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Game Theory - Applications in real life

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					    Game Theory




           Pradeeban Kathiravelu

           Advanced Topics in Distributed Systems
1
                   Agenda
    
        Games
    
        Evolutionary Game Theory
    
        Modeling Network Traffic using Game
        Theory
    
        Auctions




2
         Evolutionary Game Theory
    
        Applying Game Theory for Evolutionary
        Biology.
    
        Fitness as a result of interaction.




3
              The Body-Size Game
                       Beetle2


                       Small       Large


    Beetle1    Small   5, 5        1, 8

               Large   8, 1        3, 3




4
        Evolutionary Stable Strategies
          S      L


    S     5, 5   1, 8


    L     8, 1   3, 3
    
      Expected payoff:
    To a small beetle during a random
      encounter =
    5 * probability of meeting another small +
      1 * probability of meeting a large =
    5 (1 – x ) + 1. x = 5 – 4x.
5
        Evolutionary Stable Strategies
          S      L


    S     5, 5   1, 8


    L     8, 1   3, 3
    
      Expected payoff:
    To a large beetle during a random
      encounter =
    8 * probability of meeting a small +
    3 * probability of meeting another large =
    8 (1 – x ) + 3. x = 8 – 5x.
6
        Evolutionary Stable Strategies
           S      L


    S      5, 5   1, 8


    L      8, 1   3, 3
    
         For a small enough fraction of x,
                  • 8 – 5 x > 5 – 4 x.
    
         “Small” is not evolutionarily stable.



7
        Evolutionary Stable Strategies
          S      L


    S     5, 5   1, 8


    L     8, 1   3, 3
    
     Expected payoff:
    To a small beetle = 5.x + 1.(1 – x) = 1 + 4x




8
        Evolutionary Stable Strategies
          S      L


    S     5, 5   1, 8


    L     8, 1   3, 3
    
     Expected payoff:
    To a small beetle = 5.x + 1.(1 – x) = 1 + 4x
    To a large beetle = 8.x + 3.(1 – x) = 3 + 5x



9
         Evolutionary Stable Strategies
           S      L


     S     5, 5   1, 8


     L     8, 1   3, 3
     
       Expected payoff:
     To a small beetle = 5.x + 1.(1 – x) = 1 + 4x
     To a large beetle = 8.x + 3.(1 – x) = 3 + 5x
     3 + 5x > 1 + 4x
     “Large” is evolutionarily stable.
10
     Is S Evolutionarily Stable?
                                       S      T


                                   S   a, a   b, c


                                   T   c, b   d, d




11
     Is S Evolutionarily Stable?
                                        S      T

       
           a (1 - x) + b.x >
                                    S   a, a   b, c
              c (1 – x) + d.x
                                    T   c, b   d, d

       
           a > c or a = c & b > d




12
     Is S Evolutionarily Stable?
                                           S      T

       
           a (1 - x) + b.x >
                                      S    a, a   b, c
              c (1 – x) + d.x
                                      T    c, b   d, d

       
           a > c or a = c & b > d

       
           S, S – best response to the other player
            – Evolutionarily Stable (ES) → Nash
              Equilibrium.
            – Strict Nash Equilibrium (unique
13
              response to the other player) → ES
14
           Modeling Network Traffic
     
         Game-theoretic reasoning
            – Traveling through a
               transportation network.
            – Sending packets through the
               Internet.
     
         Evaluate routes in presence of the
         congestion resulting from the decisions
         of myself and others.


15
             Traffic at Equilibrium:
               4000 Cars, A → B




     
         Time taken at equilibrium for every driver =
     
         Equally divided between ACB and ADB =
            2000/100 + 45 = 65
16
               Nash Equilibrium
     
         Equal Balance → Nash Equilibrium.
     
         Nash Equilibrium → Equal Balance.
     
         No incentive to switch over to the other
         route.
     
         Any unequal weights will not be in Nash
         Equilibrium.




17
                Braess's Paradox
     
         Adding capacity to a network can
         sometimes actually slow down the traffic.
     
         Adding resources to a transportation
         network can sometimes hurt performance at
         equilibrium.




18
               4000 Cars, A → B




     
         The unique Nash equilibrium leads towards
         using ACDB.
     
         Time taken for every driver = 4000/100 + 0
19       + 4000/100 = 80
               4000 Cars, A → B




     
         No benefit by changing their route, for an
         individual.
     
         Changing will increase the travel time as,
20       45 + 4000/100 = 85
                 Case Study:
          Inverse of Braess’s paradox
     
         Destruction of a six-lane highway to build a
         public park actually improved travel time
         into and out of the city in Seoul, South
         Korea.




21
                    Auction
     
         Ascending-Bid Auctions (English Auctions)
     
         Descending-Bid Auctions (Dutch Auctions)
     
         First-Price Sealed-Bid Auctions
     
         Second-Price Sealed-Bid Auctions




22
     Ascending-Bid Auction




23
     10 $




24
     30 $




25
     50 $




26
     70 $




            Awarded for: 70 $ !!!


27
     Descending-Bid Auction




28
     100 $




29
     90 $




30
     80 $




31
     70 $




            Awarded for: 70 $ !!!


32
         First-Price Sealed-Bid Auction
     
         Bidders submit simultaneous sealed-bids to
         the seller.
     
         Seller opens the sealed bids together.
     
         Highest bid wins the object.
     
         Pays the value of his bid.




33
     Second-Price Sealed-Bid Auction
     
         Bidders submit simultaneous sealed-bids to
         the seller.
     
         Seller opens the sealed bids together.
     
         Highest bid wins the object.
     
         Pays the price of the second highest bid.
     
         If multiple bids share the highest position?




34
     Second-Price Sealed-Bid Auction
     
         Bidders submit simultaneous sealed-bids to
         the seller.
     
         Seller opens the sealed bids together.
     
         Highest bid wins the object.
     
         Pays the price of the second highest bid.
     
         If multiple bids share the highest position
             – one of the bidders awarded the
                 object.
             – for the same price.

35
         First-Price Sealed-Bid Auction
     
         Similar to descending bid auction.
     
         Takes the risk, without knowing the value
         of others.




36
     Second-Price Sealed-Bid Auction
     
         Similar to ascending bid auction.
     
         Others' price is known.




37
     Payoff and true value




38
                When Auctions?
     
         When the true-value of the buyer is
         unknown.
     
         Bidding one's true value is the dominant
         strategy. (Really?)
     
         Reselling?
             – Winner's curse.
     
         All-pay Auctions?



39
                 References
     Easley, David., Kleinberg, Jon, “Networks,
      Crowds, and Markets: Reasoning about a
      Highly Connected World,” Cambridge
      University Press, 2010: 155 – 268.




40
     Questions?




41
     Thank you!




42

				
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Description: Game theory, applied to the evolution (evolutionary game theory), network traffic modelling, and auctions.