+Algorithmic+Game+Theory
Playing Games with Algorithms Algorithmic Combinatorial Game Theory
a rXiv:cs /01619v2[cs.CC]22A pr28Playing Games with Algorithms:Algorithmic Combinatorial Game Theory ∗Erik D.Demaine †Robert A.Hearn ‡Abstract Combinatorial games lead to several interesting,clean problems in algorithms and complexity theory,many of which remain open.The purpose of this paper is to provide an overview of the area to encourage further research.In particular,we begin with general background in Combinatorial Game Theory,which analyzes ideal play in perfect-information games,and Constraint Logic,which provides a framework for showing hardness.Then we survey results about the complexity of determining ideal play in these games,and the related problems of solving puzzles,in terms of both polynomial-time algorithms and computational intractability results.Our review of background and survey of algorithmic results are by no means complete,but should serve as a useful primer.1Introduction Many classic games are known to be computationally intractable (assuming P =NP):one-player puzzles are often NP-complete (as in Minesweeper)or PSPACE-complete (as in Rush Hour),and two-player games are often PSPACE-complete (as in Othello)or EXPTIME-complete (as in Check-ers,Chess,and Go).Surprisingly,many seemingly simple puzzles and games are also hard.Other results are positive,proving that some games can be played optimally in polynomial time.In some cases,particularly with one-player puzzles,the computationally tractable games are still interesting for humans to play.We begin by reviewing some basics of Combinatorial Game Theory in Section 2,which gives tools for designing algorithms,followed by reviewing the relatively new theory of Constraint Logic in Section 3,which gives tools for proving hardness.In the bulk of this paper,Sections 4–6survey many of the algorithmic and hardness results for combinatorial games and puzzles.Section 7concludeswith a small sample of difficult open problems in algorithmic Combinatorial Game binatorial Game Theory is to be distinguished from other forms of game theory arising in the context of economics.Economic game theory has many applications in computer science as well,for example,in the context of auctions [dVV03]and analyzing behavior on the Internet [Pap01].2Combinatorial Game TheoryA combinatorial game typically involves two players,often called Left and Right,alternating play in well-defined moves.However,in the interesting case of a combinatorial puzzle,there is only one player,and for cellular automata such as Conway’s Game of Life,there are no players.In all cases,no randomness or hidden information is permitted:all players know all information about gameplay(perfect information).The problem is thus purely strategic:how to best play the game against an ideal opponent.It is useful to distinguish several types of two-player perfect-information games[BCG04,pp.14–15].A common assumption is that the game terminates after afinite number of moves(the game isfinite or short),and the result is a unique winner.Of course,there are exceptions:some games (such as Life and Chess)can be drawn out forever,and some games(such as tic-tac-toe and Chess) define ties in certain cases.However,in the combinatorial-game setting,it is useful to define the winner as the last player who is able to move;this is called normal play.If,on the other hand,the winner is thefirst player who cannot move,this is called mis`e re play.(We will normally assume normal play.)A game is loopy if it is possible to return to previously seen positions(as in Chess, for example).Finally,a game is called impartial if the two players(Left and Right)are treated identically,that is,each player has the same moves available from the same game position;otherwise the game is called partizan.A particular two-player perfect-information game without ties or draws can have one of four outcomes as the result of ideal play:player Left wins,player Right wins,thefirst player to move wins(whether it is Left or Right),or the second player to move wins.One goal in analyzing two-player games is to determine the outcome as one of these four categories,and tofind a strategy for the winning player to win.Another goal is to compute a deeper structure to games described in the remainder of this section,called the value of the game.A beautiful mathematical theory has been developed for analyzing two-player combinatorial games.A new introductory book on the topic is Lessons in Play by Albert,Nowakowski,and Wolfe[ANW07];the most comprehensive reference is the book Winning Ways by Berlekamp, Conway,and Guy[BCG04];and a more mathematical presentation is the book On Numbers and Games by Conway[Con01].See also[Con77,Fra96]for overviews and[Fra07]for a bibliography. The basic idea behind the theory is simple:a two-player game can be described by a rooted tree, where each node has zero or more left branches corresponding to options for player Left to move and zero or more right branches corresponding to options for player Right to move;leaves correspond tofinished games,with the winner determined by either normal or mis`e re play.The interesting parts of Combinatorial Game Theory are the several methods for manipulating and analyzing such games/trees.We give a brief summary of some of these methods in this section.2.1Conway’s Surreal NumbersA richly structured special class of two-player games are John H.Conway’s surreal numbers1[Con01, Knu74,Gon86,All87],a vast generalization of the real and ordinal number systems.Basically,a surreal number{L|R}is the“simplest”number larger than all Left options(in L)and smaller than all Right options(in R);for this to constitute a number,all Left and Right options must be numbers,defining a total order,and each Left option must be less than each Right option.See [Con01]for more formal definitions.For example,the simplest number without any larger-than or smaller-than constraints,denoted{|},is0;the simplest number larger than0and without smaller-than constraints,denoted{0|}, is1;and the simplest number larger than0and1(or just1),denoted{0,1|},is2.This method can be used to generate all natural numbers and indeed all ordinals.On the other hand,the simplest number less than0,denoted{|0},is−1;similarly,all negative integers can be generated. Another example is the simplest number larger than0and smaller than1,denoted{0|1},which is1ω, etc.)As such,surreal numbers are useful in their own right for cleaner forms of analysis;see,e.g., [All87].What is interesting about the surreals from the perspective of combinatorial game theory is that they are a subclass of all two-player perfect-information games,and some of the surreal structure, such as addition and subtraction,carries over to general games.Furthermore,while games are not totally ordered,they can still be compared to some surreal numbers and,amazingly,how a game compares to the surreal number0determines exactly the outcome of the game.This connection is detailed in the next few paragraphs.First we define some algebraic structure of games that carries over from surreal numbers;see Table1for formal definitions.Two-player combinatorial games,or trees,can simply be represented as{L|R}where,in contrast to surreal numbers,no constraints are placed on L and R.The negation of a game is the result of reversing the roles of the players Left and Right throughout the game.The(disjunctive)sum of two(sub)games is the game in which,at each player’s turn,the player has a binary choice of which subgame to play,and makes a move in precisely that subgame.A partial order is defined on games recursively:a game x is less than or equal to a game y if every Left option of x is less than y and every Right option of y is more than x.(Numeric)equality is defined by being both less than or equal to and more than or equal to.Strictly inequalities,as used in the definition of less than or equal to,are defined in the obvious manner.Note that while{−1|1}=0={|}in terms of numbers,{−1|1}and{|}denote different games(lasting1move and0moves,respectively),and in this sense are equal in value but not identical symbolically or game-theoretically.Nonetheless,the games{−1|1}and{|}have the same outcome:the second player to move wins.Amazingly,this holds in general:two equal numbers represent games with equal outcome(under ideal play).In particular,all games equal to0have the outcome that the second player to move wins.Furthermore,all games equal to a positive number have the outcome that the Left player wins;more generally,all positive games(games larger than0)have this outcome.Symmetrically, all negative games have the outcome that the Right player wins(this follows automatically by the negation operation).Examples of zero,positive,and negative games are the surreal numbers themselves;an additional example is described below.There is one outcome not captured by the characterization into zero,positive,and negative games:thefirst player to move wins.Tofind such a game we must obviously look beyond the surreal numbers.Furthermore,we must look for games G that are incomparable with zero(none of G=0,G<0,or G>0hold);such games are called fuzzy with0,denoted G 0.An example of a game that is not a surreal number is{1|0};there fails to be a number strictly between1and0because1≥0.Nonetheless,{1|0}is a game:Left has a single move leading to game1,from which Right cannot move,and Right has a single move leading to game0,from whichLet x={x L|x R}be a game.•x≤y precisely if every x L<y and every y R>x.•x=y precisely if x≤y and x≥y;otherwise x=y.