Based on a plenary talk presented at the 1996 Summer Meetings of the Econometric Society. We thank Ken Binmore and Al Roth for helpful comments and discussion.
Drew Fudenberg and David K. Levine
June 26, 1996; revised July 30, 1996
This essay has two goals. First, it is an
overview of the field to give an idea of what has been going on
recently. Second, it is an editorial for our own views of how
future research ought to proceed.
The basic research agenda is to explore
non-equilibrium explanations of equilibrium in games; to view
equilibrium as the long-run outcome of a dynamic process of
adjustment or learning. In other words, equilibrium concepts
should be interpreted as steady states or ergodic distributions
of a dynamic process. This essay focuses on theoretical models of
learning, but is informed by and addressed to experimenters as
well, and we have suggestions about how each group
"ought" to proceed.
In our view, game theorists ought to pay more
attention to game theory experiments; for example, the literature
on refinements of Nash equilibrium proceeded based almost
entirely on introspection and theoretical concerns. Laboratory
experiments may not be perfect tests of theories, but they are
much better than no tests at all. Our second message to theorists
is that more attention should be paid to the "medium
run" properties of learning models: Most work, including our
own, has focused on asymptotic results, which tend to be easier
to obtain.
Of course, any experiment in which subjects
play more than one round at least implicitly involves learning,
and indeed most experiments focus on play in the last few rounds
of the experiment, when subjects have had the most time to learn.
Given this heavy implicit reliance on the idea of learning, we
feel that experimenters ought to pay more attention to the
relevant learning theory. For example, a variety of learning
models have been studied by experimental economists without much
concern for the extent to which the models do a good job of
learning. Although we are prepared to be convinced otherwise by
sufficiently strong data, our prior belief is that in
environments where simple learning rules can do well, agents will
tend not to use rules that do poorly. For this reason we are
skeptical of claims that very simple models fit the data well
unless those models are compared with medium-complexity models
that do a better job of learning.
From a theorist's perspective, there are three
criteria for good experiments: interesting questions, appropriate
theoretical foundations, and careful experimental design. In our
opinion, too many experiments focus on careful experimental
design, changing only a few treatment variables from a past
experiment in order to isolate their effect, and too few
experiments try to answer questions of theoretical interest. Of
course, this procedure is useful for identifying the reasons that
outcomes differ between the treatments, but we believe that more
attention to theoretical issues would help experimental economics
move beyond laundry-lists of treatment effects in various games.
Instead of drawing a t distinction between models of "learning" and other forms of adjustment, it is easier to make a distinction between models that describe behavior of individual agents and those that start at the aggregate level. Consequently, the real taxonomy is not between "learning" and "evolutionary" models, but rather between "individual level" models and "aggregate" models. This essay focuses on individual level models, but we would like to make a few comments on aggregate models before we begin.
An aggregate model starts with a description of
aggregate behavior of a population of agents. An example of this
is the "replicator dynamic" that postulates that the
fraction of the population playing a strategy increases if the
utility received from that strategy is above average. There are
two main reasons that economists are interested in the replicator
and related models.
First, although the replicator dynamics was
originally motivated by biological evolution, it can be derived
from various sorts of individual-level models. One such model is
the stimulus-response model of Borgers and Sarin [1994] which
postulates that actions that perform well are "positively
reinforced" and so more likely to be played in the future.
An alternative model is the "emulation" model in which
players copy their more successful predecessor, as in Binmore and
Samuelson [1995], Bjonerstedt and Weibull [1995], or Schlag
[1995]. Note, however, that the "emulation" stories are
not applicable to most experiments, because subjects are not told
what happens in other matches. These emulation models are very
similar to non-equilibrium models of "social learning,"
where players are trying to learn what technology, crop or brand
is best, such as Kirman [1993] and Ellison and Fudenberg
[1993,1995]. The links between emulation and social learning
models needs further exploration.
A second reason the replicator dynamic is of
interest is because some of its the properties of the replicator
dynamic extend to classes of more general processes that
correspond to other sorts of learning or emulation.
