"Learning,
Matching and Aggregation," Games and Economic Behavior, January, 1999.
Fictitious play and reinforcement learning are examined in the context of a large population where agents are repeatedly randomly matched. We show that the aggregation of this learning behaviour can be qualitatively different from learning at the level of the individual. This aggregate dynamic belongs to the same class of simply defined dynamic as do several formulations of evolutionary dynamics. We obtain sufficient conditions for convergence and divergence which are valid for the whole class of dynamics. These results are therefore robust to most specifications of adaptive behaviour.
"A Note
on Best Response Dynamics," Games and Economic Behavior,
October, 1999.
We
investigate the relationship between the continuous time best response dynamic, its
perturbed version and evolutionary dynamics in relation to mixed
strategy equilibria. We find that as the level of
noise approaches zero, the perturbed best response dynamic has the same
qualitative properties
as a broad class of evolutionary dynamics. That is, stability properties of
equilibria are robust across learning dynamics of
quite different
origins and motivations.
"The Stability
of Price Dispersion under Seller and Consumer Learning", with Robert M. Seymour. International Economic Review, November,
2002.
In many markets it is possible to find rival
sellers charging different prices for the same good. Earlier research has
attempted to
explain this phenomenon by demonstrating the existence of dispersed
price equilibria when consumers must make use of
costly search
to discover prices. We ask whether such equilibria
can be learnt
when sellers adjust prices adaptively in response to current
market conditions. With consumer behaviour fixed, convergence
to a dispersed price equilibrium is possible in some
cases.
However, once consumer learning is introduced, the monopoly outcome first
found by Diamond (1971) is the only stable equilibrium.
"Two Competing Models of How People Learn in Games". Econometrica, November, 2002.
Reinforcement learning and stochastic fictitious play are apparent rivals as models of human learning. They embody quite different assumptions about the processing of information and optimisation. This paper compares their properties and finds that they are far more similar than were thought. In particular, the expected motion of stochastic fictitious play and reinforcement learning with experimentation can both be written as a perturbed form of the evolutionary replicator dynamics. Therefore they will in many cases have the same asymptotic behaviour. In particular, they have identical local stability properties at mixed equilibria. The main identifiable difference between the two models is speed: stochastic fictitious play gives rise to faster learning.
“Adaptive Learning Models of Consumer Behavior”.
Journal of Economic Behavior and Organization, November, 2007.
This
paper applies recent advances in the theory of learning to the analysis of
consumer behavior in a dynamic duopoly. Nash
equilibrium play is characterized when consumers learn adaptively about the
relative quality of the two products. A
contrast is made between belief-based and reinforcement/familiarity-based
learning. In the latter case, consumers
can become locked into the habit of purchasing inferior goods. Such lock-in
permits the existence of multiple history-dependent asymmetric steady states in
which one firm dominates. In contrast, belief-based learning rules must lead
asymptotically to correct beliefs about the relative quality of the two brands
and so in this case there is a unique steady state. However, if consumers'
initial estimate of a firm's quality is high (low), a firm has an incentive to
charge above (below) the myopic duopoly price in order to slow (speed up)
learning.
“Learning in Perturbed Asymmetric Games”, with Josef Hofbauer. Games and Economic Behavior, July, 2005.
We investigate the stability of mixed strategy equilibria in 2 person (bimatrix) games under perturbed best response dynamics. A mixed equilibrium is asymptotically stable under all such dynamics if and only if the game is linearly equivalent to a zero sum game. In this case, the mixed equilibrium is also globally asymptotically stable. Global convergence to the set of perturbed equilibria is shown also for (rescaled) partnership games (also known as games of identical interest). Some applications of these results to stochastic learning models are given.
“Learning, Information and Sorting in Market Entry Games: Theory and Evidence”, with John Duffy. Games and Economic Behavior, April, 2005.
Previous
data from experiments on market entry games, N-player games where
each player faces a choice between entering a market and staying
out, appear inconsistent with either mixed or pure Nash equilibria. Here we show that, in this class of game,
learning theory predicts sorting, that is, in the long run, agents play a pure strategy equilibrium with some agents permanently in
the market, and some permanently out. We conduct experiments with a larger
number of repetitions than in previous work in order to test this prediction.
We find that
when subjects are given minimal information, only after close to 100
periods do subjects begin to approach equilibrium. In contrast, with
full information, subjects learn to play a pure
strategy
equilibrium relatively quickly. However, the information which permits
rapid convergence, revelation of the individual play of all
opponents, is not predicted to have any effect by existing models of
learning.
“Running to Keep in the Same Place: Consumer Choice as a Game of Status”, with Tatiana Kornienko.
American
Economic Review, September 2004.
If individuals
care about their status, defined as their rank in the distribution of
consumption of one ``positional'' good, then the consumer's problem is
strategic as her utility depends on the consumption choices of others. In the
symmetric Nash equilibrium, each individual spends an inefficiently high amount
on the status good. Using techniques from auction theory, we analyze the
effects of exogenous changes in the distribution of income. In a richer
society, almost all individuals spend more on conspicuous consumption, and
individual utility is lower at each income level. In a more equal society, the
poor are worse off.
“Attainability
of Boundary Points under Reinforcement Learning” with Martin Posch.
Games
and Economic Behavior, October, 2005.
“Inequality
and Growth in the Presence of Competition for Status”, with Tatiana
Kornienko. Economics Letters, October, 2006.
“Cross
and Double Cross: Comparative Statics in First Price Auctions”, with Tatiana Kornienko.
The B.E. Journal of Theoretical
Economics, 2007.
“Status, Affluence, and Inequality:
Rank-Based Comparisons in Games of Status”, with Tatiana Kornienko.
“Learning in Games with Unstable Equilibria”, with Josef Hofbauer and Michel Benaim.
“Job
Market Signalling of Relative Position, or Becker Married to Spence”
“Which Inequality? The Inequality of
Endowments Versus the Inequality of Rewards”, with Tatiana Kornienko.