•x<y precisely if x≤y and x=y,or equivalently,x≤y and x≥y.•−x={−x R|−x L}.•x+y={x L+y,x+y L|x R+y,x+y R}.•x is impartial precisely if x L and x R are identical sets and recursively everyposition(∈x L=x R)is impartial.•A one-pile Nim game is defined by∗n={∗0,...,∗(n−1)|∗0,...,∗(n−1)},together with∗0=0.Table1:Formal definitions of some algebra on two-player perfect-information games.In particular,all of these notions apply to surreal numbers.Left cannot move.Thus,in either case,thefirst player to move wins.The claim above implies that {1|0} 0.Indeed,{1|0} x for all surreal numbers x,0≤x≤1.In contrast,x<{1|0}for all x<0and{1|0}<x for all1<x.In general it holds that a game is fuzzy with some surreal numbers in an interval[−n,n]but comparable with all surreals outside that interval.Another example of a game that is not a number is{2|1},which is positive(>0),and hence Right wins, but fuzzy with numbers in the range[1,2].For brevity we omit many other useful notions in Combinatorial Game Theory,such as ad-ditional definitions of summation,super-infinitesimal games∗and↑,mass,temperature,thermo-graphs,the simplest form of a game,remoteness,and suspense;see[BCG04,Con01].2.2Sprague-Grundy TheoryA celebrated result in Combinatorial Game Theory is the characterization of impartial two-player perfect-information games,discovered independently in the1930’s by Sprague[Spr36]and Grundy [Gru39].Recall that a game is impartial if it does not distinguish between the players Left and Right(see Table1for a more formal definition).The Sprague-Grundy theory[Spr36,Gru39,Con01, BCG04]states that everyfinite impartial game is equivalent to an instance of the game of Nim, characterized by a single natural number n.This theory has since been generalized to all impartial games by generalizing Nim to all ordinals n;see[Con01,Smi66].Nim[Bou02]is a game played with several heaps,each with a certain number of tokens.A Nim game with a single heap of size n is denoted by∗n and is called a nimber.During each move a player can pick any pile and reduce it to any smaller nonnegative integer size.The game ends when all piles have size0.Thus,a single pile∗n can be reduced to any of the smaller piles∗0,∗1,...,∗(n−1).Multiple piles in a game of Nim are independent,and hence any game of Nim is a sum of single-pile games∗n for various values of n.In fact,a game of Nim with k piles of sizes n1,n2, ...,n k is equivalent to a one-pile Nim game∗n,where n is the binary XOR of n1,n2,...,n k.As a consequence,Nim can be played optimally in polynomial time(polynomial in the encoding size of the pile sizes).Even more surprising is that every impartial two-player perfect-information game has the samevalue as a single-pile Nim game,∗n for some n.The number n is called the G-value,Grundy-value,or Sprague-Grundy function of the game.It is easy to define:suppose that game x has koptions y1,...,y k for thefirst move(independent of which player goesfirst).By induction,we cancompute y1=∗n1,...,y k=∗n k.The theorem is that x equals∗n where n is the smallest naturalnumber not in the set{n1,...,n k}.This number n is called the minimum excluded value or mex of the set.This description has also assumed that the game isfinite,but this is easy to generalize[Con01,Smi66].The Sprague-Grundy function can increase by at most1at each level of the game tree,andhence the resulting nimber is linear in the maximum number of moves that can be made in the game;the encoding size of the nimber is only logarithmic in this count.Unfortunately,computing the Sprague-Grundy function for a general game by the obvious method uses time linear in the number of possible states,which can be exponential in the nimber itself.Nonetheless,the Sprague-Grundy theory is extremely helpful for analyzing impartial two-playergames,and for many games there is an efficient algorithm to determine the nimber.Examples in-clude Nim itself,Kayles,and various generalizations[GS56b];and Cutcake and Maundy Cake[BCG04,pp.24–27].In all of these examples,the Sprague-Grundy function has a succinct charac-terization(if somewhat difficult to prove);it can also be easily computed using dynamic program-ming.The Sprague-Grundy theory seems difficult to generalize to the superficially similar case ofmis`e re play,where the goal is to be thefirst player unable to move.Certain games have been solved in this context over the years,including Nim[Bou02];see,e.g.,[Fer74,GS56a].Recently a general theory has emerged for tackling mis`e re combinatorial games,based on commutative monoids called“mis`e re quotients”that localize the problem to certain restricted game scenarios. This theory was introduced by Plambeck[Pla05]and further developed by Plambeck and Siegel [PS07].For good descriptions of the theory,see Plambeck’s survey[Plaa],Siegel’s lecture notes [Sie06],and a webpage devoted to the topic[Plab].2.3Strategy StealingAnother useful technique in Combinatorial Game Theory for proving that a particular player must win is strategy stealing.The basic idea is to assume that one player has a winning strategy,and prove that in fact the other player has a winning strategy based on that strategy.This contradiction proves that the second player must in fact have a winning strategy.An example of such an argument is given in Section4.1.Unfortunately,such a proof by contradiction gives no indication of what the winning strategy actually is,only that it exists.In many situations,such as the one in Section4.1, the winner is known but no polynomial-time winning strategy is known.2.4PuzzlesThere is little theory for analyzing combinatorial puzzles(one-player games)along the lines of the two-player theory summarized in this section.We present one such viewpoint here.In most puzzles, solutions subdivide into a sequence of moves.Thus,a puzzle can be viewed as a tree,similar to a two-player game except that edges are not distinguished between Left and Right.With the view that the game ends only when the puzzle is solved,the goal is then to reach a position from which there are no valid moves(normal play).Loopy puzzles are common;to be more explicit,repeated subtrees can be converted into self-references to form a directed graph,and losing terminal positions can be given explicit loops to themselves.A consequence of the above view is that a puzzle is basically an impartial two-player game except that we are not interested in the outcome from two players alternating in moves.Rather,questions of interest in the context of puzzles are(a)whether a given puzzle is solvable,and(b)finding the solution with the fewest moves.An important open direction of research is to develop a general theory for resolving such questions,similar to the two-player theory.3Constraint LogicCombinatorial Game Theory provides a theoretical framework for giving positive algorithmic results for games,but does not naturally accommodate puzzles.In contrast,negative algorithmic results—hardness and completeness within computational complexity classes—are more uniform:puzzles and games have analogous prototypical proof structures.Furthermore,a relatively new theory called Constraint Logic attempts to tie together a wide range of hardness proofs for both puzzles and games.Proving that a problem is hard within a particular complexity class(like NP,PSPACE,or EX-PTIME)almost always involves a reduction to the problem from a known hard problem within the class.For example,the canonical problem to reduce from for NP-hardness is Boolean Satisfiability (SAT)[Coo71].Reducing SAT to a puzzle of interest proves that that puzzle is NP-hard.Similarly, the canonical problem to reduce from for PSPACE-hardness is Quantified Boolean Formulas(QBF) [SM73].Constraint Logic[DH08]is a useful tool for showing hardness of games and puzzles in a variety of settings that has emerged in recent years.Indeed,many of the hardness results mentioned in this survey are based on reductions from Constraint Logic.Constraint Logic is a family of games where players reverse edges on a planar directed graph while satisfying vertex in-flow constraints. Each edge has a weight of1or2.Each vertex has degree3and requires that the sum of the weights of inward-directed edges is at least2.Vertices may be restricted to two types:And vertices have incident edge weights of1,1,and2;and Or vertices have incident edge weights of2,2,and2.A player’s goal is to eventually reverse a given edge.This game family can be interpreted in many game-theoretic settings,ranging from zero-player automata to multiplayer games with hidden information.In particular,there are natural versions of Constraint Logic corresponding to one-player games(puzzles)and two-player games,both of bounded and unbounded length.