子非鱼之逻辑“博弈论”
博弈论的意义-Read
博弈论（Game Theory）博弈论简介博弈论(Game Theory)，有时也称为对策论，或者赛局理论，是研究具有斗争或竞争性质现象的理论和方法，它是应用数学的一个分支，既是现代数学的一个新分支，也是运筹学的一个重要学科。
目前在生物学、经济学、国际关系学、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。
主要研究公式化了的激励结构（游戏或者博弈（Game））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，也是运筹学的一个重要学科。
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。
表面上不同的相互作用可能表现出相似的激励结构(incentive structure)，所以他们是同一个游戏的特例。
其中一个有名有趣的应用例子是囚徒困境悖论(Prisoner's dilemma)。
具有竞争或对抗性质的行为成为博弈行为。
在这类行为中，参加斗争或竞争的各方各自具有不同的目标或利益。
为了达到各自的目标和利益，各方必须考虑对手的各种可能的行动方案，并力图选取对自己最为有利或最为合理的方案。
比如日常生活中的下棋，打牌等。
博弈论就是研究博弈行为中斗争各方是否存在着最合理的行为方案，以及如何找到这个合理的行为方案的数学理论和方法。
生物学家使用博弈理论来理解和预测进化论的某些结果。
例如：John Maynard Smith和George R. Price在1973年发表于Nature上的论文中提出的―evolutionarily stable strategy‖的这个概念就是使用了博弈理论。
还可以参见演化博弈理论（evolutionary game theory）和行为生态学（behavioral ecology）。
博弈论也应用于数学的其他分支，如概率、统计和线性规划等。
博弈论的发展博弈论思想古已有之，我国古代的《孙子兵法》就不仅是一部军事著作，而且算是最早的一部博弈论专著。
In the beginning was game semantics
a r X i v :c s /0507045v 3 [c s .L O ] 24 O c t 2008In the beginning was game semanticsGiorgi Japaridze ∗AbstractThis chapter presents an overview of computability logic —the game-semantically constructed logic of interactive computational tasks and resources.There is only one non-overview,technical section in it,devoted to a proof of the soundness of affine logic with respect to the semantics of computability logic.Invited book chapter to appear in Games:Unifying Logic,Language and Philosophy.O.Majer,A.-V.Pietarinen and T.Tulenheimo,eds.,Springer.Contents1Introduction 22Constant games73Games in general,and nothing but games 104Game operations144.1Negation ................................................144.2Choice operations ...........................................154.3Parallel operations ..........................................174.4Reduction ...............................................194.5Blind operations ...........................................224.6Recurrence operations ........................................235Static games 336Winnability 357Validity378Uniform validity 399Logic CL44010Applied systems based on CL 4311Affine logic4712Soundness proof for affine logic4912.1CL4-derived validity lemmas ....................................5012.2Closure lemmas ............................................5112.3More validity lemmas ........................................5412.4Iteration principle for branching recurrence ............................5812.5Finishing the soundness proof for affine logic ...........................6413What could be next?651IntroductionIn the beginning was Semantics,and Semantics was Game Semantics,and Game Semantics was Logic.1 Through it all concepts were conceived;for it all axioms are written,and to it all deductive systems should serve...This is not an evangelical story,but the story and philosophy of computability logic(CL),the recently introduced[12]mini-religion within logic.According to its philosophy,syntax—the study of axiomatizations or any other,deductive or nondeductive string-manipulation systems—exclusively owes its right on existence to semantics,and is thus secondary to it.CL believes that logic is meant to be the most basic,general-purpose formal tool potentially usable by intelligent agents in successfully navigating the real life.And it is semantics that establishes that ultimate real-life meaning of logic.Syntax is important,yet it is so not in its own right but only as much as it serves a meaningful semantics,allowing us to realize the potential of that semantics in some systematic and perhaps convenient or efficient way.Not passing the test for soundness with respect to the underlying semantics would fully disqualify any syntax,no matter how otherwise appealing it is.Note—disqualify the syntax and not the semantics.Why this is so hardly requires any explanation: relying on an unsound syntax might result in wrong beliefs,misdiagnosed patients or crashed spaceships.Unlike soundness,completeness is a desirable but not necessary condition.Sometimes—as,say,in the case of pure second-order logic,orfirst-order applied number theory with+and×—completeness is impossible to achieve in principle.In such cases we may still benefit from continuing working with various reasonably strong syntactic constructions.A good example of such a“reasonable”yet incomplete syntax is Peano arithmetic.Another example,as we are going to see later,is affine logic,which turns out to be sound but incomplete with respect to the semantics of CL.And even when complete axiomatizations are known,it is not fully unusual for them to be sometimes artificially downsized and made incomplete for efficiency,simplicity,convenience or even esthetic considerations.Ample examples of this can be found in applied computer science.But again,while there might be meaningful trade-offs between(the degrees of)completeness,efficiency and other desirable-but-not-necessary properties of a syntax,the underlying semantics