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Transcript of Game Theory Report
CONTENTS
ABSTRACT I
LIST OF TABLES II
LIST OF SYMBOLS III
CHAPTER TITLE
1 INTRODUCTION TO GAME THEORY
1.1 DEFINITION OF GAME THEORY
1.2 HISTORY
2 BASICS OF GAME THEORY
2.1 GAME
2.2 MOVE
2.3 STRATEGY
2.4 PLAYERS
2.5 TIMING
2.6 CONFLICTING GOALS
2.7 REPETITION
2.8 PAYOFF
2.9 INFORMATION AVAILIBLITY
2.10 EQUILIBRIUM
3 TYPES OF GAMES
3.1 CO-OPERATIVE AND NON-CO-OPERATIVE
3.2 SYMMETRIC AND ASYMMETRIC
3.3 ZERO SUM AND NON ZERO SUM
3.4 SIMULTANEOUS AND SEQUENTIAL
3.5 PERFECT INFORMATION AND IMPERFECT
INFORMATION
3.6 COMBINATIONAL GAMES
3.7 INFINITELY LONG GAMES
3.8 DISCREATE AND CONTINUOUS GAMES
3.9 MANY PLAYER AND POPULATION GAME
MCOE, T.E. Computer Science 20121
3.10 METAGAMES
4 REPRESENTATION OF GAMES
4.1 EXTENSIVE FORM
4.2 NORMAL FORM
4.3 CHARACTERISTIC FUNCTION FORM
4.4 PARTITION FUNCTION FORM
5 NASH EQUILIBRIUM
5.1 INTRODUCTION
5.2 HISTORY
5.3 INFORMAL DEFINITION
5.4 FORMAL DEFINITION
5.5 APPLICATION
5.6 STABILITY
5.7 OCCURRENCES
5.8 COMPUTING NASH EQUILIBRIUM
5.9 PROOF OF EXISTENCE
5.10 PURE AND MIXED STRATEGIES
5.11 MIXED STRATEGY
6 POPULAR GAMES ON GAME THEORY
6.1 PRISONER’S DILEMMA
6.2 CHICKEN GAME
7 GENERAL AND APPLIED USES OF GAME THEORY
7.1 ECONOMICS AND BUSINESS
7.2 POLITICAL SCIENCE
7.3 BIOLOGY
7.4 COMPUTER SCIENCE AND LOGIC
7.5 PHILOSOPHY
8 CONCLUSION
MCOE, T.E. Computer Science 20122
28/02/2012
GAME THEORY
Game theory is a method of studying strategic decision making. More formally, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. It’s the study of rational behaviour broken down into strategic decisions, is all about equations. If you do this, I will decide to do this, and then you will probably decide to do that -- expressed in the language of math. Game theorists try to find math equations that describe a problem as completely as possible in order to predict the outcome that benefits each individual in a group. The most universally beneficial outcome is considered the logically best outcome, and the solution. But the central paradox of game theory is that it seeks to mathematically explain decisions that are frequently made in the grip of intense emotion. Game theory sets out to analyse and explain rational behaviour. Plants, evolving mutely as they do, are rational. People, who screw up, break hearts and move markets, aren't always.
The difficulty with game theory is that its attempt to explain everything in one unified theory results in a patchwork of math that is possibly too ugly to be elegantly unified, and one that cannot possibly explain everything. A chess game, maybe. The worth tomorrow of the mutual fund in your retirement account, no .As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviours. It was initially developed in economics to understand a large collection of economic behaviours, including behaviours of firms, markets, and consumers. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviours as well.
In addition to being used to describe, predict, and explain behaviour, game theory has also been used to develop theories of ethical or normative behaviour and to prescribe such behaviour. In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behaviour. Game-theoretic arguments of this type can be found as far back as Plato.
Soumyashree Bilwar
MCOE, T.E. Computer Science 20123
List of Tables:
1. Extensive form of Game representation
2. Normal form of Game representation
3. Pure co-ordination game
4. Prisoner’s Dilemma
5. Chicken Game
List of Symbols:ααα
1. - symmetric difference
2. ∑ - summation
3. ¶ - product
4. Ω -omega
5. δ- delta
6. α – alpha
7. + - addition
8. - - subtraction
9. * - multiplication
10. > - greater
11. < -smaller
12. = - equal
MCOE, T.E. Computer Science 20124
CHAPTER 1
INTRODUCTION TO GAME THEORY
1.1 Definition of Game Theory
The study of mathematical models of conflict and cooperation between
intelligent rational decision-makers.
Game theory is a method of studying strategic decision making. More
formally, it is "the study of mathematical models of conflict and cooperation
between intelligent rational decision-makers." An alternative term suggested "as
a more descriptive name for the discipline" is interactive decision theory. Game
theory is mainly used in economics, political science, and psychology, as well as
logic and biology. The subject first addressed zero-sum games, such that one
person's gains exactly equal net losses of the other participant(s). Today,
however, game theory applies to a wide range of class relations, and has
developed into an umbrella term for the logical side of science, to include both
human and non-humans, like computers. Classic uses include a sense of balance
in numerous games, where each person has found or developed a tactic that
cannot successfully better his results, given the other approach.
Modern game theory began with the idea regarding the existence of mixed-
strategy equilibrium in two-person zero-sum games and its proof by John von
Neumann. Von Neumann's original proof used Brouwer's fixed-point theorem on
continuous mappings into compact convex sets, which became a standard method
in game theory and mathematical economics. His paper was followed by his 1944
book Theory of Games and Economic Behaviour, with Oskar Morgenstern,
which considered cooperative games of several players. The second edition of
this book provided an axiomatic theory of expected utility, which allowed
MCOE, T.E. Computer Science 20125
mathematical statisticians and economists to treat decision-making under
uncertainty.
This theory was developed extensively in the 1950s by many scholars. Game
theory was later explicitly applied to biology in the 1970s, although similar
developments go back at least as far as the 1930s. Game theory has been widely
recognized as an important tool in many fields. Eight game-theorists have won
the Nobel Memorial Prize in Economic Sciences, and John Maynard Smith was
awarded the Crafoord Prize for his application of game theory to biology.
1.2 History
The Danish mathematician Zeuthen proved that a mathematical model has a
winning strategy by using Brouwer's fixed point theorem. In his 1938
bookApplications aux Jeux de Hasard and earlier notes, Émile Borel proved a
minimax theorem for two-person zero-sum matrix games only when the pay-off
matrix was symmetric. Borel conjectured that non-existence of a mixed-strategy
equilibria in two-person zero-sum games would occur, a conjecture that was
proved false.
Game theory did not really exist as a unique field until John von
Neumann published a paper in 1928. His paper was followed by his 1944
book Theory of Games and Economic Behavior, with Oskar Morgenstern, which
considered cooperative games of several players. Von Neumann's work in game
theory culminated in the 1944 book Theory of Games and Economic Behavior by
von Neumann and Oskar Morgenstern. This foundational work contains the
method for finding mutually consistent solutions for two-person zero-sum games.