Samuelson-Zhang[1992] and Hofbauer and Weibull [1995] have
results along these lines, giving sufficient conditions for a
dynamic process to eliminate all strategies that are eliminated
by iterated strict dominance.
The focus of the remainder of the essay is
individual level models, and more specifically on variants of
"fictitious play" in two players games This is not a
comprehensive survey and many relevant papers have been left out
).. Moreover, we will suppose that players observe the strategies
played by their opponents at the end of each round. If the game
has a non-trivial extensive form, this implies that players see
how opponents would have played at information sets that were not
reached.
The first question a learning model must
address is what individuals know before the game starts, and what
it is that they are learning about. Are they learning about
exogenous data and choosing best responses, or are they learning
how to play? A closely connected issue is how much rationality to
attribute to the agents? We believe that for most purposes the
right models involve neither full rationality nor the extreme
naïveté of most stimulus-response models;
"better" models have agents consciously but perhaps
imperfectly trying to get a good payoff.
In this essay we will simplify matters by
assuming that players know the extensive form of the game and
their own payoff functions. They may or may not know their
opponents payoffs, but they are concerned with it only insofar as
they can use it to deduce how their opponents will play. In this
setup the only thing players are learning about from one
iteration of the game to the next one is the distribution of
strategies used by opponents.
From this point of view, if players do have
information about opponents' payoffs they use it to form priors
beliefs about how their opponents are likely to play. For
example, they might think it is unlikely that opponents will play
strategies that are dominated given the presumed payoff
functions.
These two cases fit most laboratory experiments, but in other applications players might have less information. At the other extreme, players might not know anything about their opponents, so that players do not even know if they are playing a game, and are only aware of a subset of their own possible strategies. They might observe only the history of payoffs and their own strategies, and discover new strategies by experimentation or deduction. So far only the genetic algorithm model of learning (for example Holland [1975] has attempted to explore a setting like this.
There are several ways of modeling learning
about opponents' strategies. In the Bayes model of learning
agents have priors, likelihood functions, and update according to
Bayes rule. Such agents can be "Savage rational," but
they need not be fully rational in the sense of basing their
inference on a correct model of the world. Indeed, if correct
model of the world means consistent with a common prior, then
there the issue of whether play converges to an equilibrium is
moot, since play must be an equilibrium (either Bayesian
correlated, or Bayesian Nash) from the outset, as shown by Aumann
[1987], and Brandenburger and Dekel [1987]. So it is not sensible
to impose a common prior when trying to understand how learning
leads to equilibrium.
Since we are trying to understand how common
knowledge might arise from common experience, and is not clear
what we might mean by a model being correct without common
knowledge, we look at learning processes where the players'
models of the environment are "sensible" as opposed to
"correct." (If we focus not on players' models of the
environment, but on the "behavior rules" by which they
choose their actions , then we are interested in sensible rules.)
This means that players are trying to get a good payoff.
Moreover, sensible rules should do reasonably well in a
reasonably broad set of circumstances. Rules that do poorly even
in simple environments are not likely to be used for important
decisions.
There are two levels of sophistication in
learning. One is simply to forecast how opposing players will
play. However, if two people repeatedly play a two person game
against each other, they ought to consider the possibility that
their current play may influence the future play of their
opponent. For example, players might think that if they are nice
they will be rewarded by their opponent being nice in the future,
or that they can teach their opponent to play a best response to
a particular action by playing that action over and over. Most of
learning theory abstracts from these repeated game considerations
by explicitly or implicitly relying on a model in which the
incentive to try to alter the future play of opponents is small
enough to be negligible. Lock-in with small discount factors is
one such model. A second class of models that makes repeated play
considerations negligible is that of a large number of players,
who interact relatively anonymously, with the population size
large compared to the discount factor.
There are a variety of large-population models,
depending on how players meet, and what information is revealed
at the end of each round of play.