remains untouchable,and the condition of soundness unnegotiable.It is that very untouchable core that should be the point of departure for logic as a fundamental science.A separate question,of course,is what counts as a semantics.The model example of a semantics with a capital‘S’is that of classical logic.But in the logical literature this term often has a more generous meaning than what CL is ready to settle for.As pointed out,CL views logic as a universal-utility tool.So,a capital-‘S’-semantics should be non-specific enough,and applicable to the world in general rather than some very special and artificially selected(worse yet,artificially created)fragment of it.Often what is called a semantics is just a special-purpose apparatus designed to help analyze a given syntactic construction rather than understand and navigate the outside world.The usage of Kripke models as a derivability test for intuitionistic formulas,or as a validity criterion in various systems of modal logic is an example.An attempt to see more than a technical,syntax-serving instrument(which,as such,may be indeed very important and useful)in this type of lowercase‘s’semantics might create a vicious circle:a deductive system L under question is“right”because it derives exactly the formulas that are valid in a such and such Kripke semantics; and then it turns out that the reason why we are considering the such and such Kripke semantics is that... it validates exactly what L derives.This was about why in the beginning was Semantics.Now a few words about why Semantics was Game Semantics.For CL,game is not just a game.It is a foundational mathematical concept on which a powerful enough logic(=semantics)should be based.This is so because,as noted,CL sees logic as a“real-life navigational tool”,and it is games that appear to offer the most comprehensive,coherent,natural,adequate and convenient mathematical models for the very essence of all“navigational”activities of agents:their interactions with the surrounding world.An agent and its environment translate into game-theoretic terms as two players;their actions as moves;situations arising in the course of interaction as positions;and success or failure as wins or losses.It is natural to require that the interaction strategies of the party that we have referred to as an“agent”be limited to algorithmic ones,allowing us to henceforth call that player a machine.This is a minimum conditionthat any non-esoteric game semantics would have to satisfy.On the other hand,no restrictions can or should be imposed on the environment,who represents‘the blind forces of nature,or the devil himself’([12]). Algorithmic activities being synonymous to computations,games thus represent computational problems—interactive tasks performed by computing agents,with computability meaning winnability,i.e.existence of a machine that wins the game against any possible(behavior of the)environment.In the1930s mankind came up with what has been perceived as an ultimate mathematical definition of the precise meaning of algorithmic solvability.Curiously or not,such a definition was set forth and embraced before really having attempted to answer the seemingly more basic question about what com-putational problems are—the very entities that may or may not have algorithmic solutions in thefirst place.The tradition established since then in theoretical computer science by computability simply means Turing computability of functions,as the task performed by every Turing machine is nothing but receiving an input x and generating the output f(x)for some function f.Turing[34]himself,however,was more cautious about making overly broad philosophical conclusions,acknowledging that not everything one would potentially call a computational problem might necessarily be a function,or reducible to such.Most tasks that computers and computer networks perform are interactive.And nowadays more and more voices are being heard[8,15,30,36]pointing out that true interaction might be going beyond what functions and hence ordinary Turing machines are meant to capture.Two main concepts on which the semantics of CL is based are those of static games and their winnability (defined later in Sections5and6).Correspondingly,the philosophy of CL relies on two beliefs that,together, present what can be considered an interactive version of the Church-Turing thesis:Belief1.The concept of static games is an adequate formal counterpart of our intuition of(“pure”, speed-independent)interactive computational problems.Belief2.The concept of winnability is an adequate formal counterpart of our intuition of algorithmic solvability of such problems.As will be seen later,one of the main features distinguishing the CL games from more traditional game models is the absence of procedural rules([2])—rules strictly regulating which player is to move in any given situation.Here,in a general case,either player is free to move.It