On the Rationality of Escalation

On the Rationali t y of Escalation Pierre Lescanne and Matthieu P errinel Univ ersit y of Ly on, ENS de Ly on, CNRS (LIP), U. Claude Bernard de Lyon 46 all´ ee d’Italie, 6936 4 Ly on, F rance No v em b er 15 , 2018 Abstract Escalation is a t yp ical feature of infinite games. Theref ore to ols con- ceive d for studying infinite mathematical structu res, namely those deriv- ing from c oi nduction are essen tial. Here w e use coinduction, or back- w ard coinduction (to sho w its connection with the same concept for finite games) to study carefully and formally the infi nite games esp ecially t hose called do l lar auctions , whic h are considered as the paradigm of escalation. Unlike what is commonly admitted, w e show t hat, provided one assumes that t he other agen t will alwa ys stop, bidding is rational, b ecause it re- sults in a subgame p erfect eq u ilibrium. W e show that this is not the only rational strategy profile (the only subgame p erfect equilibrium). I ndeed if an agent stops and will stop at every step, w e claim that h e is ratio- nal as w ell, if one admits th at his opp onent will never stop, b ecause this corresponds to a subgame p erfect eq uilibrium. Amazingly , in the infinite dollar au ct ion game, th e b eh a vior in whic h b oth agents stop at each step is not a N ash equilibrium, hence is n ot a subgame p erfect equilibrium, hence is n ot rational. The right notion of rationalit y we obtain fits with common sense and exp erience and remov e all feeling of paradox. Keyword: escalation, rationality , extensive form, backw ard induction. JEL C o de: C72, D 44, D 74. 1 In tro duc tion Escalatio n takes place in spe c ific sequential games in which players contin ue al- though their pay off decr eases on the whole. The dol lar auction game has b een presented by Shubik [1971] as the par a digm o f es calation. He noted that, even though their cost (the opp osite o f the payoff ) basically incr eases, players may keep bidding. This attitude is considered as inadequate and when ta lk ing abo ut escalation, Shubik [19 71] says this is a par adox, O’Neill [1986] a nd Leining er [1989] co ns ider the bidders as irr ational, Gintis [200 0] sp eaks o f il lo gic c onflict of esc alation and Colman [199 9] calls it Macb eth effe ct after Shakesp eare’s play . 1 INTR ODUCTION Nov ember 15, 2018 – 2 In contrast with these author s, in this pap er, we prov e using a r e asoning c on- ceived for infinite structures that escalation is lo g ic a nd that agents ar e ra tional, therefore this is not a paradox and we are le d to ass ert that Macb eth is some- what ra tional. This escalation pheno menon occur s in infinite sequential games and only there. Therefore it must b e studied w ith adeq uate to ols, i.e., in a fra mework designed for mathematical infinite structur es. Like Shubik [19 71] w e limit our - selves to tw o play ers only . In auctions , this consists in the tw o play er s bidding forever. This statement is bas ed on the co mmo n a ssumption that a pla yer is rational if he ado pts a s trategy which cor r esp onds to a sub game p erfe ct e quilib- rium . T o characterize this equilibrium the ab ov e cited a utho r s consider a finite restriction of the game for which they c ompute the subgame p erfect equilib- rium by b ackwar d induction 1 . In pra c tice, they add a new hypothesis o n the amount of money the bidder s are ready to pay , a lso called the limited ba nkroll. In the a mputated ga me, they conclude that there is a unique subgame per fect equilibrium. This consists in bo th a gents giving up immediately , not star ting the auc tio n and a dopting the sa me choice at ea ch step. In o ur formalization in infinite games, we show that extending that case up to infinity is not a sub- game p e rfect equilibr ium a nd we found t wo s ubgame p erfect equilibria , namely the ca ses when one age n t co ntin ues at each step and the other leav es a t e a ch step. Thos e equilibria which c orresp ond to rationa l a ttitudes account for the phenomenon o f e scalation. The origin of the misco nception that co ncludes the irrationality of e scalation is the belief that prop erties of infinite mathematical ob jects can b e extrap olated from prop erties of finite ob jects. This do es not work. As F agin [1 993] recalls, “most o f the classica l theorems of log ic [for infinite structures] fail for finite structures” (see Ebbinghaus and Flum [1995] for a full developmen t of the fi- nite mo del theor y). The recipro cal holds o bviously “mo st o f the results whic h hold for finite s tr uctures, fail for infinite structures”. This has b een bea utifully evidenced in mathematics, when