Random Matching Model: Each period, all players
are randomly matched. At the end of each round each player
observes only the play in his own match. The way a player acts
today will influence the way his current opponent plays tomorrow,
but the player is unlikely to be matched with his current
opponent or anyone who has met the current opponent for a long
time. Myopic play is approximately optimal if the population is
finite but large compared to the discount factors. Most game
theory experiments use this design. Note, however, that myopic
play need not be optimal for a fully rational player (who knows
that others play myopically) even in a large population if that
player is sufficiently patient; see Ellison [1995]).
Aggregate Statistic Model: Each period, all
players are randomly matched. At the end of the round, the
population aggregates are announced. If the population is large
each player has little influence on the population aggregates,
and consequently little influence on future play, so players have
no reason to depart from myopic play. Some, but too few,
experiments use this design.
Note that with either form of a large population, repeated game effects are small but not nonzero with the typical population size of 20 to 30 subjects. Since the aggregate statistic
model seems to approximate about as many
real-world situations of interest as the random-matching model,
and has the additional virtue of promoting faster learning, it is
unfortunate that it has not been used more often. This may be due
to a fear that repeated game effects would prove larger in the
aggregate statistic model, but we are unaware of theoretical or
empirical evidence that this is the case. Our guess is that the
preponderance of the random-matching model is due rather to the
desire to maintain comparability with past experiments.
Although the large population stories explain naïve or myopic play; they only apply if the relevant population is large. Such cases may be more prevalent than it first appears, as players may extrapolate between similar games. Most non-equilibrium models in the literature either explicitly or implicitly have a large population model in mind; Kalai and Lehrer [1993] is one notable exception.
Fictitious play is a kind of quasi-Bayesian
model: Agents behave as if they think they are facing an
exogenous, stationary, unknown, distribution of opponents'
strategies. The standard formal model with a single agent in each
player role. There is some recent work by Kanivkoski and Young
[1994] that examines the dynamics of fictitious play in
population of players.
In the standard formal model denote a strategy
by player i by 
, and the set of available strategies by 
. As
usual, we will use 
to denote the player(s) other than i. The
best response correspondence is denoted by 
. In
fictitious play, player i has an exogenous initial weight
function 
. This weight is updated by adding 1 to the weight of
each opponent strategy each time it is played. The probability
that player i assigns to player -i playing 
at date
t is given by
,
Fictitious play itself is then defined as any
rule that assigns 
. This method corresponds to Bayesian
inference when player i believes that his opponents' play
corresponds to a sequence of i.i.d. multinomial random variables
with a fixed but unknown distribution, and player i's prior
beliefs over that unknown distribution take the form of a
Dirichlet distribution.
Proposition (Fudenberg and Kreps 1993): (a) If
s is a strict Nash equilibrium, and s is played at date t in the
process of fictitious play, s is played at all subsequent dates.
That is, strict Nash equilibria are absorbing for the process of
fictitious play. (b) Any pure strategy steady state of fictitious
play must be a Nash equilibrium. (c) If the empirical marginal
distributions converge, the strategy profile corresponding to the
product of these distributions is a Nash equilibrium.
The early literature on fictitious play viewed
the process as describing pre-play calculations players might use
to coordinate their expectations on a particular Nash equilibrium
(hence the name "fictitious" play.) From this
viewpoint, or when using fictitious play as a means of
calculating Nash equilibria, the identification of a cycle with
its time average is not problematic, and the early literature on
fictitious play accordingly focused on finding conditions that
guaranteed the empirical distributions converge.
