Static Games and Classical Mechanism Design

Transcription

1 CSCI : Optmzaton and Control of Network Statc Game and Clacal Mechanm Degn Ljun Chen 01/12/2016

2 Outlne Strategc game and ther oluton concept q Strategc form game and domnated tratege q Nah equlbrum and correlated equlbrum Clacal mechanm degn q Incomplete nformaton game q Incentve-compatble mechanm q VCG mechanm 2

3 Strategc game q Def: a game n trategc form a trple q Notaton q q q N G = { N, S N, u N } q the et of player (agent q S the player trategy pace q u : S R the player payoff functon S = S : the et of all profle of player tratege 1 S2! S N = ( 1, 2,!, N : profle of tratege = ( 1, 2,!, 1, +1,!, N : the profle of tratege other than player 3

4 Implctly aume that player have preference over dfferent outcome, whch can be captured by agnng payoff to the outcome The bac model of ratonalty that of a payoff maxmzer Frt conder pure trategy, wll conder mxed trategy later 4

5 Example: fnte game L column M R U 4,3 5,1 6,2 row M 2,1 8,4 3,6 D 3,0 9,6 2,8 5

6 Example: Contnuou trategy game Cournot competton q Two player: frm 1 and frm2 q Strategy produce [ 0, ] : the amount of wdget that frm q The payoff for each frm the net revenue p u ( 1, 2 p( = c c where the prce, the unt cot for frm 6

7 Domnated tratege How to predct the outcome of a game? Proner Dlemma D C D -2,-2-5,-1 C -1,-5-4,-4 Two proner wll play (C,C Def: a trategy (weakly domnated for player f there ext ʹ uch that u ( ʹ, S u (, for all S 7

8 Iterated elmnaton of domnated tratege Iterated elmnaton of domnated tratege L M R U 4,3 5,1 6,2 M 2,1 8,4 3,6 D 3,0 9,6 2,8 However, mot of game are not olvable by terated elmnaton of domnated tratege 8

9 Nah equlbrum Def: a trategy profle a Nah equlbrum, f for all, For any, defne bet repone functon Then a trategy profle a Nah equlbrum ff 9 * S u u for all, (, ( * * * S }., (, ( { ( S u u S B ʹ ʹ = *. ( * * B

10 Example Battle of the Sexe Ballet Soccer Ballet 2,1 0,0 Soccer 0,0 1,2 Two Nah equlbra (Ballet, Ballet and (Soccer, Soccer 10

11 Cournot Competton Suppoe a prce functon Suppoe cot c = c = c Then, the bet repone functon p ( = max{0,1 ( 1 + 2} 1 B B 1 2 ( 2 ( 1 = = (1 (1 2 1 c / 2 c / 2 Nah equlbrum atfe 1 2 = B = B 1 ( 2 2 ( 1,.e., 1 2 = (1 c / 3 = (1 c / 3 11

12 Second prce aucton An object to be old to a player n N Each player ha a valuaton v of the object. We further aume v v >! > v 0 1 > 2 N > The player multaneouly ubmt bd b 1,!,b N The object gven to the player wth hghet bd. The wnner pay the econd hghet bd. The payoff of the wnner h valuaton of the object mnu the prce he pay. All other player payoff zero. 12

13 ( b 1,!, bn = ( v1,!, vn Nah equlbrum q Player 1 receve the object and pay, and ha payoff v1 v2 > 0. Player 1 ha no ncentve to devate, nce h payoff can only decreae q For other player, the payoff zero. In order to change h payoff, he need to bd more than v 1, but that wll reult n negatve payoff. So, no player ha ncentve to change Queton: are they more Nah equlbra? v 2 13

14 Not all game have (pure Nah equlbrum Matchng Penne Head Tal Head 1,-1-1,1 Tal -1,1 1,-1 14

15 Mxed tratege Σ Let player denote the et of probablty dtrbuton over trategy pace A mxed trategy σ over pure tratege Σ a probablty ma functon The payoff of a mxed trategy the expected value of the pure trategy profle S S S u = ( σ ( u ( j N j j 15

16 Mxed trategy Nah equlbrum Def: a mxed trategy profle Nah equlbrum f for all * * * u ( σ, σ u ( σ, σ A mxed trategy profle Nah equlbrum f for all u ( σ *, σ * u ( * σ, σ a (mxed trategy a (mxed trategy * * q The payoff u (, σ the ame for all upp( σ * * q The payoff u (, σ for each upp( σ not larger * σ * for allσ Σ for all S 16

