Table Of ContentLecture Notes in Economics
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Gerald A. Heuer Ulrike Leopold-Wildburger
Silverman's Game
A Special Class of
Two-Person Zero-Sum Games
Springer
Authors
Prof. Dr. Gerald A. Heuer
Concordia College
Department of Mathematics and Computer Science
Moorhead, MN 56562, USA
Prof. Dr. Ulrike teopold-Wildburger
Karl-Franzens-University
Department of Statistics,
Econometrics and Operations Research
Herdergasse 11, A-8010 Graz, Austria
ISBN-13: 978-3-540-59232-7 e-ISBN-13: 978-3-642-46819-3
DOl: 10.1007/978-3-642-46819-3
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FOREWORD
The structure of a Silverman game can be explained very quickly:
Each of two players independently selects a number out of a prede-
termined set, not necessarily the same one for both of them. The
higher number wins unless it is at least k times as high as the other
one; if this is the case the lower number wins. The game ends in a
draw if both numbers are equal. k is a constant greater than 1.
The simplicity of the rules stimulates the curiosity of the the-
orist. Admittedly, Silverman games do not seem to have a direct
applied significance, but nevertheless much can be learnt from their
study. This book succeeds to give an almost complete overview over
the structure of optimal strategies and it reveals a surprising wealth
of interesting detail.
A field like game theory does not only need research on broad
questions and fundamental issues, but also specialized work on re-
stricted topics. Even if not many readers are interested in the
subject matter, those who are will appreciate this monograph.
Finally let me point out that Silverman games have the potential
to become a fruitful area of experimental research. The rules are
easy to explain to experimental subjects and interesting behavioral
questions can be raised in the context of two-person games with
only three possible outcomes, one, zero or minus one. Silverman
games may be a useful tool to study game learning. In this respect
the results presented in this book may prove to be valuable for
experimentalists.
Prof. Dr. Reinhard Selten
v
TABLE OF CONTENTS
Foreword v
Table of Contents VB
Preface IX
Chapter 1.· Introduction. 1
Survey of prior work. 2
The payoff function and expected payoffs. 3
The sequences {Pk} and {Ilk}. 4
The sequences {ltk} and {Uk}. 6
Equivalent variations. 7
Ghapter 2. Silverman's game on intervals:
preliminaries. 9
The key mixed strategies. 11
Chapter 3. Intervals with equal left endpoints or equal
right endpoints. 20
The regions LAn, and equal right endpoints. 25
Case 1. [(1, B)] x [(1, B)]. ~~ 45
Case 2. [(1, B)] x [(1, D)], 1 < B < D. l ) 47
Case 3. [(1, B)] x [(A, B)], 1 < A < B. ~ 50
Chapter 4. Intervals with no common endpoints. 56
Case 4. [(1, B)] x [(A, D)], 1 < A < B < D. ( ) 56
Case 5. [(1, D)] x [(A,B)], 1 < A < B < D. ( ~ 59
Case 6. [(1, B)] x [(A, D)], 1 < B :5 A < D. ~ ~ 72
Appendix. Multisimilar distributions. 74
Chapter 5. Reduction by dominance. 77
Type A dominance. 77
Type B dominance. 81
Type C dominance. 83
Type D dominance. 85
Semi-reduced games. 86
VB
Chapter 6. The further reduction of semi-reduced
games. 92
Games with IMI = 1. (Reduction to 2 x 2.) 95
Games with M = 0 which reduce to odd order. 99
Games with M = 0 which reduce to even order. 118
Chapter 7. The symmetric discrete game. 133
The symmetric game with 1/ ~ 1. 133
The symmetric game with 1/ < I/(n) 140
Chapter 8. The disjoint discrete game. 156
The disjoint game with 1/ ~ 1. 156
The disjoint game with 1/ < 1. 177
Chapter 9. Irreducibility and solutions of the
odd-order reduced games. 185
The reduced game matrix A and the associated matrix B. 185
The polynomial sequences. 190
The odd-order game of type (i). 194
The odd-order game of type (ii). 202
The odd-order game of-type (iii). 209
The odd-order game of type (iv). 210
Chapter 10. Irreducibility and solutions of the
even-order reduced games. 218
The reduced game matrix A and the associated matrix B. 218
Further polynomial identities. 222
The even-order game of type (i). 224
The even-order games of types (ii) and (iii). 249
The even-order game of type (iv). 257
Chapter 11. Explicit solutions. 264
The game on intervals. 264
The symmetric discrete game. 265
The disjoint discrete game. 266
The reduced discrete game. 266
Semi-reduced balanced discrete games with no changes
of sign on the diagonal. 266
Maximally eccentric games. 274
References 279
Index 281
Vlll
PREFACE
In our 1991 monograph [8J, Balanced Silverman Games on Gen-
eral Discrete Sets, Number 365 in this Lecture Notes series, we
began the first systematic analysis of Silverman's game on arbi-
trary discrete sets. A major advance in the simplification of such a
game, introduced in that monograph, was the reduction by means
of four special kinds of dominance, for games with penalty greater
than or equal to 1, to a finite game with payoff matrix having a
simple characteristic form. A great many of these "semi-reduced"
games were found to be further reducible, and at that time we
found it necessary to restrict our attention to a subclass which we
called balanced games, in order to make the analysis manageable.
