Some of Mark's research has been supported by the National Science Foundation. The NSF's support has been instrumental in many of the papers listed below, and several data sets have been generated from this research. These data sets are available for other researchers to use.
"Mixed Strategy Equilibrium", written with John Wooders, is in The New Palgrave Dictionary of Economics. The paper provides a brief introduction to mixed strategy equilibrium (MSE) as well as recent research investigating whether MSE describes actual behavior.
"Stripped-Down Poker", written with David Reiley and Mike Urbancic, appeared in the Journal of Economic Education. The paper describes a simple poker game that works great as a classroom demonstration of mixed strategy equilibrium, bluffing, signalling games, and Bayesian updating.
Recent Research Papers
A Simple Market-Like Allocation Mechanism for Public Goods
Matthew Van Essen and Mark Walker
Working paper (2015).
Full text Abstract:
We argue that since allocation mechanisms will not always be in equilibrium,
their out-of-equilibrium properties must be taken into account along with
their properties in equilibrium. For economies with public goods, we define
a simple market-like mechanism in which the strong Nash equilibria yield
the Lindahl allocations and prices. The mechanism satisfies
critical out-of-equilibrium desiderata that previously-introduced mechanisms
fail to satisfy, and always (weakly) yields Pareto improvements, whether in
equilibrium or not. The mechanism requires participants to communicate
prices and quantities, and turns these into outcomes according to a natural
and intuitive outcome function. Our approach first exploits
the equivalence, when there are only two participants, between the
private-good and public-good allocation problems to obtain a two-person
public-good mechanism, and then we generalize the public-good mechanism
to an arbitrary number of participants. The results and the intuition
behind them are illustrated in the familiar Edgeworth Box and Kolm
Triangle diagrams.
Out-of-Equilibrium Performance of Three Lindahl Mechanisms: An Experiment
Matthew Van Essen, Natalia Lazzati, and Mark Walker Games and Economic Behavior, Vol. 74 (2012), 366-381.
Full text Abstract:
We describe an experimental comparison of the out-of-equilibrium performance of three allocation mechanisms designed to achieve Lindahl outcomes as Nash equilibria: the mechanisms due to Walker (1981), Kim (1993), and Chen (2002). We find that Chen's mechanism, which is supermodular, converges closest and most rapidly to its equilibrium. However, we find that the properties that move subjects toward equilibrium in Chen's mechanism typically generate sizeable taxes and subsidies when not in equilibrium, and correspondingly large budget surpluses and deficits, which typically far outweigh the surplus created by providing the public good. The Kim mechanism, on the other hand, converges relatively close to its equilibrium and exhibits much better out-of-equilibrium efficiency properties.
Behavior in Second-Price Auctions by Highly Experienced eBay Buyers and Sellers
Rod Garratt, Mark Walker and John Wooders Journal of Experimental Economics, Vol. 15 (2012), 44-57.
Full text Abstract:
When second-price auctions have been conducted in the laboratory, most of
the observed bids have been "overbids" (bids that exceed the bidder's
value) and there are very few underbids. Few if any of the subjects in those
experiments had any prior experience bidding in auctions. We report on
sealed-bid second-price auctions that we conducted on the Internet using
subjects with substantial prior experience: they were highly experienced
participants in eBay auctions. Unlike the novice bidders in previous
(laboratory) experiments, the experienced bidders exhibited no greater
tendency to overbid than to underbid. However, even subjects with
substantial prior experience tend not to bid their values, suggesting that
the non-optimal bidding of novice subjects is robust to substantial
experience in non-experimental auctions. A key determinant of bidding
behavior was whether a subject had ever been a seller on eBay.
Equilibrium Play in Matches: Binary Markov Games
Mark Walker, John Wooders, and Rabah Amir Games and Economic Behavior, Vol. 71 (2011), 487-502.