is exactly this feature that makes players’strategies no longer definable as functions(functions from positions to moves).And it is this highly relaxed nature that makes the CL games apparently mostflexible and general of all two-player,two-outcome games.Trying to understand strategies as functions would not be a good idea even if the type of games we consider naturally allowed us to do so.Because,when it comes to long or infinite games,functional strategies would be disastrously inefficient,making it hardly possible to develop any reasonable complexity theory for interactive computation(the next important frontier for CL or theoretical computer science in general).To understand this,it would be sufficient to just reflect on the behavior of one’s personal computer.The job of your computer is to play one long—potentially infinite—game against you.Now,have you noticed your faithful servant getting slower every time you use it?Probably not.That is because the computer is smart enough to follow a non-functional strategy in this game.If its strategy was a function from positions (interaction histories)to moves,the response time would inevitably keep worsening due to the need to read the entire—continuously lengthening and,in fact,practically infinite—interaction history every time before responding.Defining strategies as functions of only the latest moves(rather than entire interaction histories)in Abramsky and Jagadeesan’s[1]tradition is also not a way out,as typically more than just the last move matters.Back to your personal computer,its actions certainly depend on more than your last keystroke.Computability in the traditional Church-Turing sense is a special case of winnability—winnability restricted to two-step(input/output,question/answer)interactive problems.So is the classical concept of truth,which is nothing but winnability restricted to propositions,viewed by CL as zero-step problems,i.e. games with no moves that are automatically won or lost depending on whether they are true or false.This way,the semantics of CL is a generalization,refinement and conservative extension of that of classical logic.Thinking of a human user in the role of the environment,computational problems are synonymous to computational tasks—tasks performed by a machine for the user/environment.What is a task for a machine is then a resource for the environment,and vice versa.So the CL games,at the same time, formalize our intuition of computational resources.Logical operators are understood as operations on suchtasks/resources/games,atoms as variables ranging over tasks/resources/games,and validity of a logical formula as being“always winnable”,i.e.as existence—under every particular interpretation of atoms—of a machine that successfully accomplishes/provides/wins the corresponding task/resource/game no matter how the environment behaves.With this semantics,‘computability logic is a formal theory of computability in the same sense as classical logic is a formal theory of truth’([13]).Furthermore,as mentioned,the classical concept of truth is a special case of winnability,which eventually translates into classical logic’s being nothing but a special fragment of computability logic.CL is a semantically constructed logic and,at this young age,its syntax is only just starting to develop, with open problems and unverified conjecture prevailing over answered questions.In a sense,this situation is opposite to the case with some other non-classical traditions such as intuitionistic or linear logics where, as most logicians would probably agree,“in the beginning was Syntax”,and really good formal semantics convincingly justifying the proposed syntactic constructions are still being looked for.In fact,the semantics of CL can be seen to be providing such a justification,although,at least for linear logic,this is only in a limited sense explained below.The set of valid formulas in a certain fragment of the otherwise more expressive language of CL forms a logic that is similar to but by no means the same as linear logic.The two logics typically agree on short and simple formulas(perhaps with the exception for those involving exponentials,where disagreements may start already on some rather short formulas).For instance,both logics reject P→P∧P and accept P→P⊓P, with classical-shape propositional connectives here and later understood as the corresponding multiplicative operators of linear logic,and square-shape operators as additives(⊓=“with”,⊔=“plus”).Similarly,both logics reject P⊔¬P and accept P∨¬P.On the other hand,CL disagrees with linear logic on many more evolved formulas.E.g.,CL validates the following two principles rejected even by affine logic AL—linear logic with the weakening rule:((P∧Q)∨(R∧S))→((P∨R)∧(Q∨S));(P∧(R⊓S))⊓(Q∧(R⊓S))⊓((P⊓Q)∧R)⊓((P⊓Q)∧S)→(P⊓Q)∧(R⊓S).Neither the similarities nor the discrepancies are a surprise.The philosophies of CL and linear logic overlap in their