(Here we refer to whether the length of the game is bounded by a polynomial function of the board size.Typically,bounded games are nonloopy while unbounded games are loopy.)These games have the expected complexities:one-player bounded games are NP-complete;one-player unbounded games and two-player bounded games are PSPACE-complete; and two-player unbounded games are EXPTIME-complete.What makes Constraint Logic specially suited for game and puzzle reductions is that the prob-lems are already in form similar to many games.In particular,the fact that the games are played on planar graphs means that the reduction does not usually need a crossover gadget,whereas historically crossover gadgets have often been the complex crux of a game hardness proof.Historically,Constraint Logic arose as a simplification of the“Generalized Rush-Hour Logic”of Flake and Baum[FB02].The resulting one-player unbounded setting,called Nondeterministic Constraint Logic[HD02,HD05],was later generalized to other game categories[Hea06b,DH08].4Algorithms for Two-Player GamesMany bounded-length two-player games are PSPACE-complete.This is fairly natural because games are closely related to Boolean expressions with alternating quantifiers (for which deciding satisfiability is PSPACE-complete):there exists a move for Left such that,for all moves for Right,there exists another move for Left,etc.A PSPACE-completeness result has two consequences.First,being in PSPACE means that the game can be played optimally,and typically all positions can be enumerated,using possibly exponential time but only polynomial space.Thus such games lend themselves to a somewhat reasonable exhaustive search for small enough sizes.Second,the games cannot be solved in polynomial time unless P =PSPACE,which is even “less likely”than P equaling NP.On the other hand,unbounded-length two-players games are often EXPTIME-complete.Such a result is one of the few types of true lower bounds in complexity theory,implying that all algorithms require exponential time in the worst case.In this section we briefly survey many of these complexity results and related positive results.See also[Epp]for a related survey and [Fra07]for a bibliography.4.1HexFigure 1:A 5×5Hex board.Hex [BCG04,pp.743–744]is a game designed by Piet Hein andplayed on a diamond-shaped hexagonal board;see Figure 1.Play-ers take turns filling in empty hexagons with their color.Thegoal of a player is to connect the opposite sides of their color withhexagons of their color.(In the figure,one player is solid and theother player is dotted.)A game of Hex can never tie,because if allhexagons are colored arbitrarily,there is precisely one connecting path of an appropriate color between opposite sides of the board.John Nash [BCG04,p.744]proved that the first player to move can win by using a strategy-stealing argument (see Section 2.3).Suppose that the second player has a winning strategy,and assume by symmetry that Left goes first.Left selects the first hexagon arbitrarily.Now Right is to move first and Left is effectively the second player.Thus,Left can follow the winning strategy for the second player,except that Left has one additional hexagon.But this additional hexagon can only help Left:it only restricts Right’s moves,and if Left’s strategy suggests filling the additional hexagon,Left can instead move anywhere else.Thus,Left has a winning strategy,contradicting that Right did,and hence the first player has a winning strategy.However,it remains open to give a polynomial characterization of a winning strategy for the first player.In perhaps the first PSPACE-hardness result for “interesting”games,Even and Tarjan [ET76]proved that a generalization of Hex to graphs is PSPACE-complete,even for maximum-degree-5graphs.Specifically,in this graph game,two vertices are initially colored Left,and players take turns coloring uncolored vertices in their own color.Left’s goal is to connect the two initially Left vertices by a path,and Right’s goal is to prevent such a path.Surprisingly,the closely related problem in which players color edges instead of vertices can be solved in polynomial time;this game is known as the Shannon switching game [BW70].A special case of this game is Bridgit or Gale ,invented by David Gale [BCG04,p.744],in which the graph is a square grid and Left’s goal is to connect a vertex on the top row with a vertex on the bottom row.However,if the graph in Shannon’s switching game has directed edges,the game again becomes PSPACE-complete [ET76].A few years later,Reisch [Rei81]proved the stronger result that determining the outcome of a position in Hex is PSPACE-complete on a normal diamond-shaped board.The proof is quitedifferent from the general graph reduction of Even and Tarjan[ET76],but the main milestone is to prove that Hex is PSPACE-complete for planar graphs.4.2More Games on Graphs:Kayles,Snort,Geography,Peek,and InteractiveHamiltonicityThe second paper to prove PSPACE-hardness of“interesting”games is by Schaefer[Sch78].This work proposes over a dozen games and proves them PSPACE-complete.Some of the games involve propositional formulas,others involve collections of sets,but perhaps the most interesting are those involving graphs.Two of these games are generalizations of“Kayles”,and another is a graph-traversal game called Edge Geography.Kayles[BCG04,pp.81–82]is an impartial game,designed independently by Dudeney and Sam Loyd,in which bowling pins are lined up on a line.Players take turns bowling with the property that exactly one or exactly two adjacent pins are knocked down(removed)in each move.Thus, most moves split the game into a sum of two subgames.Under normal play,Kayles can be solved in polynomial time using the Sprague-Grundy theory;see[BCG04,pp.90–91],[GS56b].Node Kayles is a generalization of Kayles to graphs in which each bowl“knocks down”(removes) a desired vertex and all its neighboring vertices.(Alternatively,this game can be viewed as two playersfinding an independent set.)Schaefer[Sch78]proved that deciding the outcome of this game is PSPACE-complete.The same result holds for a partizan version of node Kayles,in which every node is colored either Left or Right and only the corresponding player can choose a particular node as the primary target.Geography is another graph game,or rather game family,that is special from a techniques point of view:it has been used as the basis of many other PSPACE-hardness reductions for games described in this section.The motivating example of the game is players taking turns naming distinct geographic locations,each starting with the same letter with which the previous name ended.More generally,Geography consists of a directed graph with one node initially containing a token.Players take turns moving the token along a directed edge.In Edge Geography,that edge is then erased;in Vertex Geography,the vertex moved from is then erased.(Confusingly,in the literature,each of these variants is frequently referred to as simply“Geography”or“Generalized Geography”.)Schaefer[Sch78]established that Edge Geography(a game suggested by R.M.Karp)is PSPACE-complete;Lichtenstein and Sipser[LS80]showed that Vertex Geography(which more closely matches the motivating example above)is also PSPACE-complete.Nowakowski and Poole[NP96] have solved special cases of Vertex Geography when the graph is a product of two cycles.One may also consider playing either Geography game on an undirected graph.Fraenkel, Scheinerman,and Ullman[FSU93]show that Undirected Vertex Geography can be solved in poly-nomial time,whereas Undirected Edge Geography is PSPACE-complete,even for planar graphs with maximum degree3.If the graph is bipartite then Undirected Edge Geography is also solvable in polynomial time.One consequence of partizan node Kayles being PSPACE-hard is that deciding the outcome in Snort is PSPACE-complete on general graphs[Sch78].Snort[BCG04,pp.145–147]is a game designed by S.Norton and normally played on planar graphs(or planar maps).In any case,players take turns coloring vertices(or faces)in their own color such that only equal colors are adjacent.Generalized hex(the vertex Shannon switching game),node Kayles,and Vertex Geography have also been analyzed recently in the context of parameterized complexity.Specifically,the problem of deciding whether thefirst player can win within k moves,where k is a parameter to the problem,is AW[∗]-complete[DF97,ch.14].Stockmeyer and Chandra[SC79]were thefirst to prove combinatorial games to be EXPTIME-hard.They established EXPTIME-completeness for a class of logic games and two graph games. Here we describe an example of a logic game in the class,and one of the graph games;the other graph game is described in the next section.One logic game,called Peek,involves a box containing several parallel rectangular plates.Each plate(1)is colored either Left or Right except for one ownerless plate,(2)has circular holes carved in particular(known)positions,and(3)can be slid to one of two positions(fully in the box or partially outside the box).Players take turns either passing or changing the position of one of their plates.The winner is thefirst player to cause a hole in every plate to be aligned along a common vertical line.A second game involves a graph in which some edges are colored Left and some edges are colored Right,and initially some edges are“in”while the others are“out”.Players take turns either passing or changing one edge from “out”to“in”or vice versa.The winner is thefirst player to cause the graph of“in”edges to have a Hamiltonian cycle.