Game Theory: introduction and applications to computer networks Game Theory: introduction and applications to computer networks Introduction Giovanni Neglia INRIA – EPI Maestro 3 February 2014 Part of the slides are based on a previous course with D. Figueiredo (UFRJ) and H. Zhang (Suffolk University)

Always an equilibrium with small Loss of Efficiency?  Consider only affine cost functions, i.e. c α (x)= a α +b α x r We will use the potential to derive a bound on the social cost of a NE m P(f) <= C S (f) <= 2 P(f) fαfα C α (x ) x P α (f α ) C α (f α )

Always an equilibrium with small Loss of Efficiency? r Consider only affine cost functions i.e. c α (x)= a α +b α x r We will use the potential to derive a bound on the social cost of a NE m P(f) <= C S (f) <= 2 P(f) r P(f) = Σ  ε E P  =Σ  ε E Σ t=1,…f  c  (t) <= <=Σ  ε E Σ t=1,…f  c  (f  )=Σ  ε E f  c  (f  )=C S (f) r P(f) = Σ  ε E P  =Σ  ε E Σ t=1,…f  (a α +b α t)= =Σ  ε E f α a α +b α f α (f α +1)/2 >= Σ  ε E f α (a α +b α f α )/2 =C S (f)/2

Always an equilibrium with small Loss of Efficiency? r Consider only affine cost functions i.e. c α (x)= a α +b α x r P(f) <= C S (f) <= 2 P(f) r Let’s imagine to start from routing f Opt with the optimal social cost C S (f Opt ), r Applying the BR dynamics we arrive to a NE with routing f NE and social cost C S (f NE ) r C S (f NE ) <= 2 P(f NE ) <= 2 P(f Opt ) <= 2 C S (f Opt ) r The LoE of this equilibrium is at most 2

Same technique, different result r Consider a network with a routing at the equilibrium r Add some links r Let the system converge to a new equilibrium r The social cost of the new equilibrium can be at most 4/3 of the previous equilibrium social cost (as in the Braess Paradox)

Loss of Efficiency, Price of Anarchy, Price of Stability r Loss of Efficiency (LoE) m given a NE with social cost C S (f NE ) m LoE = C S (f NE ) / C S (f Opt ) r Price of Anarchy (PoA) [Koutsoupias99] m Different settings G (a family of graph, of cost functions,...) m X g =set of NEs for the setting g in G  PoA = sup g ε G sup NE ε Xg {C S (f NE ) / C S (f Opt )} => “worst” loss of efficiency in G r Price of Stability (PoS) [Anshelevish04]  PoS = sup g ε G inf NE ε Xg {C S (f NE ) / C S (f Opt )} => guaranteed loss of efficiency in G

Stronger results for affine cost functions r We have proven that for unit-traffic routing games the PoS is at most 2 r For unit-traffic routing games and single- source pairs the PoS is 4/3 r For non-atomic routing games the PoA is 4/3 m non-atomic = infinite players each with infinitesimal traffic r For other cost functions they can be much larger (even unbounded)

Performance Evaluation Sponsored Search Markets Giovanni Neglia INRIA – EPI Maestro 4 February 2013

Google r A class of games for which there is a function P(s 1,s 2,…s N ) such that m For each i U i (s 1,s 2,…x i,…s N )>U i (s 1,s 2,…y i,…s N ) if and only if P(s 1,s 2,…x i,…s N )>P(s 1,s 2,…y i,…s N ) r Properties of potential games: Existence of a pure-strategy NE and convergence to it of best-response dynamics r The routing games we considered are particular potential games

How it works r Companies bid for keywords r On the basis of the bids Google puts their link on a given position (first ads get more clicks) r Companies are charged a given cost for each click (the cost depends on all the bids)

Some numbers r ≈ 95% of Google revenues (46 billions$) from ads m investor.google.com/financial/tables.html m 87% of Google-Motorola revenues (50 billions$) r Costs m "calligraphy pens" $1.70 m "Loan consolidation" $50 m "mesothelioma" $50 per click r Click fraud problem