W eierstrass [1 872] ha s exhibited his function: f ( x ) = ∞ X n =0 b n cos( a n xπ ) . Every finite sum is differentiable and the limit, i.e., the infinite sum, is no t. T o give another pictur e, infinite g ames are to finite games what fractal curves a re to smo oth curves [E dgar, 2 008]. In ga me theo r y the erro r done by the nineti- eth century mathematicians (W eierstrass quotes Ca uchy , Dirichlet and Gauss) would lead to the same issue: wr ong assumptions. With what we a re co ncerned, a r esult tha t ho lds o n finite ga mes do es not hold necessa rily on infinite games and vice-versa. More sp ecifically equilibria on finite ga mes are not preserved at the limit on infinite games. I n par ticular, w e cannot conclude that, wherea s the only rational attitude in finite do llar auction would b e to stop immediately , it is ir rational to es calate in the cas e of an infinite auction. W e hav e to keep 1 What is called “bac kward induction” in game theory is roughly what is called “induction” in logic. 1 INTR ODUCTION Nov ember 15, 2018 – 3 in mind that in the case o f esca lation, the game is infinite, therefo re rea son- ing made for finite ob jects ar e inappropria te a nd to o ls sp ecifica lly conceived for infinite ob jects s ho uld b e adopted. Lik e W eierstr ass’ discov er y led to the developmen t of function series, logicians hav e in ven ted metho ds for deductions on infinite structures a nd the r ight framework for r easoning logically o n infinite mathematical ob jects is ca lle d c oinduction . Like induction, c oinduction is bas ed on a fix p o int, but wherea s induction is based on the leas t fixp oint, coinduction is based on the g reatest fixp oint, for an ordering we are not going to describ e here a s it would go b eyond the sco pe of this pap er. A ttached to induction is the concept of inductive definition, w hich char- acterizes ob jects lik e finite lists, finite trees, finite ga mes, finite strategy profiles, etc. Similarly attached to coinduction is the concept of coinductive definition which characterizes streams (infinite lists ), infinite trees, infinite games , infinite strategy profiles etc. An inductiv e definition yields the least set that s atisfies the definition and a coinductive definition yields the greatest set that satisfies the definition. As s o ciated with these definitions we have inference principles . F or induction there is the famous induction principle used in ba ckw ard induction. On coinductively defined s ets of o b jects there is a principle like induction prin- ciple whic h uses the fact that the set s atisfies the definition (pr o ofs by case or by pattern) and that it is the lar gest set with this prop erty . Since coinductive defi- nitions allow us building infinite ob jects, one can ima gine co nstructing a sp ecific category of ob jects with “ lo ops”, like the infinite word ( abc ) ω (i.e., ab cabcabc... ) which is made by repea ting the sequence abc infinitely many times (o ther exam- ples with trees are given in Sec tio n 2, with infinite g ames and strategy profiles in Section 6). Suc h an ob ject is a fixp oint, this means that it contains an ob ject like itself. F o r instance ( abc ) ω = abc ( abc ) ω contains itse lf. W e say that suc h an ob ject is defined as a cofixp oint. T o pr ove a prop er t y P on a cofixp oint o = f ( o ), one assumes P holds o n o (the o in f ( o )), considered as a sub-o b ject of o . If one can prov e P on the whole ob ject (on f ( o )), then one has prov ed that P holds on o . This is called the c oinduction principle a concept which comes from Park [1981], Milner a nd T ofte [1991], and Aczel [198 8] (see also [Pair, 1970]) and was int ro duced in the framework we are consider ing by Co quand [19 9 3]. Sangio rgi [2009] gives a go o d survey with a complete historic al a ccount. T o b e sur e no t be entangled, it is advis a ble to use a pro of a ssistant that implements coinduc- tion to build and chec k the pro o f, but r easoning with coinduction is sometimes so counter-in tuitive that the use of a pro o f ass istant is not only advis a ble but compulsory . F or instance, we w er e, at first, convinced that the strateg y profile consisting in bo th a gents stopping at every step was a Nash equilibrium, like in the finite case, and only failing in pr oving it mechanically convinced us of the contrary and we w ere able to prove the opp osite. In our ca s e we ha ve c hecked every statement using Coq and in what follows a sentence like “we ha ve prov er that ...” means tha t we hav e succeeded in building a formal pro of in Coq . 