striving to develop a logic of resources.But the ways this philosophy is materialized are rather different.CL starts with a mathematically strict and intuitively convincing semantics,and only after that,as a natural second step,asks what the corresponding logic and its possible axiomatizations are.On the other hand,it would be accurate to say that linear logic started directly from the second step.Even though certain companion semantics were provided for it from the very beginning,those are not quite what we earlier agreed to call capital-‘S’.As a formal theory of resources(rather than that of phases or coherent spaces),linear logic has been motivated and introduced syntactically rather than semantically,essentially by taking classical sequent calculus and deleting the rules that seemed unacceptable from a certain intuitive, naive resource point of view.Hence,in the absence of a clear formal concept of resource-semantical truth or validity,the question about whether the resulting system was complete could not even be meaningfully asked.In this process of syntactically rewriting classical logic some innocent,deeply hidden principles could have easily gotten victimized.CL believes that this is exactly what happened,with the above-displayed formulas separating it from linear logic—and more such formulas to be seen later—viewed as babies thrown out with the bath water.Of course,many retroactive attempts have been made tofind semantical (often game-semantical)justifications for linear logic.Technically it is always possible to come up with some sort of a formal semantics that matches a given target syntactic construction,but the whole question is how natural and meaningful such a semantics is in its own rights,and how adequately it corresponds to the logic’s underlying philosophy and ambitions.‘Unless,by good luck,the target system really is“the right logic”,the chances of a decisive success when following the odd scheme from syntax to semantics could be rather slim’([12]).The natural scheme is from semantics to syntax.It matches the way classical logic evolved and climaxed in G¨o del’s completeness theorem.And,as we now know,this is exactly the scheme that computability logic,too,follows.Intuitionistic logic is another example of a syntactically conceived logic.Despite decades of efforts,no fully convincing semantics has been found for it.Lorenzen’s game semantics[5,29],which has a concept of validity without having a concept of truth,has been perceived as a technical supplement to the existing syntax ratherthan as having independent importance.Some other semantics,such as Kleene’s realizability[25]or G¨o del’s Dialectica interpretation[7],are closer to what we might qualify as capital-‘S’.But,unfortunately,they validate certain principles unnegotiably rejected by intuitionistic logic.From our perspective,the situation here is much better than with linear logic though.In[19],Heyting’sfirst-order intuitionistic calculus has been shown to be sound with respect to the CL semantics.And the propositional fragment of Heyting’s calculus has also been shown to be complete([20,21,22,35]).This signifies success—at least at the propositional level—in semantically justifying intuitionistic logic,and a materialization of Kolmogorov’s [26]well known yet so far rather abstract thesis according to which intuitionistic logic is a logic of problems. Just as the resource philosophy of CL overlaps with that of linear logic,so does its algorithmic philosophy with the constructivistic philosophy of intuitionism.The difference,again,is in the ways this philosophy is materialized.Intuitionistic logic has come up with a“constructive syntax”without having an adequate underlying formal semantics,such as a clear concept of truth in some constructive sense.This sort of a syntax was essentially obtained from the classical one by banning the offending law of the excluded middle. But,as in the case of linear logic,the critical question immediately springs out:where is a guarantee that together with excluded middle some innocent principles would not be expelled as well?The constructivistic claims of CL,on the other hand,are based on the fact that it defines truth as algorithmic solvability.The philosophy of CL does notfind the term constructive syntax meaningful unless it is understood as soundness with respect to some constructive semantics,for only a semantics may or may not be constructive in a reasonable sense.The reason for the failure of P⊔¬P in CL is not that this principle...is not included in its axioms.Rather,the failure of this principle is exactly the reason why this principle,or anything else entailing it,would not be among the axioms of a sound system for CL.Here“failure”has a precise semantical meaning.It is non-validity,i.e.existence of a problem A for which A⊔¬A is not algorithmically solvable.It is also worth noting that,while intuitionistic logic irreconcilably defies classical