During this time period, work on game theory was primarily focused
on cooperative game theory, which analyses optimal strategies for groups of
individuals, presuming that they can enforce agreements between them about
proper strategies.
MCOE, T.E. Computer Science 20126
In 1950, the first discussion of the prisoner's dilemma appeared, and an
experiment was undertaken on this game at the RAND Corporation. Around this
same time, John Nash developed a criterion for mutual consistency of players'
strategies, known as Nash equilibrium, applicable to a wider variety of games
than the criterion proposed by von Neumann and Morgenstern. This equilibrium
is sufficiently general to allow for the analysis of non-cooperative games in
addition to cooperative ones.
In the 1970s, game theory was extensively applied in biology, largely as a
result of the work of John Maynard Smith and his evolutionarily stable strategy.
In addition, the concepts of correlated equilibrium, trembling hand perfection,
and common knowledge were introduced and analysed.
In 2005, game theorists Thomas Schelling and Robert Aumann followed
Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic
models, early examples of evolutionary game theory. Aumann contributed more
to the equilibrium school, introducing an equilibrium coarsening, correlated
equilibrium, and developing an extensive formal analysis of the assumption of
common knowledge and of its consequences.
In 2007, Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was
awarded the Nobel Prize in Economics "for having laid the foundations
of mechanism design theory." Myerson's contributions include the notion
of proper equilibrium, and an important graduate text: Game Theory, Analysis of
Conflict (Myerson 1997). Hurwicz introduced and formalized the concept
of incentive compatibility.
MCOE, T.E. Computer Science 20127
CHAPTER 2
BASICS OF GAME THEORY
Game theory is the process of modelling the strategic interaction between two
or more players in a situation containing set rules and outcomes. While used in a
number of disciplines, game theory is most notably used as a tool within the study
of economics. The economic application of game theory can be a valuable tool to aide
in the fundamental analysis of industries, sectors and any strategic interaction
between two or more firms. Here, we'll take an introductory look at game theory and
the terms involved, and introduce you to a simple method of solving games, called
backwards induction.
2.1 Game
A conflict in interest among n individuals or groups (players). There
exists a set of rules that define the terms of exchange of information and
pieces, the conditions under which the game begins, and the possible legal
exchanges in particular conditions. The entirety of the game is defined by all
the moves to that point, leading to an outcome.
2.2 Move
The way in which the game progresses between states through exchange of
pieces and information. Moves are defined by the rules of the game and can be made
in either alternating fashion ,occur simultaneously for all players. Moves may be
choice or by chance. For example , choosing a card from a deck or rolling a die is a
MCOE, T.E. Computer Science 20128
chance move with known probabilities. On the other hand ,asking for cards in
blackjack is a choice move.
2.3 Strategy
A strategy is a set of best choices for a player for an entire game. It is an
overlying plan that cannot be upset by occurrences in the game first.
2.4 Players
The number of participants may be two or more. A player can be a single
individual or a group with the same objective.
2.5 Timings
The conflicting parties decide simultaneously.
2.6 Conflicting goals
Each party is interested in maximizing his or her goal at the expense of the
other.
2.7 Repetition
Most instances involve repetitive solutions.
MCOE, T.E. Computer Science 20129
2.8 Payoff
The payoffs for each combination of decisions are known by all parties.
2.9 Information Availability
All parties are aware of all pertinent information. Each player knows all
possible courses of action open to the opponent as well as anticipated payoffs.
2.10 Equilibrium
The point in a game where both players have made their decisions and an
outcome is reached.
MCOE, T.E. Computer Science 201210
CHAPTER 3
TYPES OF GAMES
3.1 Cooperative or non-cooperative
A game is cooperative if the players are able to form binding
commitments. For instance the legal system requires them to adhere to their
promises. In non-cooperative games this is not possible.Often it is assumed
that communication among players is allowed in cooperative games, but not in
non-cooperative ones. However, this classification on two binary criteria has
been questioned, and sometimes rejected (Harsanyi 1974).
Of the two types of games, non-cooperative games are able to model
situations to the finest details, producing accurate results. Cooperative games
focus on the game at large. Considerable efforts have been made to link the
two approaches. The so-called Nash-programme has already established many
of the cooperative solutions as non-cooperative equilibria. Hybrid games
contain cooperative and non-cooperative elements. For instance, coalitions of
players are formed in a cooperative game, but these play in a non-cooperative
fashion.
3.2 Symmetric and asymmetric
A symmetric game is a game where the payoffs for playing a particular
strategy depend only on the other strategies employed, not on who is playing
them. If the identities of the players can be changed without changing the
payoff to the strategies, then a game is symmetric. Many of the commonly
studied 2×2 games are symmetric. The standard representations of chicken,
the prisoner's dilemma, and the stag hunt are all symmetric games. Some
MCOE, T.E. Computer Science 201211
scholars would consider certain asymmetric games as examples of these
games as well. However, the most common payoffs for each of these games
are symmetric.
Most commonly studied asymmetric games are games where there are
not identical strategy sets for both players. For instance, the ultimatum
game and similarly the dictator game have different strategies for each player.
It is possible, however, for a game to have identical strategies for both players,
yet be asymmetric. For example, the game pictured to the right is asymmetric
despite having identical strategy sets for both players.
3.3 Zero-sum and non-zero-sum
Zero-sum games are a special case of constant-sum games, in which
choices by players can neither increase nor decrease the available resources.
In zero-sum games the total benefit to all players in the game, for every
combination of strategies, always adds to zero (more informally, a player
benefits only at the equal expense of others). Poker exemplifies a zero-sum
game (ignoring the possibility of the house's cut), because one wins exactly
the amount one's opponents lose. Other zero-sum games include matching
pennies and most classical board games including Go and chess.
Many games studied by game theorists (including the
infamous prisoner's dilemma) are non-zero-sum games, because
some outcomes have net results greater or less than zero. Informally, in non-
zero-sum games, a gain by one player does not necessarily correspond with a
loss by another.
Constant-sum games correspond to activities like theft and gambling,
but not to the fundamental economic situation in which there are
potential gains from trade. It is possible to transform any game into a
(possibly asymmetric) zero-sum game by adding an additional dummy player
(often called "the board"), whose losses compensate the players' net winnings.
MCOE, T.E. Computer Science 201212
3.4 Simultaneous and sequential
Simultaneous games are games where both players move
simultaneously, or if they do not move simultaneously, the later players are
unaware of the earlier players' actions (making them effectively
simultaneous). Sequential games (or dynamic games) are games where later
players have some knowledge about earlier actions. This need not be perfect
information about every action of earlier players; it might be very little
knowledge. For instance, a player may know that an earlier player did not
perform one particular action, while he does not know which of the other
available actions the first player actually performed.
