Mobile Computing Group

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Mobile Computing Group
An Introduction to Game Theory part I
Vangelis Angelakis
14 / 10 / 2004
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Introduction
Game theory is the mathematically founded study
of conflict and cooperation. Game theoretic
concepts apply when the decisions/actions of
independent agents affect the interests of
others Agents may be individuals, groups, firms,
intelligent devices Game theory provides them
with a methodology for structuring and analyzing
problems of strategic choice. By formally
modeling a situation as a game requires to
enumerate the players, their preferences and
their strategic options, The decision maker is
given a clearer and broader view of the situation
in hand.
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A little history lesson
1838 The first formal game theoretic analysis
the study of a duopoly by A. Cournot. 1921 E.
Borel suggests a formal theory of
games 1928 Theory of parlour games by von
Neumann 1944 Theory of Games and Economic
behaviour by von Neumann O. Morgenstern
basic terminology problem setup is
standardized in this book. 1951 J. Nash proves
that finite games always have an
equilibrium 1950s-60s applications to war
and (international) politics 1970s revolutionali
zed the economic theory, applications in
sociology,psychology, evolutionary
biology 1990s E/M spectrum auctions
design 1994 Nash is awarded the Nobel prize in
economics
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Defining Games

• A Game is a formal model of an interactive
  situation
• The formal definition of a game declares
• The players
• Their preferences
• Their information
• The strategic actions that are available to them
• How these actions influence the outcome
• Games with single players are characterized
  decision problems
• Two schools of game theory are formed depending
  on the focus of games and the granularity of the
  game description

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Cooperative Games
A high level description declaring -payoffs of
each potential coalition that can play a
game -not the formation process of the
coalition Hence, it is not defined as a game in
which players actually do cooperate, but as a
game in which any cooperation is enforced by an
outside party. This outside party involvement and
the possible bargaining process that takes place
do not belong to the cooperative game
description. Cooperative game theory
investigates coalitional games with respect to
relative power held by the players and how a
successful coalition should divide its gains.
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Non-cooperative Games
Players make choices of their own
interest Order and timing may be crucial to
determine the outcome of a game In a
non-cooperative game players are unable to make
enforceable contracts outside of those
specifically modelled in the game. Hence, it is
not defined as games in which players do not
cooperate, but as games in which any cooperation
must be self-enforcing. Cooperation arises,
when it is in the best interest of players
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Assumptions
Strategic Behavior Being aware of your
opponents existence and act trying to
anticipate/counter their moves is called
strategic behavior Rationality rational players
always chose actions that give them an outcome
they most prefer, given what their opponents
will do. Predict how a game will be played
or Advice how to best play in a game against
rational opponents Rationality is an assumption
under common knowledge Some define games as The
interaction among a group of rational agents who
behave strategically
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Strategic Extensive Form Games
Strategic form (normal form) A strategic form
game can be a single round of a repeated
game Time invariant (no turns) game is played in
the blink of an eye List each players
strategies List outcomes that result from each
possible combination of moves The outcome of a
game is the payoff each player gets The Payoff is
a numerical value, also called Utility. Extensive
form (game tree) The complete description of how
a game is played as time goes by Order of
players Information of players at each point in
time More on extensive forms in part II
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Dominance
Assume a rational player having strategies A and
B. If A has a higher utility than B no mater
what strategic combination other players make,
then strategy A is said to dominate strategy
B. Rational players do not play dominated
strategies. In some games examining available
strategies and eliminating dominated strategies
results in only one credible strategy for a
rational player.
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Prisoners Dilemma
A bank is robbed and the robbers escape the
police A pair of suspects are arrested later in
the day and are held in separate cells. Each is
told that -if one alone confesses the robbery
he will be granted pardon and the other goes in
prison for a long time. -if both confess then
both go in prison for a short time -if both
deny then both go in prison for a very short time
(lets say they will be charged only with arms
possession) What action will each prisoner take?
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Prisoners Dilemma
Elimination of dominated strategies
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Prisoners Dilemma
The individually rational outcome is worse for
both players Arms races, environmental
pollution, etc are modeled by prisoners
dilemmas The solution to a single game is a
max-min strategy, a better-play-it-safe
strategy In a repeated game patterns for
cooperation among the players arise (suspects
will actually play the deny-deny move) A
repeated game allows for a strategy to be
dependent on past moves, thus allowing for
reputation effects and retribution. In
infinitely repeated games, trigger strategies
such as tit-for-tat encourage cooperation.
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Quality Selection
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The Nash Equilibrium

• A set of strategies such that no player has
  incentive to unilaterally change her action.
• Players are in equilibrium if a change in
  strategies by any one of them would lead that
  player to earn less than if she remained with her
  current strategy.

