Accounting for Mistakes
by
Economic Agents
Thesis submitted for the degree
“Doctor of Philosophy”
Written under the supervision of
Prof. Ariel Rubinstein
Kfir Eliaz
Submitted: March 2001
Approved: December 2001
Acknowledgements
This work could not have been written without the help, guidance and
support of some very special people. I was one of the very few lucky students
to have had Ariel Rubinstein as their advisor. This page is much too short
for me to write the many ways in which Ariel has helped me throughout my
struggle with this thesis. I wish to thank him with all of my heart. This work
could not have been written without the tremendous help of Ran Spiegler,
who provided me with excellent professional advice, as well as with psychological support. I owe him great many thanks. I am also very grateful for
the help and support of two very special people: Eddie Dekel and Leonardo
Felli.
While I worked on this project I had the pleasure of discussing my
work with some of the top researchers in the field today. Their contribution to this work is enormous. I wish to thank (in alphabetical order) J.P.
Benoit, Elchanan Ben-Porath, Markus Brunnermeier, Tilman Börgers, Kim
Sau Chung, Jeff Ely, Jacob Glazer, Yoram Hamo, Matthew Jackson, Philippe
Jehiel, Michele Piccione, Andrew Postlewaite, and Ilya Segal.
Parts of this work have been presented in several forums. I have benefitted
from the comments made by some of the people who attended these forums.
I therefore wish to thank seminar participants at the University of Bristol,
Cambridge, Columbia, The Institute for Advance Study in Princeton, LSE,
the University of Michigan (Ann Arbor), Northwestern University, NYU,
Princeton University, and UCL. I also wish to thank audiences at the EDP
99 Jamboree in Barcelona, Young Economists Meeting 99 in Amsterdam,
the 13th annual Conference In Game Theory and Applications (CITG) in
Bolognia, Summer in Tel-Aviv 99 and the First World Congress of the Game
Theory Society in Bilbao (Games 2000).
The work on this project has been supported by various institutions. I
appreciate the hospitality extended to me by STICERD at the London School
of Economics where part of this research was undertaken. I am grateful to
the Europian Commission for sponsoring my stay at the LSE through the
Marie Currie Fellowship (TMR grant). This research was partially supported
by the Israel Science Foundation founded by the Israel Academy of Sciences
and Humanities. I am also grateful to the Eitan Berglas School of Economics
at Tel-Aviv University for supporting me throughout my doctoral studies. I
owe many thanks to Oved Yosha and Phyllis Klein-Avni.
1
I also wish to thank my family and friend for bearing with me while I
worked on this project. I am grateful to my parents for their tremendous
support throughout my studies. I thank my sister for being proud of me,
even when there was no apparent reason for it. I could never have survived
my graduate studies without the support and coffee of Michael Orenstein.
Finally, I would like to dedicate this work to the one person to whom I
owe everything: Noa.
2
Abstract
When economists study the steady state of a model, the standard assumption is that the economic agents do not make mistakes in equilibrium:
their actions are optimal and their forecasts are flawless. This work investigates the robustness of game-theoretic economic models to the possibility
that individuals make mistakes. In particular, I study three different models
in which some individuals either do not choose the actions which are best for
them, or they rely on incorrect forecasts.
In the first chapter I focus on the classical theory of implementation. This
theory studies the problem of a planner who faces a set of agents and wishes
to associate a set of outcomes with each possible profile of the agents’ preferences (the correspondence that assigns a set of outcomes to each profile of the
agents’ preferences is called a choice rule). The standard approach implicitly
assumes that each agent is able to correctly choose his most preferred action.
A question arises as to how robust are the conclusions reached by standard
models to slight deviations from the full rationality assumption. If we believe that decision makers might err, we may be interested in constructing
mechanisms that are immune to possible mistakes that some of the players
might make.
The first model therefore explores the question of implementation that
is robust to the potential of having a limited number of agents who make
mistakes. An agent who makes mistakes is viewed as a decision maker who
has well defined preferences that are known to others, but who fails for one
reason or another to behave optimally. We refer to such an agent as being
f aulty. The planner and the non-faulty players only know that there can
be at most k faulty players in the population. However, they know neither
the identity of faulty players, their exact number nor how faulty players
behave. We define a solution concept which requires a player to optimally
respond to the nonfaulty players regardless of the identity and actions of the
faulty players. We introduce a notion of fault-tolerant implementation, which
unlike standard notions of full implementation, also requires robustness to
deviations from the equilibrium.
