Algoritmisk Spilteori Peter Bro Miltersen dPersp, Uge 5, 2. forelæsning.

Algoritmisk Spilteori

Peter Bro Miltersen

dPersp, Uge 5, 2. forelæsning

Game theoretic solution concepts

• GTSC: Well-defined (good?) ways of playing a game.

• Examples:– Dominant strategy– Nash equilibrium– Minimax strategy

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.

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

Pr[b 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.

Nash equilibrium

• Nash equilibrium = Stable situation = Possible suggested behavior.

• Nobel prize…

• Not necessarily “good”, just “stable”.

John F. Nash Jr.,1928 -

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!

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…

Advantages of second price auctions

• Easier for bidders

• More predictable results for seller

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.

Example

100 clicks/h

200 clicks/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 2

I bid 1

I bid 3Valuations

Example

100 clicks/h

200 clicks/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 2

I bid 1

I bid 3

Example

500 $/h

1400 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 2

I bid 1

I bid 3

What happens next?

Example

500 $/h

1400 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 4

I bid 1

I bid 3

Example

700 $/h

600 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 4

I bid 1

I bid 3

Example

700 $/h

600 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 4

I bid 1

I bid 5

Example

300 $/h

1000 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 4

I bid 1

I bid 5

Example

300 $/h

1000 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 6

I bid 1

I bid 5

No, wait a minute..

I bid 2 again!

Example

500 $/h

1000 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 6

I bid 1

I bid 5

No, wait a minute..

I bid 2 again!

Example

500 $/h

1000 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 6

I bid 1

Well, then I bid 3.Again!

No, wait a minute..

I bid 2 again!

Example

500 $/h

1400 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 6

I bid 1

Well, then I bid 3.Again!

No, wait a minute..

I bid 2 again!

Example

500 $/h

1400 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

Sigh. Then I bid 4.Again!

I bid 1

Well, then I bid 3.Again!

Real Data

Why is this bad?

• Bad for advertisers. – Their bidding strategies have to be continuously updated.– They are forced to collect data about other people’s 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

Game Theory

• Like Rock-Scissors-Paper the Overture advertising game has no pure strategy Nash Equilibrium.

• Nash equilibrium = Stable situation = Possible suggested behavior.

Google’s 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!

GSP and Vickrey and Nobel

• Google web page: “Google’s unique auction model uses Nobel Prize-winning economic theory to eliminate … that feeling that you’ve paid too much”

Example

100 clicks/h 500 $/h

200 clicks/h 600 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 7

I bid 2

I bid 10

No, wait a minute..

I bid 3!

Example

100 clicks/h 800 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 7

I bid 2

I bid 10

No, wait a minute..

I bid 3! 200 clicks/h 800 $/h

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.

Efficient Equilibrium

100 clicks/h 500 $/h

200 clicks/h 1000 $/h

A click-throughis worth $10

A click-throughis worth $7

A click-throughis worth $2

I bid 5

I bid 2

I bid 10

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!

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.

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!

Courses

• dOpt (Optimization) – Compulsory undergraduate course

• Algorithmic Game Theory – Graduate course for enthusiasts