The convergence result above supposes players ignore cycles, even if, because of these cycles, the empirical joint distribution of the two players' play is correlated, and does not equal the product of the empirical marginals. Consider the following example, from Fudenberg and Kreps [1990], [1993]. (Similar examples can be found in Jordan [1993] and Young [1993])
| A | B | |
| A | 0,0 | 1,1 |
| B | 1,1 | 0,0 |
Suppose this game is played according to the
process of fictitious play, with initial weights (1, sqrt (2))
for each player. In the first period, both players think the
other will play B, so both play A. The next period the weights
are (2,sqrt (2)) and both play B; the outcome is the alternating
sequence ((B.,B),(A,A),(B,B), etc.) The empirical frequencies of
each player's choices do converge to 1/2, 1/2, which is the Nash
equilibrium, but the realized play is always on the diagonal, and
both players receive payoff 0 in every period. The empirical
joint distribution on pairs of actions does not equal the product
of the two marginal distributions, so that the empirical joint
distribution corresponds to correlated as opposed to independent
play. As a model of players learning how their opponents behave,
this is not a very satisfactory notion of "converging to an
equilibrium." Moreover, even if the empirical joint
distribution does converge to the product of the empirical
marginals, so that the players will end up getting the payoffs
they would expect to get from maximizing against i.i.d. draws
from the long-run empirical distribution of their opponents'
play, one might still wonder if players would ignore persistent
cycles in their opponents play.
One response to this is to use a more demanding
convergence notion, and only say that a player's behavior
converges to a mixed strategy if his intended actions in each
period converge to that mixed strategy. The standard fictitious
play process cannot converge to a mixed strategy in this sense
(for generic payoffs). Another response is to think that
fictitious play does describe the long-run outcome, and players
will not notice that their fictitious play model is badly off,
provided that their payoffs are not worse than the model
predicts.
Fudenberg and Levine [1994] and Monderer, Samet, and Sela [1994] show that the realized time-average payoffs under fictitious play cannot be much below what the players expect if the time path of play exhibits "infrequent switches."
We now wish to contrast fictitious play with
the partial best-response model of learning in a large
population. The partial adjustment model has a state variable 
giving
the fractions of the population playing each 
. In
discrete time, if a fraction of the population 
is
picked at random switch to the best response to their opponents
current play, and the rest of the population continues their
current play, the partial best-response dynamic is given by
.
If the time periods are very short, and the fraction of the population adjusting is very small, this may be approximated by the continuous time adjustment process
.
In contrast, the fictitious play process moves
more and more slowly over time, because the ratio of new
observations to old observations becomes smaller and smaller. Let 
denote
the historical frequency with which strategies have been played
at time t. Then in fictitious play
.
This is of course very much like the partial best-response dynamic, except that the weights on the past is converging to one, and the weight on the best response to zero.
Discrete-time fictitious play asymptotically
approximately shares the same limiting behavior as the continuous
time best-response dynamic. More precisely, the set of limit
points of discrete time fictitious play is an invariant subset
for the continuous time best- response dynamics, and the path of
the discrete-time fictitious play process starting from some
large T remains close to that of the corresponding continuous
time best- response process until some time 
, where 
can be
made arbitrarily large by taking T arbitrarily large. This was
shown by Hofbauer [1995].
In fictitious play, players can only randomize
if exactly indifferent; they cannot learn to play mixed
strategies. To develop a sensible model of learning to play mixed
strategies, one should start with an explanation for mixing. One
such explanation is Harsanyi's [1973] purification theorem, which
explains a mixed distribution over actions as the result of
unobserved payoff perturbations.
A second explanation for mixing is that it is
descriptively realistic: some psychology experiments show that
choices between alternatives that are perceived as similar tend
to be relatively random.
A third motivation for supposing that players
randomize is that stochastic fictitious play allows behavior to
avoid the discontinuity inherent in standard fictitious play;
consequently it allows much good long-run performance in a much
wider range of environments.
All three motivations lead to models where the
best response function 
are replaced by "smooth best response
functions" 
, with
The notion of a Nash equilibrium in games with
smoothed best response functions is
Definition: The profile 
is a Nash distribution if 
for
all i.
This distribution may be very different from
any Nash equilibria of the original game if the smoothed best
response functions are very far from the original ones. But in
the context of randomly perturbed payoffs, Harsanyi's
purification theorem shows that ("generically") the
Nash distributions of the perturbed game approach the Nash
equilibria of the original game as the support of the payoff
perturbations become concentrated about 0.