The difference between simultaneous and sequential games is captured
in the different representations discussed above. Often, normal form is used to
represent simultaneous games, and extensive form is used to represent
sequential ones. The transformation of extensive to normal form is one way,
meaning that multiple extensive form games correspond to the same normal
form. Consequently, notions of equilibrium for simultaneous games are
insufficient for reasoning about sequential games; see sub game perfection.
3.5 Perfect information and imperfect information
An important subset of sequential games consists of games of perfect
information. A game is one of perfect information if all players know the
moves previously made by all other players. Thus, only sequential games can
be games of perfect information, since in simultaneous games not every player
knows the actions of the others. Recreational games of perfect information
games include chess, go, and mancala. Many card games are games of
imperfect information, for instance poker or contract bridge.
MCOE, T.E. Computer Science 201213
Perfect information is often confused with complete information,
which is a similar concept. Complete information requires that every player
know the strategies and payoffs available to the other players but not
necessarily the actions taken. Games of incomplete information can be
reduced, however, to games of imperfect information by introducing "moves
by nature" .
3.6 Combinatorial games
Games in which the difficulty of finding an optimal strategy stems
from the multiplicity of possible moves are called combinatorial games.
Examples include chess and go. Games that involve imperfect or incomplete
information may also have a strong combinatorial character, for
instance backgammon. There is no unified theory addressing combinatorial
elements in games. There are, however, mathematical tools that can solve
particular problems and answer some general questions.
Games of perfect information have been studied in combinatorial
game theory, which has developed novel representations, e.g. surreal numbers,
as well as combinatorial and algebraic (and sometimes non-constructive)
proof methods to solve games of certain types, including some "loopy" games
that may result in infinitely long sequences of moves. These methods address
games with higher combinatorial complexity than those usually considered in
traditional (or "economic") game theory. A typical game that has been solved
this way is hex. A related field of study, drawing from computational
complexity theory, is game complexity, which is concerned with estimating
the computational difficulty of finding optimal strategies.
Research in artificial intelligence has addressed both perfect and
imperfect (or incomplete) information games that have very complex
combinatorial structures (like chess, go, or backgammon) for which no
provable optimal strategies have been found. The practical solutions involve
MCOE, T.E. Computer Science 201214
computational heuristics, like alpha-beta pruning or use of artificial neural
networks trained by reinforcement learning, which make games more tractable
in computing practice.
3.7 Infinitely long games
Games, as studied by economists and real-world game players, are
generally finished in finitely many moves. Pure mathematicians are not so
constrained, and set theorists in particular study games that last for infinitely
many moves, with the winner (or other payoff) not known until after all those
moves are completed.
The focus of attention is usually not so much on what is the best way
to play such a game, but simply on whether one or the other player has
a winning strategy. (It can be proven, using the axiom of choice, that there are
games—even with perfect information, and where the only outcomes are
"win" or "lose"—for which neither player has a winning strategy.) The
existence of such strategies, for cleverly designed games, has important
consequences in descriptive set theory.
3.8 Discrete and continuous games
Much of game theory is concerned with finite, discrete games, that
have a finite number of players, moves, events, outcomes, etc. Many concepts
can be extended, however. Continuous games allow players to choose a
strategy from a continuous strategy set. For instance, Cournot competition is
typically modeled with players' strategies being any non-negative quantities,
including fractional quantities.
3.9 Many-player and population games
MCOE, T.E. Computer Science 201215
Games with an arbitrary, but finite, number of players are often called
n-person games (Luce & Raiffa 1957). Evolutionary game theory considers
games involving a population of decision makers, where the frequency with
which a particular decision is made can change over time in response to the
decisions made by all individuals in the population. In biology, this is
intended to model (biological)evolution, where genetically programmed
organisms pass along some of their strategy programming to their offspring.
In economics, the same theory is intended to capture population changes
because people play the game many times within their lifetime, and
consciously (and perhaps rationally) switch strategies (Webb 2007).
3.10 Metagames
These are games the play of which is the development of the rules for
another game, the target or subject game. Metagames seek to maximize the
utility value of the rule set developed. The theory of metagames is related
to mechanism design theory.
The term metagame analysis is also used to refer to a practical
approach developed by Nigel Howard (Howard 1971) whereby a situation is
framed as a strategic game in which stakeholders try to realise their objectives
by means of the options available to them. Subsequent developments have led
to the formulation of Confrontation Analysis.
MCOE, T.E. Computer Science 201216
CHAPTER 4
REPRESENTATION OF GAMES
4.1 Type 1: Extensive form
Fig 4.1
The extensive form can be used to formalize games with a time
sequencing of moves. Games here are played on trees (as pictured to the left).
Here each vertex (or node) represents a point of choice for a player. The
player is specified by a number listed by the vertex. The lines out of the vertex
represent a possible action for that player. The payoffs are specified at the
bottom of the tree. The extensive form can be viewed as a multi-player
generalization of adecision tree.
In the game pictured to the left, there are two players. Player 1 moves
first and chooses either F or U. Player 2 sees Player 1's move and then
chooses A or R. Suppose that Player 1 chooses U and then Player
2 chooses A, then Player 1 gets 8 and Player 2 gets 2.
The extensive form can also capture simultaneous-move games and
games with imperfect information. To represent it, either a dotted line
connects different vertices to represent them as being part of the same
information set (i.e., the players do not know at which point they are), or a
closed line is drawn around them.
MCOE, T.E. Computer Science 201217
4.2 Type 2: Normal form
Fig 4.2
The normal (or strategic form) game is usually represented by
a matrix which shows the players, strategies, and pay-offs (see the example to
the right). More generally it can be represented by any function that associates
a payoff for each player with every possible combination of actions. In the
accompanying example there are two players; one chooses the row and the
other chooses the column. Each player has two strategies, which are specified
by the number of rows and the number of columns. The payoffs are provided
in the interior. The first number is the payoff received by the row player
(Player 1 in our example); the second is the payoff for the column player
(Player 2 in our example). Suppose that Player 1 plays Up and that Player 2
plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.
Every extensive-form game has an equivalent normal-form game,
however the transformation to normal form may result in an exponential blow
up in the size of the representation, making it computationally impractical.
4.3 Type 3: Characteristic function form
MCOE, T.E. Computer Science 201218
Player 2
chooses Left
Player 2
chooses Right
Player 1
chooses Up4, 3 –1, –1
Player 1
chooses Down0, 0 3, 4
Normal form or payoff matrix of a 2-player, 2-strategy
In games that possess removable utility separate rewards are not given;
rather, the characteristic function decides the payoff of each unity. The idea is
that the unity that is 'empty', so to speak, does not receive a reward at all.
The origin of this form is to be found in John von Neumann and Oskar
Morgenstern's book; when looking at these instances, they guessed that when
a union C appears, it works against the fraction (N/C) as if two individuals
were playing a normal game. The balanced payoff of C is a basic function.
Although there are differing examples that help determine coalitional amounts
from normal games, not all appear that in their function form can be derived
from such.
Such characteristic functions have expanded to describe games where
there is no removable utility.
