Mobile Computing Group

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Mobile Computing Group
An Introduction to Game Theory part II
Vangelis Angelakis
21 / 10 / 2004
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Previously on Part I
Game theory is the mathematically
founded study of conflict and cooperation. In
non-cooperative games cooperation arises, when it
is in the best interest of players Rational
players do not play dominated strategies. A
strategic form game can be a single round of a
repeated game Players are in a Nash equilibrium
if a unilateral change in strategies by any one
of them would lead that player to earn less than
if she remained with her current strategy. Under
mixed strategies any game in strategic-form has
an equilibrium
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Extensive games
Games in strategic form have no temporal
component. Players choose their strategies
simultaneously not knowing how other players
moved. An extensive form game or game tree is a
more detailed model than a normal form game. It
represents a formalization of interactions where
players can over time be informed about the
actions of others. Each player moves at a
designated time order and no simultaneous moves
are allowed. In an extensive game of perfect
information each player is aware of all the
previous choices of all other players.
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Remember this Quality Selection
1
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A third variant to the same game
Assume the VoIP provider moves first , say by
announcing the quality level he will provide and
commits himself to that move
Customer
( 2, 2 )
Buy
High
Dont buy
( 0, 1 )
Low
( 3, 0 )
VoIP Provider
Buy
Dont buy
( 1, 1 )
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Backward Induction
Customer
( 2, 2 )
Buy
High
Dont buy
( 0, 1 )
Low
( 3, 0 )
VoIP Provider
Buy
Dont buy
( 1, 1 )
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Strategies in extensive games
Backward induction gives a complete plan of what
to do at each point in the game when the player
can make a move, even though that point may never
arise in the course of a game Such a plan is
called a strategy of a player. Eg.
A Customer strategy would be Buy if High,
Dont buy if Low Where in a rational context,
only the first will come into effect.
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Strategic forms of extensive games
The strategic form of a game tree tabulates all
possible strategies of the players All move
combinations of the second player must be
distinguished as strategies since any two of them
could lead to different results depending on the
moves of player 1.
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Nash equilibria in extensive games
Backward induction always defines a Nash
equilibrium But not all possible Nash equilibria
of an extensive game arise by backward induction.
Dont/High, Dont/Low Is a Nash
equilibrium, Given the knowledge that player 1
has chosen low. This is a suboptimal solution
to a subgame. Backward induction derived Nash
equilibria are subgame perfect
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First mover advantage
Many games in strategic form exhibit what is
called first-mover advantage A player in a game
becomes the first mover or leader when he can
commit to a strategy and inform other players
about it. First mover advantage states that a
player who can become a leader will not be worse
off than an original game where all players act
simultaneously. Also known as Stackelberg
leadership If a player has the power to commit
he should do so.
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First mover advantage
If a player has the power to commit he should
do so. CAUTION what happens in the case where
more than one players can commit to a move? It
is not necessarily best go first What if player
2 was to be leader?
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Chip Duopoly
Solution by dominance gives (M, m) with a
utility of 16 for each player.
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Chip Duopoly with commitment
H
M L N
h m l n
h m l n
h m l n
h m l n
0, 0 12, 8 18, 9 36, 0
8, 12 16, 16 20, 15 32, 0
9, 18 15, 20 18,18 27, 0 0, 36
0, 32 0, 27 0, 0
Player I has an incentive to commit to strategy
H, gaining a utility of 18 and leaving II with 9.
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Imperfect information
Players do not always have full access to all
information that is relevant to the choices they
make. Situations like this are modeled by
extensive games with imperfect information. Eg.
Small startup announces the development of a key
technological product. A big and well
established company that dominates the market is
known to have a large RD dept But others dont
know what the RD has come up with The large
company can announce the future release of a
competitive product or cede the market. The first
is a real or bluff option, where the second will
not be in the case the product is within reach.
The small company after an announcement can
chose either stay in or sell out
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Imperfect information
Chance
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Imperfect information
Players can not distinguish among discrete nodes
in an information set. He must make the same
move at each node in an information set.
Backward induction can no longer work so Games
with imperfect information do not have an
equilibrium in pure strategies Randomization is
in order and the way to do it is by taking the
strategic form of the game with expected payoffs
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Zero sum games
The case of players playing fully opposed
interest games embodies the class of two-player
zero sum games. (rock-paper-scissors, chess,
poker, pistol duels etc) In zero sum games the
sum of all player utilities at each game solution
is 0. An over-class are constant-sum games,
where the sum of all utilities in each game
solution is constant. Mixed strategies are
intuitively the best solution for constant sum
games with imperfect information. Usage of zero
sum games along with randomized algorithms in
online computation analysis.
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Auctions bidding
English (open ascending bid) auction Do I
have fifty?Yes!Do I have fifty-five? Simplifie
d version Second price auction offers made
once over secure channels Highest bidder wins
and pays the value of the second bidder. How to
bid in a second price auction ? Use your private
value of the object
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Bidding private value
Private value for an object is derived from
needs and potential resell value estimated by
each bidder. Bidding the private value in a
second-price auction is a weakly dominant
strategy. ie regardless how others bid, no
other strategy will get the bidder of private
value better off. Examine what happens in
different strategies Bid lower than your private
price Bid higher than your private price Same
rationale works for the English auction
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Common values
Private values is most auctions a non realistic
assumption. Art objects / memorabilia may be
bought as investments but radio spectrum license
is bought for business. The license value
depends on market forces such as demand for
mobile telephony. Such forces have a more or
less common impact to all bidders. So we say
that such auctions have common value
aspects. What happens when we have pure common
value?
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The Winners Curse
Uncertainty about the value jumps in. For our
spectrum auction each company will conducted its
own private market research to estimate retail
demand and each will come with slightly different
results The results of these estimates is
correct on average, but the highest (winning)
estimate and the lowest will deviate from the
real value.. So playing like this and winning is
bad news the value has been overestimated and
this is the winners curse
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Further Reading

• www.gametheory.net
• R. Gibbons, Game Theory for Applied Economists
• Osborne, An Introduction to Game Theory
• Osborn and Rubinstein, A Course in Game Theory
• R. Axelrod, The Evolution of Cooperation
• Te???a ?a?????? st? ????µ??
• http//www.soc.uoc.gr/petrakis/8ewria.php
• Marcus M. Mobius Game Theory Course
• http//www.courses.fas.harvard.edu/ec1052/
• The Role of Game Theory in Ad hoc Networks
• http//www.cs.ucsb.edu/ebelding/courses/595/s04_g
  ametheory/
• Juha Leino
• Applications of Game Theory in Ad Hoc Networks

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