The smoothed best-response function for the game of matching pennies is shown below
The "threshold perception" literature
in psychology literature shows that when asked to discriminate
between two alternatives, behavior is random, becoming more
reliable as the alternatives become more distinct. In choosing
between two different strategies, one measure of the distinctness
of the strategies is the difference in utility between the
strategies. With this interpretation, Thurstone's [1927] law of
comparative judgment becomes similar to Harsanyi's random utility
model. Indeed, the picture of the smooth best response curve
drawn above is very similar to behavior curves that have been
derived empirically in psychological experiments.
While smooth fictitious play is random, the time average of many independent random variables has very little randomness. Asymptotically the dynamics resembles that of the continuous time "near best response" dynamics
.
Consequently, if the stochastic system
eventually converges to a point or a cycle, the point or cycle
should be a closed orbit of the continuous-time dynamics.
Moreover, if a point or cycle is an unstable orbit of the
continuous time dynamics, we expect that the noise would drive
the system away, so that the stochastic system can only converge
to stable orbits of the continuous-time system.
The literature on "stochastic approximation" gives conditions that allow the asymptotic limit points of stochastic discrete-time systems to be inferred from the stability of the steady states of a corresponding deterministic system in continuous time.
Proposition: (Benaim and Hirsch [1996])
Consider a two-player smooth fictitious play in which every
strategy profile has positive probability at any state 
. If 
is an
asymptotically stable equilibrium of the continuous time process,
then regardless of the initial conditions 
.
Definition: A rule 
is 
-universally consistent if
for any 
a.s.
In words, using as the criterion the time
average payoff, the utility realized cannot be much less than
that from the time invariant strategy.
This objective differs from the Bayesian
approach of specifying prior beliefs about strategies used by
opponents choosing a best response to those beliefs. However, any
Bayesian expects to be consistent almost surely.
Existence of such rules originally shown by
Hannan [1957], and by Blackwell [1956]. The result has been lost
and rediscovered several times since then by many authors. In the
computer science literature this is called the "on-line
decision problem."
Fudenberg and Levine [1995] show that universal
consistency can be accomplished by a smooth fictitious play
procedure in which 
is derived from maximizing a function of
the form 
, and at each date players play 
applied
to their current beliefs. For example, if
, then
This is "exponential fictitious
play": each strategy is played in proportion to an
exponential function of the utility it has historically yielded;
it corresponds to the logit decision model that has been
extensively used in empirical work. Notice that as 
the
probability that any strategy that is not a best response is
played goes to zero.
Note also that this is not a very complex rule. Since it is not difficult to implement universally consistent rules universal consistency may be a useful benchmark for evaluating learning rules, even if it is not exactly descriptive of experimental data in moderate size samples and even if individual players do not implement the particular procedures discussed here.
The stimulus-response model was developed largely in response to observations by psychologists about human behavior and animal behavior: choices that lead to good outcomes are more likely to be repeated
behavior is random. All of these properties are
shared by many learning processes. Stimulus-response has received
a fair bit of recent attention, but we fell that researchers
would do better to investigate models more in the style of smooth
fictitious play.
First consider stimulus response with only
positive reinforcements, as in Borgers-Sarin [1994]. Agents at
each date use a mixed strategy, and the state of the system 
is the
vector of mixed actions played at time t. Payoffs are normalized
to lie between zero and one, so that they have the same scale as
probabilities. The state evolves in the following way: if player
i's payoff at date t is 
, then
Because all payoffs are nonnegative, the
propensity to play any action increases each time that action is
played.
Narendra and Thatcher [1974] showed that
against an i.i.d. opponent, as the reinforcement parameter 
goes to
zero, the time average utility converges to the maximum that
could be obtained against the distribution of opponents play. But
this weak property is satisfied by many learning rules that are
not consistent, including "pure" fictitious play;
smooth fictitious play retains its consistency property
regardless of opponents play.
