(Mechanical) Reasoning on Infinite Extensive Games

(Mec hanical) Reasoning on Infinite E xtensiv e Games LIP rep ort RR20 08-16 Pierre L escanne Univ ersit ´ e de Lyon, ENS de Ly on, CNRS (LIP), 46 all´ ee d’Italie, 69364 Lyon, F rance No ve mber 21, 2018 Abstract In order to better understand reasoning in volved in analyzing infinite games in exten siv e form, we p erformed the exp eriments in proof assistan t Coq that are reported h ere. 1 In tro duction One of the main aims of game theory is to unders tand how agents reason. Psychologists [14, 4, 17] would say agents a re h uman and tries to ans wer the question o f how hu ma n agent s reason. In this pap er, we take a radically differ e n t v iew; for us, ag en ts are ideal abstract en tities with unlimited forma l deductio n pow e r and we attempt to answer the question of what a full reaso ning can be . F or that we decided to analyze the pro cess in full detail on a mechanical device, namely a pro of as sistant run on a computer which p erfor ms at the extreme level of detail all the steps of reasoning. This wa y , we hope to be able to highlight conce pts and deductio ns that are neces sary and that a human would do more or less. Among the p ossible concepts inv olved in a reasoning , we claim that some are forgotten whereas irrelev ant others are considered. In the first class is tempo ral reasoning (“always”, “even tually” ), in the second is the use of “excluded middle” or “double nega tion”. Our exp eriments hav e shown that far fro m easy deductions are used 1 , in par ticular with infinite games. This co mplexit y may expla in why human reasoning depar ts fr o m wha t is exp ected. Among the pa rts of game theor y that have b een ov er loo k ed, is this of infinite games, on whic h fo r mal reasoning is rather subtle. In this pap er we r epor t resear c h ab out the concepts underlying infinite ga mes and experiments on the pr oo f ass istan t Coq [1] to ma k e formal reasoning effectiv e. In this paper , we are pres en ting mechanical re a soning that is deductions that, unlike human ones, are based on the fundamental principles of logic. This means that ev er y piece of re a soning has to be justified by a rule of logic and therefore the whole pro cess can be chec ked by a machine, i.e., a computer, the Coq softw are in our case. The do main of exp erience is this of games in extensive form ([10] Chap. 5), that a re sequential games w er e each situation are owned b y a play er, whose turn co mes one after the others. More sp ecifically , w e fo cus on infinite suc h games, where there ma y be infinitely many s ituations. 2 COQ and the Constructiv e L ogic Around 1 980 a new concept called Curry-H owar d c orr esp ondenc e emerged. It relies on t yp e theory and lambda-calculus [8] and says basically that pr o ofs ar e pr o gr ams . In this theory , all ob jects ha ve a t yp e, that 1 The reader is invite d to ha ve a lo ok at the scripts. is a n annotation that limits its use. F or instance, an ob ject f of t y p e A → B , written f : A → B represents a function and can only b e applied to an ob ject of type A to pro duce an ob ject of type B . In our developmen t we will write, for instance, a : Ag ent to say that a is an A gent and s : I nf S trateg y to say that s is an infinite strategy . In w ha t follows, a no de (written iNo de ) in a infinite stra tegy is so mething that takes an ag en t, a choice, a finite stra tegy and an infinite str ategy a nd pro duces an infinite stra teg y has a t y pe, in mor e prec ise words it has the t yp e, iNo de : Ag ent → C hoice → F inS trateg y → I nf S trat eg y → I nf S tra teg y . The Curry- Ho ward corresp ondence sa ys a lso that typ es ar e pr op ositio n s and insists essentially on the co m- putational con tent of pro ofs and establishes the bases of the so-calle d co nstructive logic. Indeed since a pro of works as a co mputation, an ob ject can be taken into consider a tion only if it can b e constructed and an existential pro of is acc e pted if it allows constructing the ob ject it cla ims the exis tence of. F or instance, in an infinite extens ive g ame we assert that there exists a utility for a n a gen t