Outline r Preliminaries m Auctions m Matching markets r Possible approaches to ads pricing r Google mechanism r References m Easley, Kleinberg, "Networks, Crowds and Markets", ch.9,10,15

Types of auctions r 1 st price & descending bids r 2 nd price & ascending bids

Game Theoretic Model r N players (the bidders) r Strategies/actions: b i is player i’s bid r For player i the good has value v i r p i is player i’s payment if he gets the good r Utility: m v i -p i if player i gets the good m 0 otherwise r Assumption here: values v i are independent and private m i.e. very particular goods for which there is not a reference price

Game Theoretic Model r N players (the bidders) r Strategies: b i is player i’s bid r Utility: m v i -b i if player i gets the good m 0 otherwise r Difficulties: m Utilities of other players are unknown! m Better to model the strategy space as continuous m Most of the approaches we studied do not work!

2 nd price auction r Player with the highest bid gets the good and pays a price equal to the 2 nd highest bid r There is a dominant strategies m I.e. a strategy that is more convenient independently from what the other players do m Be truthful, i.e. bid how much you evaluate the good (b i =v i ) m Social optimality: the bidder who value the good the most gets it!

b i =v i is the highest bid bids bibi bkbk bhbh bnbn Bidding more than v i is not convenient U i =v i -b k >v i -b i =0 bids b i ’>b i bkbk bhbh bnbn U i ’=v i -b k

b i =v i is the highest bid bids bibi bkbk bhbh bnbn Bidding less than v i is not convenient (may be unconvenient) U i =v i -b k >v i -b i =0 bids b i ’<b i bkbk bhbh bnbn U i ’=0

b i =v i is not the highest bid bids bkbk bibi bhbh bnbn Bidding more than v i is not convenient (may be unconvenient) U i =0 bids b i ’>b i bkbk bhbh bnbn U i ’=v i -b k <v i -b i =0

b i =v i is not the highest bid bids bkbk bibi bhbh bnbn Bidding less than v i is not convenient U i =0 bids b i ’<b i bkbk bhbh bnbn U i ’=0

Seller revenue r N bidders r Values are independent random values between 0 and 1 r Expected i th largest utility is (N+1-i)/(N+1) r Expected seller revenue is (N-1)/(N+1)

1 st price auction r Player with the highest bid gets the good and pays a price equal to her/his bid r Being truthful is not a dominant strategy anymore! r How to study it?

1 st price auction r Assumption: for each player the other values are i.i.d. random variables between 0 and 1 m to overcome the fact that utilities are unknown r Player i’s strategy is a function s() mapping value v i to a bid b i m s() strictly increasing, differentiable function  0≤s(v)≤v  s(0)=0  We investigate if there is a strategy s() common to all the players that leads to a Nash equilibrium

1 st price auction r Assumption: for each player the other values are i.i.d. random variables between 0 and 1 r Player i’s strategy is a function s() mapping value v i to a bid b i  Expected payoff of player i if all the players plays s():  U i (s,…s,…s) = v i N-1 (v i -s(v i )) prob. i wins i’s payoff if he/she wins

1 st price auction  Expected payoff of player i if all the players play s():  U i (s,…s,…s) = v i N-1 (v i -s(v i ))  What if i plays a different strategy t()?  If all players playing s() is a NE, then :  U i (s,…s,…s) = v i N-1 (v i -s(v i )) ≥ v i N-1 (v i -t(v i )) = = U i (s,…t,…s)  Difficult to check for all the possible functions t() different from s()  Help from the revelation principle

The Revelation Principle  All the strategies are equivalent to bidder i supplying to s() a different value of v i s() vivi bibi t() vivi bi'bi' s() vi'vi' bi'bi'

1 st price auction  Expected payoff of player i if all the players plays s():  U i (v 1,…v i,…v N ) = U i (s,…s,…s) = v i N-1 (v i -s(v i ))  What if i plays a different strategy t()?  By the revelation principle:  U i (s,…t,…s) = U i (v 1,…v,…v N ) = v N-1 (v i -s(v))  If v i N-1 (v i -s(v i )) ≥ v N-1 (v i -s(v)) for each v (and for each v i )  Then all players playing s() is a NE