4.4 Type 4: Partition function form
The characteristic function form ignores the possible externalities of
coalition formation. In the partition function form the payoff of a coalition
depends not only on its members, but also on the way the rest of the players
are partitioned (Thrall & Lucas 1963).
MCOE, T.E. Computer Science 201219
CHAPTER 5
NASH EQUILLIBRIUM
5.1 Introduction
In game theory, Nash equilibrium (named after John Forbes Nash,
who proposed it) is a solution concept of a game involving two or more
players, in which each player is assumed to know the equilibrium strategies of
the other players, and no player has anything to gain by changing only his
own strategy unilaterally. If each player has chosen a strategy and no player
can benefit by changing his or her strategy while the other players keep theirs
unchanged, then the current set of strategy choices and the corresponding
payoffs constitute a Nash equilibrium. The practical and general implication is
that when players also act in the interests of the group, then they are better off
than if they acted in their individual interests alone.
Stated simply, Amy and Phil are in Nash equilibrium if Amy is
making the best decision she can, taking into account Phil's decision, and Phil
is making the best decision he can, taking into account Amy's decision.
Likewise, a group of players are in Nash equilibrium if each one is making the
best decision that he or she can, taking into account the decisions of the
others. However, Nash equilibrium does not necessarily mean the best payoff
for all the players involved; in many cases, all the players might improve their
payoffs if they could somehow agree on strategies different from the Nash
equilibrium: e.g., competing businesses forming a cartel in order to increase
their profits.
5.2 History
A version of the Nash equilibrium concept was first used by Antoine
Augustin Cournot in his theory of oligopoly (1838). In Cournot's theory, firms MCOE, T.E. Computer Science 2012
20
choose how much output to produce to maximize their own profit. However,
the best output for one firm depends on the outputs of others. A Cournot
equilibrium occurs when each firm's output maximizes its profits given the
output of the other firms, which is a pure strategy Nash Equilibrium.
The modern game-theoretic concept of Nash Equilibrium is instead
defined in terms of mixed strategies, where players choose a probability
distribution over possible actions.
Since the development of the Nash equilibrium concept, game
theorists have discovered that it makes misleading predictions (or fails to
make a unique prediction) in certain circumstances. Therefore they have
proposed many related solution concepts (also called 'refinements' of Nash
equilibrium) designed to overcome perceived flaws in the Nash concept. One
particularly important issue is that some Nash equilibria may be based on
threats that are not 'credible'.
5.3 Informal definition
Informally, a set of strategies is a Nash equilibrium if no player can do
better by unilaterally changing his or her strategy. To see what this means,
imagine that each player is told the strategies of the others. Suppose then that
each player asks himself or herself: "Knowing the strategies of the other
players, and treating the strategies of the other players as set in stone, can I
benefit by changing my strategy?"
If any player would answer "Yes", then that set of strategies is not a
Nash equilibrium. But if every player prefers not to switch (or is indifferent
between switching and not) then the set of strategies is a Nash equilibrium.
Thus, each strategy in a Nash equilibrium is a best response to all other
strategies in that equilibrium.
The Nash equilibrium may also have non-rational consequences in
sequential games because players may "threaten" each other with non-rational
MCOE, T.E. Computer Science 201221
moves. For such games the subgame perfect Nash equilibrium may be more
meaningful as a tool of analysis.
5.4 Formal definition
Let (S, f) be a game with n players, where Si is the strategy set for
player i, S=S1 × S2 ... × Sn is the set of strategy profiles and f=(f1(x), ...,
fn(x)) is the payoff function for x S. Let xi be a strategy profile of
player i and x-i be a strategy profile of all players except for player i. When
each player i 1, ..., n chooses strategy xi resulting in strategy profile x =
(x1, ..., xn) then player i obtains payofffi(x). Note that the payoff depends on the
strategy profile chosen, i.e., on the strategy chosen by player i as well as the
strategies chosen by all the other players. A strategy profile x* S is a Nash
equilibrium (NE) if no unilateral deviation in strategy by any single player is
profitable for that player, that is
When the inequality above holds strictly (with instead of ) for all
players and all feasible alternative strategies, then the equilibrium is classified
as a strict Nash equilibrium. If instead, for some player, there is exact
equality between and some other strategy in the set , then the equilibrium
is classified as a weak Nash equilibrium.
5.5 Applications
Game theorists use the Nash equilibrium concept to analyze the
outcome of the strategic interaction of several decision makers. In other
words, it provides a way of predicting what will happen if several people or
several institutions are making decisions at the same time, and if the outcome MCOE, T.E. Computer Science 2012
22
depends on the decisions of the others. The simple insight underlying John
Nash's idea is that we cannot predict the result of the choices of multiple
decision makers if we analyze those decisions in isolation. Instead, we must
ask what each player would do, taking into account the decision-making of
the others.
Nash equilibrium has been used to analyze hostile situations
like war and arms races(see Prisoner's dilemma), and also how conflict may
be mitigated by repeated interaction (see Tit-for-tat). It has also been used to
study to what extent people with different preferences can cooperate
(see Battle of the sexes), and whether they will take risks to achieve a
cooperative outcome (see Stag hunt). It has been used to study the adoption
of technical standards, and also the occurrence of bank runs and currency
crises (see Coordination game). Other applications include traffic flow
(see Wardrop's principle), how to organize auctions (see auction theory), the
outcome of efforts exerted by multiple parties in the education process, [3] and
even penalty kicks in soccer (see Matching pennies).
5.6 Stability
The concept of stability, useful in the analysis of many kinds of
equilibria, can also be applied to Nash equilibria
A Nash equilibrium for a mixed strategy game is stable if a small
change (specifically, an infinitesimal change) in probabilities for one
player leads to a situation where two conditions hold:
1. the player who did not change has no better strategy in the new circumstance
2. the player who did change is now playing with a strictly worse strategy.
If these cases are both met, then a player with the small change in his mixed-
strategy will return immediately to the Nash equilibrium. The equilibrium is said
to be stable. If condition one does not hold then the equilibrium is unstable. If
MCOE, T.E. Computer Science 201223
only condition one holds then there are likely to be an infinite number of optimal
strategies for the player who changed. John Nash showed that the latter situation
could not arise in a range of well-defined games.
In the "driving game" example above there are both stable and unstable
equilibria. The equilibria involving mixed-strategies with 100% probabilities are
stable. If either player changes his probabilities slightly, they will be both at a
disadvantage, and his opponent will have no reason to change his strategy in turn.
The (50%,50%) equilibrium is unstable. If either player changes his probabilities,
then the other player immediately has a better strategy at either (0%, 100%) or
(100%, 0%).
Stability is crucial in practical applications of Nash equilibria, since the
mixed-strategy of each player is not perfectly known, but has to be inferred from
statistical distribution of his actions in the game. In this case unstable equilibria
are very unlikely to arise in practice, since any minute change in the proportions
of each strategy seen will lead to a change in strategy and the breakdown of the
equilibrium.
The Nash equilibrium defines stability only in terms of unilateral deviations.