a tha t ca n b e asso ciated with an infinite strategy , but this w orks if w e prov e it exists, i.e., pr o v ide a w ay to construct this utility , sinc e assuming the existence without the co nstruction is not enough. The following formal development in Coq has been built in this framework and c heck ed on a computer and is a ttac hed to pro of scr ipts av ailable on the web site of the a uthor 2 . In this article we try to des cribe and comment the conten t of the scripts without entering in their full technicalit y , but the reader is a n yway strongly enco uraged to have a lo ok a t the scripts to convince himself o f the materiality of the pro ofs and moreov e r , if he has a ccess to a Coq implemen tatio n, he should try to run them on a co mputer. 2.1 Natural deduction The pro of systems w e describ e is based on natural deduction, which is o ne of the main system to formalize logic. Natura l deduction has b een formaliz ed by Gerhard Gentzen [6] in 1 935 (see a lso [11, 1 5]). Its adjective “natural” comes for m the fact that its creator s considered that it is just the natural wa y to formalize logic . It is based on the fact that to conduct a pro of, one works under h yp otheses and one tries to draw a conclusion. One considers that a theo rem has b een prov ed when all the hypothese s hav e been discharged. The bas ic concept of natural deduction is this of se quent . A sequen t is a pair o f a cont e x t Γ and a propo sition ϕ , whic h is written Γ ⊢ ϕ a nd which mea ns tha t ϕ is a lo gical c o nsequence o f the set of hypotheses Γ. A con text is a set of prop ositions and when we write Γ , ϕ w e mean that the sequent Γ is enriched by the propo sition ϕ , in other terms this is a notation fo r the union of s ets of prop ositions. In this calculus there is one ax iom a nd several rules. The axiom is Γ ⊢ ϕ if ϕ ∈ Γ There are tw o kinds of r ules for each connector, namely intr o duction ru le and elimination rule . F or ins ta nce, for the implication → , the introduction rule is Γ , ϕ ⊢ ψ Γ ⊢ ϕ → ψ whereas the elimination rule is Γ ⊢ ϕ → ψ Γ ⊢ ϕ Γ ⊢ ψ also known as the mo dus p onens . W e are now giving only the rule for the universal quantifier, the introduction rule Γ , x : A ⊢ P ( x ) Γ ⊢ ∀ x : A, P ( x ) 2 http://p erso.ens- lyon.fr/pierre.lescanne/COQ/INFGAMES/ and the elimination rule: Γ ⊢ a : A Γ ⊢ ∀ x : A, P ( x ) Γ ⊢ P ( a ) Through the Curry-Howard cor resp ondence, natural deduction is strong ly connected to type theory a nd computation. 2.2 COQ In this paper we are not descr ibing all the details of the pro of a ssistant Coq a nd the b o ok [3] is ex cellen t source of information. Basically in Coq , pro ofs are ma thematical ob jects in natural de ductio n and ar e considered as first clas s citizen that can be printed, be exc hange d as ob jects betw e e n p eople, and ov er all be c hecked b y a pro of chec ker (a sp ecific softw a re) that examines it in full detail. As building proo fs is a tedious pro cess, the softw are Coq o ffer s to ols to build them. Basically a Coq development is made of several not fully separ ated phases : the user defines data structures (in o ur case: games a nd strateg ies), then he defines predicates, relations and functions on those structures (in our case: conv er s ion, utilities and equilibria) and finally he prov es theor ems ab out those structures a nd predicates (in our ca se: a theorem that says that subgame per fect equilibria are Nash eq uilibria). T ypically a script presen ts a set of sections in which a sub dev elopment is pre s en ted, this section may inv oke other sections ; it is a sequence of declaration of v ariables, axioms, or hypothesis, follow ed b y definitions and theorems with their pro ofs. T o c heck that a script is correct, it is highly reco mmended to run it on a Coq implementation. 