17 Example Ballet Soccer Ballet 2,1 0,0 Soccer 0,0 1,2 Aume row (column player chooe ballet wth probablty p ( q and occer wth probablty 1 ( 1 q p = 2 / 3 Mxed trategy Nah equlbrum q = 1/ 3 2 q 1 p + 0 (1 + 0 (1 q p = 0 q = 0 p + 1 (1 + 2 (1 q p p 17

18 Extence of Nah equlbrum Theorem (Nah 50: Every fnte trategc game ha a mxed trategy Nah equlbrum. Example: Matchng Penne game ha a mxed trategy Nah equlbrum (½, ½; ½, ½ Head Tal Head 1,-1-1,1 Tal -1,1 1,-1 Proof: ung Kakutan fxed pont theorem. See ecton of the book by Fudenburg & Trole 18

19 Contnuou trategy game Theorem (Debreu 52; Glckberg 52; Fan 52: Conder a trategc game { N, S N, u N } wth contnuou trategy pace. A pure trategy Nah equlbrum ext f q S nonempty compact convex et q u contnuou n S and qua-concave n Theorem (Glckberg 52: Conder a trategc game { N, S N, u N } wth nonempty compact trategy pace. A mxed trategy Nah equlbrum ext f u contnuou. S 19

20 Correlated equlbrum In Nah equlbrum, player chooe tratege ndependently. How about player obervng ome common gnal? Traffc nterecton game Stop Go Stop 2,2 1,3 Go 3,1 0,0 q Two pure Nah equlbra: (top, go and (go, top q One mxed trategy equlbrum: (½, ½; ½, ½ q If there a traffc gnal uch that wth probablty ½ (red lght player play (top, go and wth probablty ½ (green lght player play (go, top. Th a correlated equlbrum. 20

21 Def: correlated equlbrum a probablty dtrbuton over the pure trategy pace uch that for all p(, [ u (, u ( t, ] 0 for all, A mxed trategy Nah equlbrum a correlated equlbrum The et of correlated equlbra convex and contan the convex hull of mxed trategy Nah equlbra t S p( 21

22 Dynamc n game Nah equlbrum a very trong concept. It aume player tratege, payoff and ratonalty are common knowledge Game theory lack a general and convncng argument that a Nah outcome wll occur One jutfcaton that equlbra are a a reult of adaptaton (learnng q Conder repeated play of the trategc game q Player are myopc, and adjut ther tratege baed on the tratege of other player n prevou round. 22

23 Bet repone ( t + 1 = B ( ( t Fcttou play, regret-baed heurtc, etc Many f not mot network algorthm are repeated and adaptve, and achevng ome equlbra. Wll dcu thee and networkng game later n th coure 23

24 Clacal mechanm degn (MD Mechanm: Protocol to mplement an outcome (equlbrum wth dered ytem-wde properte depte the elf-nteret and prvate nformaton of agent Mechanm degn: the degn of uch mechanm Provde an ntroducton to game theoretc approach to mechanm degn 24

25 Game theoretc approach to MD Start wth a trategc model of agent behavor Degn rule of a game, o that when agent play a aumed the outcome wth dered properte wll happen nduce the dered outcome among elf-ntereted agent 25

26 Incomplete nformaton game Player have prvate type Strategy a functon of a player type q Player of dfferent type may react dfferently n face of the ame tuaton Payoff u (θ S ( ( θ, θ R ( θ 1, θ2,!, θ N Θ a functon of player type q Player of dfferent type may have dfferent preference over the ame trategy profle All nformaton except actual type of player common knowledge q If a player type known, t payoff known 26

27 Incomplete nformaton game Aume type are drawn from ome objectve dtrbuton p( θ 1, θ2,!, θ N * Defnton: a trategy profle a Bayean- Nah equlbrum f every player play a bet repone to maxmze expected payoff gven t belef about dtrbuton p( θ θ,.e., * (θ argmax p(θ θ u (, * (θ,θ θ 27

28 Example: Varant of Battle of the Sexe Two type: ether want to meet the other or doe not Aume row player want to meet column player, but not ure f column player want to meet her or not (agn ½ probablty to each cae; and column player know row player type If column player want to meet row player, the payoff are Ballet Soccer Ballet 2,1 0,0 Soccer 0,0 1,2 28

29 If column player doe not want to meet row player, the payoff are Ballet Soccer Ballet Soccer 2,0 0,1 0,2 1,0 The Bayean-Nah equlbrum? (Ballet, (Ballet, Soccer q E[Ballet, (Ballet, Soccer]= ½x2+ ½x0=1 q E[Soccer, (Ballet, Soccer]= ½x0+ ½x1= ½ 29

