Learning by Similarity in Coordination Problems1
job market paper
Abstract
We study a learning process in which subjects extrapolate their experience from similar
past strategic situations to the current decision problem. When applied to coordination
games, this learning process leads to contagion of behavior from problems with extreme
payoffs and unique equilibria to very dissimilar problems. In the long-run, contagion results
in unique behavior even though there are multiple equilibria when the games are analyzed in
isolation. Characterization of the long-run state is based on a formal parallel to rational
equilibria of games with subjective priors. The results of contagion due to learning share the
qualitative features of those from contagion due to incomplete information, but quantitatively
they differ.
Keywords: Similarity, learning, contagion, case-based reasoning, global games,
coordination, subjective priors.
1 Introduction
In standard models of learning, players repeatedly interact in the same game, and use their
experience from the history of play to myopically optimize in each period. In many cases of
interest, decision-makers are faced with many different strategic situations, and the number
of possibilities is so vast that a particular situation is virtually never experienced twice. The
history of play may nonetheless be informative when choosing an action, as previous
situations, though different, may be similar to the current one. A tacit assumption of standard
learning models is that players extrapolate their experience from previous interactions
similar to the current one.
1 We are grateful to Philippe Jehiel, George Mailath, Stephen Morris, Ben Polak, Larry Samuelson, Avner
Shaked, organizers of the VI Trento Summer School in Adaptive Economic Dynamics, and seminar participants
at the University of Edinburgh, PSE Paris, Stanford University, Yale University, and the Econometric Society
meetings in Minneapolis and Vienna. Jakub Steiner benefited from the grant “Stability of the Global Financial
System: Regulation and Policy Response” during his research stay at LSE.
1
, The central message of this paper is that such extrapolation has important effects:
similarity-based learning can lead to contagion of behavior across very different strategic
situations. Two situations that are not directly similar may be connected by a chain of
intermediate situations, along which each is similar to the neighboring ones. One effect of this
contagion is to select a unique long-run action in situations that would allow for multiple
steady states if analyzed in isolation. For this to occur, the extrapolations at each step of the
similarity-based learning process need not be large; in fact, the contagion effect remains even
in the limit as extrapolation is based only on increasingly similar situations.
We focus here on the application of similarity-based learning to coordination games.
Consider, as an example, the class of 2×2 games Γ( θ) in Table 1 parameterized by a
fundamental, θ. Action I, interpreted as investing, is strategically risky, as its payoff depends
on the action of the opponent. The safe action, NI, gives a constant payoff of 0. For extreme
values of θ, the game Γ(θ) has a unique equilibrium as investing is dominant for θ > 1, and the
safe action is dominant for θ < 0. When θ lies in the interval (0,1), the game has two strict
pure
strategy equilibria.
The contagion effect can be sketched without fully specifying the learning process, which
we postpone to Section 3. Two myopic players interact in many rounds in a game Γ( θt), with
θt selected at random in each round. Roughly, we assume that players estimate payoffs for the
game Γ(θ) on the basis of past experience with fundamentals similar to θ, and that two games
Γ(θ) and Γ(θ′) are viewed by players as similar if the difference |θ − θ′| is small.
I NI
I θ,θ θ − 1,0
NI 0,θ − 1 0,0
Table 1: Payoffs in the Example of Section 2.
Since investing is dominant for all sufficiently high fundamentals, there is some θ above
which players eventually learn to invest. Now consider a fundamental just below θ, say
2
, θ − ε. At θ − ε, investing may not be dominant, but players view some games with values of θ
above θ as similar. Since the opponent has learned to invest in these games, strategic
complementarities in payoffs increase the gain from investing. When ε is small, this increase
outweighs the potential loss from investing in games below θ, where the opponent may not
invest. Thus players learn to invest in games with fundamentals below, but close to θ, giving
a new threshold θ′ above which both players invest.
Repeating the argument with θ replaced by θ′, investment continues to spread to games
with smaller fundamentals, even though these are not directly similar to games in the
dominance region. The process continues until a threshold fundamental θ is reached at which
the gain from investment by the opponent above θ is exactly balanced by the loss from
noninvestment by the opponent below θ. Not investing spreads contagiously beginning from
low values of the fundamental by a symmetric process. These processes meet at the same
threshold, giving rise to a unique long-run outcome, provided that similarity drops off quickly
in
distance.2
Contagion effects have previously been studied in local interaction and incomplete
information games. In local interaction models, actions may spread contagiously across
members of a population because each has an incentive to coordinate with her neighbors in a
social network (e.g. Morris (2000)). In incomplete information games with strategic
complementarities (global games), actions may spread contagiously across types because
private information gives rise to uncertainty about the actions of other players (Carlsson and
van Damme 1993). Unlike these models, contagion through learning depends neither on any
network structure nor on high orders of reasoning about the beliefs of other players. The
contagion is driven solely by a natural solution to the problem of learning one’s own payoffs
when the strategic situation is continually changing. This problem is familiar from
econometrics, where one often wishes to estimate a function of a continuous variable using
only a finite data set. The similarity-based payoff estimates used by players in our model have
2 In other words, players place much more weight on values of the fundamental very close to the present
one when forming their payoff estimates.
