Sarani SahaBhattacharya, HSS

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Game Theory and its Applications

• Sarani SahaBhattacharya, HSS
• Arnab Bhattacharya, CSE
• 07 Jan, 2009

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Prisoners Dilemma

• Two suspects arrested for a crime
• Prisoners decide whether to confess or not to
  confess
• If both confess, both sentenced to 3 months of
  jail
• If both do not confess, then both will be
  sentenced to 1 month of jail
• If one confesses and the other does not, then the
  confessor gets freed (0 months of jail) and the
  non-confessor sentenced to 9 months of jail
• What should each prisoner do?

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Battle of Sexes

• A couple deciding how to spend the evening
• Wife would like to go for a movie
• Husband would like to go for a cricket match
• Both however want to spend the time together
• Scope for strategic interaction

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Games

• Normal Form representation Payoff Matrix

Prisoner 2
Confess Not Confess
Confess -3,-3 0,-9
Not Confess -9,0 -1,-1
Prisoner 1
Husband
Movie Cricket
Movie 2,1 0,0
Cricket 0,0 1,2
Wife
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Nash equilibrium

• Each players predicted strategy is the best
  response to the predicted strategies of other
  players
• No incentive to deviate unilaterally
• Strategically stable or self-enforcing

Prisoner 2
Confess Not Confess
Confess -3,-3 0,-9
Not Confess -9,0 -1,-1
Prisoner 1
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Mixed strategies

• A probability distribution over the pure
  strategies of the game
• Rock-paper-scissors game
• Each player simultaneously forms his or her hand
  into the shape of either a rock, a piece of
  paper, or a pair of scissors
• Rule rock beats (breaks) scissors, scissors
  beats (cuts) paper, and paper beats (covers) rock
• No pure strategy Nash equilibrium
• One mixed strategy Nash equilibrium each player
  plays rock, paper and scissors each with 1/3
  probability

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Nashs Theorem

• Existence
• Any finite game will have at least one Nash
  equilibrium possibly involving mixed strategies
• Finding a Nash equilibrium is not easy
• Not efficient from an algorithmic point of view

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Dynamic games

• Sequential moves
• One player moves
• Second player observes and then moves
• Examples
• Industrial Organization a new entering firm in
  the market versus an incumbent firm a
  leader-follower game in quantity competition
• Sequential bargaining game - two players bargain
  over the division of a pie of size 1 the
  players alternate in making offers
• Game Tree

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Game tree example Bargaining
Period 2B offers x2. A responds.
(x1,1-x1)
(x3,1-x3)
1
1
1
Y
Y
x3
x1
N
(0,0)
B
B
N
x2
A
B
A
A
N
Y
0
0
0
Period 1A offers x1. B responds.
Period 3A offers x3. B responds.
(x2,1-x2)
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Economic applications of game theory

• The study of oligopolies (industries containing
  only a few firms)
• The study of cartels, e.g., OPEC
• The study of externalities, e.g., using a common
  resource such as a fishery
• The study of military strategies
• The study of international negotiations
• Bargaining

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Auctions

• Games of incomplete information
• First Price Sealed Bid Auction
• Buyers simultaneously submit their bids
• Buyers valuations of the good unknown to each
  other
• Highest Bidder wins and gets the good at the
  amount he bid
• Nash Equilibrium Each person would bid less than
  what the good is worth to you
• Second Price Sealed Bid Auction
• Same rules
• Exception Winner pays the second highest bid
  and gets the good
• Nash equilibrium Each person exactly bids the
  goods valuation

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Second-price auction

• Suppose you value an item at 100
• You should bid 100 for the item
• If you bid 90
• Someone bids more than 100 you lose anyway
• Someone bids less than 90 you win anyway and pay
  second-price
• Someone bids 95 you lose you could have won by
  paying 95
• If you bid 110
• Someone bids more than 11o you lose anyway
• Someone bids less than 100 you win anyway and
  pay second-price
• Someone bids 105 you win but you pay 105, i.e.,
  5 more than what you value

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Mechanism design

• How to set up a game to achieve a certain
  outcome?
• Structure of the game
• Payoffs
• Players may have private information
• Example
• To design an efficient trade, i.e., an item is
  sold only when buyer values it as least as seller
• Second-price (or second-bid) auction
• Arrows impossibility theorem
• No social choice mechanism is desirable
• Akin to algorithms in computer science

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Inefficiency of Nash equilibrium

• Can we quantify the inefficiency?
• Does restriction of player behaviors help?
• Distributed systems
• Does centralized servers help much?
• Price of anarchy
• Ratio of payoff of optimal outcome to that of
  worst possible Nash equilibrium
• In the Prisoners Dilemma example, it is 3

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Network example

• Simple network from s to t with two links
• Delay (or cost) of transmission is C(x)
• Total amount of data to be transmitted is 1
• Optimal ½ is sent through lower link
• Total cost 3/4
• Game theory solution (selfish routing)
• Each bit will be transmitted using the lower link
• Not optimal total cost 1
• Price of anarchy is, therefore, 4/3

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Do high-speed links always help?

• ½ of the data will take route s-u-t, and ½ s-v-t
• Total delay is 3/2
• Add another zero-delay link from u to v
• All data will now switch to s-u-v-t route
• Total delay now becomes 2
• Adding the link actually makes situation worse

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Other computer science applications

• Internet
• Routing
• Job scheduling
• Competition in client-server systems
• Peer-to-peer systems
• Cryptology
• Network security
• Sensor networks
• Game programming

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Bidding up to 50

• Two-person game
• Start with a number from 1-4
• You can add 1-4 to your opponents number and bid
  that
• The first person to bid 50 (or more) wins
• Example
• 3, 5, 8, 12, 15, 19, 22, 25, 27, 30, 33, 34, 38,
  40, 41, 43, 46, 50
• Game theory tells us that person 2 always has a
  winning strategy
• Bid 5, 10, 15, , 50
• Easy to train a computer to win

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Game programming

• Counting game does not depend on opponents
  choice
• Tic-tac-toe, chess, etc. depend on opponents
  moves
• You want a move that has the best chance of
  winning
• However, chances of winning depend on opponents
  subsequent moves
• You choose a move where the worst-case winning
  chance (opponents best play) is the best
  max-min
• Minmax principle says that this strategy is equal
  to opponents min-max strategy
• The worse your opponents best move is, the
  better is your move

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Chess programming

• How to find the max-min move?
• Evaluate all possible scenarios
• For chess, number of such possibilities is
  enormous
• Beyond the reach of computers
• How to even systematically track all such moves?
• Game tree
• How to evaluate a move?
• Are two pawns better than a knight?
• Heuristics
• Approximate but reasonable answers
• Too much deep analysis may lead to defeat

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Conclusions

• Mimics most real-life situations well
• Solving may not be efficient
• Applications are in almost all fields
• Big assumption players being rational
• Can you think of unrational game theory?
• Thank you!
• Discussion