(Both of these games can be rephrased under normal play by defining there to be no valid moves from positions having aligned holes or Hamiltonian cycles.)4.3Games of Pursuit:Annihilation,Remove,Capture,Contrajunctive,Block-ing,Target,and Cops and RobbersThe next suite of graph games essentially began study in1976when Fraenkel and Yesha[FY76] announced that a certain impartial annihilation game could be played optimally in polynomial time.Details appeared later in[FY82];see also[Fra74].The game was proposed by John Conway and is played on an arbitrary directed graph in which some of the vertices contain a token.Players take turns selecting a token and moving it along an edge;if this causes the token to occupy a vertex already containing a token,both tokens are annihilated(removed).The winner is determined by normal play if all tokens are annihilated,except that play may be drawn out indefinitely.Fraenkel and Yesha’s result[FY82]is that the outcome of the game can be determined and(in the case of a winner)a winning strategy of O(n5)moves can be computed in O(n6)time,where n is the number of vertices in the graph.A generalization of this impartial game,called Annihilation,is when two(or more)types of tokens are distinguished,and each type of token can travel along only a certain subset of the edges.As before,if a token is moved to a vertex containing a token(of any type),both tokens are annihilated.Determining the outcome of this game was proved NP-hard[FY79]and later PSPACE-hard[FG87].For acyclic graphs,the problem is PSPACE-complete[FG87].The precise complexity for cyclic graphs remains open.Annihilation has also been studied under mis`e re play [Fer84].A related impartial game,called Remove,has the same rules as Annihilation except that when a token is moved to a vertex containing another token,only the moved token is removed.This game was also proved NP-hard using a reduction similar to that for Annihilation[FY79],but otherwise its complexity seems open.The analogous impartial game in which just the unmoved token is removed,called Hit,is PSPACE-complete for acyclic graphs[FG87],but its precise complexity remains open for cyclic graphs.A partizan version of Annihilation is Capture,in which the two types of tokens are assigned to corresponding players.Left can only move a Left token,and only to a position that does not contain a Left token.If the position contains a Right token,that Right token is captured(removed). Unlike Annihilation,Capture allows all tokens to travel along all edges.Determining the outcome of Capture was proved NP-hard[FY79]and later EXPTIME-complete[GR95].For acyclic graphs。
UniversityofWisconsin-Madison（
University of Wisconsin-Madison(UMW)周玉龙1101213442 计算机应用UMW简介美国威斯康辛大学坐落于美国密歇根湖西岸的威斯康辛州首府麦迪逊市，有着风景如画的校园，成立于1848年, 是一所有着超过150年历史的悠久大学。
威斯康辛大学是全美最顶尖的三所公立大学之一，是全美最顶尖的十所研究型大学之一。
在美国，它经常被视为公立的常青藤。
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在本科教育方面，它列于伯克利加州大学和密歇根大学之后，排在公立大学的第三位。
除此之外，它还在本科教育质量列于美国大学的第八位。
按美国全国研究会的研究结果，威斯康辛大学有70个科目排在全美前十名。
在上海交通大学的排行中，它名列世界大学的第16名。
威斯康辛大学是美国大学联合会的60个成员之一。
特色专业介绍威斯康辛大学麦迪逊分校设有100多个本科专业，一半以上可以授予硕士、博士学位，其中新闻学、生物化学、植物学、化学工程、化学、土木工程、计算机科学、地球科学、英语、地理学、物理学、经济学、德语、历史学、语言学、数学、工商管理（MBA）、微生物学、分子生物学、机械工程、哲学、西班牙语、心理学、政治学、统计学、社会学、动物学等诸多学科具有相当雄厚的科研和教学实力，大部分在美国大学相应领域排名中居于前10名。
学术特色就学术方面的荣耀而言，威斯康辛大学麦迪逊校区的教职员和校友至今共获颁十七座诺贝尔奖和二十四座普立兹奖；有五十三位教职员是国家科学研究院的成员、有十七位是国家工程研究院的成员、有五位是隶属于国家教育研究院，另外还有九位教职员赢得了国家科学奖章、六位是国家级研究员（Searle Scholars）、还有四位获颁麦克阿瑟研究员基金。
威斯康辛大学麦迪逊校区虽然是以农业及生命科学为特色，但是令人注目，同时也是吸引许多传播科系学子前来留学的最大诱因，则是当前任教于该校新闻及传播研究所、在传播学界有「近代美国传播大师」之称的杰克·麦克劳（Jack McLauld）。
第三节博弈论（GameTheory）
第三节博弈论(Game Theory)在国际关系的研究过程中，我们时常会运用到博弈论这样一个工具。
博弈论在英语中称之为“Game Theory”。
很多人会认为这是一种所谓的游戏理论，其实不然，我们不能把Games 与Fun 同论，而应该将博弈论称之为是一种“Strategic interaction”（策略性互动）。
“博弈”一词现如今在我们的生活中出现的已经很频繁，我们经常会听说各种类型的国家间博弈（如：中美博弈），“博弈论”已经深刻的影响了世界局势和地区局势的发展。
在iChange创设的危机联动体系中，博弈论将得到充分利用，代表也将有机会运用博弈论的知识来解决iChange 核心学术委员会设计的危机。
在这一节中，我将对博弈论进行一个初步的介绍与讨论，代表们可以从这一节中了解到博弈论的相关历史以及一些经典案例的剖析。
（请注意：博弈论的应用范围非常广泛，涵盖数学、经济学、生物学、计算机科学、国际关系、政治学及军事战略等多种学科，对博弈论案例的一些深入分析有时需要运用到高等数学知识，在本节中我们不会涉及较多的数学概念，仅会通过一些基本的数学分析和逻辑推理来方便理解将要讨论的经典博弈案例。
）3.1 从“叙利亚局势”到“零和博弈”在先前关于现实主义理论的讨论中，我们对国家间博弈已经有了初步的了解，那就是国家是有目的的行为体，他们总为了实现自己利益的最大化而选择对自己最有利的战略，其次，政治结果不仅仅只取决于一个国家的战略选择还取决于其他国家的战略选择，多种选择的互相作用，或者策略性互动会产生不同的结果。
因此，国家行为体在选择战略前会预判他国的战略。
在这样的条件下，让我们用一个简单的模型分析一下发生在2013年叙利亚局势1：叙利亚危机从2011年发展至今已经将进入第四个年头。
叙利亚危机从叙利亚政府军屠杀平民和儿童再到使用化学武器而骤然升级，以2013年8月底美国欲对叙利亚动武达到最为紧张的状态，同年9月中旬，叙利亚阿萨德政府以愿意向国际社会交出化学武器并同意立即加入《禁止化学武器公约》的态度而使得局势趋向缓和。
斯坦福大学人工智能所有课程介绍
List of related AI Classes CS229covered a broad swath of topics in machine learning,compressed into a sin-gle quarter.Machine learning is a hugely inter-disciplinary topic,and there are many other sub-communities of AI working on related topics,or working on applying machine learning to different problems.Stanford has one of the best and broadest sets of AI courses of pretty much any university.It offers a wide range of classes,covering most of the scope of AI issues.Here are some some classes in which you can learn more about topics related to CS229:AI Overview•CS221(Aut):Artificial Intelligence:Principles and Techniques.Broad overview of AI and applications,including robotics,vision,NLP,search,Bayesian networks, and learning.Taught by Professor Andrew Ng.Robotics•CS223A(Win):Robotics from the perspective of building the robot and controlling it;focus on manipulation.Taught by Professor Oussama Khatib(who builds the big robots in the Robotics Lab).•CS225A(Spr):A lab course from the same perspective,taught by Professor Khatib.•CS225B(Aut):A lab course where you get to play around with making mobile robots navigate in the real world.Taught by Dr.Kurt Konolige(SRI).•CS277(Spr):Experimental Haptics.Teaches haptics programming and touch feedback in virtual reality.Taught by Professor Ken Salisbury,who works on robot design,haptic devices/teleoperation,robotic surgery,and more.•CS326A(Latombe):Motion planning.An algorithmic robot motion planning course,by Professor Jean-Claude Latombe,who(literally)wrote the book on the topic.Knowledge Representation&Reasoning•CS222(Win):Logical knowledge representation and reasoning.Taught by Profes-sor Yoav Shoham and Professor Johan van Benthem.•CS227(Spr):Algorithmic methods such as search,CSP,planning.Taught by Dr.Yorke-Smith(SRI).Probabilistic Methods•CS228(Win):Probabilistic models in AI.Bayesian networks,hidden Markov mod-els,and planning under uncertainty.Taught by Professor Daphne Koller,who works on computational biology,Bayes nets,learning,computational game theory, and more.1Perception&Understanding•CS223B(Win):Introduction to computer vision.Algorithms for processing and interpreting image or camera information.Taught by Professor Sebastian Thrun, who led the DARPA Grand Challenge/DARPA Urban Challenge teams,or Pro-fessor Jana Kosecka,who works on vision and robotics.•CS224S(Win):Speech recognition and synthesis.Algorithms for large vocabu-lary continuous speech recognition,text-to-speech,conversational dialogue agents.Taught by Professor Dan Jurafsky,who co-authored one of the two most-used textbooks on NLP.•CS224N(Spr):Natural language processing,including parsing,part of speech tagging,information extraction from text,and more.Taught by Professor Chris Manning,who co-authored the other of the two most-used textbooks on NLP.•CS224U(Win):Natural language understanding,including computational seman-tics and pragmatics,with application to question answering,summarization,and inference.Taught by Professors Dan Jurafsky and Chris Manning.Multi-agent systems•CS224M(Win):Multi-agent systems,including game theoretic foundations,de-signing systems that induce agents to coordinate,and multi-agent learning.Taught by Professor Yoav Shoham,who works on economic models of multi-agent interac-tions.•CS227B(Spr):General game playing.Reasoning and learning methods for playing any of a broad class of games.Taught by Professor Michael Genesereth,who works on computational logic,enterprise management and e-commerce.Convex Optimization•EE364A(Win):Convex Optimization.Convexity,duality,convex programs,inte-rior point methods,algorithms.Taught by Professor Stephen Boyd,who works on optimization and its application to engineering problems.AI Project courses•CS294B/CS294W(Win):STAIR(STanford AI Robot)project.Project course with no lectures.By drawing from machine learning and all other areas of AI, we’ll work on the challenge problem of building a general-purpose robot that can carry out home and office chores,such as tidying up a room,fetching items,and preparing meals.Taught by Professor Andrew Ng.2。
爱丁堡大学计算机科学本科专业
留学监理服务网
算法和数据结构 编译技术 可计算性和棘手问题 计算机系统结构 计算机设计 数据库系统 语言的语义和执行 操作系统 对象和组件的软件工程 计算机安全 计算机通信与网络 初级应用机器学习 逻辑编程 第4年 荣誉学位的项目（情报学） 选8门 高级数据库 通信和并发 计算复杂性 计算机代数 计算机图形学（10 级） 并行编程语言和系统（10 级） 并行结构（10 级） 查询和存储 XML 生物信息学 2 应用数据库 生物信息学 1 算法博弈论及其应用 分布式数据库 并行算法设计与分析 概率建模和推理
院系介绍
爱丁堡大学分设三大学院（College），分别是：人文与社会科学院、科学与工程学院、医学 与兽医学院，三大学院下设有22个小学院（School），包括商学院、工程学院、经济学院、 医学院、法学院、社科学院、信息学院、生物 化学学院等。其中，经济学 、人类学、社会学、 建筑学、土木工程、电子工程学、化学、计算机科学、法律、地质 学、数学和统计学、物理 学、银行与风险学、细胞和分子生物学是爱丁堡大学的热门专业。
建筑系
£11200 约合11万 (人民币)
园林建筑学 Landscape Architecture
文学学 3,4
士,硕士
爱丁堡艺术 每年 学院 秋季 建筑与园林
建筑系
£11200 约合11万 (人民币)
情报学（5 年本硕连读） MInf Informatics
£15850
每年
5 硕士
信息学院 约合16万
园林建筑学本硕连读 Hons Landscape Architecture
人工智能 Artificial Intelligence 认知科学 Cognitive Science 计算机科学 Computer Science 计算机科学工程 Computer Science 软件工程 Software Engineering 数学与统计 Mathematics & Statistics 经济与统计学本硕连读 Hons Economics and Statistics 图形设计 Graphic Design 室内设计 Interior Design 产品设计 Production Design 会计与金融 Accounting and Finance 经济学
新一代人工智能发展规划
《新一代人工智能发展规划》解读—人工智能的过去、现在和未来前言：人工智能的概念及发展历史一）人工智能的概念人工智能：以机器为载体的智能，是相对于人类智能和动物智能，也叫机器智能。
2017年7月20日发布《新一代人工智能发展规划》(国发〔2017〕35号)一）人工智能的概念。
人工智能的迅速发展将深刻改变人类社会生活、改变世界。
三个阶段的目标：2020年：与世界先进水平同步2025年：部分达到世界领先水平2030年：总体达到世界领先水平（二）世界各国高度重视人工智能的发展方向美国：2016年10月，美国连续发布两个重要战略文件《为人工智能的未来做好准备》和《国家人工智能研究与发展战略规划》，将人工智能上升到国家战略层面。
美国有很多著名的IT跨国企业，如谷歌、Facebook、微软、IBM等，都将人工智能技术作为企业的核心战略，持续投入巨资并招聘领军人才，强力涉足该领域。
在技术方向上，美国将机器人技术列为警惕技术，主攻军用机器人技术。
人工智能技术使五角大楼重新调整了人和机器在战场上的位置，这些新武器的速度和精确度都会大大提高，可以大幅减少士兵伤亡。
日本：日本政府将人工智能定位为增长战略的支柱，提出“机器人驱动的新工业革命”。
日本文部科学省计划在今后10年投入1000亿日元，用于人工智能的研发，在东京建立研究基地。
日本在2017年度预算中，对人工智能的研究是924亿日元，是2016年预算的9倍。
欧洲：欧盟2013年启动人脑计划，为期10年，欧盟和参与国投入近12亿欧元经费，在2024年设计出能够模拟人脑运作原理的超级计算机。
英国：2012年，英国政府把人工智能及机器人技术列为国家重点发展的八大技术之一。
2015年出台了《英国机器人及自主系统发展图景》。
2016年，英国政府科学办公室发布了《人工智能：未来决策制定的机遇与影响》。
目前，英国已经把人工智能列为战略性和尖端科技的重中之重，力图抢占人工智能发展的制高点。
博弈论又被称为对策论(Game Theory)既是现代数学的一个新分支 ...