logic,computability logic comes up with a peaceful solution acceptable for everyone.The key is the expressiveness of its lan-guage,that has(at least)two versions for each traditionally controversial logical operator,and particularly the two versions∨and⊔of disjunction.As will be seen later,the semantical meaning of∨conservatively extends—from moveless games to all games—its classical meaning,and the principle P∨¬P survives as it represents an always-algorithmically-solvable combination of problems,even if solvable in a sense that some constructivistically-minded might fail—or pretend to fail—to understand.And the semantics of ⊔,on the other hand,formalizes and conservatively extends a different,stronger meaning which apparently every constructivist associates with disjunction.As expected,then P⊔¬P turns out to be semantically invalid.CL’s proposal for settlement between classical and constructivistic logics then reads:‘If you are open(=classically)minded,take advantage of the full expressive power of CL;and if you are constructivis-tically minded,just identify a collection of the operators whose meanings seem constructive enough to you, mechanically disregard everything containing the other operators,and put an end to those fruitlessfights about what deductive methods or principles should be considered right and what should be deemed wrong’([12]).Back to linear—more precisely,affine—logic.As mentioned,AL is sound with respect to the CL semantics,a proof of which is the main new technical contribution of the present paper.This is definitely good news from the“better something than nothing”standpoint.AL is simple and,even though incomplete, still reasonably strong.What is worth noting is that our soundness proof for AL,just as all other soundness proofs known so far in CL,including that for the intuitionistic fragment([19]),or the CL4fragment([18]) that will be discussed in Section9,is constructive.This is in the sense that,whenever a formula F is provable in a given deductive system,an algorithmic solution for the problem(s)represented by F can be automatically extracted from the proof of F.The persistence of this phenomenon for various fragments of CL carries another piece of good news:CL provides a systematic answer not only to the theoretical question “what can be computed?”but,as it happens,also to the more terrestrial question“how can be computed?”.The main practical import of the constructive soundness result for AL(just as for any other sublogic of CL)is related to the potential of basing applied theories or knowledge base systems on that logic,the latter being a reasonable,computationally meaningful alternative to classical logic.The non-logical axioms—or knowledge base—of an AL-based applied system/theory would be any collection of(formulas representing) problems whose algorithmic solutions are known.Then our soundness result for AL guarantees that every theorem T of the theory also has an algorithmic solution and that,furthermore,such a solution,itself,canbe effectively constructed from a proof of T.This makes AL a systematic problem-solving tool:findinga solution for a given problem reduces tofinding a proof of that problem in an AL-based theory.The incompleteness of AL only means that,in its language,this logic is not as perfect/strong as as a formaltool could possibly be,and that,depending on needs,it makes sense to continue looking for further soundextensions(up to a complete one)of it.As pointed out earlier,when it comes to applications,unlike soundness,completeness is a desirable but not necessary condition.With the two logics in a sense competing for the same market,the main—or perhaps only—advantage of linear logic over CL is its having a nice and simple syntax.In fact,linear logic is(rather than has)abeautiful syntax;and computability logic is(rather than has)a meaningful semantics.At this point it isnot clear what a CL-semantically complete extension of AL would look like syntactically.As a matter of fact,while the set of valid formulas of the exponential-free fragment of the language of linear logic has beenshown to be decidable([18]),so far it is not even known whether that set in the full language is recursivelyenumerable.If it is,finding a complete axiomatization for it would most likely require a substantially new syntactic approach,going far beyond the traditional sequent-calculus framework within which linear logic isconstructed(a possible candidate here is cirquent calculus,briefly discussed at the end of this section).And, in any case,such an axiomatization would hardly be as simple as that of AL,so the syntactic simplicityadvantage of linear logic will always remain.Well,CL has one thing to say:simplicity is good,yet,if it ismost important,then nothing can ever beat...the empty logic.The rest of this paper is organized as follows.Sections2-8provide a detailed introduction to the basicsemantical concepts of computability