1 INTR ODUCTION Nov ember 15, 2018 – 4 Bac kwa rd coinduction as a metho d for pro ving inv arian ts In infinite stra tegy pr o files, the coinduction principles can be seen as follows: a pr op erty which holds o n a stra tegy pr ofile o f a n infinite extens ive game is an invariant , i.e., a prop erty that s tays always true, alo ng the temp ora l line and to prov e that this is an inv ariant one pro c e e ds back to the past. There fo re the name b ackwar d c oinduction is a ppr opriate, since it pro ceeds backw ard the time, from future to past. Bac kwa rd induction vs bac kw ard coinduction One may wonder the difference betw een the class ical metho d, whic h we call b ackwar d induction and the new metho d we prop ose, which we call b ackwar d c oinduction . The main difference is that backw ard induction star ts the reas on- ing from the leav es, works o nly on finite ga mes and do es no t work on infinite games (or on finite stra tegy profiles), b ecause it r equires a w ell- fo undedness to work prop er ly , wherea s b ackwar d c oinduction works o n infinite g a mes (or on infinite str ategy pro files). Coinduction is unav oidable on infinite games, since the metho ds that consis ts in “c utting the ta il” to get a finite g ame o r a finite strategy profile canno t so lve the problem or even a ppr oximate it. Using back- ward induction to a game which is in trinsica lly infinite like the escalation in the dollar a uction was a mistake. It is indeed the same erroneous rea soning as this of the predecess ors of W eier strass who concluded that since: ∀ p ∈ N , f ( x ) = p X n =0 b n cos( a n xπ ) , is differentiable everywhere then f ( x ) = ∞ X n =0 b n cos( a n xπ ) . is differentiable everywhere wher eas it is differentiable nowhere. Much ea rlier, during the IV th centu r y B C , the improp er use of inductive reasoning a llows Parmenides a nd Zeno to neg ate motion and leads to Zeno’s p ar adox of A chil les and t he tortoise . This pa radox w a s r e p o r ted b y Aristotle as follows: “In a ra c e, the quickest ru nner c an n ever overtake t he slow- est, sinc e the pursuer must first r e ach the p oint whenc e the pursue d starte d, so that the slower m ust always hold a le ad.” Aristotle , Physics VI:9, 239b15 Zeno’s rea soning is corr ect, b ecause by induction, one ca n prov e that Achilles will never ov ertake the tortoise, but w e k now by exp erience that this is not the case, hence the par adox. Zeno error w as to apply induction to an infinite ob ject, he should have used coinduction if he would have known this concept. 1 INTR ODUCTION Nov ember 15, 2018 – 5 V on Neumann and coinduction As one knows, von Neumann [von Neumann, 1928, von Neumann and Morgens ter n, 1944] is the creato r o f g ame theory , whereas extensive games and equilibr ium in non co op erative games are due to Kuhn [195 3] and Nas h, Jr. [1950]. In the spir it of their creator s all those games are finite and backw ar d induction is the basic principle for c o mputing subgame p er fect equilibria [Selten, 1965]. This is not surprising since von Neumann [1 925] is also a t the orig in o f the ro le of well-foundedness in set theor y despite he left a do or op en for a not well- founded mem b ers hip relation. As explained by Sangiorg i [200 9], resea r ch on anti-foundation initiated b y Mir imanoff [1 917] are at the orig in of coinduction and were no t well known until the work of Aczel [198 8]. Wh y in infinite pla ys, agen ts do not ha v e a utility ? In our framework, in an infinite play (a play that runs forever, i.e., that does not lead to a leaf ) no age n t has a utility . People might say that this a n anomaly , but w e claim that this is p er fectly sensible. Let us affirm that in a r bitrary long plays, which lead to a leaf, all ag ent s have a utility . Only in plays that diverge, it is the case that a gents hav e no utility . This fits well with Binmore [1988] statements “The use of c omputing m achines (automata) to mo del players in an evolutiv e c ont ext is pr esumably un c ontro versial ... machi nes ar e also appr opriate for mo deling players in an eductiv e c ontext ” . Here we a r e co ncerned by the eductive c o nt ex t where “e qu ilibrium is achie ve d thr ough c ar eful r e asoning by the agents b efor e and during the play of t he game” [Binmor e, 1988, lo c. cit]. By automaton, we mean any mo del of computation 2 , since all the mode ls of computation a re equiv alent by Ch urch thesis. If an agent is mo deled by an automaton, this means also that the function that computes the utility for this agent is also mo deled b y a n automaton. It seems then sensible that one ca nno t compute the utility of a n agent for an infinite play , since computing is a finite pro cess working on finite da ta (or at le ast data that a re finitely describ ed). Since the agent cannot co mpute the utilit y o f an infinite play , no sensible v alue can b e attributed to him. If one wan ts absolutely to a ssign a v alue to an infinite play , one must abandon the automaton framework and mor eov er this v alue should be the limit of a sequence of v alues, whic h do es not exist in most of the cas e s . 