Smooth Games, Price of Anarchy and Composability of Auctions - a Quick Tutorial

Transcription

1 Smooth Games, Prce of Anarchy and Composablty of Auctons - a Quck Tutoral Abhshek Snha Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology, Cambrdge, MA Emal: snhaa@mt.edu Abstract In ths tutoral we revew the concept of smooth games as ntroduced by Roughgarden n [1]. We show how to upper bound the Prce of Anrchy of dfferent games usng ths general tool. Then along the lnes of [2], we extend the dea of smooth games to smooth mechansms and weakly smooth mechansms and prove that many of the common auctons fall under ths category. We then show how effcency results of these mechansms n a Bayes-Nash equlbra may be derved from smoothness arguments. Then we prove that smultaneous composton of smooth mechansm preserves the smoothness property. Hence all the prevous results about effcency of smooth mechansms carry over to smultaneous compostons. I. INTRODUCTION Selfsh behavor by ndvdual decson makers generally leads to an neffcent result, an outcome whch could be mproved upon gven the ndvduals coordnate ther actons together. The Prce of Anarchy (POA) measures the sub-optmalty caused by selfsh behavors. Gven a game, a noton of an equlbrum (such as pure Nash equlbra), and a non-negatve objectve functon (such as the sum of players costs), the POA of the game s defned as the rato between the largest cost of an equlbrum and the cost of an optmal outcome. We compute POA usng a smoothness argument n ths tutoral. We also consder extenson of the smoothness framework n the settng of mechansm desgn and show that a number of popular auctons fall n ths category. It follows that smoothness of auctons ensures effcency for Bayesan Nash Equlbrum of the underlyng game. Fnally, we prove that smultaneous compostons of smooth mechansms are smooth. Ths result s partcularly mportant for onlne market settngs where each bdders partcpate n a multtude of auctons smultaneously (e.g. n ebay). Hence the above result mples that smooth auctons wth good effcency bounds may be run smultaneously wthout compromsng ther overall effcency guarantee. II. SMOOTH GAMES The concept of smooth games was frst ntroduced by Roughgarden n [1]. He defned t frst n the context of costmnmzaton games. A cost-mnmzaton game s defned by a 3-tuple (N, S, C), where N s the set of players, A = A, where A s the set of strateges avalable for the th player and C = C, where C : S R + s the cost functon for the th player. The socal-cost for the game for the strategy-vector s S s gven by the separable functon A. Smooth Games [1] C(s) = N C (s) (1) Formally, Roughgarden defned a cost-mnmzaton game to be (, µ) smooth f the followng holds for any strategy-pars (s, s): C (s, s ) C(s ) + µc(s) (2) Intutvely, gven two strateges s and s, a game s called smooth f the sum of the costs of the players due to unlateral devaton from s to s can be upper bounded by a lnear combnaton of the socal-costs at strateges s and s. Defnton 1 (Prce of Anarchy): For a gven cost-mnmzaton game, let s be an optmal acton-vector for the player and let ŝ be any Nash-equlbrum. The Prce-of-Anarchy (PoA) s defned as the supremum of the rato C(ŝ) C(s), where the supremum s taken over all Nash-Equlbrums of the underlyng game. The above defnton of smoothness mmedately yelds the followng upper-bound on the Prce-of-anarchy : Theorem 1: If a cost-mnmzaton game s (, µ) smooth wth > 0 and µ < 1, then Prce-of-anarchy (PoA) s upperbounded by 1 µ. Proof: Let s be a socally-optmal strategy vector,.e. : s arg mn C(s) (3) s S