1 st price auction  If v i N-1 (v i -s(v i )) ≥ v N-1 (v i -s(v)) for each v (and for each v i )  Then all players playing s() is a NE  f(v)=v i N-1 (v i -s(v i )) - v N-1 (v i -s(v)) is minimized for v=v i  f’(v)=0 for v=v i,  i.e. (N-1) v i N-2 (v i -s(v)) + v i N-1 s’(v i ) = 0 for each v i  s’(v i ) = (N-1)(1 – s(v i )/v i ), s(0)=0  Solution: s(v i )=(N-1)/N v i

1 st price auction r All players bidding according to s(v) = (N-1)/N v is a NE r Remarks m They are not truthful m The more they are, the higher they should bid r Expected seller revenue m ((N-1)/N) E[v max ] = ((N-1)/N) (N/(N+1)) = (N- 1)/(N+1) m Identical to 2 nd price auction! m A general revenue equivalence principle

Outline r Preliminaries m Auctions m Matching markets r Possible approaches to ads pricing r Google mechanism r References m Easley, Kleinberg, "Networks, Crowds and Markets", ch.9,10,15

Matching Markets v 11, v 21, v 31 v 12, v 22, v 32 How to match a set of different goods to a set of buyers with different evaluations v ij : value that buyer j gives to good i goods buyers

Matching Markets , 4, 2 8, 7, 6 7, 5, 2 How to match a set of different goods to a set of buyers with different evaluations p 1 =2 p 2 =1 p 3 =0 Which goods buyers like most? Preferred seller graph

Matching Markets , 4, 2 8, 7, 6 7, 5, 2 p 1 =2 p 2 =1 p 3 =0 Which goods buyers like most? Preferred seller graph r Given the prices, look for a perfect matching on the preferred seller graph r There is no such matching for this graph

Matching Markets , 4, 2 8, 7, 6 7, 5, 2 p1=3p1=3 p 2 =1 p 3 =0 Which goods buyers like most? Preferred seller graph r But with different prices, there is

Matching Markets , 4, 2 8, 7, 6 7, 5, 2 p1=3p1=3 p 2 =1 p 3 =0 Which goods buyers like most? Preferred seller graph r But with different prices, there is r Such prices are market clearing prices

Market Clearing Prices r They always exist m And can be easily calculated if valuations are known r They are socially optimal in the sense that they maximize the sum of all the payoffs in the network (both sellers and buyers)

Outline r Preliminaries m Auctions m Matching markets r Possible approaches to ads pricing r Google mechanism r References m Easley, Kleinberg, "Networks, Crowds and Markets", ch.9,10,15

Ads pricing v1v1 v2v2 v3v3 How to rank ads from different companies v i : value that company i gives to a click Ads positions companies r1r1 r2r2 r3r3 r i : click rate for an ad in position i (assumed to be independent from the ad and known a prior i)

Ads pricing as a matching market v 1 r 1, v 1 r 2, v 1 r 3 v i : value that company i gives to a click Ads positions companies r1r1 r2r2 r3r3 r i : click rate for an ad in position i (assumed to be independent from the ad and known a prior i) v 2 r 1, v 2 r 2, v 2 r 3 v 3 r 1, v 3 r 2, v 3 r 3 r Problem: Valuations are not known! r … but we could look for something as 2 nd price auctions

The VCG mechanism r The correct way to generalize 2 nd price auctions to multiple goods r Vickrey-Clarke-Groves r Every buyers should pay a price equal to the social value loss for the others buyers m Example: consider a 2 nd price auction with v 1 >v 2 >…v N With 1 present the others buyers get 0 Without 1, 2 would have got the good with a value v 2 then the social value loss for the others is v 2