logic:games and operations on them,two equivalent models of inter-active computation(algorithmic strategies),and validity.The coverage of most of these concepts is more detailed here than in the earlier survey-style papers[12,15]on CL,and is supported with ample examplesand illustrations.Section9provides an overview,without a proof,of the strongest technical result obtained so far in computability logic,specifically,the soundness and completeness of system CL4,whose logical vo-cabulary contains negation¬,parallel(“multiplicative”)connectives∧,∨,→,choice(“additive”)connectives⊓,⊔with their quantifier counterparts⊓,⊔,and blind(“classical”)quantifiers∀,∃.Section10outlines potential applications of computability logic in knowledge base systems,systems for planning and action,and constructive applied theories.There the survey part of the paper ends,and the following two sectionsare devoted to a formulation(Section11)and proof(Section12)of the new result—the soundness of affine logic with respect to the semantics of CL.Thefinal Section13outlines some possible future developments in the area.This paper gives an overview of most results known in computability logic as of the end of2005,by thetime when the main body of the text was written.The present paragraph is a last-minute addition made atthe beginning of2008.Below is a list of the most important developments that,due to being very recent, have received no coverage in this chapter:•As already mentioned,the earlier conjecture about the completeness of Heyting’s propositional intu-itionistic calculus with respect to the semantics of CL has been resolved positively.A completenessproof for the implicative fragment of intuitionistic logic was given in[20],and that proof was later extended to the full propositional intuitionistic calculus in[21].With a couple of months’delay, Vereshchagin[35]came up with an alternative proof of the same result.•In[22],the implicative fragment of affine logic has been proven to be complete with respect to thesemantics of computability logic.The former is nothing but implicative intuitionistic logic without the rule of contraction.Thus,both the implication of intuitionistic logic and the implication of affine logic have adequate interpretations in CL—specifically,as the operations◦–and→,respectively.Intuitively,as will be shown later in Section4,these are two natural versions of the operation ofreduction,with the difference between A◦–B and A→B being that in the former A can be“reused”while in the latter it cannot.[22]also introduced a series of intermediate-strength natural versions of reduction operations.•Section4.6briefly mentions sequential operations.The recent paper[24]has provided an elaborationof this new group of operations(formal definitions,associated computational intuitions,motivations,etc.),making them full-fledged citizens of computability logic.It has also constructed a sound andcomplete axiomatization of the fragment of computability logic whose logical vocabulary,together with negation,contains three—parallel,choice and sequential—sorts of conjunction and disjunction.•Probably the most significant of the relevant recent developments is the invention of cirquent calculus in[16,23].Roughly,this is a deductive approach based on circuits instead of formulas or sequents.It can be seen as a refinement of Gentzen’s methodology,and correspondingly the methodology of linear logic based on the latter,achieved by allowing shared resources between different parts of sequents and proof trees.Thanks to the sharing mechanism,cirquent calculus,being more general andflexible than sequent calculus,appears to be the only reasonable proof-theoretic approach capable of syntactically taming the otherwise wild computability logic.Sharing also makes it possible to achieve exponential-magnitude compressions of formulas and proofs,whether it be in computability logic or the kind old classical logic.2Constant gamesThe symbolic names used in CL for the two players machine and environment are⊤and⊥,respectively.℘is always a variable ranging over{⊤,⊥},with¬℘meaning℘’s adversary,i.e.the player which is not℘.Even though it is often a human user who acts in the role of⊥,our sympathies are with⊤rather than⊥,and by just saying“won”or“lost”without specifying a player we always mean won or lost by⊤.The reason why I should be a fan of the machine even—in fact especially—when it is playing against me is that the machine is a tool,and what makes it valuable as such is exactly its winning the game,i.e. its not being malfunctional(it is precisely losing by a machine the game that it was supposed to win what in everyday language is called malfunctioning).Let me imagine myself using a computer for computing the “28%of x”function in the process of preparing my federal tax return.This is a game where thefirst move is mine,consisting in inputting a number m and meaning asking⊤the question“what is28%of m?”