2 2 Also, let ŝ be a Nash-equlbrum of the game. Then by defnton of Nash-equlbrum, we have C(ŝ) C (ŝ, ŝ ) (4) C (s, ŝ ) (5) C(s ) + µc(ŝ) (6) Here Eqn. (4) follows from the defnton of socal cost functon, Eqn. (5) follows from the defnton of Nash equlbrum and Eqn. (6) follows from the smoothness property of the game. The above nequalty mples Ths establshes the result. C(ŝ) 1 µ C(s) B. An Example of Computaton of PoA usng Smoothness Arguments 1) Congeston Games wth Affne Cost Functons: A congeston game s a cost-mnmzaton game defned by a ground set E f resources, a set k of players wth strategy sets S 1, S 2,, S k 2 E. The abstract framework may be understood as follows, Consder a graph G(V, E). Each player s assocated wth a source-destnaton par (s, t ) that they wsh to travel. The strategy set of each player s smply the set of dstnct paths from hs source to the destnaton. When x e of the users aval the edge e E, they ncur a cost of c e (x e ) = a e x e + b e. Hence, gven a strategy profle s = {s 1, s 2,..., s k }, the load nduced on the edge e s gven by x e = : e s. The cost ncurred by the player s smply C (s) = e S c e (x e ) and the total cost of the game s C(s) = C (s). Interchangng the order of summatons, t follows that C(s) = e E x e c e (x e ) (7) We have the followng theorem on the PoA of Congeston games : Theorem 2: The Prce of Anarchy of congeston games s at most 5 2. Proof: Let the strategy set (possbly randomzed) s be a Nash Equlbrum and s s the socally optmal strategy. Observe that, n the modfed strategy (s, s ), the number of players usng resource e s at most one more than that n s and ths resource contrbutes to precsely x e terms of the form C (s, s ). Hence k =1 C (s, s ) e E To proceed further, we need the followng lemma : Lemma 1: For all non-negatve ntegers y, z, the followng holds:.e., (a e (x e + 1) + b e )x e (8) y(z + 1) 5 3 y z2 (9) Proof: a) Case 1: y = 0: Trval. b) Case 2: y = 1: In ths case, we need to show that for all non-negatve ntegers z, we have whch s clearly true for all z Z +. z z2 (10) (z 1)(z 2) 0 (11)

3 3 c) Case 3:y 2: We need to show that : z 2 3yz + 5y 2 3y 0 (12) Treatng the above as a quadratc n z, t s enough to show that the dscrmnant s negatve,.e. 9y 2 4(5y 2 3y) < 0 (13) whch s true f y > 12 11,.e. f y 2. Now we return to the proof of the man result. Usng the above lemma, from Eqn. (8) we have k C (s, s ) 5 (a e x e + b e )x e + 1 (a e x e + b e )x e (14) 3 3 =1 Fnally, appealng to theorem (1), we obtan the result. e E e E = 5 3 C(s ) + 1 C(s) (15) 3 III. SMOOTH MECHANSIMS Motvated by smooth games defned by Roughgarden, Syrgkans and Tardos [2] generalzed the dea to mechansm desgn settngs. They nvestgated the settng of smultaneous mechansms and showed that f the component mechansms satsfes smoothness condtons, then the overall mechansm also satsfes smoothness condtons and hence, provng a smoothness bound on ndvdual mechansm yelds an automatc smoothness bound on the composed mechansm. 2) Mechansms wth Quaslnear Preferences : A mechansm desgn settng conssts of a set of n players and a set of outcomes X X, where X s the set of allocatons for player. Each player has a valuaton v : X R +. Gven an allocaton x X and a payment p R +, the utlty of the player s smply gven by u (x, p ) = v (x ) p (16) Gven an outcome space X, a mechansm M s a tuple (S, X, P ), where S = S s the strategy space of the players, X : S X s the allocaton functon and P : S R + n s the payment rule. Defnton 2: (SMOOTH MECHANISM) A mechansm M s (, µ) smooth, f for any valuaton profle v V and any acton profle a, there exsts a randomzed acton a (v, a ) for each player such that : u (a (v, a ), a ) OPT(v) µ P (a) (17) Smlar to theorem (1), a smooth mechansm mmedately mples that socal welfare at any Correlated equlbrum of the game s at least a constant fracton of the optmal socal welfare. Theorem 3: If a mechansm s (, µ) smooth and satsfes IR and NPT propertes then the expected socal welfare at any max{µ,1} Correlated Equlbrum of the game s at least of the optmal socal welfare. Proof: Assume that the (possbly randomzed) acton-profle a consttutes a correlated equlbrum of the game. Then we have, by the defnton of correlated equlbrum, for each player and for any other acton profle a Thus, W (a) = u (a, a ) u (a, a ) (18) (v (X (a)) (19) = u (a, a ) + P (a) (20) u (a (v, a ), a ) + P (a) (21) OPT(v) + (1 µ) P (a) (22) Where Eqn. (20) follows from the defnton of quaslnear utlty, Eqn. (21) follows from Eqn. (18) and Eqn. (22) follows from the smoothness assumpton of the mechansm. If µ < 1, the result follows because of NPT property of the mechansm (P (a) 0).