In cooperative games such a concept is not convincing enough. Strong Nash
equilibrium allows for deviations by every conceivable coalition. Formally,
a Strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the
actions of its complements as given, can cooperatively deviate in a way that
benefits all of its members. However, the Strong Nash concept is sometimes
perceived as too "strong" in that the environment allows for unlimited private
communication. In fact, Strong Nash equilibrium has to be Pareto efficient. As a
result of these requirements, Strong Nash is too rare to be useful in many
branches of game theory. However, in games such as elections with many more
players than possible outcomes, it can be more common than a stable
equilibrium.
A refined Nash equilibrium known as coalition-proof Nash
equilibrium (CPNE)[6] occurs when players cannot do better even if they are
allowed to communicate and make "self-enforcing" agreement to deviate. Every
MCOE, T.E. Computer Science 201224
correlated strategy supported by iterated strict dominance and on the Pareto
frontier is a CPNE.[8] Further, it is possible for a game to have a Nash equilibrium
that is resilient against coalitions less than a specified size, k. CPNE is related to
the theory of the core.
Finally in the eighties, building with great depth on such ideas Mertens-stable
equilibria were introduced as a solution concept. Mertens stable equilibria satisfy
both forward induction and backward induction. In a Game theory context stable
equilibria now usually refer to Mertens stable equilibria.
5.7 Occurrences
If a game has a unique Nash equilibrium and is played among players under
certain conditions, then the NE strategy set will be adopted. Sufficient conditions
to guarantee that the Nash equilibrium is played are:
1. The players all will do their utmost to maximize their expected payoff as
described by the game.
2. The players are flawless in execution.
3. The players have sufficient intelligence to deduce the solution.
4. The players know the planned equilibrium strategy of all of the other players.
5. The players believe that a deviation in their own strategy will not cause
deviations by any other players.
6. There is common knowledge that all players meet these conditions, including
this one. So, not only must each player know the other players meet the
conditions, but also they must know that they all know that they meet them,
and know that they know that they know that they meet them, and so on.
Where the conditions are not met
MCOE, T.E. Computer Science 201225
Examples of game theory problems in which these conditions are not met:
1. The first condition is not met if the game does not correctly describe the
quantities a player wishes to maximize. In this case there is no particular
reason for that player to adopt an equilibrium strategy. For instance, the
prisoner’s dilemma is not a dilemma if either player is happy to be jailed
indefinitely.
2. Intentional or accidental imperfection in execution. For example, a computer
capable of flawless logical play facing a second flawless computer will result
in equilibrium. Introduction of imperfection will lead to its disruption either
through loss to the player who makes the mistake, or through negation of
the common knowledge criterion leading to possible victory for the player.
(An example would be a player suddenly putting the car into reverse in
the game of chicken, ensuring a no-loss no-win scenario).
3. In many cases, the third condition is not met because, even though the
equilibrium must exist, it is unknown due to the complexity of the game, for
instance in Chinese chess. Or, if known, it may not be known to all players,
as when playing tic-tac-toe with a small child who desperately wants to win
(meeting the other criteria).
4. The criterion of common knowledge may not be met even if all players do, in
fact, meet all the other criteria. Players wrongly distrusting each other's
rationality may adopt counter-strategies to expected irrational play on their
opponents’ behalf. This is a major consideration in “Chicken” or an arms
race, for example.
Where the conditions are met
Due to the limited conditions in which NE can actually be observed,
they are rarely treated as a guide to day-to-day behaviour, or observed in
practice in human negotiations. However, as a theoretical concept
in economics and evolutionary biology, the NE has explanatory power. The
MCOE, T.E. Computer Science 201226
payoff in economics is utility (or sometimes money), and in evolutionary
biology gene transmission, both are the fundamental bottom line of survival.
Researchers who apply games theory in these fields claim that strategies
failing to maximize these for whatever reason will be competed out of the
market or environment, which are ascribed the ability to test all strategies.
This conclusion is drawn from the "stability" theory above.
5.8 Computing Nash Equilibrium
If a player A has a dominant strategy then there exists a Nash
equilibrium in which A plays . In the case of two players A and B, there
exists a Nash equilibrium in which A plays and B plays a best
response to . If is a strictly dominant strategy, A plays in all Nash
equilibria. If both A and B have strictly dominant strategies, there exists a
unique Nash equilibrium in which each plays his strictly dominant strategy.
In games with mixed strategy Nash equilibria, the probability of a
player choosing any particular strategy can be computed by assigning a
variable to each strategy that represents a fixed probability for choosing that
strategy. In order for a player to be willing to randomize, his expected payoff
for each strategy should be the same. In addition, the sum of the probabilities
for each strategy of a particular player should be 1. This creates a system of
equations from which the probabilities of choosing each strategy can be
derived.
5.9 Proof of existence
Proof using the Kakutani fixed point theorem
Nash's original proof (in his thesis) used Brouwer's fixed point
theorem (e.g., see below for a variant). We give a simpler proof via
MCOE, T.E. Computer Science 201227
the Kakutani fixed point theorem, following Nash's 1950 paper (he
credits David Gale with the observation that such a simplification is possible).
To prove the existence of a Nash Equilibrium, let be the best
response of player i to the strategies of all other players.
Here, , where , is a mixed strategy profile in the set
of all mixed strategies and is the payoff function for player i. Define a set-
valued function such that . The
existence of a Nash Equilibrium is equivalent to having a fixed point.
Kakutani's fixed point theorem guarantees the existence of a fixed point if the
following four conditions are satisfied.
1. is compact, convex, and nonempty.
2. is nonempty.
3. is convex.
4. is upper hemicontinuous
Condition 1. is satisfied from the fact that is a simplex and thus compact.
Convexity follows from players' ability to mix strategies. is nonempty as
long as players have strategies.
Condition 2. is satisfied because players maximize expected payoffs which is
continuous function over a compact set. The Weierstrass Extreme Value
Theorem guarantees that there is always a maximum value.
Condition 3. is satisfied as a result of mixed strategies.
Suppose , then . i.e. if two
strategies maximize payoffs, then a mix between the two strategies will yield
the same payoff.
MCOE, T.E. Computer Science 201228
Condition 4. is satisfied by way of Berge's maximum theorem. Because is
continuous and compact, is upper hemicontinuous.
Therefore, there exists a fixed point in and a Nash Equilibrium.
When Nash made this point to John von Neumann in 1949, von Neumann
famously dismissed it with the words, "That's trivial, you know. That's just
a fixed point theorem." (See Nasar, 1998, p. 94.)
Alternate proof using the Brouwer fixed-point theorem
We have a game where is the number of players
and is the action set for the players. All of the action
sets are finite. Let denote the set of mixed
strategies for the players. The finiteness of the s ensures the compactness
of .