We showed then how large classes of these balanced games were
further reducible to games of nine fundamental types, one of which
is a 2 x 2 game, four of which are infinite families of larger games
of even order, and four of which are families of games of odd order.
These "reduced" games appeared to be irreducible (meaning that
optimal strategies were unique and completely mixed; i.e., assigned
non-zero probabilities to every pure strategy), but at that time, ex-
cept for the 2 x 2 game, no proof of this irreducibility was known.
A great deal of progress on Silverman's game has been made
since that time. For the game on discrete sets, we have been able
to remove the restriction to balanced games, and at the same time
greatly simplify the reduction to the nine fundamental types (which
remain the same without the assumption of balancedness). More-
over, the irreducibility of these nine fundamental types has now
been established. Much of this work appears for the first time in
the present monograph. Earlier work on the game on discrete dis-
joint sets has meanwhile been greatly simplified and included here.
Our earlier monograph did not deal at all with Silverman's game
on intervals. At that time, the only case which had been inves-
tigated was the symmetric game, where the two players have the
same interval as strategy set. Since then we have completed a fairly
IX
thorough investigation of the game on arbitrary intervals, and the
results of that research, most of it previously unpublished, are in-
cluded here as well.
A few questions remain open. The game on discrete sets with
penalty smaller than 1 is still only incompletely understood, but
indications are that things get chaotic as the penalty approaches
zero, and that it may be impossible to obtain satisfactory general
results. For the game on intervals there are a few conditions, princi-
pally borderline cases, for which conjectured results have not been
proved in general. However, it does seem to us that the work has
reached a certain stage of near completeness. Our purpose in the
present volume is to give a unified and fairly comprehensive treat-
ment of Silverman's game, bringing together and in many cases
simplifying material which is somewhat scattered in the liturature,
and publishing for the first time significant portions of the theory.
Various parts of this work have been treated in lectures by one
or the other of the authors at the Karl-Franzens Universitat Graz,
the University of Minnesota in Minneapolis, and the Tri-College
University of Moorhead-Fargo.
We acknowledge with gratitude support from Concordia College
in the form of three Centennial Scholars Summer Research Grants,
and from the Institute for Statistics, Econometrics and Operations
Research of Graz University in the form of Visiting Professor and
Visiting Scholar stipends.
Gerald A. Heuer
Ulrike Leopold-Wild burger
x
CHAPTER 1. INTRODUCTION.
A well-known example of a two-person zero-sum game with no
optimal strategies (pure or mixed) is "higher number wins" played
on a set with no highest number. Not only are there no optimal
strategies, but all strategies are essentially equally bad. If the payoff
to the winner is 1, there is no mixed stragegy with expected payoff
better than -1 against all opposing strategies. More precisely, if U
is -0.99, or -0.999, or any other number larger than -1, and F
is any mixed strategy whatever, there is an opposing pure strategy
against which the expected payoff for F is less than U.
Perhaps in response to this situation, David Silverman in the
mid-1970s considered the following game. Each player chooses an
element from the set N of positive integers. The higher number
wins 1, unless it is at least 3 times as large as the other, in which
case it loses 1. Ties draw. From the symmetry it is obvious that if a
game value exists, it must be zero. Silverman found that this game
has a surprisingly simple solution. The unique optimal strategy
is to choose integers 1, 2 and 5 each with probability 1/3. The
reader will have no trouble verifying that this strategy assures an
expected payoff of at least 0 against every positive integer, and
hence against every mixed strategy. A similar game appeared in [3,
p. 212]. Silverman investigated some variations of this game, and
suggested analyzing such a game on a general set S.
R. J. Evans generalized the game as follows, and began calling
it Silverman's game. Let S be any set of positive real numbers.
Let parameters T > 1 (the threshold) and v > 0 (the penalty) be
given. Each player independently chooses an element of S. The
player with the larger number wins 1, unless this number is at least
T times as large as the other, in which case he loses v. Ties draw.
The game is still completely symmetric, so that an optimal strategy
for one player is optimal for the other as well, and the game value,
if it exists, must be O. Evans [1] analyzed the game for S = (A, B)
an arbitrary open interval with 0 < A < B $ 00, and Evans and
Heuer investigated the game for discrete S.
There is no reason, of course, to insist that the players ha.ve
the same strategy set, and such games with different strategy sets
1