3 Induction, Coinduction, and Fixed p oin t Induction is a to ol to reaso n o ver infinite sets of ob jects, provided those ob jects are finitely based. On the opp osite, c oinduction [2] has b een designed to rea s on on infinite ob jects, lik e games with infinite paths. How ever infinite ob jects ma y have parts that ar e finite lik e finite pa ths in infinite games and induction can also be used on infinite ob jects, sp ecifically on their finite subparts. In particular, if the paths are finite, one can compute the utility . Moreover in infinite games we can define finite relations, called c onvertibility . Therefore finite concepts are interleav ed with infinite ones. Beside games, we can define other finite or infinite ob jects, further we co nsider the following inductive (finite) ob jects: 1) finite g ames, 2) finite strategies (o r strategy pro files), 3) the predicate eventu al ly right that says when applied to a path of a g ame that the path in q uestion go es even tually to the right, 4) the t wo conv ertibility cong ruences among strategies, namely among finite str ategies, but also among infinitely strategy (w e indeed claim this latter congruence can be describ ed finitely provided we r estrict it to appropriate strategies). 5) Nash equilibrium on finite g ames, 6) backw a rd induction predicate and 7 ) Nash equilibrium on infinite games a re also finite inductive concepts. In this pap er the archet yp e of a c oinductive ob ject is an infinite g ame. Obviously the asso ciated concept of infinite stra teg y is coinductive. The function i2u that asso ciates a utility with a str ategy is a lso coinductive since the infinite strategies are. The predica te SGPE which tells whether a n infinite strategy is a sub game p erfe ct e quilibrium is a coinductive ob ject s inc e the str ategy is. In what follows we will de fine predica tes that say that a prop erty is “always” satisfied along a pa th, those proper ties on infinite ob jects are coinductively defined. Roughly spe a king inductive definitions a re like equations and co rresp ond to define the concept as le ast fixe d p oints and the pro perties of this concept are der iv ed of this minimality . On another ha nd, coinductiv e definitions cor respo nd to gr e atest fixe d p oints a nd the prop erties of this concept ar e derived fro m the max- imality . Handling infinite ob jects (a c tua l infinit y vs p otential infinity) is tr ic ky a nd the Coq user do es not escap e this rule. Notice that Coq offers to ols to verify along a pro of that the one one builds will b e ac c e pted by the chec ker. 4 Informal presen tation of infinite games In this pap er, we analyze games in ext ensive form . Informally suc h ga mes are presented as trees. Each no de of the tree is a situation in the ga me where a play er ha s to take a decision. F or reason of simplicit y , following V ester gaard [16], w e studied binary games where the play ers hav e only tw o c hoices. This seems a reasona ble design decision, since we can reduce a choice among n to a binary choice betw ee n the n th and the set { 1 , ..., n − 1 } followed by the binary c ho ic e betw een the ( n − 1) th and the set { 1 , ..., n − 2 } etc... W e do not loo se an y gene r alit y in term of mo deling, and in term of abstraction this is the same as n choices at each no de. A reader interested by an implemen ta tion based on mor e than tw o choices, i.e., on p olyadic trees, is advised to consult St ´ ephane Le Roux P hD [12]. By consider ing binary games, w e hope to be more didactic. First w e will consider finite binary games . Those games a re so mew ha t co nnec ted with games with finite horizon , but in addition to offer choices to a finite depth, they are also finitely br anch ing. Such a game is either a leaf, that is a ga me which ends and attributes the utilities, or a node wher e an agent has to take a decisio n. This lea ds typically to an inductive description. Like a natural is either 0 or the successo r of a natura l, or a bina ry tree is either a leaf with a conten t or a no de made of t wo binary subtrees, a finite game is either a leaf with a function that a s so c iates a utility to each ag en t or a no de with an agent and t wo subgames, namely a left subgame and a right subgame. With a finite game, we asso ciate strateg ies. A strategy has the same str ucture a s the ga me it is ass ocia ted with, except that to each (internal) node w e give a direction lef t or right , w hich cor resp onds to the c ho ic e made by the a gent in this pa r ticular