3 F or insta nce, in the case of the dolla r auction (Section 6), the utility asso ciated with the unique infinite pla y are the sequence ..., v + n, n + 1 , v + n + 1 , n + 2 , ... Therefore consider ing that in infinite plays, a gents hav e no utilit y is p erfectly consistent with a mo deling of agents by a utomata. By the wa y , do es an a gent care ab out a pay off he (she) receives in infinitely many years? Will he (she) adapt his (her) strategy on this? 2 Our mo del of computation i s this of the calculus of inductiv e construction, a kind of λ -calculus behind Coq [T uri ng, 1937]. 3 If utilities ar e natural num b ers, it exists only if the s equence is stationary , which is not the case in escalation. 1 INTR ODUCTION Nov ember 15, 2018 – 6 Pro of assistan ts vs automated theorem pro ve rs Coq is a pro of as sistant built by Co q development t e am [2007] , see Ber tot and Cast´ eran [2004] for a go o d intro duction and notice tha t they call it “interactive theo rem prov ers” , which is a strict synonymous. Despite b oth deal with theorems a nd their pr o ofs a nd a re mechanized using a computer, pro of as sistants a re no t a u- tomated theorem provers. In particular , they are muc h more express ive than automated theore m prov ers and this the reas on why they are interactiv e. F or in- stance, there is no automated theorem prov er implemen ting coinduction. P ro of assistants are automa ted only for elemen tar y steps and interactive for the rest. A sp ecificity of a pro of a ssistant is that it builds a ma thematical ob ject called a (formal) pr o of which can be check ed indep endently , copied, sto red and ex- changed. F ollowing Harr is on [2008] and Do wek [2007], w e can cons ide r that they are the to ols of the mathematicia ns o f the XXI th centu r y . Ther efore using a pr o of as sistant is a highly mathematical mo dern a c tivit y . The mathematical dev elopment presented here corresp onds to a Coq script 4 which c a n b e found on the following url’s : http:/ /pers o.ens- lyon.fr/pierre.lescanne/COQ/EscRat/ http:/ /pers o.ens- lyon.fr/pierre.lescanne/COQ/EscRat/SCRIPTS/ Structure of the pap er The pap er is structured as follows. In Sectio n 2 we present coinduction illus- trated by the example of infinite binary trees. In Section 3 we present infinite games. In Section 4, we intro duce the cor e concept of infinite strateg y profile which allows us presenting equilibria in Section 5. The dollar auction game is presented in Section 6 a nd the escala tion is discussed in Section 7. Readers who wan t to hav e a quick idea ab out the results of this pap er on the ra tionality o f escalation a re a dvised to read sections 6, 7 and 9. Related works T o our knowledge, the only application of coinductio n to extensive game the- ory has b een ma de by Capretta [20 07] who uses coinductio n to define only common knowledge not equilibria in infinite games. Another strongly con- nected work is this o f Coup et-Grima l [2003] on temp or al log ic . O ther applica- tions a re o n representation of rea l num b ers by infinite sequences [Bertot, 2007, Julien, 2008] and implement atio n o f str eams (infinite lists) in electronic circuits [Coup et-Grimal and Ja k ubiec, 2004]. An ancestor of our description of infi- nite games and infinite strategy profiles is the constr uctive description o f finite games, finite strateg y pr ofiles, a nd equilibria by V esterg aard [2006]. Lescanne [2009] intro duces the fra mework o f infinite ga mes with more deta il. Infinite games ar e introduced in O sb orne and Rubinstein [1994] a nd Osb or ne [20 0 4] us- ing historie s , but this is not alg orithmic and therefore not a menable to formal pro ofs and coinduction. 4 A script i s a list of commands of a pro of assistant. 2 COINDUCTION, INFINITE TRE ES Nov ember 15, 2018 – 7 Many authors have studied infinite g ames (see for instance Martin [19 98], Mazala [2 001]), but except the name “game” (an ov erlo aded one), those games hav e nothing to see with infinite extensive g ames as pres ent ed in this paper . The infiniteness of Bla ckwell games for instance is derived fro m a top ology , by adding r eal num b ers a nd pr obability . Sangiorgi [20 09] mentioned the c o n- nection be t ween E hrenfeuch t-F r a ¨ ıss´ e games [E bbinghaus and Flum, 19 95] and coinduction, but the connection with extensive games is ex tremely r emote. 