Full text Abstract:
We study two-person extensive form games, or "matches," in which the only
possible outcomes (if the game terminates) are that one player or the other is
declared the winner, and in which the winner of the match is determined by the
winning of points, in "point games." We call such matches binary Markov
games. We show that if a simple monotonicity condition is satisfied, then (a)
it is a Nash equilibrium of the match for the players, at each point, to play a
Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the
match for a player to play minimax in each point game (thus, playing minimax at
each point assures a player that the probability he will win the match is at least
as great as under equilibrium play, no matter how his opponent plays); and (c)
when the point games all have unique Nash equilibria, the only Nash equilibrium of
the binary markov game consists of minimax play at each point. The minimax result
provides a rationale for the players to play minimax (and therefore Nash
equilibrium) in every point game, even if the behavioral assumptions typically
used to justify Nash equilibrium are not satisfied. An application to tennis is
provided.
Discrete Implementation of the Groves-Ledyard Mechanism
Todd Swarthout and Mark Walker Review of Economic Design, Vol. 13 (2009), 101-114.
Full text Abstract:
When implementing an economic institution in the field or in the laboratory, the
participants’ action spaces and the institution’s outcomes are typically discrete, while our
theoretical analysis of the institution often assumes the sets are continuous. Predictions by
the continuous model generally turn out to be good approximations to the performance of
the discrete implementation. We present an example in which the continuous version has a
unique and Pareto efficient equilibrium, but in which the discrete version often has vastly
more equilibria, many of them far from efficient. We show that the same phenomenon
appears in two experiments investigating the Groves-Ledyard mechanism, and that it may
account for the experimental results.
Unobserved Heterogeneity and Equilibrium: An Experimental Study of
Bayesian and Adaptive Learning in Normal Form Games
Jason Shachat and Mark Walker Journal of Economic Theory, Vol. 114 (2004), 280-309.
Full text
A demo of the experiment is available as well. Abstract:
We describe an experiment based on a repeated two-person game of incomplete
information designed so that Jordan's Bayesian model of learning in games and
the best response model make completely opposite predictions. Econometric
analysis of the experimental data, using the maximum likelihood procedure due
to El Gamal and Grether, reveals clear heterogeneity in the subjects' learning
behavior. The heterogeneity is not diffuse, however: the subjects follow only
a few decision rules for basing their play on their information, and their
decision rules have simple cognitive interpretations. Although the repeated
game has many equilibria, including a unique pure strategy equilbrium, we find
that the only equilibrium consistent with the data is one of the mixed strategy
equilibria. This equilibrium is shown, surprisingly, to be consistent with
Jordan's Bayesian model, in a "representative player" sense, each subject using
a pure strategy, but the distribution of strategies among subjects coinciding
with the mixed strategy equilbrium.
Hide and Seek in Arizona
Robert Rosenthal, Jason Shachat, and Mark Walker International Journal of Game Theory, Vol. 32 (2003), 273-293.
Full text Abstract:
Laboratory subjects repeatedly played one of two variations of a simple
two-person zero-sum game of pursuit and evasion. Three puzzling departures
from the prescriptions of equilibrium theory are found in the data: an
asymmetry related to the player's role in the game; an asymmetry across the
game variations; and positive serial correlation in subjects' play. Possible
explanations for these departures are considered.
Minimax Play at Wimbledon
Mark Walker and John Wooders American Economic Review, Vol. 91 (2001), 1521-1539.
Full text Abstract:
We use data from classic professional tennis matches to provide an empirical
test of the minimax hypothesis. We find that the serve-and-return play of
John McEnroe, Bjorn Borg, Boris Becker, Pete Sampras and others is largely
consistent with the minimax hypothesis. The same statistical tests soundly
reject the assumption of minimax play in experimental data, including the
data from Barry O'Neill's celebrated experiment. [A more extended
abstract for the non-technical reader
is available as well.]
An Experiment to Evaluate Bayesian Learning of Nash Equilibrium
Play
James Cox, Jason Shachat, and Mark Walker Games and Economic Behavior, Vol. 34, (2001), 11-33;
Full text Abstract:
Some recent theoretical approaches to the question of how players
might converge over time to a Nash equilibrium have assumed that the
players update their beliefs about other players via Bayes' Rule.