.The machine wins iffit answers by the move/output n such that n=0.28m.Of course,I do not want the machine to tell me that27,000is28%of100,000.In other words,I do not want to win against the machine.For then I could lose the more important game against Uncle Sam.Before getting to a formal definition of games,let us agree without loss of generality that a move is always a string over the standard keyboard alphabet.One of the non-numeric and non-punctuation symbols of this alphabet,denoted♠,is designated as a special-status move,intuitively meaning a move that is always illegal to make.A labeled move(labmove)is a move prefixed with⊤or⊥,with its prefix(label)indicating which player has made the move.A run is a(finite or infinite)sequence of labmoves,and a position is a finite run.We will be exclusively using the lettersΓ,∆,Θ,Φ,Ψ,Υ,Λ,Σ,Πfor runs,α,β,γ,δfor moves,andλfor labmoves.Runs will be often delimited by“ ”and“ ”,with thus denoting the empty run.The meaning of an expression such as Φ,℘α,Γ must be clear:this is the result of appending to positionΦthe labmove ℘αand then the runΓ.We write¬Γfor the result of simultaneously replacing every label℘in every labmove ofΓby¬℘.Our ultimate definition of games will be given later in terms of the simpler and more basic class of games called constant.The following is a formal definition of constant games combined with some less formal conventions regarding the usage of certain terminology.Definition2.1A constant game is a pair A=(Lr A,Wn A),where:1.Lr A,called the structure of A,is a set of runs not containing(whatever-labeled)move♠,satisfying the condition that afinite or infinite run is in Lr A iffall of its nonemptyfinite—not necessarily proper—initial segments are in Lr A(notice that this implies ∈Lr A).The elements of Lr A are said to be legal runs of A,and all other runs are said to be illegal runs of A.We say thatαis a legal move for℘in。
子非鱼之逻辑“博弈论”
子非鱼之逻辑“博弈论”《庄子·秋水》中有一段庄子与惠施的有趣的对话。
其文如下：庄子与惠子游于濠梁之上。
庄子日：“倏鱼出游从容，是鱼之乐也。
”惠子日：“子非鱼，安知鱼之乐乎?”庄子日：“子非吾，安知吾不知鱼之乐乎?”惠子曰：“吾非子，固不知子矣；子固非鱼也，子之不知鱼之乐，全矣!”庄子日：“请循其本。
子日‘汝安知鱼乐’云者，既已知吾知之而问吾。
吾知之濠上也。
”这段对话令人感兴趣的地方在于，庄子究竟采用了什么方法，使“鱼乐”这个本来缺乏证据的观点变成了证据确凿得令惠子无法辩解的观点?要分析这个问题，我们须从简化问题入手，正如笛卡儿所说：要理解一个问题，必须从一切多余的因素中把它提取出来，让它尽可能简单。
在这段对话里，一共有三处立论的地方。
庄子确立了两个论点，一个是：“倏鱼出游从容，是鱼之乐也”，另一个是：“子非吾，安知吾求知鱼之乐”；惠子只有一个论点：“子非鱼，安知鱼之乐”。
庄子“鱼乐”的观点一开始就被提出，然而他并没有提出有力的证据来证明鱼是决乐的，这就是伊姆雷·拉卡拉斯所说的：立论忘掉了证明。
庄子的第二个论点，即“子非吾，安知吾不知鱼之乐乎?”这个论点本来是合乎逻辑的，就是说，你不是我，你就无从知道我的心里了解了些什么。
事实确实是这样，一个人肯定无法完全地了解其他人真实的内心世界。
问题在于，庄子提出这论点的目的，是要为他的第一个论点提供支持，所以，这个论点的意义就被歪曲了。
庄子是这样做的：他巧妙地把“鱼乐”这个本来缺乏证据的不合理的论点放在了第二个论点之中，把“你不是我，你根本就不可能知道我的心里想什么”这个论点变成了：“你不是我，你根本就不可能知道我心里是知晓鱼的快乐的”。
惠子的论点本来十分合乎客观现实的逻辑：子非鱼，安知鱼之乐。
不错，人与鱼不是同类，无法进行交流，特别是情感方面，更没有沟通的可能，庄子根本就不可能知道鱼的快乐与否。
值得注意的是，惠子的立论有一个隐含的逻辑前提：先不管鱼是不是快乐，只要庄子不是与它同类，就无从得知鱼的快乐与否。
子非鱼安知鱼之乐的辩论方法
子非鱼安知鱼之乐的辩论方法子非鱼的这句话“安知鱼之乐”，传达的意思是“如何知道鱼的快乐体验”，这句话包含了一个非常深刻的哲学思想。
以下是我对于这个哲学思想的个人理解和辩论。
一、从语言角度解读从语言角度上来看，“安知鱼之乐”可以理解为“如何知道鱼的快乐体验”，但是在这个语句中，“知道”这个动词背后带有一个前提条件：人类能够理解鱼的感觉和体验，这样才能知道鱼是否快乐。
但是，人类和鱼的感官、生理结构存在差异，我们不可能真正理解鱼的感受。
因此，从语言角度来看，“安知鱼之乐”这句话是一种反问句，意思是我们不可能真正理解鱼的快乐体验。
二、从哲学角度解读从哲学角度上来看，“安知鱼之乐”这句话存在多重解读。
其中，一个主要的哲学思想是“他者性”（alterity），即人类与其他物种、事物之间的差异和不同。
人类和鱼作为两种不同的生物，其思想、感觉和行为都存在不同，人类无法直接理解鱼的感觉。
因此，我们无法直接知道鱼是否快乐，也无法直接知道鱼享受什么样的快乐。
另外，从“动物权利”（Animal Rights）的角度来看，“安知鱼之乐”这句话也具有一定的哲学含义。
人类应该尊重动物的权利，不应该将它们视为我们的物品或是工具。
我们无法理解鱼的感受，但是我们应该关注和尊重它们的生命和权利。
三、辩论方法通过对“安知鱼之乐”的解读，我们不难发现，这句话背后隐含着深层次的哲学思想。
因此，在辩论时，我们可以从以下几个角度出发：1. 对于语言层面的解读，我们可以讨论“知识”的内涵及其限制，引出“知识论”的相关理论。
2. 对于哲学角度的解读，我们可以讨论人类与动物之间的差异和关系，引出“他者性”、“动物权利”等相关哲学理论。
3. 在辩论中，我们可以运用反问句的修辞手法，将话题推向更深刻的层次。
4. 此外，我们可以展开关于“幸福”的哲学思考，探讨不同动物的幸福是如何定义和体验的。
通过以上的角度解读和辩论方法的运用，我们可以深入探讨“安知鱼之乐”这个句子背后的哲学思想，并且更好地理解动物权利和人与自然的关系。
一分钟漫画哲学 逻辑学 博弈论
一分钟漫画哲学逻辑学博弈论
在一分钟内，我们将探讨漫画哲学、逻辑学和博弈论的交汇之处。
漫画哲学强调生活中的喜剧和悲剧，逻辑学则关注思维的合理性，而博弈论则涉及人们在决策中的策略和影响。
在漫画中，我们经常看到角色在面对冲突和选择时展现出逻辑思维和博弈策略。
比如，柯南·道尔在解决案件时运用逻辑推理，而蜘蛛侠则在与反派对决时采取博弈策略。
这些角色的行为反映了哲学思考，逻辑推理和博弈策略的交织。
因此，漫画不仅是娱乐，也是对生活和人性的思考。
在漫画中，我们可以发现哲学、逻辑和博弈的精彩交融，这让我们不禁思考，在现实生活中，我们是否也能像漫画中的角色一样，运用逻辑和策略来面对挑战和决策呢？或许，我们可以从漫画中汲取一些智慧，来更好地理解和应对现实生活中的各种情境。
《博弈心理学》读书笔记
《博弈心理学》读书笔记我们每天生活与工作，不可避免地需要和形形色色的人打交道。
但是所谓“子非鱼焉知鱼之乐知鱼之乐焉知鱼之痛”，同样，不同的人之间存在意识形态的差异性，我们如何体察别人的想法，如何让别人接受自己的意见，这都是我们可以在心理学中窥探到的宝藏。
“人的所有行动都是有理由的，基于这些理由来进行预测，这就是心理学。
只要弄清楚这些理由，你应该采取什么行动，自然也会明晰起来。
”这是《博弈心理学》的作者在结语中说的一句话。
在《博弈心理学》这本书中，我意识到心理博弈里最强劲的敌人是“自己的心”，因为我会为了迁就自己而编造谎言、我的记忆不靠谱、我会过度自信、我的欲望会扭曲判断……所以我们要突破瓶颈，首先要做的就是认识自己，认识自己的是否有一件看似微不足道的事深深影响了自己的心理状况，认识自己的性格给自己带来的是坦途还是翻滚着巨浪的危险海洋，认识自己的情绪对日常生活或者决策会产生怎样的影响，认识自己的思维是否给自己的脚步拷上了枷锁。
而世事的博弈，我们除了要知己，还要知彼。
书中写了很多判断心中所想的方法，例如：“表示“YES”的动作：1、手处于放松状态，并没有紧握。
2、手打开放在桌面上。
3、收拾书桌上多余碍事的物品。
4、用手蹭着下巴。
”“人在心情愉悦时才会敞开心扉聊一些私人话题”等等。
每一次与人接触，我们都不能忽视心理对事情的结果可能产生的影响，很多时候我们其实可以考虑换一种方式去说服他人，核心就是找到对方能接受的心理状态，找到对方心中坚持意见的症结在哪里。
他最关注的核心是哪里，从核心入手，尝试去引导他人的思维。
这些都是心理学为我们提供的强大而切实的法宝。
阅读完这本书，我发现心理学并不是晦涩难懂的代名词，它其实是为我们每个人量身定制的导师。
人类面前的路却永远是未知，我们的心灵也不时的萦绕在无知、无奈与恐惧之中，了解一些心理学知识，它能引导我们更了解自己的行为，让我们有能力改变自己的思想、决策方法甚至让我们在社会中以更有效的博弈方法来改变命运。
《博弈论》学习体会
生活中的博弈论-—学习《博弈论》课程的收获高中毕业那个暑假我阅读了一本书名为《非是非非：世界经典趣味悖论》的书，里面主要讲述了一些有趣的悖论和哲学以及逻辑的思维，同时该书也讲述了经典的囚徒困境的博弈，但是书是以趣味故事来讲述了,我也不知道那就是囚徒困境。