situation. In other words, a strategy is either a leaf asso ciated with a utility , exactly like for games, o r a node with an agent, a choice (left or right), and tw o substrategies. The main goal of the study prese n ted in this pape r is infinite ga mes, more sp ecifically w e describ e deductions and fo rmal r easoning o n those ga mes . Again for reason of s implicit y , w e study infinite g ames that lo o ks like “cent ip edes”, in other w ords binary games with a unique infinite path. The basic concept of those games is a no de made of three co mponents: an a g en t, a left subg ames whic h is itself an infinite subgame, and a righ t subga me whic h is a finite s ubgame. This mea ns that we “recycle” the for mal developmen t made for finite subgames for the right branch. Since the g ame is infinite its formalization relies no more on induction, but on coinduction. Like for finite binary games , the attached concept of strategy relies on a no de with four comp o nen ts: an ag en t, a choice (left or r igh t), an infinite left substrateg y , and a finite right strategy . 5 Equilibria on finite Games As an in tro duction to the Coq development, let us study equilibria on finite games . 5.1 Agen t and Utilit y First we set t wo basic concepts na mely A gent a nd Utility . In Coq , this is done b y the following declaration: V ariable A gent : Set . V ariable Utility : Set . In additio n we set a pr efer enc e as a binar y relation on Utility and we ma ke this pr efer enc e a preorder , which we write = < , Utility fun is the type of the utility fu n ctions , in o ther words the type o f the ob jects belo nging to A gent → Utility . 5.2 Finite Games A finite binary game, whic h w e call a FinGa m e , in Coq , is built b y induction, this means that this is either a le af or this is a game with t wo subgames that ar e themselv es binary games, we encapsulate these three items under the la bel gNo de . In Coq , suc h a data str ucture will b e written: Inductive FinGame : Set := — gL e af : Utility fun → FinGame — gNo de : A gent → FinGame → FinGame → FinGame . where Inductive is the key w or d to intro duce any inductive definition. FinGame is a data structure, but elsewhere the same Inductive keyw o rd will b e used to define a pre dicate. An inductiv e definition crea tes a deduction r ule that a llows us to reaso n by induction, namely in the case of FinGame , the term FinGame ind is created b y Coq : ∀ P : FinGame → Pr op , ( ∀ u : Utility fun , P ( gL e af u )) → ( ∀ ( a : A gent ) ( f0 : Fi n Game ), P f0 → ∀ f1 : FinGame , P f1 → P ( gNo de a f0 f1 ) ) → ∀ f1 : FinGame , P f1 It can be written as the rule ∀ u : Utility fun , P ( gL e af u ) [ ∀ f0 : FinGame , P f0 ∧ ∀ f1 : FinGame , P f1 ] → ∀ a : Age nt , P ( gNo de a f0 f1 ) ∀ f : FinGame , P f in other w or ds, to prov e that some prop erties holds for all the finite binary games, one has to prove it to hold for leaves and to pr o ve tha t if it ho lds for t wo games then it holds for the g ame obtained by pa iring those t wo games under the “control” of an agent. 5.3 Finite Strategies As we said, a finite s tr ategy 3 has a structure which is very similar to a finite game, the only difference is that w e add a “choice” at each no de. Inductive FinStr ate gy : Set := — sL e af : Utility fun → FinStr ate gy — sNo de : A gent → Choic e → FinStr ate gy → FinS t r ate gy → FinStr ate gy . With a stra tegy we provide a function that asso ciates with every strategy a utility function. The Coq keyw ord to define such a function is Fixpoi nt . A syntactic construction called match allows using a pa ttern matching mechanism. Fixpo int f2 u ( s : FinStr ate gy ) : Utility fun := match s with — ( sL e af uf ) ⇒ uf — ( sNo de a left sl sr ) ⇒ ( f2u sl ) — ( sNo de a right sl sr ) ⇒ ( f2 u sr ) end . This reads as  if the strategy s is a leaf, i.e., sL e af uf (where uf is a utilit y function), then one returns uf ,  otherwise o ne retur ns the utilit y function ass ocia ted with the left substrategy , if the choice is left , or the right substrategy if the c hoice is right . Now we define o n finit e s tr ategies a relatio n, which we call a - c onvertibility , and which we wr ite s << − a − >> s’ . Lab eled with a , it is as soc ia ted with the agent a and says that  Two leav es asso ciated with the same utility function ar e a -conv er tible,  Two sNo de s, ( sNo de a c s1 s2 ) and ( sNo de a c’ s1’ s2’ ), a sso ciated with the same age nt a a r e a - conv ertible if s1 < < − a − >> s1’ and s2 < < − a − >> s2’ . No tice that c and c’ do no t have to b e the same, 3 Actually we should probably say a str ate gy pr ofile , but, for conv enience and conciseness, we call this data structure just a str ate gy .  