Jordan has shown in a Bayesian model of this kind that play will
(theoretically) always converge to a complete-information Nash
equilibrium, even though individual players will not generally attain
complete information. We report on an experiment designed to evaluate
the empirical implications of Jordan's model. A finite version of the
model is constructed which generates unique predictions of subjects'
choices in nearly all periods. The experimental data reveals that the
theory does reasonably well at predicting the equilbria that subjects
eventually play, even when there are multiple equilibria. The results
thus suggest that Jordan's Bayesian model can provide an empirically
effective solution to the equilibrium selection problem when the
players have beliefs with finite support. However, the model's
predictions about the path of play over time are not consisitent with
the experimental data.
Learning to Play Cournot Duopoly Strategies
James Cox and Mark Walker Journal of Economic Behavior and Organization, Vol. 36 (1998),
141-161.
Full text. Abstract:
In most theories of out-of-equilibrium play or "learning" in games,
the stability of Cournot equilibrium depends upon how the firms'
reaction functions cross. In particular, if the firms' marginal costs
decline rapidly enough with increased output, then the interior
equilibrium will be (theoretically) unstable. We report on a series
of experiments designed to determine whether interior Cournot
equilibria are attained in such theoretically unstable environments.
The experiments include duopolies with constant and with decreasing
marginal costs, and with theoretically stable and unstable equilibria.
The experimental results reveal a sharp distinction between behavior
in the stable and the unstable duopolies: after a few early periods,
play in the stable duopolies is at or near the equilibrium, and in the
unstable duopolies it is almost never at or near the interior
equilibrium.
Two Problems in Applying Ljung's 'Projection Algorithms' to the Analysis of Decentralized Learning
Diego Moreno and Mark Walker Journal of Economic Theory, Vol. 62 (1994), 420-427.
Full text. Abstract:
We show that Ljung's projection algorithms, which have recently been used by economists to establish convergence to rational expectations equilibrium, do not seem to apply to learning or forecasting behavior that one would normally call "decentralized." If the algorithm is defined in a way that allows individuals to have differing information, then Ljung's theorem does not apply. And even if a similar theorem could be proved that would allow for differing information, there remains a Lyapunov-like condition that is central to Ljung's projection method and which requires that individual beliefs be narrowly related to the equilibrium and to one another.
Nonmanipulable Voting Schemes when Participants' Interests are Partially Decomposable
Diego Moreno and Mark Walker Social Choice & Welfare, Vol. 8 (1991), 221-233.
Full text. Abstract:
Recent papers by Barbera and Peleg and by Zhou have established that
the Gibbard-Satterthwaite Theorem remains valid when individuals are restricted
to reporting only "reasonable" preferences. We present a theorem that covers
situations in which, as in Barbera & Peleg and Zhou, preferences may be restricted
to reasonable ones, but in which, additionally, it may be known in advance that
some dimensions of the social decision do not affect all the participants
-- i.e., in which the social decisions are partially decomposable into decisions that
affect only subsets of the participants. As in the previous theorems, the conclusion
of this new theorem is that nonmanipulable voting schemes must be dictatorial.
On the Generic Non-optimality of Dominant-Strategy Allocation Mechanisms: A General Theorem that
Includes Pure Exchange Economies
Leonid Hurwicz and Mark Walker Econometrica, Vol. 58 (1990), 683-704.
Full text. Abstract:
It is shown that if an economy's participants cannot be separated into groups across
which there are no potentially conflicting interests -- i.e., if the economy is "indecomposable"
-- then every continuous truth-dominant allocation mechanism will attain nonoptimal
allocations on an open dense set of preference profiles. Classical "Edgeworth-box"
exchange economies (economies with no externalities and no production, but with
arbitrary numbers of consumers and goods), as well as economies with public goods and
economies with other kinds of externalities, are all shown via simple arguments to be
indecomposable. The results are extended to cover nonrevelation mechanisms that have
dominant-strategy equilibria.
Maximal Elements of Weakly Continuous Relations
Donald Campbell and Mark Walker Journal of Economic Theory, Vol. 50 (1990), 459-464.
Full text. Abstract:
A weaker than usual continuity property is defined for binary relations. Relations
that have this property, along with certain transitivity properties, are shown to
have maximal elements on compact sets. The results cover "interval orders," the
kind of relations that often characterize choice situations in which similar
alternatives are indistinguishable.