The VCG mechanism r The correct way to generalize 2 nd price auctions to multiple goods r Vickrey-Clarke-Groves r Every buyers should pay a price equal to the social value loss for the others buyers m If V B S is the maximum total valuation over all the possible perfect matchings of the set of sellers S and the set of buyers B, m If buyer j gets good i, he/she should be charged V B-j S - V B-j S-i

VCG example v 1 =3 v i : value that company i gives to a click Ads positions companies r 1 =10 r 2 =5 r 3 =2 r i : click rate for an ad in position i (assumed to be independent from the ad and known a prior i) v 2 =2 v 3 =1

VCG example , 15, 6 Ads positions companies 20, 10, 4 10, 5, 2

VCG example , 15, 6 Ads positions companies 20, 10, 4 10, 5, 2 r This is the maximum weight matching r 1 gets 30, 2 gets 10 and 3 gets 2

VCG example , 15, 6 Ads positions companies 20, 10, 4 10, 5, 2 r If 1 weren’t there, 2 and 3 would get 25 instead of 12, r Then 1 should pay 13

VCG example , 15, 6 Ads positions companies 20, 10, 4 10, 5, 2 r If 2 weren’t there, 1 and 3 would get 35 instead of 32, r Then 2 should pay 3

VCG example , 15, 6 Ads positions companies 20, 10, 4 10, 5, 2 r If 3 weren’t there, nothing would change for 1 and 2, r Then 3 should pay 0

The VCG mechanism r Every buyers should pay a price equal to the social value loss for the others buyers m If V B S is the maximum total valuation over all the possible perfect matchings of the set of sellers S and the set of buyers B, m If buyer j gets good i, he/she should be charged V B-j S - V B-j S-i r Under this price mechanism, truth-telling is a dominant strategy

Outline r Preliminaries m Auctions m Matching markets r Possible approaches to ads pricing r Google mechanism r References m Easley, Kleinberg, "Networks, Crowds and Markets", ch.9,10,15

Google’s GSP auction r Generalized Second Price r Once all the bids are collected b 1 >b 2 >…b N r Company i pays b i+1 r In the case of a single good (position), GSP is equivalent to a 2 nd price auction, and also to VCG r But why Google wanted to implement something different???

GSP properties r Truth-telling may not be an equilibrium

GSP example v 1 =7 v i : value that company i gives to a click Ads positions companies r 1 =10 r 2 =4 r 3 =0 r i : click rate for an ad in position i (assumed to be independent from the ad and known a prior i) v 2 =6 v 3 =1 r If each player bids its true evaluation, 1 gets a payoff equal to 10 r If 1 bids 5, 1 gets a payoff equal to 24

GSP properties r Truth-telling may not be an equilibrium r There is always at least 1 NE maximizing total advertiser valuation

GSP example v 1 =7 v i : value that company i gives to a click Ads positions companies r 1 =10 r 2 =4 r 3 =0 r i : click rate for an ad in position i (assumed to be independent from the ad and known a prior i) v 2 =6 v 3 =1 r Multiple NE m 1 bids 5, 2 bids 4 and 3 bids 2 m 1 bids 3, 2 bids 5 and 3 bids 1

GSP properties r Truth-telling may not be an equilibrium r There is always at least 1 NE maximizing total advertiser valuation r Revenues can be higher or lower than VCG m Attention: the revenue equivalence principle does not hold for auctions with multiple goods! m Google was targeting higher revenues… m … not clear if they did the right choice.

GSP example v 1 =7 Ads positions companies r 1 =10 r 2 =4 r 3 =0 v 2 =6 v 3 =1 r Multiple NE  1 bids 5, 2 bids 4, 3 bids 2  google’s revenue=48  1 bids 3, 2 bids 5, 3 bids 1  google’s revenue=34 r With VCG, google’s revenue=44

Other issues r Click rates are unknown and depend on the ad! m Concrete risk: low-quality advertiser bidding high may reduce the search engine’s revenue m Google’s solution: introduce and ad-quality factor taking into account actual click rate, relevance of the page and its ranking Google is very secretive about how to calculate it => the market is more opaque r Complex queries, nobody paid for m Usually engines extrapolate from simpler bids