Two sNo de s, ( sNo de a’ c s1 s2 ) a nd ( sNo de a’ c s1’ s2’ ), asso ciated with another agent a’ are a -con vertible if s1 << − a − > > s1’ and s2 << − a − > > s2’ . Notice that in this case c has to be the same. W e prove in Coq (a first interesting exercise) that the a -con vertibility is an eq uiv a lence relation i.e., it is reflexive, symmetric and transitive. W e are no w e quipped to define the predicate N ash e quilibrium on finite str ate gy (in Coq the predicate FinNashEq on FinS tr ate gy ): Inductive FinNashEq : FinStr ate gy → Pr op := — NE : ∀ ( s : FinStr ate gy ), ( ∀ ( a : Ag en t ) ( s’ : FinStr ate gy ), s < < − a − >> s’ → ( f2u s’ a = < f2u s a )) → FinNashEq s . It sa ys that s is a Nash equilibr ium if for all strategy s’ that is a -conv e rtible to s , f2u s’ a = < f2u s a , in other words, the utilit y for a computed for s’ is less tha n the utilit y for a computed for s . This is no thing more than the traditional definition of Nash equilibrium for extensive game, written in the for malism of Coq . Beside the predica te Nash e qu ilibri u m , we define the pr edicate BI whic h says whether the strategy s can be o btained by the so-called b ackwar d induction . In Coq , it is written: Inductive BI : FinStr ate gy → Pr op := — BIL e af : ∀ uf : Utility fun , BI ( sL e af u f ) — BINo de left : ∀ ( a : A gent ) ( sl sr : FinStr ate gy ), BI sl → BI sr → ( f2u sr a = < f2u s l a ) → BI ( sNo de a left sl sr ) — BINo de right : ∀ ( a : A gent ) ( sl sr : FinStr ate gy ), BI sl → BI sr → ( f2u sl a = < f2u sr a ) → BI ( sNo de a right sl sr ). In other words,  BI holds for leav es,  if BI holds for sl and sr and f2u sl a = < f2 u sr a then BI holds for sNo de a left sl sr ( BINo de left principle),  the BINo de right principle is sy mmetric, it s a y s, that if BI holds for sl and for sr and f2u sr a = < f2u sl a then is BI ho lds for sNo de a right sl sr . This is the formaliza tion of backward induction for finite horizon games as describ ed in textbo oks [9, 10, 7]. Then w e are able to pro ve in Coq the theorem: Theorem BI is FinNashEq : ∀ s , BI s → FinNashEq s . The theorem re lie s on the following fact: the inductiv e definition of BI is a so mewhat equational definition that says that BI is the leas t fixe d p oint of that equatio n. There fo re if we can prov e that if FinNashEq is another fixed p oint , this other fixed p oin t is implied b y BI and w e a re set. This can b e done b y three lemmas. • FinNashEq satisfies the statemen t given by BIL e af : Lemma FinNashEq L e af : ∀ ( uf : U tility fun ), FinNashEq ( sL e af uf ). • FinNashEq satisfies the statemen t given by BINo de left : Lemma FinNashEq fixpt left : ∀ ( a : A gent ) ( s1 s2 : FinStr ate gy ), FinNashEq s1 → FinNashEq s2 → ( f2u s2 a = < f2u s1 a ) → FinNashEq ( sNo de a left s1 s2 ). • FinNashEq satisfies the statemen t given by BINo de right : Lemma FinNashEq fixpt right : ∀ ( a : A gent ) ( s1 s2 : FinStr ate gy ), FinNashEq s1 → FinNashEq s2 → ( f2u s1 a = < f2u s2 a ) → FinNashEq ( sNo de a right s1 s2 ). Then we co nc lude than FinNashEq satisfies the same equation a s BI . Since BI is the least fixed p oint, BI s implies FinNashEq s . 