博弈论又被称为对策论（Game Theory）既是现代数学的一个新分支，也是运筹学的一个重要学科。
博弈论主要研究公式化了的激励结构间的相互作用。
是研究具有斗争或竞争性质现象的数学理论和方法。
博弈论考虑游戏中的个体的预测行为和实际行为，并研究它们的优化策略。
生物学家使用博弈理论来理解和预测进化论的某些结果。
博弈论已经成为经济学的标准分析工具之一。
在生物学、经济学、国际关系、计算机科学、政治学、军事战略和其他很多学科都有广泛的应用。
自从博弈论被引入经济学以来，现在经济的许多领域都发生了巨大变化。
博弈论在强调经济活动的利益主体行为所产生的相互作用和相互影响的同时，也在突出反映社会制度的本质。
人们或组织需要更多的信息在预期其他参与方行动决策的情况下做出自己的行动选择期求更大的利益。
而我们所谓的制度就是均衡行动选择的本质特征，被参与方普遍认可并与他们的行动息息相关。
下面以最近的南海争端作为案例用博弈论的知识对争端各方所认同的制度进行研究。
由于南海问题牵涉利益参与方较多，争端较为复杂，我们只考虑中国和南海诸国双边的政治博弈。
首先看南海争端的地理位置。
南沙群岛陆地面积虽然只有二平方公里，但是整个海域面积达八十二万三千平方公里，而且地理位置非常重要。
南沙群岛地处越南金兰湾和菲律宾苏比克湾两大海军基地之间，战略位置突出，扼西太平洋至印度洋海上交通要冲，通往非洲和欧洲的咽喉要道。
再次，南海的资源也成为各国关注的焦点。
南海地处中、菲、越、日、马各国交界地带，渔业矿产资源丰富，各国利益争端复杂，这也成为南海争端形成的必要条件。
二十世纪六十年代开始，越、菲、马等国以军事手段占领南沙群岛部分岛礁，在南沙群岛附近海域进行大规模的资源开发活动并提出主权要求。
众所周知，作为一个行为主体忽略和偏离制度对其而言是无利可图甚至产生消极效应。
从60年代至今，中方与南海边界小国以及美日印诸国产生了重复参与博弈的战略互动的稳定状态。
上世纪80年代末90年代初，这些国家开始分别在所占据的岛礁上修建飞机跑道，建海港、灯塔和旅游观光点，并纷纷与外国石油公司合作，开采南沙地区的油气资源。
计算机应用技术专业本科直博研究生培养方案(081203)
计算机应用技术专业本科直博研究生培养方案（081203）（信息科学技术学院）一、培养目标（一）较好地掌握马克思主义、毛泽东思想和中国特色社会主义理论体系，深入贯彻科学发展观，热爱祖国，遵纪守法，品德良好，学风严谨，身心健康，具有较强的事业心和献身精神，积极为社会主义现代化建设事业服务。
（二）在本门学科上掌握坚实宽广的基础理论和系统深入的专门知识，同时要掌握一定的相关学科知识，具有独立从事科学研究工作的能力，在科学或专门技术上做出创造性的成果。
（三）熟练掌握一门外语，能阅读本专业外文文献，具有运用外文写作和进行国际学术交流的能力。
二、培养方式与学习年限（一）培养方式本科直博研究生进入博士阶段的学习后，一方面进行必要的课程学习，夯实专业基础，拓展学术视野；另一方面开始着手科学研究。
对本科直博研究生的课程要重新设置，充分体现学科特色和培养需求；课程时间一般为一至二年。
以资格考核的结果作为能否进入下一阶段的依据。
通过资格考试的本科直博研究生进入科学研究和撰写博士学位论文阶段，学习年限一般为三到四年；未通过资格考试的可按照同专业硕士研究生的培养要求进行培养，时间一般为二到三年。
本科直博研究生，在科研能力、学位论文等方面的要求，均应高于同专业四年制博士研究生要求。
（二）学习年限本科直博研究生学习年限一般为五年至六年。
若在五年内不能完成预定的学业，可适当延长学习年限，但一般不超过六年。
三、主要研究方向1.计算机网络与通信2.网络与嵌入式系统3.普适计算与情景感知计算4.大数据分析与知识处理5.分布计算与云计算技术6.模式识别与机器学习7.模式识别与机器智能四、学分要求和课程设置（一）学分要求本科直博研究生课程包括学位公共课、学位基础课、学位专业课。
学位公共课包括政治理论、外国语等公共必修课程和公共选修课程，至少修读8学分（注：公共选修课指研究方法类课程。
如不选修则应以学位专业课相应学分抵充）；学位基础课为学位必修课程，至少选修3门，不少于8学分；学位专业课包括以学科群为单位开设的专业必修课程和指向研究方向的专业选修课程，至少选修8门，不少于17学分。
算法博弈论
算法博弈论算法博弈论（algorithmic game theory）是2018年公布的计算机科学技术名词。
是计算机科学与博弈论的交叉研究领域。
从博弈的角度、以经济学和计算理论的方法分别研究计算机科学和经济学中的计算模型。
长期以来来，经济学研究人员专注各种经济活动和各种相应的经济关系及其运行，以及身为一名理性人在经济活动中的行为；而计算机科学研究人员则专注于研究信息与计算以及计算机系统中如何实现与应用，二者互不干涉。
这一情况在上个世纪90年代得到了改变，互联网的兴起，让原来只关注自身领域的计算机研究人员和经济学研究人员走到了一起：对于计算机科学研究人员，他们开始考虑互联网上的非合作博弈（non-cooperative）特性以及相应的激励(incentive)问题；同样的，经济学研究人员也开始涉足新兴的互联网，研究其跟经济相关的问题。
就这样计算机科学（computer science）与博弈论（game theory）走到了一起，形成了一门新的学科：算法博弈论（algorithmic game theory).和传统的博弈论和计算机科学相比，算法博弈论主要关注点在互联网网络，非传统拍卖等，主要不同体现在这些方面：应用领域：算法博弈论主要研究包括Internet网络和非传统拍卖，比如社交网络里的个体行为，baidu,google等用拍卖的方式出售它的关键字广告位，或者4G频段的拍卖。
工程量化方法：从具体优化问题的角度对应用建模，寻求最优解、判断不可解问题以及研究可解优化的上下限问题。
比如，在对问题用博弈论的框架进行建模过程中，可能会得到很多个稳定的状态（纳什均衡）。
那么在在这些稳定状态中，我们会关注系统最好情况的系统状况，最坏情况下系统的状况，以及统计意义上平均的系统状况。
以经典的囚徒困境为例，很显然在均衡状态下总共的收益是-4，而当两人都选择沉默时，每个人的收益-2。
很显然在均衡状态下并不最优的（inefficient），那我们该如何去量化这种inefficiency呢，这是算法博弈化研究内容之一。
博弈论用英文怎么说英语是什么
博弈论用英文怎么说英语是什么博弈论主要研究公式化了的激励结构间的相互作用，是研究具有斗争或竞争性质现象的数学理论和方法。
那么你知道博弈论的英文怎么说吗?下面店铺为大家带来博弈论的英文说法，供大家参考学习。
博弈论的英文说法：game theory英 [ɡeim ˈθiəri]美 [ɡem ˈθiəri]博弈论相关英文表达：博弈论算法 Algorithmic Game Theory合作博弈论 Cooperative Games Theory经济博弈论 Economic Game Theory重复博弈论 repeated game approach博弈论英文说法例句：1. Game theory is a powerful weapon to decision - making of multi - person.博弈论是解决多人竞争决策问题的有利武器.2. Game theory is an equilibrium problem in decision influence.博弈论是在选择中的决策影响和均衡问题.3. Conflict and cooperation are the two fundamental issues in game theory.冲突与合作是博弈论研究的两大基本问题.4. The main research way is the analytical method of Game Theory.研究方法主要是博弈论中的博弈分析方法.5. Game theory; High - tech SMEs; principal - agent; performance management; incentive mechanism.博弈论; 高新技术中小企业; 委托—代理; 绩效管理; 激励机制.6. Textbook farce and textbook game theory – how delightful!经典的闹剧,经典的博弈论——多有趣啊 !7. Contract Theory, Information Economics, Applied Game Theory, Corporate Finance.契约理论, 信息经济学, 应用博弈论, 公司财务,政治经济学.8. This story illustrates an important distinction between ordinary decision theory and theory.这个故事说明了普通决策理论和博弈论之间的一个重要的区别.9. The backward induction is an important reasoning method in game theory.逆向归纳法(倒推法)是博弈论中的一种重要的推理方法.10. What's the definition of the game theory?[灌水]博弈论的定义是什么 ?11. Game theory studies mutual roles among rational agents.博弈论是研究理性的行动者相互作用的理论.12. So the penalty kick, for instance, is like this laboratory mile of game theory.打个比方, 罚点球, 就像博弈论里面的实验室.13. Based on principal - agency, the thesis analyzes budget management system from the angle of game theory.本文以委托代理理论为基础, 从博弈论视角对企业预算管理制度进行了分析.14. Evolutionary game theory provides a uniform frame to study the evolution of cooperation.进化博弈论为理解合作行为的演化提供了一个统一的框架.15. Proving the possibility and inevitability of the tax evasion by the basic theory of GAME.用博弈论的基本理论证明企业偷税的可能性和必然性.。
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Algorithmic Game TheoryOver the last few years,there has been explosive growth in the research done at the in-terface of computer science,game theory,and economic theory,largely motivated by the emergence of the Internet.Algorithmic Game Theory develops the central ideas and results of this new and exciting area.More than40of the top researchers in thisfield have written chapters whose topics range from the foundations to the state of the art.This book contains an extensive treatment of algorithms for equilibria in games and markets,computational auctions and mechanism design,and the“price of anarchy,”as well as applications in networks,peer-to-peer systems, security,information markets,and more.This book will be of interest to students,researchers,and practitioners in theoretical computer science,economics,networking,artificial intelligence,operations research,and discrete mathematics.Noam Nisan is a Professor in the Department of Computer Science at The Hebrew Univer-sity of Jerusalem.His other books include Communication Complexity.Tim Roughgarden is an Assistant Professor in the Department of Computer Science at Stanford University.His other books include Selfish Routing and the Price of Anarchy.´Eva Tardos is a Professor in the Department of Computer Science at Cornell University. Her other books include Algorithm Design.Vijay V.Vazirani is a Professor in the College of Computing at the Georgia Institute of Technology.His other books include Approximation Algorithms.Algorithmic Game TheoryEdited byNoam NisanHebrew University of JerusalemTim RoughgardenStanford University´Eva TardosCornell UniversityVijay V.VaziraniGeorgia Institute of Technologycambridge university pressCambridge,New York,Melbourne,Madrid,Cape Town,Singapore,S˜a o Paulo,Delhi Cambridge University Press32Avenue of the Americas,New York,NY10013-2473,USAInformation on this title:/9780521872829C Noam Nisan,Tim Roughgarden,´Eva Tardos,Vijay V.Vazirani2007This publication is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2007Printed in the United States of AmericaA catalog record for this book is available from the British Library.Library of Congress Cataloging in Publication DataAlgorithmic game theory/edited by Noam Nisan...