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player l's payoff is at most her equilibrium payoff. In a game that is not strictly competitive a player's equilibrium
strategy does not in general have these properties (consider, for example, BoS (Figure 16.1).
Exercise 24.1
Let G be a strictly
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for all
and hence
for all
maxx miny u1(x, y) and x* is a maxminimizer for player 1.
, so that
. Thus u1(x*,y*) =
An analogous argument for player 2 establishes that y* is a maxminimzer for player 2 and u2(x*,y*) = maxy minx u2
(x, y), so that u1(x
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In words, a maxminimizer for player i is an action that maximizes the payoff that player i can guarantee. A
maxminimizer for player 1 solves the problem maxx miny u1(x, y) and a maxminimizer for player 2 solves the
problem maxy minx = u2 (x, y).
I
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theorem to prove that there is an action
such that
is a Nash equilibrium of the game. (Such an
equilibrium is called a symmetric equilibrium.) Give an example of a finite symmetric game that has only
asymmetric equilibria.
2.5 Strictly Competitive
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Lemma 20.1
(Kakutani's fixed point theorem) Let X be a compact convex subset of
for which
for all
and let
be a set-valued function
the set f(x) is nonempty and convex
the graph of f is closed (i.e. for all sequences cfw_xn and cfw_yn such that
ha
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There is a continuum of citizens, each of whom has a favorite position; the distribution of favorite positions is
given by a density function f on [0,1] with f(x) > 0 for all
. A candidate attracts the votes of those citizens
whose favorite positi
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The notion of a strategic game encompasses situations much more complex than those described in the last five
examples. The following are representatives of three families of games that have been studied extensively:
auctions, games of timing, and
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Figure 17.1
The Prisoner's Dilemma (Example 16.2)
Figure 17.2
Hawk-Dove (Example 16.3).
each animal is that in which it acts like a hawk while the other acts like a dove; the worst outcome is that in which
both animals act like hawks. Each animal
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Figure 16.1
Bach or Stravinsky? (BoS) (Example 15.3).
Figure 16.2
A coordination game (Example 16.1).
Example 16.1
(A coordination game) As in BoS, two people wish to go out together, but in this case they agree on the more
desirable concert. A g
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he chooses , given that every other player j chooses his equilibrium action
deviate, given the actions of the other players.
The following restatement of the definition is sometimes useful. For any
player i's best actions given a-i:
. Briefly, no
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the basis of information about the way that the game or a similar game was played in the past (see Section 1.5). A
sequence of plays of the game can be modeled by a strategic game only if there are no strategic links between the
plays. That is, an
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Figure 13.1
A convenient representation
of a two-player strategic game
in which each player has two actions.
Under a wide range of circumstances the preference relation of player i in a strategic game can be represented
by a payoff function
(also
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The high level of abstraction of this model allows it to be applied to a wide variety of situations. A player may be
an individual human being or any other decision-making entity like a government, a board of directors, the
leadership of a revolut
Page 11
2
Nash Equilibrium
Nash equilibrium is one of the most basic concepts in game theory. In this chapter we describe it in the context of a
strategic game and in the related context of a Bayesian game.
2.1 Strategic Games
2.1.1 Definition
A strategic
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I
STRATEGIC GAMES
In this part we study a model of strategic interaction known as a strategic game, or, in the terminology of yon
Neumann and Morgenstern (1944), a 'game in normal form". This model specifies for each player a set of possible
action
Page 8
Notes
Von Neumann and Morgenstern (1944) is the classic work in game theory. Luce and Raiffa (1957) is an early
textbook; although now out-of-date, it contains superb discussions of the basic concepts in the theory. Schelling
(1960) provides a verb
Page 7
vectors of n nonnegative real numbers by . For
and
we use
to mean
. for i = 1,., n and
x > y to mean xi > yi for i = 1,.,n. We say that a function
is increasing if f (x) > f(y) whenever x > y and is
nondecreasing if
whenever x > y. A function
is co
Page 6
1.6 Bounded Rationality
When we talk in real life about games we often focus on the asymmetry between individuals in their abilities. For
example, some players may have a clearer perception of a situation or have a greater ability to analyze it. Th
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To model decision-making under uncertainty, almost all game theory uses the theories of von Neumann and
Morgenstern (1944) and of Savage (1972). That is, if the consequence function is stochastic and known to the
decision-maker (i.e. for each
the c
Page 4
the agents' activities. In a competitive analysis of this situation we look for a level of pollution consistent with the
actions that the agents take when each of them regards this level as given. By contrast, in a game theoretic analysis
of the si
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devoted to noncooperative games; it does not express our evaluation of the relative importance of the two branches.
In particular, we do not share the view of some authors that noncooperative models are more "basic" than
cooperative ones; in our op
Page 2
ical; in principle a book could be written that had essentially the same content as this one and was devoid of
mathematics. A mathematical formulation makes it easy to define concepts precisely, to verify the consistency of
ideas, and to explore th
Page 1
1
Introduction
1.1 Game Theory
Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when
decision-makers interact. The basic assumptions that underlie the theory are that decision-makers pursue welld
Page xv
and Tel Aviv University are gratefully appreciated. Special thanks are due to my friend Asher Wolinsky for endless
illuminating conversations. Part of my work on the book was supported by the United States-Israel Binational
Science Foundation (gra
Page xiv
to generic individuals. "They" has many merits as a singular pronoun, although its use can lead to ambiguities (and
complaints from editors). My preference is to use "she" for all individuals. Obviously this usage is not genderneutral, but its us
Page xiii
Exercises
Many of the exercises are challenging; we often use exercises to state subsidiary results. Instructors will probably
want to assign additional straightforward problems and adjust (by giving hints) the level of our exercises to make it
Page xii
The main interactions between the chapters. The areas of the boxes in
which the names of the chapters appear are proportional to the lengths
of the chapters. A solid arrow connecting two boxes indicates that one
chapter depends on the other; a do
Page xi
PREFACE
This book presents some of the main ideas of game theory. It is designed to serve as a textbook for a one-semester
graduate course consisting of about 28 meetings each of 90 minutes.
The topics that we cover are those that we personally wo
Page ix
15
The Nash Solution
299
15.1 Bargaining Problems
299
15.2 The Nash Solution: Definition and Characterization
301
15.3 An Axiomatic Definition
305
15.4 The Nash Solution and the Bargaining Game of Alternating
Offers
310
15.5 An Exact Implementatio
Notes
14
Stable Sets, the Bargaining Set, and the Shapley Value
274
277
14.1 Two Approaches
277
14.2 The Stable Sets of yon Neumann and Morgenstern
278
14.3 The Bargaining Set, Kernel, and Nucleolus
281
14.4 The Shapley Value
289
Notes
297