6 Coinduction Coinduction is the pa rtner of induction, but whereas the induction defines ob jects or predicates as the least fixed point of some equations and therefore spec ifie s finite o b jects and presents them fr om basic ob jects, coinduction defines infinite ob jects or infinite and finite ob jects as g reatest fixed point of some equation. If one has in mind to define infinite ob jects only , there is no need to specify basic ob jects (i.e., ob jects meant to be the basis on whic h to define finite ob ject). The typical infinite o b ject is CoInductive InfGame : S et := — igNo de : A gent → InfGame → FinGame → InfGa me . In other words, an infinite binary game is made o f a n agen t, an infinite binary subgame, and a finite binary subg ame. On the same mo del as for finite binary g a mes, w e define infinite strategies the s ame wa y , i.e., as a c o inductiv e: CoInductive InfStr ate gy : Set := — iNo de : A gent → Choic e → InfStr ate gy → FinStr ate gy → InfStr ate gy . 7 Equilibria on Infinite Games 7.1 Decomp osing Infinite Games Since the strategies are infinite we can define total functions iA gent (which giv e s the agent a t the r oo t of the game), SubGameL eft (the infinite game on the left) and SubGameRight (the finite ga me on the right) with the lemma: Lemma InfGame De c omp osition : ∀ ( g : InfGame ), igNo de ( iA gent g ) ( SubGameL eft g ) ( Su bGameR ight g ) = g . that says that a game can b e uniquely decomp osed in to an agent, a left subgame and a rig h t subgame. 7.2 Generalizing the not ion of horizon In classical extensive game theory , the concept of backw ard induction relies on this o f fin ite horizon , our exp erience in mec haniz ing the proo f has s ho wn us that finite horizon is not exac tly the right notion, we pre fer the notion of limi t e d horizon , which means that the ho rizon o f the agents is not b ounded b y a num b er, but that how e ver it canno t go to infinit y . In the fr ame of binar y games, this means that we forbid pa ths that go alwa ys to the left, we w ant to consider the paths that even tua lly go to the right so that one can compute the utilit y of each agent. Since the g a mes are now infinite, a total function that ass ocia tes, b y computation, a utilit y to a str ategy can no more b e defined, one ca n only define a r elation b etw een a strategy and a utility . If the path of the choice go es a lw ays to the left (on the backbo ne) the utility ca nnot b e defined, but if the pa th go e s “eventual ly to the righ t ” , the utilit y of a strategy makes s e nse. W e ther efore define a predicate EvtR ight which says whether the path o f the given str a tegy go es even tually to the r igh t. The basic case for this predicate is that it holds for the game iNo de a right sl sr . The induction case says that if EvtRigh t sl then EvtRight ( iNo de a left s l sr ). The predicate EvtRi ght is typical of an inductive predicate. W e g e t tw o lemmas whic h show the existence a nd the uniqueness of a utilit y for a strategy that go es ev entually to the righ t, therefor e in that case, the asso ciatio n of the utilit y to the strategy is functional. Since most of the time we w ant this functional asso ciation, w e require the predicate “ even tually right” to hold. Lemma Existenc e i2u : ∀ ( a : A gent ) ( s : InfStr ate gy ), EvtRig ht s → ∃ u : Ut ility , i2u a u s . Lemma Uniqueness i2u : ∀ ( a : A gent ) ( u v : Utility ) ( s : InfSt r ate gy ), EvtRig ht s → i2 u a u s → i2u a v s → u = v . Going event ua lly to the rig h t is no t enough. In some cas e s, when dea ling with subgames, more sp ecifically with subgame per fect equilibria , o ne wants to b e able to compute utilities in subga mes . Therefore o ne wan ts to be sure that even further in subgames one will g o e ventually to the rig h t. Hence w e define another predicate that ensures that we will a lw ays go ev entually to the right. This pre dic a te Alw EvtRight reminds us the same kind of predicate defined in the fra me of temp oral lo gic (see [3] chap.13 and [5]). Since A lwEvtRight has to tra verse the whole infinite g ame, it has to be c o inductiv e. It says that AlwEvtR ight ( iNo de a c sl sr ) holds, if AlwEvtRig ht sl and EvtR ight sr hold, in other words, a strateg y g oes a lways even tually to the r igh t if its left s ubs tr ategy go es alwa ys even tually to the r igh t a nd if its right s ubstrategy (whic h is finite) go es even tually to the r igh t. 7.3 Con ve rtibilit y of infinite strategies The definition of the con vertibility s < − − a −− > s’ of t wo stra tegies s a nd s’ is inductive. Its base c ase is the reflexivity of the con vertibilit y . In o ther w ords, s < −− a −− > s’ if:  s = s’ , o r  if s1 < −− a −− > s1’ and s2 << − a − >> s2’ and s is iNo de a c s1 s2 and s’ is iNo de a c’ s1’ s2’ ( c may be different, but a ha s to b e the same), or  if s1 < −− a −− > s1’ and s2 << − a − > > s2’ and s is iNo de a’ c s1 s2 and s’ is iNo de a ’ c s1’ s2’ (different ag e n ts, but s ame c ). 