[et al.];forewordby Christos Papadimitriou.p.cm.Includes index.ISBN-13:978-0-521-87282-9(hardback)ISBN-10:0-521-87282-0(hardback)1.Game theory.2.Algorithms.I.Nisan,Noam.II.Title.QA269.A432007519.3–dc222007014231ISBN978-0-521-87282-9hardbackCambridge University Press has no responsibility forthe persistence or accuracy of URLS for external orthird-party Internet Web sites referred to in this publicationand does not guarantee that any content on suchWeb sites is,or will remain,accurate or appropriate.ContentsForeword page xiii Preface xvii Contributors xixI Computing in Games1Basic Solution Concepts and Computational Issues3´Eva Tardos and Vijay V.Vazirani1.1Games,Old and New31.2Games,Strategies,Costs,and Payoffs91.3Basic Solution Concepts101.4Finding Equilibria and Learning in Games161.5Refinement of Nash:Games with Turns and Subgame Perfect Equilibrium181.6Nash Equilibrium without Full Information:Bayesian Games201.7Cooperative Games201.8Markets and Their Algorithmic Issues22Acknowledgments26 Bibliography26 Exercises26 2The Complexity of Finding Nash Equilibria29 Christos H.Papadimitriou2.1Introduction292.2Is the N ash Equilibrium Problem NP-Complete?312.3The Lemke–Howson Algorithm332.4The Class PPAD362.5Succinct Representations of Games392.6The Reduction412.7Correlated Equilibria452.8Concluding Remarks49Acknowledgment50 Bibliography50vvi contents3Equilibrium Computation for Two-Player Games in Strategicand Extensive Form53 Bernhard von Stengel3.1Introduction533.2Bimatrix Games and the Best Response Condition543.3Equilibria via Labeled Polytopes573.4The Lemke–Howson Algorithm613.5Integer Pivoting633.6Degenerate Games653.7Extensive Games and Their Strategic Form663.8Subgame Perfect Equilibria683.9Reduced Strategic Form693.10The Sequence Form703.11Computing Equilibria with the Sequence Form733.12Further Reading753.13Discussion and Open Problems75Bibliography76 Exercises77 4Learning,Regret Minimization,and Equilibria79 Avrim Blum and Yishay Mansour4.1Introduction794.2Model and Preliminaries814.3External Regret Minimization824.4Regret Minimization and Game Theory884.5Generic Reduction from External to Swap Regret924.6The Partial Information Model944.7On Convergence of Regret-Minimizing Strategies to NashEquilibrium in Routing Games964.8Notes99Bibliography99 Exercises101 5Combinatorial Algorithms for Market Equilibria103 Vijay V.Vazirani5.1Introduction1035.2Fisher’s Linear Case and the Eisenberg–Gale Convex Program1055.3Checking If Given Prices Are Equilibrium Prices1085.4Two Crucial Ingredients of the Algorithm1095.5The Primal-Dual Schema in the Enhanced Setting1095.6Tight Sets and the Invariant1115.7Balanced Flows1115.8The Main Algorithm1155.9Finding Tight Sets1175.10Running Time of the Algorithm1185.11The Linear Case of the Arrow–Debreu Model1215.12An Auction-Based Algorithm1225.13Resource Allocation Markets124contents vii 5.14Algorithm for Single-Source Multiple-Sink Markets126 5.15Discussion and Open Problems131 Bibliography132 Exercises133 6Computation of Market Equilibria by Convex Programming135 Bruno Codenotti and Kasturi Varadarajan6.1Introduction1356.2Fisher Model with Homogeneous Consumers1416.3Exchange Economies Satisfying WGS1426.4Specific Utility Functions1486.5Limitations1506.6Models with Production1526.7Bibliographic Notes155 Bibliography156 Exercises158 7Graphical Games159 Michael Kearns7.1Introduction1597.2Preliminaries1617.3Computing Nash Equilibria in Tree Graphical Games1647.4Graphical Games and Correlated Equilibria1697.5Graphical Exchange Economies1767.6Open Problems and Future Research1777.7Bibliographic Notes177 Acknowledgments179 Bibliography179 8Cryptography and Game Theory181 Yevgeniy Dodis and Tal Rabin8.1Cryptographic Notions and Settings1818.2Game Theory Notions and Settings1878.3Contrasting MPC and Games1898.4Cryptographic Influences on Game Theory1918.5Game Theoretic Influences on Cryptography1978.6Conclusions2028.7Notes203 Acknowledgments204 Bibliography204 II Algorithmic Mechanism Design9Introduction to Mechanism Design(for Computer Scientists)209 Noam Nisan9.1Introduction2099.2Social Choice2119.3Mechanisms with Money2169.4Implementation in Dominant Strategies222viii contents9.5Characterizations of Incentive Compatible Mechanisms2259.6Bayesian–Nash Implementation2339.7Further Models2389.8Notes239Acknowledgments240 Bibliography241 10Mechanism Design without Money243 James Schummer and Rakesh V.Vohra10.1Introduction24310.2Single-Peaked Preferences over Policies24410.3House Allocation Problem25310.4Stable Matchings25510.5Future Directions26210.6Notes and References263Bibliography264 Exercises264 11Combinatorial Auctions267 Liad Blumrosen and Noam Nisan11.1Introduction26711.2The Single-Minded Case27011.3Walrasian Equilibrium and the LP Relaxation27511.4Bidding Languages27911.5Iterative Auctions:The Query Model28311.6Communication Complexity28711.7Ascending Auctions28911.8Bibliographic Notes295Acknowledgments296 Bibliography296 Exercises298 12Computationally Efficient Approximation Mechanisms301 Ron Lavi12.1Introduction30112.2Single-Dimensional Domains:Job Scheduling30312.3Multidimensional Domains:Combinatorial Auctions31012.4Impossibilities of Dominant Strategy Implementability31712.5Alternative Solution Concepts32112.6Bibliographic Notes327Bibliography327 Exercises328 13Profit Maximization in Mechanism Design331 Jason D.Hartline and Anna R.Karlin13.1Introduction33113.2Bayesian Optimal Mechanism Design33513.3Prior-Free Approximations to the Optimal Mechanism33913.4Prior-Free Optimal Mechanism Design344contents ix13.5Frugality35013.6Conclusions and Other Research Directions35413.7Notes357Bibliography358 Exercises360 14Distributed Algorithmic Mechanism Design363 Joan Feigenbaum,Michael Schapira,and Scott Shenker14.1Introduction36314.2Two Examples of DAMD36614.3Interdomain Routing37014.4Conclusion and Open Problems37914.5Notes380Acknowledgments381 Bibliography381 Exercises383 15Cost Sharing385 Kamal Jain and Mohammad Mahdian15.1Cooperative Games and Cost Sharing38515.2Core of Cost-Sharing Games38715.3Group-Strategyproof Mechanisms and Cross-MonotonicCost-Sharing Schemes39115.4Cost Sharing via the Primal-Dual Schema39415.5Limitations of Cross-Monotonic Cost-Sharing Schemes40015.6The Shapley Value and the Nash Bargaining Solution40215.7Conclusion40515.8Notes406Acknowledgments408 Bibliography408 Exercises410 16Online Mechanisms411 David C.Parkes16.1Introduction41116.2Dynamic Environments and Online MD41316.3Single-Valued Online Domains41716.4Bayesian Implementation in Online Domains43116.5Conclusions43516.6Notes436Acknowledgments437 Bibliography437 Exercises439 III Quantifying the Inefficiency of Equilibria17Introduction to the Inefficiency of Equilibria443 Tim Roughgarden and´Eva Tardos17.1Introduction443x contents17.2Fundamental Network Examples44617.3Inefficiency of Equilibria as a Design Metric45417.4Notes456Bibliography457 Exercises459 18Routing Games461 Tim Roughgarden18.1Introduction46118.2Models and Examples46218.3Existence,Uniqueness,and Potential Functions46818.4The Price of Anarchy of Selfish Routing47218.5Reducing the Price of Anarchy47818.6Notes480Bibliography483 Exercises484 19Network Formation Games and the Potential Function Method487´Eva Tardos and Tom Wexler19.1Introduction48719.2The Local Connection Game48919.3Potential Games and a Global Connection Game49419.4Facility Location50219.5Notes506Acknowledgments511 Bibliography511 Exercises513 20Selfish Load Balancing517 Berthold V¨o cking20.1Introduction51720.2Pure Equilibria for Identical Machines52220.3Pure Equilibria for Uniformly Related Machines52420.4Mixed Equilibria on Identical Machines52920.5Mixed Equilibria on Uniformly Related Machines53320.6Summary and Discussion53720.7Bibliographic Notes538Bibliography540 Exercises542 21The Price of Anarchy and the Design of Scalable ResourceAllocation Mechanisms543 Ramesh Johari21.1Introduction54321.2The Proportional Allocation Mechanism54421.3A Characterization Theorem55121.4The Vickrey–Clarke–Groves Approach55921.5Chapter Summary and Further Directions564contents xi21.6Notes565Bibliography566 Exercises567IV Additional Topics22Incentives and Pricing in Communications Networks571 Asuman Ozdaglar and R.Srikant22.1Large Networks–Competitive Models57222.2Pricing and Resource Allocation–Game Theoretic Models57822.3Alternative Pricing and Incentive Approaches587Bibliography590 23Incentives in Peer-to-Peer Systems593 Moshe Babaioff,John Chuang,and Michal Feldman23.1Introduction59323.2The p2p File-Sharing Game59423.3Reputation59623.4A Barter-Based System:BitTorrent60023.5Currency60123.6Hidden Actions in p2p Systems60223.7Conclusion60823.8Bibliographic