snippets

A network example

We focus on the competition between banks for consumers via ATM networks studied in (Ishii, 2005). Ishii analyzes the choice of ATMs by banks and the impact of those choices on consumer banking decisions and equilibrium interest rates. In the process she provides detailed estimates of the primitives from a two period model which allows her to construct the profits of each bank conditional on any set of ATM networks for all banks in a number of markets. We take the information she provides on

Number and nature of equilibria

The first part of the analysis proceeds by simply enumerating the “limiting equilibria”: i.e., the Nash equilibria when all firms know the expected value of the cost shock. Since banks are asymmetric, there are 170,544 different allocations of up to 15 ATMs among seven banks.⁵ Table 1 lists all equilibrium allocations when firms know the expected value of the per-period cost shock. We report results for

Equilibrium selection by learning

The second part of our analysis examines the implications of assuming that different learning processes govern the adjustment of firms to changes in their environment. Here, the changes comprise a merger accompanied by a cost shock. Motivated by our application to modeling a counterfactual, we assume firms begin in the initial post merger allocation of ATMs, and adjust from there.

We assume that each firm expects its own costs to be drawn from a distribution whose mean was equal to the sample

Concluding remarks

We considered two approaches for analyzing counterfactuals in a particular applied problem in which multiple equilibria are possible. The first simply enumerated the total number of equilibria and examined their relationship to one another. To the extent that our findings here are indicative of what might happen in other applied problems, they are good news for the applied researcher. The small number of equilibria should make it feasible to do the enumeration, and the fact that the equilibria

References (8)

Allcott, H., 2009, Real Time Pricing and Electricity Markets....
FudenbergD. et al.
The Theory of Learning in Games
(1998)
Ishii, J., 2005, Interconnection Pricing, Compatibility, and Investment in Network Industries: ATM Networks in the...
JiaP.
What Happens When Wal-Mart Comes to Town: An Empirical Analysis of the Discount Industry
Econometrica
(2008)

There are more references available in the full text version of this article.

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^☆: We are grateful to Joy Ishii for supplying the necessary data.

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Multiple equilibria and selection by learning in an applied setting☆