3
, a direct parallel in the use of kernel estimators by econometricians. Moreover, the use of such
estimates for choosing actions is consistent with the case-based decision theory of Gilboa and
Schmeidler (2001), who propose similarity-weighted payoff averaging as a general theory of
decisions under un-
certainty.
While the learning model we have described is one of complete information, the same
reasoning applies when, as in the global game model, players imperfectly observe the value of
the fundamental. In order to directly compare the process of contagion through learning to
that from incomplete information, players in the general model of Section 3 observe private
signals of the fundamental that may be noisy. The fundamental and signals are independently
drawn in each round. From the history of play, players have experience with realized payoffs
for signals similar to, but different from, their current signal. They estimate the current
payoffs based on the payoffs of similar types in the past.
The main tool for understanding the result of contagion through learning is a formal
parallel to rational play in a modified version of the game. This modified game differs from the
original game only in the priors: players eventually behave as if they incorrectly believe their
own signal to be more noisy than it actually is, while holding correct beliefs about the
precision of the other players’ signals. More precisely, players learn not to play strategies that
would be serially dominated in the modified version of the game (see Theorem 3.1).
This result enables us to solve the modified game by extending the techniques of Carlsson
and van Damme (1993), further developed by Morris and Shin (2003). With complete
information, the original game has a continuum of equilibria, but contagion leads to a unique
learning outcome when similarity is concentrated on nearby fundamentals. With small noise
in observations of the fundamental, the underlying game has a unique equilibrium as a result
of contagion from incomplete information. In this case, there is also a unique learning
outcome when similarity is concentrated, but this outcome depends on the relative size of the
noise compared to the concentration of the similarity. In particular, the process of contagion
through learning does not generally coincide with that of contagion from incomplete
information. However, the qualitative features of these processes agree, as both converge to
play of symmetric threshold strategies, and give rise to comparative statics of the same sign.
4
job market paper
Abstract
We study a learning process in which subjects extrapolate their experience from similar
past strategic situations to the current decision problem. When applied to coordination
games, this learning process leads to contagion of behavior from problems with extreme
payoffs and unique equilibria to very dissimilar problems. In the long-run, contagion results
in unique behavior even though there are multiple equilibria when the games are analyzed in
isolation. Characterization of the long-run state is based on a formal parallel to rational
equilibria of games with subjective priors. The results of contagion due to learning share the
qualitative features of those from contagion due to incomplete information, but quantitatively
they differ.
Keywords: Similarity, learning, contagion, case-based reasoning, global games,
coordination, subjective priors.
1 Introduction
In standard models of learning, players repeatedly interact in the same game, and use their
experience from the history of play to myopically optimize in each period. In many cases of
interest, decision-makers are faced with many different strategic situations, and the number
of possibilities is so vast that a particular situation is virtually never experienced twice. The
history of play may nonetheless be informative when choosing an action, as previous
situations, though different, may be similar to the current one. A tacit assumption of standard
learning models is that players extrapolate their experience from previous interactions
similar to the current one.
1 We are grateful to Philippe Jehiel, George Mailath, Stephen Morris, Ben Polak, Larry Samuelson, Avner
Shaked, organizers of the VI Trento Summer School in Adaptive Economic Dynamics, and seminar participants
at the University of Edinburgh, PSE Paris, Stanford University, Yale University, and the Econometric Society
meetings in Minneapolis and Vienna. Jakub Steiner benefited from the grant “Stability of the Global Financial
System: Regulation and Policy Response” during his research stay at LSE.
1
, The central message of this paper is that such extrapolation has important effects:
similarity-based learning can lead to contagion of behavior across very different strategic
situations. Two situations that are not directly similar may be connected by a chain of
intermediate situations, along which each is similar to the neighboring ones. One effect of this
contagion is to select a unique long-run action in situations that would allow for multiple
steady states if analyzed in isolation. For this to occur, the extrapolations at each step of the
similarity-based learning process need not be large; in fact, the contagion effect remains even
in the limit as extrapolation is based only on increasingly similar situations.
We focus here on the application of similarity-based learning to coordination games.