Notes608Bibliography609 Exercises610 24Cascading Behavior in Networks:Algorithmic and Economic Issues613 Jon Kleinberg24.1Introduction61324.2A First Model:Networked Coordination Games61424.3More General Models of Social Contagion61824.4Finding Influential Sets of Nodes62224.5Empirical Studies of Cascades in Online Data62724.6Notes and Further Reading630Bibliography631 Exercises632 25Incentives and Information Security633 Ross Anderson,Tyler Moore,Shishir Nagaraja,and Andy Ozment25.1Introduction63325.2Misaligned Incentives63425.3Informational Asymmetries63625.4The Economics of Censorship Resistance64025.5Complex Networks and Topology64325.6Conclusion64625.7Notes647Bibliography648xii contents26Computational Aspects of Prediction Markets651 David M.Pennock and Rahul Sami26.1Introduction:What Is a Prediction Market?65126.2Background65226.3Combinatorial Prediction Markets65726.4Automated Market Makers66226.5Distributed Computation through Markets66526.6Open Questions67026.7Bibliographic Notes671Acknowledgments672 Bibliography672 Exercises674 27Manipulation-Resistant Reputation Systems677 Eric Friedman,Paul Resnick,and Rahul Sami27.1Introduction:Why Are Reputation Systems Important?67727.2The Effect of Reputations68027.3Whitewashing68227.4Eliciting Effort and Honest Feedback68327.5Reputations Based on Transitive Trust68927.6Conclusion and Extensions69327.7Bibliographic Notes694Bibliography695 Exercises696 28Sponsored Search Auctions699 S´e bastien Lahaie,David M.Pennock,Amin Saberi,and Rakesh V.Vohra28.1Introduction69928.2Existing Models and Mechanisms70128.3A Static Model70228.4Dynamic Aspects70728.5Open Questions71128.6Bibliographic Notes712Bibliography713 Exercises715 29Computational Evolutionary Game Theory717 Siddharth Suri29.1Evolutionary Game Theory71729.2The Computational Complexity of Evolutionarily Stable Strategies72029.3Evolutionary Dynamics Applied to Selfish Routing72329.4Evolutionary Game Theory over Graphs72829.5Future Work73329.6Notes733Acknowledgments734 Bibliography734 Exercises735 Index737ForewordAs the Second World War was coming to its end,John von Neumann,arguably the foremost mathematician of that time,was busy initiating two intellectual currents that would shape the rest of the twentieth century:game theory and algorithms.In1944(16 years after the minmax theorem)he published,with Oscar Morgenstern,his Games and Economic Behavior,thus founding not only game theory but also utility theory and microeconomics.Two years later he wrote his draft report on the EDV AC,inaugurating the era of the digital computer and its software and its algorithms.V on Neumann wrote in1952thefirst paper in which a polynomial algorithm was hailed as a meaningful advance.And,he was the recipient,shortly before his early death four years later,of G¨o del’s letter in which the P vs.NP question wasfirst discussed.Could von Neumann have anticipated that his twin creations would converge half a century later?He was certainly far ahead of his contemporaries in his conception of computation as something dynamic,ubiquitous,and enmeshed in society,almost organic–witness his self-reproducing automata,his fault-tolerant network design,and his prediction that computing technology will advance in lock-step with the economy (for which he had already postulated exponential growth in his1937Vienna Colloquium paper).But I doubt that von Neumann could have dreamed anything close to the Internet, the ubiquitous and quintessentially organic computational artifact that emerged after the end of the Cold War(a war,incidentally,of which von Neumann was an early soldier and possible casualty,and that was,fortunately,fought mostly with game theory and decided by technological superiority–essentially by algorithms–instead of the thermonuclear devices that were von Neumann’s parting gift to humanity).The Internet turned the tables on students of both markets and computation.It transformed,informed,and accelerated markets,while creating new and theretofore unimaginable kinds of markets–in addition to being itself,in important ways,a market. Algorithms became the natural environment and default platform of strategic decision making.On the other hand,the Internet was thefirst computational artifact that was not created by a single entity(engineer,design team,or company),but emerged from the strategic interaction of puter scientists were for thefirst time faced with an object that they had to feel with the same bewildered awe with which economists havexiiixiv forewordalways approached the market.And,quite predictably,they turned to game theory for inspiration–in the words of Scott Shenker,a pioneer of this way of thinking who has contributed to this volume,“the Internet is an equilibrium,we just have to identify the game.”A fascinating fusion of ideas from bothfields–game theory and algorithms–came into being and was used productively in the effort to illuminate the mysteries of the Internet.It has come to be called algorithmic game theory.The chapters of this book,a snapshot of algorithmic game theory at the approximate age of ten written by a galaxy of its leading researchers,succeed brilliantly,I think,in capturing thefield’s excitement,breadth,accomplishment,and promise.Thefirst few chapters recount the ways in which the newfield has come to grips with perhaps the most fundamental cultural incongruity between algorithms and game theory:the latter predicts the agents’equilibrium behavior typically with no regard to the ways in which such a state will be reached–a consideration that would be a computer scientist’s foremost concern.Hence,algorithms for computing equilibria(Nash and correlated equilibria in games,price equilibria for markets)have been one of algorithmic game theory’s earliest research goals.This body of work has become a valuable contribu-tion to the debate in economics about the validity of behavior predictions:Efficient computability has emerged as a very desirable feature of such predictions,while com-putational intractability sheds a shadow of implausibility on a proposed equilibrium putational models that reflect the realities of the market and the Internet better than the von Neumann machine are of course at a premium–there are chapters in this book on learning algorithms as well as on distributed algorithmic mechanism design.The algorithmic nature of mechanism design is even more immediate:This elegant and well-developed subarea of game theory deals with the design of games,with players who have unknown and private utilities,such that at the equilibrium of the designed game the designer’s goals are attained independently of the agents’utilities(auctions are an important example here).This is obviously a computational problem,and in fact some of the classical results in this area had been subtly algorithmic,albeit with little regard to complexity considerations.Explicitly algorithmic work on mechanism design has,in recent years,transformed thefield,especially in the case of auctions and cost sharing(for example,how to recover the cost of an Internet service from customers who value the service by amounts known only to them)and has become the arena of especially intense and productive cross-fertilization between game theory and algorithms;these problems and accomplishments are recounted in the book’s second part.The third part of the book is dedicated to a line of investigation that has come to be called“the price of anarchy.”Selfish rational agents reach an equilibrium.The question arises:exactly how inefficient is this equilibrium in comparison to an idealized situation in which the agents would strive to collaborate selflessly with the common goal of minimizing total