We can now define the gain functions. For a mixed strategy ,
we let the gain for player on action be
The gain function represents the benefit a player gets by unilaterally changing
his strategy. We now define where
MCOE, T.E. Computer Science 201229
for . We see that
We now use to define as follows. Let
for . It is easy to see that each is a valid mixed strategy in . It is
also easy to check that each is a continuous function of , and hence is a
continuous function. Now is the cross product of a finite number of
compact convex sets, and so we get that is also compact and convex.
Therefore we may apply the Brouwer fixed point theorem to . So has a
fixed point in , call it .
I claim that is a Nash Equilibrium in . For this purpose, it suffices to
show that
This simply states the each player gains no benefit by unilaterally changing
his strategy which is exactly the necessary condition for being a Nash
Equilibrium.
MCOE, T.E. Computer Science 201230
Now assume that the gains are not all zero. Therefore, , ,
and such that . Note then that
So let .
Also we shall denote as the gain vector indexed by actions in .
Since we clearly have that . Therefore we see
that
Since we have that is some positive scaling of the vector .
Now I claim that
. To see this, we first note that if then this is
true by definition of the gain function. Now assume that . By our previous
statements we have that
MCOE, T.E. Computer Science 201231
and so the left term is zero, giving us that the entire expression is as needed.
So we finally have that
where the last inequality follows since is a non-zero vector. But this is a
clear contradiction, so all the gains must indeed be zero. Therefore is a
Nash Equilibrium for as needed.
5.10 Pure and mixed strategies
A pure strategy provides a complete definition of how a player will
play a game. In particular, it determines the move a player will make for any
situation he or she could face. A player's strategy set is the set of pure
strategies available to that player.
MCOE, T.E. Computer Science 201232
A mixed strategy is an assignment of a probability to each pure
strategy. This allows for a player to randomly select a pure strategy. Since
probabilities are continuous, there are infinitely many mixed strategies
available to a player, even if their strategy set is finite.
Of course, one can regard a pure strategy as a degenerate case of a
mixed strategy, in which that particular pure strategy is selected with
probability 1 and every other strategy with probability 0.
A totally mixed strategy is a mixed strategy in which the player
assigns a strictly positive probability to every pure strategy. (Totally mixed
strategies are important for equilibrium refinement such astrembling hand
perfect equilibrium.)
5.11 Mixed strategy
Illustration
Consider the payoff matrix pictured to the right (known as a coordination
game). Here one player chooses the row and the other chooses a column. The row
player receives the first payoff, the column player the second. If row opts to
play A with probability 1 (i.e. play A for sure), then he is said to be playing a pure
MCOE, T.E. Computer Science 201233
A B
A 1, 1 0, 0
B 0, 0 1, 1
Pure coordination game Fig 5.1
strategy. If column opts to flip a coin and play A if the coin lands heads and B if the
coin lands tails, then she is said to be playing a mixed strategy, and not a pure
strategy.
Significance
In his famous paper, John Forbes Nash proved that there is
an equilibrium for every finite game. One can divide Nash equilibria into two
types. Pure strategy Nash equilibriaare Nash equilibria where all players are
playing pure strategies. Mixed strategy Nash equilibria are equilibria where at
least one player is playing a mixed strategy. While Nash proved that every
finite game has a Nash equilibrium, not all have pure strategy Nash equilibria.
For an example of a game that does not have a Nash equilibrium in pure
strategies, see Matching pennies. However, many games do have pure strategy
Nash equilibria (e.g. the Coordination game, the Prisoner's dilemma, the Stag
hunt). Further, games can have both pure strategy and mixed strategy
equilibria.
A disputed meaning
During the 1980s, the concept of mixed strategies came under heavy
fire for being "intuitively problematic".[2] Randomization, central in mixed
strategies, lacks behavioral support. Seldom do people make their choices
following a lottery. This behavioral problem is compounded by the cognitive
difficulty that people are unable to generate random outcomes without the aid
of a random or pseudo-random generator.[2]
In 1991, game theorist Ariel Rubinstein described alternative ways of
understanding the concept. The first, due to Harsanyi (1973), [4] is
called purification, and supposes that the mixed strategies interpretation
merely reflects our lack of knowledge of the players' information and
decision-making process. Apparently random choices are then seen as
MCOE, T.E. Computer Science 201234
consequences of non-specified, payoff-irrelevant exogeneous factors.
However, it is unsatisfying to have results that hang on unspecified factors.[3]
A second interpretation imagines the game players standing for a large
population of agents. Each of the agents chooses a pure strategy, and the
payoff depends on the fraction of agents choosing each strategy. The mixed
strategy hence represents the distribution of pure strategies chosen by each
population. However, this does not provide any justification for the case when
players are individual agents.
Later, Aumann and Brandenburger (1995), [5] re-interpreted Nash
equilibrium as an equilibrium in beliefs, rather than actions. For instance,
in Rock-paper-scissors an equilibrium in beliefs would have each
player believing the other was equally likely to play each strategy. This
interpretation weakens the predictive power of Nash equilibrium, however,
since it is possible in such an equilibrium for each player to actually play a
pure strategy of Rock.
Ever since, game theorists' attitude towards mixed strategies-based
results have been ambivalent. Mixed strategies are still widely used for their
capacity to provide Nash equilibria in games where no equilibrium in pure
strategies exists, but the model does not specify why and how players
randomize their decisions.
MCOE, T.E. Computer Science 201235
CHAPTER 6
POPULAR PROBLEMS ON GAME THEORY
6.1 Prisoner’s Dilemma
Co-operation is usually analyzed in game theory by the means of non-zero-sum game called
Prisoner’s Dilemma.
Prisoner B stays silent
(co-operates)
Prisoner B confesses
(defects)
Prisoner A stays silent
(co-operates)
Each serve one month Prisoner A: 1 year
Prisoner B: goes free
Prisoner A confesses
(defects)
Prisoner A: goes free
Prisoner B: 1 year
Each serves 3 months
Analysis of Prisoner’s Dilemma
Each player gains when both co-operate (1 month )
One player co-operates then one who defects will gain more(defects –
freed ,confesses- 1year)
If both defect both lose (or gain very little) but not as much as the “cheated”
co-operator who’s co-operation is not returned.(3 months)
Prisoner’s Dilemma has a single Nash Equilibrium.
MCOE, T.E. Computer Science 201236
6.2 Chicken Game
Chicken is a famous game where two people drive on a collision course
straight towards each other. Whoever swerves is considered a ‘chicken’ and loses, but
if nobody swerves, they will both crash.
Driver B
Swerve
Driver B
Goes Straight
Driver A
Swerve
Tie, Tie Lose , Win
Driver A
Goes Straight
Win ,Lose Crash
Analysis of Chicken’s Game
Both lose when both swerve
One player wins when one swerves and other goes straight.
If both go straight , both lose (lose more than what they would have lost when
both swerve. Because if both go straight they crash)
Chicken Game has two Nash Equilibrium.
CHAPTER 7MCOE, T.E. Computer Science 2012
37
GENERAL AND APPLIED USES OF GAME THEORY
7.1 Economics and Business
Game theory is a major method used in mathematical economics and
business for modelling competing behaviours of interacting agents.