Consider, as an example, the class of 2×2 games Γ( θ) in Table 1 parameterized by a
fundamental, θ. Action I, interpreted as investing, is strategically risky, as its payoff depends
on the action of the opponent. The safe action, NI, gives a constant payoff of 0. For extreme
values of θ, the game Γ(θ) has a unique equilibrium as investing is dominant for θ > 1, and the
safe action is dominant for θ < 0. When θ lies in the interval (0,1), the game has two strict
pure
strategy equilibria.
The contagion effect can be sketched without fully specifying the learning process, which
we postpone to Section 3. Two myopic players interact in many rounds in a game Γ( θt), with
θt selected at random in each round. Roughly, we assume that players estimate payoffs for the
game Γ(θ) on the basis of past experience with fundamentals similar to θ, and that two games
Γ(θ) and Γ(θ′) are viewed by players as similar if the difference |θ − θ′| is small.
I NI
I θ,θ θ − 1,0
NI 0,θ − 1 0,0
Table 1: Payoffs in the Example of Section 2.
Since investing is dominant for all sufficiently high fundamentals, there is some θ above
which players eventually learn to invest. Now consider a fundamental just below θ, say
2
, θ − ε. At θ − ε, investing may not be dominant, but players view some games with values of θ
above θ as similar. Since the opponent has learned to invest in these games, strategic
complementarities in payoffs increase the gain from investing. When ε is small, this increase
outweighs the potential loss from investing in games below θ, where the opponent may not
invest. Thus players learn to invest in games with fundamentals below, but close to θ, giving
a new threshold θ′ above which both players invest.
Repeating the argument with θ replaced by θ′, investment continues to spread to games
with smaller fundamentals, even though these are not directly similar to games in the
dominance region. The process continues until a threshold fundamental θ is reached at which
the gain from investment by the opponent above θ is exactly balanced by the loss from
noninvestment by the opponent below θ. Not investing spreads contagiously beginning from
low values of the fundamental by a symmetric process. These processes meet at the same
threshold, giving rise to a unique long-run outcome, provided that similarity drops off quickly
in
distance.2
Contagion effects have previously been studied in local interaction and incomplete
information games. In local interaction models, actions may spread contagiously across
members of a population because each has an incentive to coordinate with her neighbors in a
social network (e.g. Morris (2000)). In incomplete information games with strategic
complementarities (global games), actions may spread contagiously across types because
private information gives rise to uncertainty about the actions of other players (Carlsson and
van Damme 1993). Unlike these models, contagion through learning depends neither on any
network structure nor on high orders of reasoning about the beliefs of other players. The
contagion is driven solely by a natural solution to the problem of learning one’s own payoffs
when the strategic situation is continually changing. This problem is familiar from
econometrics, where one often wishes to estimate a function of a continuous variable using
only a finite data set. The similarity-based payoff estimates used by players in our model have
2 In other words, players place much more weight on values of the fundamental very close to the present
one when forming their payoff estimates.
3
, a direct parallel in the use of kernel estimators by econometricians. Moreover, the use of such
estimates for choosing actions is consistent with the case-based decision theory of Gilboa and
Schmeidler (2001), who propose similarity-weighted payoff averaging as a general theory of
decisions under un-
certainty.
While the learning model we have described is one of complete information, the same
reasoning applies when, as in the global game model, players imperfectly observe the value of
the fundamental. In order to directly compare the process of contagion through learning to
that from incomplete information, players in the general model of Section 3 observe private
signals of the fundamental that may be noisy. The fundamental and signals are independently
drawn in each round. From the history of play, players have experience with realized payoffs
for signals similar to, but different from, their current signal. They estimate the current
payoffs based on the payoffs of similar types in the past.
The main tool for understanding the result of contagion through learning is a formal
parallel to rational play in a modified version of the game. This modified game differs from the
original game only in the priors: players eventually behave as if they incorrectly believe their
own signal to be more noisy than it actually is, while holding correct beliefs about the
precision of the other players’ signals. More precisely, players learn not to play strategies that
would be serially dominated in the modified version of the game (see Theorem 3.1).
This result enables us to solve the modified game by extending the techniques of Carlsson
and van Damme (1993), further developed by Morris and Shin (2003). With complete
information, the original game has a continuum of equilibria, but contagion leads to a unique
learning outcome when similarity is concentrated on nearby fundamentals. With small noise
in observations of the fundamental, the underlying game has a unique equilibrium as a result
of contagion from incomplete information. In this case, there is also a unique learning
outcome when similarity is concentrated, but this outcome depends on the relative size of the
noise compared to the concentration of the similarity. In particular, the process of contagion
through learning does not generally coincide with that of contagion from incomplete
information. However, the qualitative features of these processes agree, as both converge to
play of symmetric threshold strategies, and give rise to comparative statics of the same sign.
4
