ࡱ> t1(0 D/ 0|DArialr Net,o0,0$DTimes New Roman,o0,0$ DSymbolew Roman,o0,0$0DVerdanaw Roman,o0,0$"@DCourier Newman,o0,0$1@ .  @n?" dd@  @@``  2!  C 9  d+ !"#p$%&()*+,. 0AA0 3ff3@3ʚ;ʚ;g4GdGdD0ppp@ <4ddddp~0s 80___PPT10 ?  %=+Best-Reply Mechanisms0Brian Thompson Syeda Arzoo Zehra April 27, 2009 1Z",Best-reply dynamics Practical examplescInternet protocols Auctions Economic markets When are best-reply dynamics strategically justified?:b6.6(c  EnvironmentsSynchronous Strategy update: One player Delayed/lost messages: No Simultaneous Strategy update: Many players Delayed/lost messages: No Asynchronous Strategy update: Many players Delayed/lost messages: Yes  6 8 9 6 8 9Max-dominated strategiesA strategy is max-dominated if it is not a best-reply to any strategy-profile of the other players. Every strictly-dominated strategy is max-dominated.>d0b4 H4  Max-solvable gamesA max-solvable game is a game in which the iterated elimination of max-dominated strategies results in a single strategy-profile.(0b t  "Convergence of best-reply dynamics##&Any max-solvable game has a unique pure Nash equilibrium. Best-reply dynamics converge in n(Si mi ) steps for an n-player max-solvable game in which the size of the strategy space of a player i is mi.:0b Z  (    (      N    ( .Z  j  Asynchronous convergence In any max-solvable game, best-reply dynamics converge for every asynchronous schedule within Si mi asynchronous phases. An asynchronous phase is a period of time in which every player gets activated at least once receives at least one update message from every neighboryb@V^( (  "  Incentive compatibilityAMax-solvable games are not guaranteed to be incentive compatible. $Universally-max-dominated strategies%%%A set T of strategies for some player i is universally-max-dominated if the utility of its best strategy is strictly less than that of all other strategies (not in it).He Universally-max-solvable gamesA universally-max-solvable game is a game in which the iterated elimination of a universally-max-dominated strategy-set results in a single strategy-profile.$ Convergence results(A universally-max-solvable game has a unique pure Nash equilibrium. The pure Nash equilibrium is Collusion-proof: No group of players (including all) can change strategies without hurting at least one member. Pareto optimal: no one could be made better off without making someone else worse off. Raa_G(r ^X Incentive compatibility.Universally-max-solvable games with private information are incentive compatible in ex-post Nash equilibrium for any best-reply mechanism that consists of at least n(Si mi ) steps. Ex-post Nash: if all players play best-reply then a player cannot gain by not playing best-replya  (    (    a*  aBest-reply mechanismBest-reply dynamics + penalty. Assumptions: Sufficient number of rounds for best-reply dynamics to converge. Convergence is in the best interest of every player.&,v,vExamples of univ-max-solv games ' NRouting Congestion control Iterated 1st-price auction Stable roommates etc. 0N%(Routing: settingA model of interdomain routing in the Internet. Assume a network graph with a fixed target node d. Each node wishes to establish a path to d.  z Routing: gameEach node is a player. Every player has strict preferences over all possible paths (these are private), must choose a single outgoing edge. Utility of a player: rank of its assigned route given its preferences.6$hG$hGRouting: mechanism#Go over the players in round-robin order, repeatedly updating their chosen outgoing edge. If in some round no one changes their edge, then stop and output the route assignment. If after |V|3 - 2|V|2 + |V| rounds players still did not converge, then stop. Penalty: no traffic is sent at all.V$ZHHXRouting: best-reply strategiesGiven choices of other nodes, choose an edge s.t. the induced route is the most preferred. If no such route is induced then the node does not choose an outgoing edge at all.-~!Convergence resultsIf No Dispute Wheel condition holds then the prescribed best strategies converge to a unique pure Nash equilibrium. Incentive compatibility holds. Theorem: Routing games are universally-max-solvable.$#Iterated 1st-price auction$ YA single item is sold. People submit bids. Highest bidder pays his bid and gets the item.$ Auction: game~There are n bidders (players). Each player i has a private value, vi, for the item, must submit a bid bi T {1,2& ,k} for the item. Utility of a player: 0 if he did not win, otherwise vi - bi B-U'  H"H Ha    H  %Auction: mechanismGo over the players in round-robin order, repeatedly updating their bids. If in some round no one changes their bid, then stop and output the highest bid. If after n2k + n rounds players still did not converge, then stop. Penalty: item is given to no one.<HV& Auction: best-reply strategiesIf not the highest bidder and the highest current bid < vi then increase bi (less than vi) to beat all others. If highest current bid > vi then bid at most vi.5@HH HHH'!Convergence resultsTheorem: 1st-price auction games are universally-max-solvable. Therefore, The prescribed best-reply strategies converge to pure Nash equilibrium. The strategies are incentive compatible. :@ { " Questions?  ( Thank you)" Beyond universal-max-solvability!!(,Iterative auctions with unit demand bidders  Better-reply dynamicscPlayers are required to select only better-replies to the current strategies of the other players. cbdc   Potential gamesvAll games for which better reply dynamics always converge. Defined using better-reply dynamics [Monderer and Shapley] vwP! &/ :  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴` 3Sf3f` 3Sf3f>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> h(  h h 6lڥ " `}  T Click to edit Master title style! !$ h 0ݥ " `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S h 0 "^ `  >* h 0p "^   @* h 0 "^ `  @*H h 0޽h ? 3Sf3f80___PPT10.ޫ Default Design0 (    Nkk s$   n*  J%%JJnn  Nkk  /$  p*  J%%JJnnd  c $ ?NH  4  N@kk  x!  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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H  0jB ? 3380___PPT10.i'r B_E`LJj6%,G/1HS|1Q}Z'1*;09 X2y^acmNp{<~3)OOh+'0 hp   Best-Reply MechanismsArzooArzoo22Microsoft PowerPoint@@bM@$)Zѫ@ptGg    --$--'@Arial-. 3 2 PBest)"."System-@Arial-. 3 2 P%-.-@Arial-. 32 P9Reply Mechanisms,""3""""3.-@Arial-. 2 dBrian Thompson  !.-@Arial-. 2 KSyeda .-@Arial-. 2 Arzoo   .-@Arial-. 2 5Zehra  .-@Arial-. 2 ]April 27, 2009 .-՜.+,0    On-screen Show $ArialTimes New RomanSymbolVerdana Courier NewDefault DesignBest-Reply MechanismsBest-reply dynamicsPractical examples EnvironmentsMax-dominated strategiesMax-solvable games#Convergence of best-reply dynamicsAsynchronous convergenceIncentive compatibility%Universally-max-dominated strategiesUniversally-max-solvable gamesConvergence resultsIncentive compatibilityBest-reply mechanism Examples of univ-max-solv gamesRouting: settingRouting: gameRouting: mechanismRouting: best-reply strategiesConvergence resultsIterated 1st-price auctionAuction: gameAuction: mechanismAuction: best-reply strategiesConvergence results Slide 26 Thank you!Beyond universal-max-solvabilityBetter-reply dynamicsPotential games  Fonts UsedDesign Template Slide Titles_׈ ArzooArzoo  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Current UserSummaryInformation(PowerPoint Document(DocumentSummaryInformation8