Borgers-Sarin [1995] consider a more general
model in which reinforcements can be either positive or negative,
depending on whether the realized payoff is greater or less than
the agent's "aspiration level." The simplest case is a
constant but nonzero aspiration level: if the aspiration level is
greater than 1, (all outcomes are disappointing) play cannot lock
on to a pure action. Less obviously, if there are only two
strategies H and T, all payoff realizations are either 0 or 1,
the probability that H yields 1 is p, and the probability that T
yields 1 is 1-p, ( as if the agent is playing against an i.i.d.
strategy in the game matching pennies) and the aspiration level
is constant at ½, then probability of playing H converges to
p. This is referred to as probability matching. This is not
optimal, but at one time psychologists believed it was
characteristic of human behavior; we do not believe that they
still do.
Roth and Er'ev [1994],[1996] develop a closely
related but more highly parameterized variation of the
stimulus-response model and use it to study experimental data on
2-player matrix games with mixed-strategy equilibria. They assume
that the aspiration level follows a linear adjustment process
that ensures that the system moves away from probability matching
in the long run, and add a minimum probability on every strategy.
Their motivation is to see how far they can get with a
"simple" model with low cognition.
They have the nice idea of looking at
experiments with very long run length. This is important because
learning theory suggests that many more trials are needed for
players to learn the distribution of opponents' play when it
corresponds to a mixed strategy than when it corresponds to a
pure one.
The Roth and Er'ev model fits the aggregate
data (pooled over subjects) well, but it has a large number of
free parameters. Most learning models with enough flexibility in
functional form are going to fit the aggregate data relatively
well, because of its high degree of autocorrelation.
Roth and Er'ev compare their stimulus-response
model to a simple (0-parameter) deterministic fictitious play on
individual data, and argue that this comparison makes a
"case for low-rationality game theory." In my opinion,
this case is far from proven: they looked only at deterministic
fictitious play, and we would expect a priori that a
deterministic model will not fit the individual data nearly as
well as a stochastic one. (This is acknowledged in Roth and Er'ev
[1996], who say that stochastic modifications of fictitious play
are beyond the scope of the paper.) Also, Er'ev and Roth specify
a fictitious play with zero prior weights, which implies that the
players will be very responsive to their first few observations,
while actual behavior is known to be more "sluggish"
than that.
In contrast, Cheung and Friedman [1994] have
had some success in fitting modified fictitious play models to
experimental data. Their model (in games with 2 actions) has
three parameters per player: the probability that a player uses
his first action is a linear function (2 parameters) of the
exponentially-weighted history (1 more parameter for the
exponential weight.) The median exponential weighting over
players and experiments is about .3 or .4; best response
corresponds to weight zero and classic fictitious play is weight
one.
Reflecting on these papers suggests to us that, as a group, theorists and experimenters should develop, characterize, and test models of "intermediate-cognition" procedural rationality; by this we mean models in which players are trying to maximize something in an approximate or loosely interpreted sort of way.
A simple experiment shows that prior
information can have an important impact on the course of play.
This is the Prasnikar and Roth [1992] experiment on the
"best-shot" game, in which two players sequentially
decide how much to contribute to a public good. The
backwards-induction solution to this game is for player 1 to
contribute nothing and player 2 to contribute 4; there is also an
imperfect Nash equilibrium in which player 1 contributes 4 and
player 2 contributes nothing. There is no Nash equilibrium where
player 1 mixes.
Prasnikar and Roth ran two treatments of this
game. In the first, players were informed of the function
determining opponents' monetary payoffs. Here, by the last few
rounds of the experiment the first movers had stopped
contributing. In the second treatment, subjects were not given
any information about the payoffs of their opponents. In this
treatment even in the later rounds of the experiment a
substantial number of player 1's contributed 4 to the public
good, while others played 0.
This is not consistent with backwards induction
or even Nash equilibrium, but it is consistent with an
(approximate, heterogeneous) self-confirming equilibrium
(Fudenberg and Levine [1996]). These experiments provide evidence
that information about other players' payoffs is used as if
players maximized.
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