本科上了微观经济学课程才恍然大悟，我原来早就对这个经典博弈有了一定程度上的感性认识。
电影《美丽心灵》也让我认识到了纳什传奇的一生。
本科的学习中一直对博弈论很好奇，但是也没有自己去学习。
在学习之前我认为是一门有很强的实用性而且很有趣的课程，但是实际上的博弈论与我的想象又说区别,数学知识的要求较高,而且理解起来有一定的困难。
但是博弈论课程让我全面系统的了解了博弈论这么学科外,也让我认识到博弈论的深奥,需要自己去领悟。
通过不断努力学习和思考，博弈论也给我带来了许多的收获和启发。
一、了解了四种博弈论和四种均衡博弈论本是数学的分支，博弈论也是一种分析问题的工具，它不仅仅运用于经济学也运用于军事、政治学、生物学等多个学科。
中国古代的许多经典故事也是博弈论的模型，例如田忌赛马.当博弈论被引入经济学研究，为经济学研究方法开启了一扇崭新的大门,改变了经济学的轮廓。
在分析经济学的各种问题时都有所运用,特别是产业组织理论方面，在研究市场结构时学者运用了各种博弈模型分析市场参与者的行为和策略.根据博弈的行动顺序和信息的完全与否将博弈分为四种：完全信息静态博弈；完全信息动态博弈；不完全信息静态博弈；不完全信息动态博弈。
对应的四种均衡：纳什均衡;子博弈精炼纳什均衡；贝叶斯均衡;精炼贝叶斯均衡。
在课程的学习中我们了解了多种经典的博弈模型:智猪博弈，市场进入阻挠，蜈蚣博弈等等。
不同的博弈模型都用其简单的表述，描述了复杂的经济学问题,从中要仔细体会才能得到答案。
在本科的学校中囚徒困境是无法解决的，但是在现在看来，如果将囚徒困境放到无限次的重复博弈中，结果就会不一样了，即如果囚徒困境不再是完全信息静态博弈,而变成完全信息动态博弈均衡结果会不一样，囚徒困境得以解决。
博弈论(第七、八章)
第一节静态贝叶斯博弈和贝叶斯纳什均衡
海萨尼（Harsanyi）转换
高成本(p) B 不进入 进入 不进入 N 低成本(1-p) B 进入 A 默许 斗争
(0,200)
默许
A (0, 200) (0,2 斗争
(30,40)
(-10,0)
(20,70)
(-10,80)
1994年度经济学诺奖得主---海萨尼
1 max[(vi − bi ) P{bi > b j } + (vi − bi ) P{bi = b j }] bi 2
第二节 典型不完全信息静态博弈
暗标拍卖
假设博弈方的策略是风险中性，标价符合线性函数： bi=civi，其中, ci≥0。有：
1 max[(v1 − b1 ) P{b1 > c2 v2 } + (v1 − b1 ) P{b1 = b2 }] b1 2 = max[(v1 − b1 ) P{b1 > c2 v2 }]
a − q1* − cH q (c H ) = 2
* 2
a − q1* − cL q (c L ) = 2
* 2 * * θ [ a − q2 (cH ) − c1 ] + (1 − θ )[a − q2 (cL ) − c1 ] q = 2 * 1
第二节 典型不完全信息静态博弈
不完全信息的古诺模型
b1
= max[(v1 − b1 ) P{v2 <
b1
b1 }] c2
= max[(v1 − b1 )
b1
b1 ] c2
式中P｛bi＝bj｝＝0，上式等于：
第二节 典型不完全信息静态博弈
暗标拍卖
第三节不完全信息与混合策略均衡
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子非鱼之逻辑“博弈论”
《庄子·秋水》中有一段庄子与惠施的有趣的对话。
其文如下：庄子与惠子游于濠梁之上。
庄子日：“倏鱼出游从容，是鱼之乐也。
”惠子日：“子非鱼，安知鱼之乐乎?”庄子日：“子非吾，安知吾不知鱼之乐乎?”惠子曰：“吾非子，固不知子矣；子固非鱼也，子之不知鱼之乐，全矣!”庄子日：“请循其本。
子日‘汝安知鱼乐’云者，既已知吾知之而问吾。
吾知之濠上也。
”这段对话令人感兴趣的地方在于，庄子究竟采用了什么方法，使“鱼乐”这个本来缺乏证据的观点变成了证据确凿得令惠子无法辩解的观点?要分析这个问题，我们须从简化问题入手，正如笛卡儿所说：要理解一个问题，必须从一切多余的因素中把它提取出来，让它尽可能简单。
在这段对话里，一共有三处立论的地方。
庄子确立了两个论点，一个是：“倏鱼出游从容，是鱼之乐也”，另一个是：“子非吾，安知吾求知鱼之乐”；惠子只有一个论点：“子非鱼，安知鱼之乐”。
庄子“鱼乐”的观点一开始就被提出，然而他并没有提出有力的证据来证明鱼是决乐的，这就是伊姆雷·拉卡拉斯所说的：立论忘掉了证明。
庄子的第二个论点，即“子非吾，安知吾不知鱼之乐乎?”这个论点本来是合乎逻辑的，就是说，你不是我，你就无从知道我的心里了解了些什么。
事实确实是这样，一个人肯定无法完全地了解其他人真实的内心世界。
问题在于，庄子提出这论点的目的，是要为他的第一个论点提供支持，所以，这个论点的意义就被歪曲了。
庄子是这样做的：他巧妙地把“鱼乐”这个本来缺乏证据的不合理的论点放
在了第二个论点之中，把“你不是我，你根本就不可能知道我的心里想什么”这个论点变成了：“你不是我，你根本就不可能知道我心里是知晓鱼的快乐的”。
惠子的论点本来十分合乎客观现实的逻辑：子非鱼，安知鱼之乐。
不错，人与鱼不是同类，无法进行交流，特别是情感方面，更没有沟通的可能，庄子根本就不可能知道鱼的快乐与否。
值得注意的是，惠子的立论有一个隐含的逻辑前提：先不管鱼是不是快乐，只要庄子不是与它同类，就无从得知鱼的快乐与否。
从全部对话来看，似乎庄子最后赢得了辩论，他得出的结论是：“子日：‘汝安知鱼之乐’云者，既已知吾知之而问吾，吾知之濠上也。
”就是说，不管惠子你如何进行辩解，你旱就知道我是知晓鱼之乐的，你从一开始就支持了我庄子的观点：鱼乐。
庄子与惠施是终生的好友，两人思维方式完全不同，但却棋逢对手。
惠施死后，庄子哀叹“我再也找不到对话者了”。
这段公案十分著名，由于是庄子或其弟子记录的，表面上看似乎庄子在辩论中胜利了，事实上庄子完全是强词夺理，他的逻辑破绽百出。
庄子曾批评中国有史以来最伟大的逻辑学家公孙龙的坚白同异论、白马非马论“能胜人之口，不能服人之心”，至少在这场影响深远的辩论中，他对公孙龙的批评完全可以用在他自己身上。
而且，“胜人之口”的根本原因也不是他把惠施驳得体无完肤、哑口无言，而是因为这段故事是由他或他的弟子记述的，他们运用“话语权”让庄子说了最后一句，至于惠施接下来还说了什么，后人就不得而知了。
这就如同街头相骂者
一样，“讼此而不决，以后息者为胜”（韩非），谁说最后一句话，谁就算胜者，所以街头相骂者往往已经辞穷，但还是翻来覆去地重复滥调，没人肯率先闭嘴。
我相信如果这一事件由惠施或其弟子来写，情形就会完全不同。
但是即便从庄子自我偏袒的记述来看，庄子也根本没有胜利。
分析一下这两人从立论到辩论的整个过程，我们实在不难看出，庄子的所谓结论只不过是建立在诡辩之上的。
这种诡辩方法正是庄子所认同的，他以为有力的辩论，当然，就是我们所说的诡辩，是可以掩盖本来错误的命题和观点的，《庄子·盗跖》篇中有一句话对此作了很好的注解：“辩足以饰非”。
我们却不上庄子的当，我们有足够的证据证明庄子的这种诡辩，实际上可以归结到以形式主义为防护堤的逻辑实证主义中去。
形式主义大师卡尔纳普认为：哲学应当被科学的逻辑所取代，而科学的逻辑不是别的，只是科学语言的逻辑语法。
根据这种理论，只要用他们这个学派所认可的逻辑语法组织起所谓的科学语言，就能建立科学的逻辑，也就足以建立起哲学体系；并且，一个直觉推论的有效性还要依赖于怎么把这个推论从普通语言翻译成逻辑语言，也就是依赖于采取什么样的翻译。
庄子在《秋水》篇这一段对话中采用的逻辑思维方式，就是逻辑实证主义的最好写照。
上文说过，庄子抓住了惠子立论时不明确的前提（这种不明确，我们甚至可以视为忘记的前提），反驳道：“子非吾，安知吾不知鱼之乐乎”，这时的庄子，已经悄然地把惠子忘掉的隐含前提，即“我们
根本无法知道鱼乐与不乐”彻底抹去了，变成了：鱼是决乐的，只不过我知道了，而你不是我，所以你不知道我是知晓鱼的快乐的。
惠子果然上了当，顺着庄子设下的逻辑陷阱跳了下去，就是：我不是你，本来就不知道你（的心理活动），你也不是鱼，你不知道鱼的快乐的原因，就在这里了。
这时的惠子已经在话语中帮助了庄子，庄子乘机抓住了惠子话中对自己的无意识的支持：“请循其本，子既已知吾知之而问吾”，庄子把惠子辩论中的逻辑漏洞加在了自己的立论之上，双管齐下，使得惠子无从再进行辩解。
庄子由是轻松得出结论：鱼就是快乐的。
正如卡尔纳普所希望的那样，庄子选择了极好的语言，并在组织语言逻辑的过程中确定了他心目中的科学逻辑，使我们惊诧不已地发现：给特定对象作的一切证明，可以立即转用于新的对象，一切都不用改，连用词也不用改，因为它们的名称已经变成一样的了，也就是说，经过庄子的语言逻辑组织之后，惠子话语中本来隐含着的逻辑前提“我们根本无法知道鱼乐与不乐”被彻底抹杀，惠子的话语因此不再指它原先想指的东西了，而是被庄子翻译成了：你已经承认鱼是快乐的，只不过你不是我，你就不知道我内心的活动（即知晓鱼的快乐）。
我们之所以说庄子的这种以形式主义为基础的逻辑实证主义的辩论是一种诡辩，也是因为它完全违背了马克思主义哲学原理，是彻头彻尾的主观唯心主义的表现。
马克思主义认为，逻辑思维是人在认识过程中借助了概念、判断、推理，反映现实的思维方式，以抽象性为特征，撇开具体形象，揭示事物的本质属性，也叫抽象思维。
（《现
代汉语词典》修订本，商务印书馆，1998年。
）而“观念的东西不外是移入人的头脑并在人的头脑中改造过的物质的东西而已。
”（《马克思恩格斯选集》第二卷，第217页。
）列宁说：“物质是标志客观实在的哲学范畴，这种客观实在是人通过感觉感知的，它不依赖于我们的感觉而存在，为我们的感觉所复写、摄影、反映。
”物质无非是各种实物的总和，概念就是从这一总和中抽象出来的；意识反映客观存在，这是整个唯物主义的一般原理。
在科学的逻辑思维过程中，人们所借助的概念必须是从客观实在的物质总和中抽象出来的，逻辑思维虽然抽象，但其合理的生存基础是以客观存在为根本的概念、判断和推理。
象我们前面分析过的，鱼快乐与否，庄子肯定不知道，因为，他的意识反映不了这种客观事实，感觉是人类对于客观世界的主观映象，而在人力所能及的范围内，“鱼是否快乐”这个客观世界自身的映象并没有明确地体现出来。
庄子给“鱼乐”下的定义是亚里士多德式的：定义是一个有关本质的不可证明的陈述。
（《后分析篇》）鱼是否快乐，人类根本无法去加以证明，它们在水里游来游去并不表明它们的快乐，因为，这只不过是它们无法更改的生存方式，由此可以看出，庄子下的定义就成为一个无法证明的定义。
恩格斯说过：“一切观念都来自经验，都是现实的反映一一正确的或者歪曲的反映。
”（《马克思恩格斯选集》第二十卷，第661页。
）庄子的这种观念甚至都谈不上是对现实的歪曲反映，而是纯粹从主观的角度出发，按照自己所认为的世界的面目去描绘世界，这种反映自然也就失去了它的客观存在的基础，根本就是错误的。
可惜的是，惠子的论点，即：“子非鱼，安知鱼之乐”缺少一个合理有力的逻辑前提的提出作为证明的支撑，如果惠子改变一下提问的方式，加上必要的逻辑前提，变成这样：鱼乐与不乐，其自在也，子非鱼，安知鱼乐与不乐乎?那么，庄子的诡辩就无从下手。
当然，如果《庄子·秋水》果真这样写，庄子的论点也就站不住，写庄子这部书的人绝不至于愚蠢如是，所以，我们今天不得不花费笔墨和时间来揭穿庄子的诡辩。
当然，关于庄子是否诡辩，不同的人也有其他的不同看法，但至少以今天的逻辑学特别是西方逻辑学来看，的确有诡辩的性质。