4 4 On the other hand, f µ > 1, snce at any correlated equlbrum, every player derves a non-negatve utlty (due to the mechansm beng IR). Ths mples v (X (a)) P (a),.e. W (a) P (a). Hence, from Eqn. (22), we have Thus, we have W (a) OPT(v) + (1 µ)w (a) (23) W (a) OPT(v) (24) µ Ths proves the result. A. Smoothness of Popular Auctons In ths secton we show that two popular auctons, namely, Frst Prce and All-Pay auctons are smooth. The dervaton conssts of dervng a smoothness nequalty of the form (17), by consderng a hypothetcal devaton from any bddng strategy b. 1) Frst Prce Aucton: Descrpton: In frst prce aucton there s one tem to be auctoned off. Each bdder has a prvate value v and submts a sealed-bd b. We allocate the the tem to the bdder wth hghest bd and pays an amount equal to hs bd. The rest of the bdder pays zero. Theorem 4: The Frst-Prce aucton s (1 1/e, 1) smooth. Proof: Consder a valuaton profle v and bddng profle b. Then we have OPT(v) = max v and P (a) = max b. Now consder the followng devaton from the bddng profle b : The bdder wth the hghest valuaton max v v max submts a bd randomly sampled from the dstrbuton f(b ) = 1 v max b and the support [0, (1 1/e)v max ] 1 and the rest of the bdders submt zero. Let us denote the hghest bdder by. When the rest of the bdders play b and he samples b = b, he derves the followng utlty : { v max b f b > max k b k u (b, b ) = (25) 0 o.w. Thus the expected utlty of the player s smply u (b, b ) = (1 1/e)vmax max k b k v max b v max b db (1 1/e)v max max k = (1 1/e)OPT(v) b k P (a) (26) Snce, utlty of the rest of the bdders are non-negatve, the above nequalty shows that frst prce aucton s (1 1/e, 1) smooth. 2) All-Pay Aucton: Descrpton: Lke the frst prce aucton, n an All-Pay aucton there s only one tem to be auctoned off and each bdder has prvate value v for the tem. Every bdder submts a sealed bd b. The tem s allocated to the hghest bdder, but unlke the frst prce aucton, every bdder must pay hs bd, rrespectve of whether he s allocated the tem or not. Note that the All-Pay aucton s not IR. Theorem 5: The All-Pay aucton s an ( 1 2, 1) smooth mechansm. Proof: For a submtted bd-profle b, we consder smlar devaton as n the Frst-Pay aucton. Ths tme, the hghest valued bdder submts a bd b sampled unformly at random from the nterval [0, v max ] and all bdders submt zero. We have OPT(v) = v max and P (a) = b. The utlty of the hghest valued bdder, denoted by, s wrtten as follows: { v max b f b > max k b k u (b, b ) = (27) 0 o.w. 1 It can be easly checked that t s a vald probablty dstrbuton.