The main result of this model establishes that under symmetric information any choice rule that satisfies two properties - k monotonicity and
no veto power - can be implemented by a strategic game form if there are
3
at least three players and the number of faulty players is less than 12 n − 1.
As an application of our result we present examples of simple mechanisms
that implement the constrained Walrasian function and a choice rule for the
efficient allocation of an indivisible good.
In the second chapter I investigate the robustness of Nash equilibrium,
one of the basic notions in game theory, to the possibility that players may
use incorrect forecasts. Nash equilibrium is a steady state in which each
player chooses a best response to his opponents’ strategies. One of the most
common interpretations of this equilibrium concept, is the following: each
player forecasts the strategies of his opponents, and then chooses a best
response to that forecast; in a steady state, each player correctly forecasts
his opponents’ strategies. I argue that this interpretation may not be valid
when players are concerned about the complexity of their forecasts.
The model I study consists of a two-player strategic game in which each
player chooses a finite machine to implement a strategy in an infinitely repeated 2 × 2 symmetric game. The players preferences are lexicographic such
that their primary concern is to obtain the highest discounted sum of payoffs, and their secondary concern is the complexity of their forecast. This
means that a player prefers to have the simplest forecast which rationalizes
his payoff-maximizing strategy. The final component of the model is a partial ordering of machines, which I interpret to be a measure of the simplicty
of pure-strategy forecasts. I then propose two alternative notions of Nash
equilibria for this model. According to the first definition, an equilibrium
is a pair of machines with the following properties: (1) each player machine
is a best response to his opponent’s machine and (2) the simplest forecast
which rationalizes each player’s machine is the opponent’s actual machine.
According to the second definition, an equilibrium is a pair of machines with
the following properties: (1) each player machine is a best response to his
opponent’s machine and (2) each player has no other best response to his
opponent, which can be rationalized by a forecast which is simpler than the
opponent’s actual machine.
I compare each of the proposed equilibria with a classical result in game
theory known as the folk theorem. This result shows that any outcome of
the repeated game, in which none of the players is driven below the minimal
payoff which he can secure for himself, can be sustained as a Nash equilibrium. I show that according to my first definition of Nash equilibrium many
outcomes cannot be sustained in equilibrium. For example, in the repeated
4
game of chicken, the Pareto superior path cannot be sustained. The set of
equilibria shrinks further when we use the second definition of equilibrium
which I propose. For example, in the repeated game of chicken every equilibrium generates only combinations of the one-shot Nash equilibria. I interpret
these results as demonstrating that when players account for the complexity
of their forecast, we cannot interpret Nash equilibrium as a steady state in
which each player’s forecast is correct.
In the third chapter I consider the possibility that decision makers may
make mistakes in processing information. In particular, I study games in
which players rely on the advice of experts who are concerned that their
advice may be misunderstood. The model consists of two-player games in
which each player’s forecast is given to him by an advisor who wants the
player to obtain the highest payoff in the game. However, each advisor is
uncertain as to how his forecast will be understood. The advisor to each
player has a belief function that associates with every possible forecast a
probability distribution over the possible understandings of that forecast.
Given a pair of belief functions, an equilibrium of this model is a pair of
stratgies such that there exists a pair of forecasts with the following properties: (1) each player’s strategy is a best response to a possible understanding
of his forecast (an understanding which is assigned a positive probability by
the belief assigned to the forecast), and (2) given his opponent’s strategy, the
forecast of each player maximizes his expected utility. I apply this equilibrium notion to a few simple examples, where I show the existence of steady
states which differ from the ones predicted by the standard Nash equilibrium. In particular, I demonstrate that under certain conditions players may
choose to continue in the centipede game. I also show that under certain
conditions every equilibrium in the Nash demand game, in which each player
receives a positive share of the surplus, is inefficient. I argue that these result
demonstrate the sensitivity of the standard game-theoretic framework to the
assumption of perfect understanding by the players.
5