30 Stronger oluton concept * Defnton: a trategy profle ex pot Nah equlbrum f every player trategy bet repone whatever the type of other * ( θ arg max u ( Defnton: a trategy profle domnant trategy equlbrum f every player trategy bet repone whatever the type and whatever the trategy of other, * ( θ, θ for allθ * ( θ arg max u (, ( θ, θ for all - *, θ 30

31 Domnant trategy equlbrum Very robut oluton concept Make no aumpton about nformaton avalablty Do not requre an agent to beleve other wll behave ratonally A domnant trategy mplementaton much more derable than Nah equlbrum 31

32 Example: econd prce aucton The type player valuaton Each player ubmt bd b v A domnant trategy to bd Player don t need to know valuaton (type, or tratege of other ( v * b ( v = v 32

33 Model of Mechanm Degn Set of alternatve outcome O Player ha prvate nformaton (type Type defne a value functon v outcome o O for each player ( o; θ Player payoff u ( o; θ = v ( o; θ p for outcome and payment p The dered properte are encapulated n the ocal choce functon f : Θ O θ R for q e.g., chooe o to maxmze ocal welfare,.e., o O f ( θ = arg max u ( o; θ o 33

34 The goal to mplement ocal choce functon 1 ( θ 1 N ( θ N Mechanm M = { g, p} o = g( ( p,! 1, p = N p( f (θ q A mechanm defned by an outcome rule n and a payment rule p : S R q A mechanm M mplement ocal choce functon f (θ * * f g( 1( θ,, ( (, where the trategy profle 1! N θ N = f θ * * an equlbrum oluton of the game ( 1,!, N nduced by M g : S O 34

35 Properte of ocal choce functon and mechanm Pareto optmal: f for every a f( θ, u (a,θ > u (cf ( θ, θ j u j ( a, θ < u j ( cf ( θ, θ Effcent: f f ( θ argmax v ( a, θ a Budget-balance: f p ( θ = 0 A mechanm that mplement the correpondng ocal choce functon called Pareto optmal, effcent, or budget-balanced mechanm, repectvely 35

36 Incentve-compatble mechanm Revelaton prncple: any mechanm can be tranformed nto an ncentve compatble, drectrevelaton mechanm that mplement the ame ocal choce functon Drect-revelaton mechanm a mechanm n whch player trategy pace retrcted to ther type θ 1 Mechanm o = g(θ θ N M = { g, p} ( p,! 1, p = N p( θ 36

37 Incentve-compatble mean the equlbrum trategy to report truthful nformaton about ther type (truth-revelaton q Frt prce aucton not ncentve-compatble. In frt prce aucton, the buyer wth hghet bd get the object and pay h bd q The econd prce aucton ncentve compatble, drect-revelaton mechanm Capture the eence of degnng a mechanm to overcome the elf-nteret of agent q Report t prvate nformaton truthfully, out of t own elf-nteret 37

38 Truthful mechanm Truthful (aka trategy-proof mechanm: truthrevelaton a domnant trategy equlbrum. q Domnant trategy mplementaton remove game theoretc complexty from mechanm degn q Very robut to aumpton about agent ratonalty and nformaton about each other q An agent can compute t optmal trategy wthout modelng the type and tratege of other 38

39 Vckrey-Clarke-Grove mechanm VCG mechanm: q Collect θ = θ, θ,!, θ ( 1 2 N from agent q g(θ : elect an outcome o O * o argmax v ( o; θ q p(θ : agent pay v j ( o ; θ j v j ( o * ; θ j, where j j o arg max v j ( o; θ j o O j 39

40 Theorem: VCG mechanm effcent and truthful Proof: VCG mechanm the only mechanm that effcent and trategy-proof among drect-revelaton mechanm 40 + = j j j j j j o v o v o v u ; ( ; ( ; (, ( * * θ θ θ θ θ

41 Combnatoral aucton Good P Outcome: allocaton A = ( A 1,!, A N, where A P and are not overlapped Agent valuaton v for ( A ; θ A P Goal: allocate good to maxmze Applcaton: wrele pectrum aucton, coure chedulng, v ( ; θ A 41

42 Two tem A and B; 3 agent Valuaton A B AB Outcome? agent 3 wn AB and pay 10-0=10 42

43 Another valuaton A B AB Outcome? agent 1 and 2 wn and each pay 7-5=2 43

44 Remark Only conder the ncentve ue: to overcome the elf-nteret of agent Not dcu computatonal and nformatonal ue q Tractablty (algorthmc MD q Dtrbuted computaton (dtrbuted MD q Mnmal nformaton revelaton q Bounded-ratonal agent q 44

45 Problem feature ncentve contrant computatonal contrant nformatonal contrant

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