5 5 Thus expected utlty of the player s smply gven by u (b, b ) = vmax max k b k 1 v max (v max b)db v max max b k 1 k 2 v max 1 2 OPT(v) P (b) (28) The above shows that All-Pay aucton s ( 1 2, 1) smooth. 3) Frst Prce Publc Project Aucton: Descrpton: There are n bdders and m publc projects. Each player has value v j f project j s mplemented. The bdders submt ther bds b j and the mechansm chooses the project j whch maxmzes b j. If project j s selected, player pays b j. Hence P (b) = b j Theorem 6: The Frst Prce publc project aucton s ( 1 e n n, 1 ) smooth. Proof: Let j be the optmal project for a gven valuaton profle v = {v j }. Suppose each player devates to bddng b [0, (1 e n )v j ] wth probablty densty functon f(t) = 1/n v j b to the project j and zero otherwse. Let j(b) be the project chosen accordng to the submtted bd b. Clearly, f b > b j(b), then the project j would be chosen nstead and player would derve an utlty of v j b. Thus Summng over all bdders, we obtan n Ths proves the result. B. Weak Smoothness u (b, b ) (1 e n )v j (v j b 1/n b ) v j(b) j b db (29) 1 n (1 e n )v j 1 n b j(b) n (30) =1 =1 u (b, b ) 1 n (1 e n )OPT(v) P (b) (31) In ths secton we gve a generalzaton of the smoothness framework to capture mechansms that produce hgh-effcency under a no-overbddng refnement. The most promnent example among these mechansms s the celebrated Second Prce Aucton. In the second-prce aucton the bd of a player s hs maxmum wllngness to pay when he wns. We start wth the followng defntons : Defnton 3 (Wllngness to Pay): Gven a mechansm (A, X, P) a player s maxmum wllngness-to-pay for an allocaton x when usng strategy a s defned as the maxmum he could ever pay condtonal on allocaton x : B (a, x ) = max a :X (a)=x P (a) (32) Defnton 4 (Weakly Smooth Mechansm): A mechansm s weakly (, µ 1, µ 2 )-smooth for, µ 1, µ 2 0 f for any type profle v V and for any acton profle a there exsts a randomzed aucton a (v, a ) for each player, s.t. u v (a (v, a ), a ) OPT(v) µ 1 P (a) µ 2 B (a, X (a)) (33) Defnton 5 (No-overbddng): A randomzed stategy profle a satsfes the no-overbddng assumpton f: E[B (a, X (a))] E[v (X (a))] (34) Under the assumpton of no-overbddng and Weakly smooth mechansm, we have an effcency result smlar to (3). Theorem 7: If a mechansm s weakly (, µ 1, µ 2 )-smooth then any Correlated Equlbrum n the full nformaton settng and any mxed Bayes-Nash Equlbrum n the Bayesan Settng that satsfes the no-overbddng assumpton acheves effcency at least µ 2+max{1,µ 1} of the expected optmal.

6 6 C. Examples of Weakly Smooth Mechansms 1) γ-hybrd aucton: In γ-hybrd aucton, bdders submt ther bds for procurement of a sngle tem. The bdder submttng the hghest bd wns and pays hs bd wth probablty γ and the second hghest bd wth probablty (1 γ). Theorem 8: The γ-hybrd aucton s (γ(1 1/e) + (1 γ) 2, 1, (1 γ) 2 ) smooth. Proof: Consder a valuaton profle v and a bd profle b. Let v max and b max be the hghest value of the bdders and and ther bds respectvely. Consder the followng devaton to the bd profle b. The hghest value bdder bds v max wth probablty 1 γ, and wth bds b [0, (1 1/e)v max ] wth densty f(b 1 ) = v max b. All remanng bdders devate to bddng zero n the modfed profle. Clearly, f b > b max, the hghest valued bdder wns. Hence hs utlty may be wrtten as follows: Case 1: bds hs true value Then f v max b max he wns the tem and gets an utlty of u (b, b ) = v max γv max (1 γ)b max = (1 γ)(v max b max ) (35) If v max < b max, he loses and gets zero utlty. Hence hs utlty n ths case s at least (1 γ)(v max b max ) (36) Case 2: bds accordng to the random strategy u (b, b ) (1 1/e)vmax ( vmax γb ) 1 (1 γ)b max b max v max b db (37) (1 1/e)vmax ( vmax b ) 1 b max v max b db (38) = (1 1/e)v max b max (39) Thus hs overall utlty s at least γ ( (1 1/e)v max b max ) + (1 γ) ( (1 γ)(vmax b max ) ) T he (40) The lemma follows by just observng that the payment under bd profle b s just γb max. D. The Composton Framework Mechansms rarely run n solaton but rather, several mechansms take place smultaneously and players typcally have valuatons that are functons on the outcomes of dfferent mechansms. Consder the followng general settng: there