Applications include a wide array of economic phenomena and approaches,
such as auctions, bargaining, fair division, duopolies, oligopolies, social
network formation, agent-based computational economics, general
equilibrium, mechanism design, andvoting systems, and across such broad
areas as experimental economics, behavioral economics, information
economics, industrial organization, and political economy.
This research usually focuses on particular sets of strategies known
as equilibrium in games. These "solution concepts" are usually based on what
is required by norms of rationality. In non-cooperative games, the most
famous of these is the Nash equilibrium. A set of strategies is a Nash
equilibrium if each represents a best response to the other strategies. So, if all
the players are playing the strategies in a Nash equilibrium, they have no
unilateral incentive to deviate, since their strategy is the best they can do given
what others are doing.
The payoffs of the game are generally taken to represent the utility of
individual players. Often in modeling situations the payoffs represent money,
which presumably corresponds to an individual's utility. This assumption,
however, can be faulty.
A prototypical paper on game theory in economics begins by
presenting a game that is an abstraction of some particular economic situation.
One or more solution concepts are chosen, and the author demonstrates which
strategy sets in the presented game are equilibria of the appropriate type.
Naturally one might wonder to what use should this information be put.
Economists and business professors suggest two primary uses (noted
above): descriptive and prescriptive.
MCOE, T.E. Computer Science 201238
7.2 Political science
The application of game theory to political science is focused in the
overlapping areas of fair division, political economy, public choice, war
bargaining, positive political theory, and social choice theory. In each of these
areas, researchers have developed game-theoretic models in which the players
are often voters, states, special interest groups, and politicians.
For early examples of game theory applied to political science, see the
work of Anthony Downs. In his book An Economic Theory of
Democracy (Downs 1957) he applies the Hotelling firm location model to the
political process. In the Downsian model, political candidates commit to
ideologies on a one-dimensional policy space. The theorist shows how the
political candidates will converge to the ideology preferred by the median
voter.
A game-theoretic explanation for democratic peace is that public and
open debate in democracies send clear and reliable information regarding their
intentions to other states. In contrast, it is difficult to know the intentions of
nondemocratic leaders, what effect concessions will have, and if promises will
be kept. Thus there will be mistrust and unwillingness to make concessions if
at least one of the parties in a dispute is a non-democracy (Levy &
Razin 2003).
7.3 Biology
Unlike economics, the payoffs for games in biology are often
interpreted as corresponding to fitness. In addition, the focus has been less
on equilibria that correspond to a notion of rationality, but rather on ones that
would be maintained by evolutionary forces. The best known equilibrium in
biology is known as theevolutionarily stable strategy (or ESS), and was first
introduced in (Smith & Price 1973). Although its initial motivation did not
MCOE, T.E. Computer Science 201239
involve any of the mental requirements of the Nash equilibrium, every ESS is
a Nash equilibrium.
In biology, game theory has been used to understand many different
phenomena. It was first used to explain the evolution (and stability) of the
approximate 1:1sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a
result of evolutionary forces acting on individuals who could be seen as trying
to maximize their number of grandchildren.
Additionally, biologists have used evolutionary game theory and the
ESS to explain the emergence of animal communication (Harper & Maynard
Smith 2003). The analysis of signaling games and other communication
games has provided some insight into the evolution of communication among
animals. For example, the mobbing behavior of many species, in which a
large number of prey animals attack a larger predator, seems to be an example
of spontaneous emergent organization. Ants have also been shown to exhibit
feed-forward behavior akin to fashion, see Butterfly Economics.
Biologists have used the game of chicken to analyze fighting behavior
and territoriality.
Maynard Smith, in the preface to Evolution and the Theory of Games,
writes, "paradoxically, it has turned out that game theory is more readily
applied to biology than to the field of economic behaviour for which it was
originally designed". Evolutionary game theory has been used to explain
many seemingly incongruous phenomena in nature.
One such phenomenon is known as biological altruism. This is a
situation in which an organism appears to act in a way that benefits other
organisms and is detrimental to itself. This is distinct from traditional notions
of altruism because such actions are not conscious, but appear to be
evolutionary adaptations to increase overall fitness. Examples can be found in
species ranging from vampire bats that regurgitate blood they have obtained
from a night's hunting and give it to group members who have failed to feed,
to worker bees that care for the queen bee for their entire lives and never mate,
to Vervet monkeys that warn group members of a predator's approach, even MCOE, T.E. Computer Science 2012
40
when it endangers that individual's chance of survival. All of these actions
increase the overall fitness of a group, but occur at a cost to the individual.
Evolutionary game theory explains this altruism with the idea of kin
selection. Altruists discriminate between the individuals they help and favor
relatives. Hamilton's rule explains the evolutionary reasoning behind this
selection with the equation c<b*r where the cost ( c ) to the altruist must be
less than the benefit ( b ) to the recipient multiplied by the coefficient of
relatedness ( r ). The more closely related two organisms are causes the
incidences of altruism to increase because they share many of the same
alleles. This means that the altruistic individual, by ensuring that the alleles of
its close relative are passed on, (through survival of its offspring) can forgo
the option of having offspring itself because the same number of alleles are
passed on. Helping a sibling for example (in diploid animals), has a
coefficient of ½, because (on average) an individual shares ½ of the alleles in
its sibling's offspring. Ensuring that enough of a sibling’s offspring survive to
adulthood precludes the necessity of the altruistic individual producing
offspring. Similarly if it is considered that information other than that of a
genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time
the playing field becomes larger still, and the discrepancies smaller.
7.4 Computer science and logic
Game theory has come to play an increasingly important role
in logic and in computer science. Several logical theories have a basis in game
semantics. In addition, computer scientists have used games to
model interactive computations. Also, game theory provides a theoretical
basis to the field of multi-agent systems.
Separately, game theory has played a role in online algorithms. In
particular, the k-server problem, which has in the past been referred to
as games with moving costs and request-answer games (Ben David, Borodin
& Karp et al. 1994). Yao's principle is a game-theoretic technique for proving
MCOE, T.E. Computer Science 201241
lower bounds on the computational complexity of randomized algorithms, and
especially of online algorithms.
The emergence of the internet has motivated the development of
algorithms for finding equilibria in games, markets, computational auctions,
peer-to-peer systems, and security and information markets.Algorithmic game
theory and within it algorithmic mechanism design combine
computational algorithm design and analysis of complex systems with
economic theory.
7.5 Philosophy
Game theory has been put to several uses in philosophy. Responding
to two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game theory
to develop a philosophical account of convention. In so doing, he provided the
first analysis of common knowledge and employed it in analyzing play
in coordination games. In addition, he first suggested that one can
understand meaning in terms of signaling games. This later suggestion has
been pursued by several philosophers since Lewis (Skyrms (1996), Grim,
Kokalis, and Alai-Tafti et al. (2004)). Following Lewis (1969) game-theoretic
account of conventions, Ullmann Margalit (1977) and Bicchieri (2006) have
developed theories of social norms that define them as Nash equilibria that
result from transforming a mixed-motive game into a coordination game.