cost?The ratio of these quantities(the cost of an equilibrium over the optimum cost)has been estimated successfully in various Internet-related setups,and it is often found that“anarchy”is not nearly as expensive as one might have feared.For example,in one celebrated case related to routing with linear delays and explained in the“routing games”chapter,the overhead of anarchy is at most33%over the optimum solution–in the context of the Internet such a ratio is rather insignificantforeword xv and quickly absorbed by its rapid growth.Viewed in the context of the historical development of research in algorithms,this line of investigation could be called“the third compromise.”The realization that optimization problems are intractable led us to approximation algorithms;the unavailability of information about the future,or the lack of coordination between distributed decision makers,brought us online algorithms;the price of anarchy is the result of one further obstacle:now the distributed decision makers have different objective functions.Incidentally,it is rather surprising that economists had not studied this aspect of strategic behavior before the advent of the Internet.One explanation may be that,for economists,the ideal optimum was never an available option;in contrast,computer scientists are still looking back with nostalgia to the good old days when artifacts and processes could be optimized exactly.Finally,the chapters on“additional topics”that conclude the book(e.g.,on peer-to-peer systems and information markets)amply demonstrate the young area’s impressive breadth, reach,diversity,and scope.Books–a glorious human tradition apparently spared by the advent of the Internet–have a way of marking and focusing afield,of accelerating its development.Seven years after the publication of The Theory of Games,Nash was proving his theorem on the existence of equilibria;only time will tell how this volume will sway the path of algorithmic game theory.Paris,February2007Christos H.PapadimitriouPrefaceThis book covers an area that straddles twofields,algorithms and game theory,and has applications in several others,including networking and artificial intelligence.Its text is pitched at a beginning graduate student in computer science–we hope that this makes the book accessible to readers across a wide range of areas.We started this project with the belief that the time was ripe for a book that clearly develops some of the central ideas and results of algorithmic game theory–a book that can be used as a textbook for the variety of courses that were already being offered at many universities.We felt that the only way to produce a book of such breadth in a reasonable amount of time was to invite many experts from this area to contribute chapters to a comprehensive volume on the topic.This book is partitioned into four parts:thefirst three parts are devoted to core areas, while the fourth covers a range of topics mostly focusing on applications.Chapter1 serves as a preliminary chapter and it introduces basic game-theoretic definitions that are used throughout the book.Thefirst chapters of Parts II and III provide introductions and preliminaries for the respective parts.The other chapters are largely independent of one another.The authors were requested to focus on a few results highlighting the main issues and techniques,rather than provide comprehensive surveys.Most of the chapters conclude with exercises suitable for classroom use and also identify promising directions for further research.We hope these features give the book the feel of a textbook and make it suitable for a wide range of courses.You can view the entire book online at/us/9780521872829username:agt1userpassword:camb2agtMany people’s efforts went into producing this book within a year and a half of itsfirst conception.First and foremost,we thank the authors for their dedi-cation and timeliness in writing their own chapters and for providing importantxviixviii prefacefeedback on preliminary drafts of other chapters.Thanks to Christos Papadimitriou for his inspiring Foreword.We gratefully acknowledge the efforts of outside review-ers:Elliot Anshelevich,Nikhil Devanur,Matthew Jackson,Vahab Mirrokni,Herve Moulin,Neil Olver,Adrian Vetta,and several anonymous referees.Thanks to Cindy Robinson for her invaluable help with correcting the galley proofs.Finally,a big thanks to Lauren Cowles for her stellar advice throughout the production of this volume.Noam NisanTim Roughgarden´Eva TardosVijay V.VaziraniContributorsRoss AndersonComputer Laboratory University of CambridgeMoshe BabaioffSchool of Information University of California,Berkeley Avrim BlumDepartment of Computer Science Carnegie Mellon UniversityLiad BlumrosenMicrosoft ResearchSilicon ValleyJohn ChuangSchool of Information University of California,Berkeley Bruno CodenottiIstituto di Informatica e Telematica,Consiglio Nazionale delle Ricerche Yevgeniy DodisDepartment of Computer Science Courant Institute of Mathematical Sciences,New York UniversityJoan FeigenbaumComputer Science DepartmentYale UniversityMichal FeldmanSchool of Business Administrationand the Center for the Study of Rationality Hebrew University of JerusalemEric FriedmanSchool of Operations Researchand Information EngineeringCornell UniversityJason D.HartlineMicrosoft ResearchSilicon ValleyKamal JainMicrosoft ResearchRedmondRamesh JohariDepartment of Management Scienceand EngineeringStanford UniversityAnna R.KarlinDepartment of Computer Scienceand EngineeringUniversity of Washingtonxixxx contributorsMichael KearnsDepartment of Computerand Information Science University of PennsylvaniaJon KleinbergDepartment of Computer Science Cornell UniversityS´e bastien LahaieSchool of Engineeringand Applied SciencesHarvard UniversityRon LaviFaculty of Industrial Engineering and Management,The Technion Israel Institute of Technology Mohammad MahdianYahoo!ResearchSilicon ValleyYishay MansourSchool of Computer ScienceTel Aviv UniversityTyler MooreComputer Laboratory University of Cambridge Shishir NagarajaComputer Laboratory University of CambridgeNoam NisanSchool of Computer Science and EngineeringHebrew University of Jerusalem Asuman Ozdaglar Department of Electrical Engineering and Computer Science,MITAndy OzmentComputer Laboratory University of Cambridge Christos H.Papadimitriou Computer Science Division University of California,Berkeley David C.ParkesSchool of Engineeringand Applied SciencesHarvard UniversityDavid M.PennockYahoo!ResearchNew YorkTal RabinT.J.Watson Research CenterIBMPaul ResnickSchool of InformationUniversity of MichiganTim RoughgardenDepartment of Computer Science Stanford UniversityAmin SaberiDepartment of Management Science and EngineeringStanford UniversityRahul SamiSchool of InformationUniversity of MichiganMichael SchapiraSchool of Computer Scienceand EngineeringThe Hebrew University of Jerusalem James SchummerM.E.D.S.Kellogg School of Management Northwestern Universitycontributors xxiScott ShenkerEECS DepartmentUniversity of California,BerkeleyR.SrikantDepartment of Electrical and Computer Engineering and Coordinated Science Laboratory,University of Illinois at Urbana-ChampaignSiddharth SuriDepartment of Computer Science Cornell University´Eva TardosDepartment of Computer Science Cornell UniversityKasturi VaradarajanDepartment of Computer Science University of Iowa Vijay V.VaziraniCollege of ComputingGeorgia Institute of TechnologyBerthold V¨o cking Department of Computer Science RWTH Aachen UniversityRakesh V.VohraM.E.D.S.Kellogg School of Management Northwestern University Bernhard von Stengel Department of Mathematics London School of EconomicsTom WexlerDepartment of Computer Science Cornell University。