are n bdders and m mechansms. Each mechansm M j has ts own outcome space X j and conssts of a tuple (A j, X j, P j ). We assume that a player has a valuaton over vectors of outcomes from dfferent mechansms : v : X R +. The player s overall utlty for the outcome x = (x 1, x2,..., xm ) and payment p = (p 1, p2,..., pm ) s gven by the quaslnear functon u (x, p ) = v (x (a)) j p j (41) The smultaneous composton of such mechansms may be vewed as a global mechansm M = (A, X, P ), where A = j A j, X = jx j and P (a) = j pj. The socal-welfare of a strategy-profle a has the followng separable form : W (a) = v (x (a)) (42) For any valuaton profle v, there exsts an allocaton x (v) whch maxmzes v (x ) over all X. We denote the resultng optmal socal welfare by OPT(v). We restrct ourselves to a class of valuaton functons known as XOS valuaton defned below. Defnton 6 (XOS Valuaton): A valuaton s XOS f there exst a set L of addtve valuatons vj l (x j), such that v(x) = max l L vj(x l j ) j

7 7 We have the followng theorem regardng the smoothness of composed mechansms Theorem 9 (Smultaneous Composton): Consder the smultaneous composton of m mechansms. Suppose that each mechansm M j s (, µ) smooth when the mechansm restrcted valuatons of the players come from a class (V j ) [n]. If the valuaton v : X R + of each player across mechansms s XOS and can be expressed by component valuatons v l j, then the global mechansm s also (, µ)-smooth. Proof: Consder a valuaton profle v and an acton profle a. Let x be the optmal allocaton for the profle v. Let v j be the representatve addtve valuaton for x,.e. v(x ) = j v j(x j) (43) Also, from the defnton of XOS valuaton, for any x X v (x ) j v j(x j ) (44) To prove the theorem, we wll show that there exsts a devaton a = a (v, a ) of the global mechansm such that u v (a, a ) v (x ) µ P (a) (45) To defne such a devaton we use the fact that each mechansm M j s (, µ) smooth. Suppose that we run mechansm M j and each player has valuaton vj on X j. By the smoothness property of the mechansm M j, for any acton profle a j, there exsts a randomzed acton a j = a (v j, aj ) such that the sum of the utltes of the agents when each agent unlaterally devates to t, s at least v j (x j ) µ P j (aj ). For the global mechansm, we consder a randomzed devaton a = a (v, a ), that conssts of ndependent randomzed devatons a j = a j (v j, aj ). By the propertes of the representatve addtve valuatons, we have v (X (a, a ) j v j(x j (a j, a j ) (46) Hence, u v (a, a ) j, By smoothness of mechansm M j : u v (a, a ) j E[vj(X j (a j, a j ) P j (a j, a j )] (47) = ( v j(x j) µ v (x ) µ P j (a j)) (48) P (a) (49) Corollary 1: Smultaneous compostons of Frst Prce Auctons, All-Pay auctons and Publc project auctons are [(1 1/e), 1], [1/2, 1], [(1 e n )/n, 1])-smooth respectvely. IV. CONCLUSION In ths tutoral, we have ntroduced the basc dea of smoothness of games and extenson of ths concept to mechansm desgn. We showed that smoothness s a general tool whch may be used effectvely n determnng the Prce of Anarchy of games and effcency of mechansms. We also showed that smultaneous composton of (, µ) smooth mechansms are (, µ)-smooth. Future drecton would be to analyze PoA of more games usng the smoothness framework and obtan better bounds of effcency of new mechansms. REFERENCES [1] T. Roughgarden, Intrnsc robustness of the prce of anarchy, n Proceedngs of the forty-frst annual ACM symposum on Theory of computng. ACM, 2009, pp [2] V. Syrgkans and E. Tardos, Composable and effcent mechansms, n Proceedngs of the forty-ffth annual ACM symposum on Theory of computng. ACM, 2013, pp [3] G. Aggarwal and J. D. Hartlne, Knapsack auctons, n Proceedngs of the seventeenth annual ACM-SIAM symposum on Dscrete algorthm. Socety for Industral and Appled Mathematcs, 2006, pp [4] M. Feldman, H. Fu, N. Gravn, and B. Lucer, Smultaneous auctons are (almost) effcent, n Proceedngs of the forty-ffth annual ACM symposum on Theory of computng. ACM, 2013, pp [5] B. Lucer and A. Borodn, Prce of anarchy for greedy auctons, n Proceedngs of the twenty-frst annual ACM-SIAM symposum on Dscrete Algorthms. Socety for Industral and Appled Mathematcs, 2010, pp