Algoritmisk Spilteori

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Algoritmisk Spilteori

• Peter Bro Miltersen
• dPersp, Uge 5, 2. forelæsning

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Game theoretic solution concepts

• GTSC Well-defined (good?) ways of playing a
  game.
• Examples
• Dominant strategy
• Nash equilibrium
• Minimax strategy

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Mauritz auctions, Phase 1/2

• First price, open bid auction with fixed
  deadline.
• Timing (and location) is everything
• War between Snipers and Flooders
• Not much game theory can do (more like
  Counterstrike than Poker), but still possible to
  do non-trivial stuff.
• More strategic (less real time) if late bids
  extend the deadline.
• In the experiments, the groups knew their
  valuation. In real life/more realistic models,
  other bids help you learn your valuation.

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Mauritz auctions, Phase 3

• First price, sealed bid auction.
• Central question How much to underbid?
• It depends on what other people are bidding!
• Reasonable approach
• Continously update statistics of other bids
  (Bayesian model)
• Play a best reply to this model (an optimal bid)
  The bid b maximizing Prb is
  highest bid (v b).
• In game theory, playing rationally is defined
  as playing a best reply to your beliefs about the
  plays of other parties.
• Suppose everybody follows this rational
  approach and continuously update their model.
• A stable situation in such play (with accurate
  statistics) is also known as a Nash equilibrium.

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Nash equilibrium

• Nash equilibrium Stable situation Possible
  suggested behavior.
• Nobel prize
• Not necessarily good, just stable.

John F. Nash Jr., 1928 -
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Mauritz auction, Phase 4

• Second price, sealed bid auction (Vickrey
  auction)
• Bidding your valuation is optimal (a best reply)
  no matter what other parties are bidding!

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Applying game theory to auctions

• William Vickrey, 1914-1996
• Invented second-price auctions
• His Nobel Prize in economics was announced three
  days before his death

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Advantages of second price auctions

• Easier for bidders
• More predictable results for seller

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Prehistory of AdWords

• Pre-1997 Search engine advertising based on
  large contracts
• 1997 Auction Revolution by Overture (then GoTo,
  now Yahoo!)
• Advertisers submit a bid for each keyword.
• Highest bidders get displayed. Ads arranged in
  descending order of bids.
• Advertisers obtaining a click-through pay their
  bid.

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Example
A click-through is worth 10
I bid 3
Valuations
200 clicks/h
A click-through is worth 7
I bid 2
100 clicks/h
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
I bid 3
200 clicks/h
A click-through is worth 7
I bid 2
100 clicks/h
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
I bid 3
1400 /h
A click-through is worth 7
I bid 2
500 /h
I bid 1
A click-through is worth 2
What happens next?
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Example
A click-through is worth 10
I bid 3
1400 /h
A click-through is worth 7
I bid 4
500 /h
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
I bid 3
600 /h
A click-through is worth 7
I bid 4
700 /h
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
I bid 5
600 /h
A click-through is worth 7
I bid 4
700 /h
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
I bid 5
1000 /h
A click-through is worth 7
I bid 4
300 /h
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
I bid 5
1000 /h
A click-through is worth 7
I bid 6
300 /h
No, wait a minute..
I bid 2 again!
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
I bid 5
1000 /h
A click-through is worth 7
I bid 6
500 /h
No, wait a minute..
I bid 2 again!
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
Well, then I bid 3. Again!
1000 /h
A click-through is worth 7
I bid 6
500 /h
No, wait a minute..
I bid 2 again!
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
Well, then I bid 3. Again!
1400 /h
A click-through is worth 7
I bid 6
500 /h
No, wait a minute..
I bid 2 again!
I bid 1
A click-through is worth 2
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Example
A click-through is worth 10
Well, then I bid 3. Again!
1400 /h
A click-through is worth 7
Sigh. Then I bid 4. Again!
500 /h
I bid 1
A click-through is worth 2
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Real Data
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Why is this bad?

• Bad for advertisers.
• Their bidding strategies have to be continuously
  updated.
• They are forced to collect data about other
  peoples bid. May not be possible.
• They may want to spend their intellectual
  resources elsewhere
• Bad for search engine company.
• Unhappy advertises may go to other search engine
  company.
• It is hard to predict what will actually happen
  (including revenue) and plan accordingly.
• May not optimize revenue

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Game Theory

• Like Rock-Scissors-Paper the Overture advertising
  game has no pure strategy Nash Equilibrium.
• Nash equilibrium Stable situation Possible
  suggested behavior.

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Googles generalized second-prize auction (GSP)

• Ads arranged in descending order of bids. Bidders
  pay the bid of the ad below them.
• Adopted by Google in 2002 and soon also adopted
  by Overture/Yahoo!

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GSP and Vickrey and Nobel

• Google web page Googles unique auction model
  uses Nobel Prize-winning economic theory to
  eliminate that feeling that youve paid too
  much

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Example
A click-through is worth 10
I bid 10
No, wait a minute..
200 clicks/h 600 /h
I bid 3!
A click-through is worth 7
I bid 7
100 clicks/h 500 /h
I bid 2
A click-through is worth 2
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Example
A click-through is worth 10
I bid 10
No, wait a minute..
200 clicks/h 800 /h
I bid 3!
A click-through is worth 7
I bid 7
100 clicks/h 800 /h
I bid 2
A click-through is worth 2
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GSP vs. Vickrey and Nobel

• Truth telling is not a dominant strategy in GSP.
• Truth telling might not even be
  a Nash Equilibrium.
• But Unlike the Overture game,
  GSP always has some pure Nash equilibrium.

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Efficient Equilibrium
A click-through is worth 10
I bid 10
200 clicks/h 1000 /h
A click-through is worth 7
I bid 5
100 clicks/h 500 /h
I bid 2
A click-through is worth 2
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GSP redeemed?

• Auction theory for GSP only developed in 2006
  In three independent papers, two by economists
  and one by computer scientists.
• GSP has pure Nash equilibra but
• Lacks truthfulness
• Admits inefficient equilibria (that may be more
  credible than the efficient ones!)
• What about revenue?
• Also, what about including in the model
• The fact that bidders have budgets.
• The fact that budgets have to be allocated to
  various search terms.
• The fact that bidders participate in a sequence
  of auctions, not just one.
• The fact that the sequence of search terms is not
  known in advance.
• 100 papers on game theoretic analysis of
  sponsored search in 2005-2007!

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Maximin strategies in Paper Rules (and other
Poker-like games)

• Maximin Play in a randomized fashion so as to
  maximize your winnings, assuming that your
  opponent knows your source code and will play so
  as to minimize your winnings.
• In a two-player zero-sum games (your winnings are
  paid by your opponent and the winnings of your
  opponent are paid by you), if everyone plays
  maximin, we have a (mixed strategy) Nash
  Equilibrium.

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Poker bots

• Two approaches
• Game theoretic Play a maximin strategy. If the
  game is like Play-with-fire, we will win if the
  opponent makes mistakes.
• Non-game theoretic Do not play maximin try to
  be smarter than your opponent. Problem Your
  source code cannot be released!

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Courses

• dOpt (Optimization) Compulsory undergraduate
  course
• Algorithmic Game Theory Graduate course for
  enthusiasts

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