Conflicting Interests, Decomposability, and Comparative Statics
Mark Walker Mathematical Social Sciences, Vol. 18 (1989), 57-79.
Full text. Abstract:
When optimizing an aggregate of several individual objective functions, it may be possible to
decompose the set of individual objectives into groups across which there are no conflicting
interests. It is shown that changes in an individual objective will affect those individuals, and only
those individuals, whose objective is in potential conflict with the changed objective. Thus, in
particular, each individual can affect every other individual if and only if the optimization problem
is indecomposable -- i.e. if and only if it is impossible to separate the individuals into groups
across which there are no conflicting interests.
A Simple Auctioneerless Mechanism with Walrasian Properties
Mark Walker Journal of Economic Theory, Vol. 32 (l984), 111-127.
Full text. Abstract:
A simple mechanism for reallocating holdings is described, in which no
auctioneer is required: outcomes are determined solely from traders’ actions and
without any requirement that the mechanism be in equilibrium. The mechanism is
shown to exactly duplicate the performance of the Walrasian auctioneer (both in its
equilibria and in its disequilibrium path) if individuals are price takers, and, if the
number of individuals is large, to approximately duplicate the auctioneer’s performance
even when individuals behave strategically, each taking account of his own
influence on prices.
The Free Rider Problem: Experimental Evidence
Oliver Kim and Mark Walker Public Choice, Vol.43 (l984), 3-24.
A Simple Incentive-Compatible Scheme for Attaining Lindahl Allocations
Mark Walker Econometrica, Vol. 49 (l98l), 65-7l.
Full text. Abstract:
A simple scheme is presented for making decisions about the production and financing of public goods. The "competitive" equilibria under the scheme are Pareto optimal; more important, they are Lindahl equilibria. Thus, it is never in any individual's interest to refuse to participate (no one will be worse off at the equilibrium than at his initial holding); moreover, the existence of equilibria is assured in the usual classical public-goods economies.
On the Nonexistence of a Dominant-Strategy Mechanism for Making Optimal Public Decisions
Mark Walker Econometrica, Vol. 48 (l980), l52l-l540.
Full text. Abstract:
In a broad class of cases not covered by the Gibbard-Satterthwaite Theorem it is shown that one cannot design a strategy-proof choice mechanism which attains Pareto optimal outcomes. The results are shown to be generic in character -- i.e., any nonmanipulable mechanism will attain nonoptimal outcomes virtually everywhere -- and the results cover, in particular, certain problems in allocating public and private goods. The analysis is carried out in transferable-utility environments, and makes extensive use of the mechanisms recently introduced by Groves.
A Generalization of the Maximum Theorem
Mark Walker International Economic Review, Vol. 19 (l979), 267-272.
On the Characterization of Mechanisms for the Revelation of Preferences
Mark Walker Econometrica, Vol. 46 (l978), l47-l52.
Full text. Abstract:
Green and Laffont have characterized certain appealing dominant-strategy revelation mechanisms as precisely the mechanisms introduced by Groves, but they have established the characterization only for unstructured sets of public alternatives: if the set has some natural structure, their proof generally requires that pathological preferences be admissible -- for example, discontinuous and/or non-convex preferences. It is shown here that the same characterization holds on sets in R^{n}, even if we rule out pathological preferences. This greatly extends the usefulness of the characterization.
A Note on the Existence of Maximal Elements
Mark Walker Journal of Economic Theory, Vol. l6 (l977), 470-474.
Mark Walker Journal of Economic Theory, Vol. l5 (l977), 366-375.
Full text. Abstract:
In a recent paper, Mount and Reiter established that, in a certain sense, the
competitive mechanism is an "informationally most efficient" procedure for
allocating resources. This result depends of course upon the way we
characterize the notion of informational efficiency. Several alternative
characterizations, and the relationships among them, are given here, and it is shown
under which characterizations the above result is true and under which ones it is false.
It is shown that there is an intuitively appealing "best" characterization for which
the result is true.