7.4 Equilibria T o define Nash Equilibria on infinite g ames, one m ust b e able to compare utilities. F or that, one must b e able to compute thos e utilities. Hence one restricts to s tr ategies tha t go eventually to the right. Inductive InfNashEq : InfStr ate gy → Pr op := — INE : ∀ ( s : InfStr ate gy ), EvtRig ht s → ( ∀ ( a : Ag en t ) ( s’ : InfStr ate gy ) ( u u’ : Utility ), EvtRig ht s’ → s’ < −− a −− > s → ( i2u a u’ s’ ) → ( i2u a u s ) → ( u’ = < u )) → InfNashEq s . On infinite strategie s one defines a predicate Sub game Perfe ct Equilibrium . Because one has to be able to compute utilities on s ubgames, the Subgame P er fect Equilibrium predica te, written SGPE , is defined on strategies whic h go alw ays ev entually to the right. Notice tha t Subgame Perfect E quilibria are defined on a smaller class of strategies than Nash Equilibria, but also that the definition of SGPE is co inductiv e, since its definition requires to traverse the whole infinite subgame. CoInductive SGPE : InfStr ate gy → Pr op := — SGPEno de left : ∀ ( a : A gent )( u : Utility ) ( sl : InfStr ate gy ) ( sr : FinStr ate gy ), AlwE vt R ight sl → SGPE sl → BI sr → i2u a u sl → ( f2u sr a = < u ) → SGPE ( iNo de a left sl sr ) — SGPEno de right : ∀ ( a : A gent ) ( u : Utility ) ( sl : In fStr ate gy ) ( sr : FinStr ate gy ), AlwE vt R ight sl → SGPE sl → BI sr → i2u a u sl → ( u = < f2u sr a ) → SGPE ( iNo de a right sl sr ). One can sho w that InfNashEq is a fixed p oin t of this definition. T o prove the lemma SGPE s → InfNashEq s , one needs to p erform a pr oo f by induction, this cannot b e an induction on the definition SGPE which is c o inductiv e. T o enable a pro of b y induction, we imp ose an additional requirement, namely that somewhere in the game a maxima l utility is reached for all agen ts. W e g iv e a statemen t of this requir emen t through the follo w ing predicate EvtMaxU : Inductive EvtMaxU : InfSt r ate gy → Pr op := — EUM b asis : ∀ ( s : InfStr ate gy ), ( ∀ ( a : Ag en t )( u u’ : Utility )( s’ : InfStra t e gy ), s < −− a −− > s’ → i2u a u’ s’ → i2u a u s → ( u’ = < u )) → EvtMaxU s — EUM gen : ∀ ( a : A gent )( c : Choic e )( sl : InfStr ate gy )( sr : FinStr ate gy ), EvtMaxU sl → EvtMaxU ( iNo de a c s l sr ). It has t wo parts.  The game fulfills the maximal utility r e q uiremen t for every agent,  The left subgame do es. Then w e can set and pro ve the main theorem: Theorem SGPE is InfNashEq : ∀ s : InfStr ate gy , EvtMaxU s → SGPE s → InfNashEq s . If every agent reaches a maximal utility somewher e in the ga me, then a subgame p erfect equilibrium is a Nash equilibrium. 8 The “Illogic of Conflict Escalation” revisited Now we may w ant to apply those general results on a sp ecific infinite subgame. Co ns ider the following game prop osed b y Sh ubik [13]. Reca ll its principle. Two a gent s A lic e and Bob comp ete in an auction for an ob ject of a v alue, say 2 ¢ , in this statement. The tw o agent s bid 2 ¢ , one after the other. If one agent gives up, the highest bidder g ets the ob ject, but the second bidder pa y s a lso for his bid. As Shubik no ted this game may never ceas e. Let us tak e as the utility order ing, the order ≥ on the nat (the natural num b ers or no n negative integers), in other w ords , if u a nd v are tw o utilities, u and v are of type nat a nd u = < v means u ≥ v , i.e., the la r ger the bid, the smaller the utility . 8.1 Strategy Never gi ve up Let us consider the function enlar ge left left n on strategies , that takes a strateg y and returns another one and that can be describ ed b y the following picture, where the thic k arr ows g iv e the choice at each no de. Ali ce B ob 2 n +1 , 2 n 2 n +1 , 2 n +2 W e define the strategy never give up (in shor t n gu ) as the strategy where the agents always keep bidding. It requires to define first a strategy that starts with the v alue n as the solution of ngu n = enlar ge left left (ngu (S n)). ngu n is an infinite strategy and w e can prov e tha t for n , (ngu n) is no t a Nash equilibrium. In this pro of, reasoning has a flav o r of temp or al lo gic . 