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Revisiting Quality Selection
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No dominant Strategies ! Two Nash Equilibria
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Equilibrium selection

• In games with multiple Nash equilibria a theory
  of strategic interaction should guide players to
  the most reasonable equilibrium
• Large number of papers focused with equilibrium
  refinements
• that attempt to derive conditions making an
  equilibrium more convincing than the another.

For example, in the previous game the High-buy
equilibrium yields higher utility for both
players, so it can easily be argued that they
will be self-coordinated to it.
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Prisoners Dilemma Nash Equilibrium
Single Strategy combination rising from the
process of dominated strategies elimination is a
Nash Equilibrium
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Mixed Strategies
So far a player chose deterministically among
one of her strategies. Such selections are named
selections from pure strategies. Randomizing
ones choice, by selecting among her pure
strategies with a certain probability is called a
mixed strategy. Equilibrium is now defined by a
mixed strategy for each player so that none can
gain on average by unilaterally deviating. Nash
proved (1951) that under mixed strategies any
game in strategic-form has an equilibrium Such
is therefore the game theorists recommendation in
the case when an equilibrium in pure strategies
does not exist.
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Compliance Inspections
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Compliance Inspections
Mixed strategy for I chose Inspect with a
probability. (giving a sufficient change to
getting caught should deter II from choosing
Cheat)

• Inspect with p 0.01 then II receives a utility
  of
• 0 under Comply
• 0.99 x 10 0.01 x (-90) 9 under Cheat
• (has incentive to Cheat inspection not too
  often)
• Inspect with p 0.2 then II receives a utility
  of
• 0 under Comply
• 0.8 x 10 0.2 x (-90) -10 under Cheat
• (incentive to always Comply inspections too
  often)

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Compliance Inspections
So if player I randomizes poorly she leads
player II to selecting a pure strategy. Player I
must make player II indifferent to achieve
equilibrium

• Inspect with p 0.1 then II receives a utility
  of
• 0 under Comply
• 0.9 x 10 0.1 x (-90) 0 under Cheat
• Player II is rational so he knows player I will
  mix strategies only if I is indifferent too.
• How can II make I indifferent?
• Cheat with p 0.2 then I receives a utility of
• 0.8 x 0 0.2 x (-10) -2 under Dont Inspect
• 0.8 x (-1) 0.2 x (-6) -2 under Inspect

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Compliance Inspections
Equilibrium yields an expected payoff of 0 for
player II -2 for player I

Does randomizing make sense for II ??? ok, why
dont I just always chose comply since I got
nothing to win anyways? Lets assume player II
always chooses Comply under this rationale, then
I needs not randomize, just purely chose Dont
Inspect so player IIs strategy becomes
sub-optimal (irrational). With no incentive to
select one strategy over the other a player can
mix strategies and only thus is the equilibrium
reached
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Compliance Inspections
We saw that the probabilities for mixing ones
strategies depend on opponent's payoffs

It could be reasoned by I that We just have
the penalty for Cheating disgustingly high and
II will be deterred to select Cheat We saw
that IIs payoffs determine the probabilities
that will make I be indifferent
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Compliance Inspections
One could see this game as an evolutionary
game It is the interaction between an
organization who chooses Dont Inspect and
Inspect for certain fractions of a large number
of people.

Player IIs actions Comply and Cheat are each
chosen by a certain fraction of people involved
in these interactions. If these fractions would
deviate from the equilibrium probabilities, the
group whose strategies do better would increase.
For example if player I chooses Inspect too
often (relative to the penalty for a cheater who
is caught), the fraction of cheaters will
decrease, which in turn makes Dont Inspect a
better strategy In this dynamic process, the
long-term averages of the fractions approximate
the equilibrium probabilities
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TNL - Mobile Computing Group Angelakis
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