In ethics, some authors have attempted to pursue the project, begun
by Thomas Hobbes, of deriving morality from self-interest. Since games like
the Prisoner's dilemma present an apparent conflict between morality and self-
interest, explaining why cooperation is required by self-interest is an
important component of this project. This general strategy is a component of
the general social contractview in political philosophy (for examples,
see Gauthier (1986) and Kavka (1986).
MCOE, T.E. Computer Science 201242
Other authors have attempted to use evolutionary game theory in order
to explain the emergence of human attitudes about morality and corresponding
animal behaviors. These authors look at several games including the Prisoner's
dilemma, Stag hunt, and the Nash bargaining game as providing an
explanation for the emergence of attitudes about morality (see, e.g.,
Skyrms (1996, 2004) and Sober and Wilson (1999)).
Some assumptions used in some parts of game theory have been
challenged in philosophy; psychological egoism states that rationality reduces
to self-interest—a claim debated among philosophers.
MCOE, T.E. Computer Science 201243
CHAPTER 8
CONCLUSION
"Managers have much to learn from game theory - provided they use it
to clarify their thinking, not as a substitute for business experience" .
FOR old-fashioned managers, business was a branch of warfare - a
way of 'capturing markets' and 'making a killing'. Today, however, the
language is all about working with suppliers, building alliances, and thriving
on trust and loyalty. Management theorists like to point out that there is such a
thing as 'win-win', and that business feuds can end up hurting both parties.
But this can be taken too far. Microsoft's success has helped Intel, but
it has been hell for Apple Computer. Instead, business needs a new way of
thinking that makes room for collaboration as well as competition, for mutual
benefits as well as trade-offs. Enter game theory.
Stripped to its essentials, game theory is a tool for understanding how
decisions affect each other. Until the theory came along, economists assumed
that firms could ignore the effects of their behaviour on the actions of rivals,
which was fine when competition was perfect or a monopolist held sway, but
was otherwise misleading. Game theorists argue that firms can learn from
game players: no card player plans his strategy without thinking about how
other players are planning theirs.
Economists have long used game theory to illuminate practical
problems, such as what todo about global warming or about fetuses with
Down's syndrome. Now business people have started to wake up to the
theory's possibilities. McKinsey, a consultancy, is setting up a practice in
game theory. Firms as diverse as Xerox, an office-equipment maker, Bear
Stearns, an investment bank, and PepsiCo, a soft-drinks giant, are all
MCOE, T.E. Computer Science 201244
interested. They will no doubt seize on 'Co-petition' (Doubleday, $ 24.95),
because it is written by two of the leading names in the field, Adam
Brandenburger, of Harvard Business School, and Barry Nalebuff, of the Yale
School of Management. It also helps by using readable case studies rather
than complex mathematics.
The main practical use of game theory, say the authors, is to help a
firm decide when to compete and when to co-operate. Broadly speaking, the
time to co-operate is when you are increasing the size of the pie, and the time
to compete is when you are dividing it up. The authors also argue that, to get a
full picture of their business, managers need to think about a new category of
firms, 'complementers', which lead your customers to value your products
more highly than if they had only your product. Hot-dog makers and
Colman'smustard are complementers: buy one and you are more likely to buy
the other. So are Intel and Microsoft.
The most important thing to know about a game is who the players
are. A small changein the number of players can have unexpected
consequences. NutraSweet managed to keep the predator out, but only after
Coca-Cola and Pepsi used the threat of competition to force NutraSweet to
lower its prices.
When competition between two players benefits third parties in this
way, there is scope for the beneficiary to split its gains. Holland Sweetener in
effect gave up its share of the gains that it had helped Coke and Pepsi to win.
BellSouth, a telephone company, was wiser: it insisted on being paid to play.
The firm said that it would bid against CraigMcCaw for control of LIN
Broadcasting Corporation only if LIN paid it $ 54m for entering the fray and a
further $ 15m in expenses if it lost the bid.
MCOE, T.E. Computer Science 201245
One way for a player to do well in a game is to make itself
indispensable. Nintendo built its video-games business in the late 1980s by
restricting software developers to making five games each, keeping retailers
on short rations, and doing much of the developmentin-house. Nobody else
had any bargaining power. By contrast, IBM stored up trouble for itself in
personal computers by allowing Microsoft and Intel to establish a lock on the
two most valuable bits of the business.
A second technique is to tempt lots of competing players into the game
- for instance by increasing the prize. That is what American Express did in
1994 when it organised acoalition with other big companies to purchase health
care. The potential contract was so large that a host of health-care providers
got into a bidding war.
A third technique is to make intelligent use of a resource which is
worth more to your customer than to you. In1993 TWA lifted itself off the
bottom of the airline league by tearing out several rows of seats that were
usually empty because the carrier was so unpopular, giving passengers more
leg-room - and making the airline popular once more.
MCOE, T.E. Computer Science 201246
Summary
Game theory is exciting because although the principles are simple, the
applications are far reaching.
Game theory is the study of co-operative and non-co-operative approaches
to games and social situations in which participants must choose between
individual benefits and collective benefits.
Gam theory can be used to design credible commitments, threats, or
promises, or to assess propositions and statements offered by others.
Non-cooperative game theory … has brought a fairly flexible language to
many issues, together with a collection of notions of "similarity" that has
allowed economists to move insights from one context to another and to
probe the reach of these insights. But too often it, and in particular
equilibrium analysis, gets taken too seriously at levels where its current
behavioural assumptions are inappropriate. We (economic theorists and
economists more broadly) need to keep a better sense of proportion about
when and how to use it. And we (economic and game theorists) would do
well to see what can be done about developing formally that senses of
proportion.
MCOE, T.E. Computer Science 201247
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Title: Existence of Equilibrium in Discrete Market Games
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Year of Publication :2005
Title: The Compleat Strategyst: Being a Primer on the Theory of
Games of Strategy
Publisher : MIT press
Author: Adam M. Brandenburger
Year of Publication : 2004
Title: Co-Opetition : A Revolution Mindset That Combines
Competition and Cooperation : The Game Theory Strategy That's
Changing the Game of Business Publisher : Mcgraw-hill Companies
MCOE, T.E. Computer Science 201248
Author: Tom Siegfried
Year of Publication : 2002
Title : A Beautiful Math: John Nash, Game Theory, and the Modern
Quest for a Code of Nature
Publisher : British Press
URL:
http://en.wikipedia.org/wiki/Game_theory
http://faculty.lebow.drexel.edu/mccainr/top/eco/game/game-toc.html
http://www2.owen.vanderbilt.edu/mike.shor/courses/gametheory/
quiz/problems2.html
http://en.wikipedia.org/wiki/Nash_equilibrium
https://class.coursera.org/gametheory/auth/
http://www.dklevine.com/general/whatis.htm
MCOE, T.E. Computer Science 201249