8.2 Strategy A lways give up Here w e consider the function enlar ge right right n on strategies. Ali ce B ob 2 n +1 , 2 n 2 n +1 , 2 n +2 Again agu n is an infinite stra tegy and we c a n prov e the lemma: Lemma SGPEA GU : ∀ ( n : nat ), SGPE A g12 nat ge ( agu n ). which says tha t that this strategy is a Sub game Perfe ct Equilibrium. 9 Conclusion The exper imen t pr e s en ted in this pap er is in a v er y stag e . Its goal is to ma k e clear that, in g ames, especia lly in infinite ga mes, the reasoning can b e mechanized despite it is not ob vio us; it evidences with no s urprise some subtlety . T o mimic it, humans elab orate rather complex deductions. O ne o f the main statemen t w e c a n make is that true classic al lo gic is never u se d . More sp ecifically , we noticed no use of the ex cluded middle or pro ofs b y double negations. W e ho pe this expe rimen t op ens a disc us sion and shows how far h umans a r e from the somewhat idea l mechanized reasoning . W e wish to pursue this res e a rch by tr ying other mo delings and concepts a nd g o ing further in the pro ofs, for instance finding other conditions for SGPE s → In fNashEq s and deeper results on the example of “ Illogic Conflict o f Escalation”. References [1] Bruno Barr as, Samuel Boutin, Cristina Cornes, Judica¨ el Courant, Y ann Coscoy , David Delahay e, Daniel de Rauglaudre , J ean-Christophe Filliˆ atre, Eduardo Gim´ enez, Hugo Herb elin, G´ erard Huet, Henri Laulh ` ere, C´ esar Mu˜ noz, Chetan Murthy , Catherine P a rent - Vigouroux, P atrick Lo iseleur, Christine Paulin-Mohring, Amokrane Sa ¨ ıbi, and Benjamin W er ner. The Co q Pr o of Assistant R efer en c e Manual . INRIA, v er sion 6.3 .11 edition, May 20 00. [2] Y. Bertot. Coinduction in Co q. Op en a r c hives, March 2006. ht tp:/ / cel.archiv es- o uv er tes.fr/inria-00001174/fr/ . [3] Yv es Berto t and P ie r re Cast´ eran. In ter active The or em Pr oving and Pr o gr am Development Co q’A r t : The Calculus of Inductive Constructions . Springer-V e r lag, 200 4. [4] A.M. C o lman. Depth of strategic reaso ning in games. T r ends Co gn. Sci. , 7:1–2, 2003. [5] Solange Coup et-Grimal. An axio ma tization of linear temporal logic in the calculus of inductiv e con- structions. J L o gic Computation , 13(6):8 0 1–813, 20 03. [6] G. Gentzen. Untersuch ungen ¨ uber das logische Schließen. Mathematische Zeitschrift , 39:1 7 6–210, 405– 431, 1935. [7] Herbert Gintis. Game The ory Evolvi n g: A Pr oblem-Center e d Intr o duction to Mo deli ng Str ate gic Inter- action . P rinceton Univ ersity Press, 2 0 00. [8] J.-Y. Gira rd, Y. Lafont, and P . T aylor. Pr o ofs and T yp es , v olume 7 o f Cambridge T r acts in The or et Computer Scienc e . Camb ridge Univ ersity Press, 1 9 89. [9] M. J. Osb orne and A. Rubinstein. A Course in Game The ory . The MIT Pr e ss, Cam br idge, Mas- sach usetts, 1994. [10] Martin J. O s bor ne. An Intr o duction to Game The ory . Oxford, 2004. [11] Dag Prawitz. Natur al De duction: A Pr o of The or et ic al S tudy . Almqvist and Wiksell, Stockholm, 1965 . [12] St ´ ephane Le Roux. Abstr action and F ormalization in Game The ory . PhD thesis, ´ Ecole norma le sup´ er ieure de Lyon (F ra nc e ), January 2 008. [13] Martin Shubik. The dolla r auction game: A paradox in nonco op erative behavior and e s calation. J ournal of Conflict R esolution , 15(1):109– 111, 1971 . [14] D. Stahl and P Wilson. On play er s mo dels o f other play ers: theo ry and exp erimental evidence. Games and Ec onomic Behavi or , 10:21 8–254, 19 9 5. [15] D. v an Dalen. L o gic and Structur e . Spr inger, Berlin, Heidelb erg, 3 edition, 1997. [16] Ren ´ e V esterg aard. A constructive a ppro c ah to seq ue ntial Nash equilibria. In f. Pr o c ess. L ett. , 9 7 :46–51, 2006. [17] Jun Zhang a nd T rey Hedden. Two paradigms for depth of strategic reasoning in games respo nse to Colman. T r ends Co gn. S ci. , 7(1):4–5, Jan uar y 2003.