2 Coinduction and infin ite binary trees As an example of a co inductiv e definition co nsider this of lazy binary tre es , i.e., finite a nd infinite binary trees.  •   • •    . . . • • •   •  • •  •  •  •  •  . . . •  • •   • •   Backb one Zig Figure 1: Coinductive binary trees A coinduc tive binary t r e e (or a la zy binary tree or a finite-infinite binary tree) is • either the empty binar y tree  , 2 COINDUCTION, INFINITE TRE ES Nov ember 15, 2018 – 8 • or a binar y tree of the form t · t ′ , where t and t ′ are binary trees. By the keyw ord co inductive w e mean that we define a coinductive set of ob jects, hence w e a ccept infinite ob jects. Some coinductive binary tre es ar e given o n Fig. 1. W e define on a co inductive binary tree a predicate whic h ha s also a coinductive definition: A bina ry tr ee is infinite if (coinductively) • either its left subtree is infinite • or its right subtree is infinite . W e define tw o trees that we call zig and zag . zig and zag are defined toge ther as c ofixp oint s as follows: • zig has  as left subtree and zag as right subtree, • zag has zi g as left subtree and  as right subtree. This s ays that zig and zag are the greatest solutions 5 of the tw o s imultaneous equations: zig =  · zag zag = zig ·  • •  • zig  ⇒ • •  • zig  zig Figure 2: How cofix works on zig for is infinite ? It is common sens e that zig a nd zag are infinite, but to pr ov e that “ zig is infinite” using the cofix tactic 6 , we do as follows: ass ume “ zig is infi nite” , then zag is infinite, fro m which we g et that “ zig is infi nite” . Since we use the assumption on a strict subtree of zig (the direct subtree of zag , which is itself a 5 In this case, the least solutions are unin teresting as they are ob jects nowhere defined. Indeed there is no basic case in the inductive definition. 6 The cofix tact ic i s a metho d proposed by the proof assistant Coq which implements coinduction on cofixp oint ob jects. Roughly s peaking, it attempts to prov e that a property is an invariant , by proving it is preserved along the infinite ob ject. Here “ is infinite” is such an inv ariant on zig . 3 FINITE AND INFINITE GAMES Nov ember 15, 2018 – 9 direct s ubtree o f zig ) we can co nclude that the c ofix ta ctic has b een used prop erly and that the prop erty ho lds, na mely that “ zig is infin ite” . This is pictured on Fig.2, wher e the square b ox r epresents the predicate is infinite . Ab ov e the rule, there is the step of co induction and b elow the rule the conclusion, namely that the whole zig is infinite. W e let the reader prove that backb o ne is infinite, whe r e backb one is the gre atest fixp oint of the equa tion: backb one = backb one ·  and is an infinite tree that lo o ks like the s keleton of a infinite ce n tip ede ga me as s hown on Fig.1 (see Section 8). Int er ested reader s may have a lo ok at Co upet- Gr imal [2003], Coup et-Gr ima l and Jak ubiec [2004], Le s canne [2009], Be r tot [2005, 20 07] and esp ecially Bertot and Ca st´ eran [2004, c hap. 13] for other exa mples of cofix reasoning . 3 Finite and infinite games As an intermediary b e t ween histories and str ategy pro files, let us define finite and infinite ga mes. T ra ditionally , games a r e defined through tre e s asso ciated with utility function at the leav es. Another appro ach which Osb orne [2 004] at- tributes to Rubinstein us es histor ies. A third appr oach prop osed b y V esterga a rd [2006] which fits well with inductive reaso ning is to g ive an inductive definition of ga mes. T o handle infinite games we prop ose a coinductive definition. The type of Game s is defined as a coi nductiv e as follows: • a Ut ility function ma kes a Game , • an A gent and t wo Game s make a Game . A Game is either a leaf (a ter minal no de) or a comp osed game made of an agent (the agent who has the turn) and tw o subg a mes (the for mal definition in the Coq vernacular is g iven in the a ppe ndix A). W e use the expressio n gL e af f to denote the leaf game ass o ciated with the utilit y function f and the expr ession gNo de a g l g r to denote the ga me with ag ent a at the r o ot and tw o subgames g l and g r . Hence o ne builds a finite ga me in t wo wa ys: either a g iven utilit y function f is encapsulated by the oper ator gL e af to make the game ( gL e af f ), or an a gent a a nd tw o games g l and g r are given to make the game ( gNo de a g l g r ). Notice that in such games, it can b e the case that the sa me a gent a has the turn twice in a row, like in the g ame ( gNo de a ( gNo de a g 1 g 2 ) g 3 ). Concerning co mparisons of utilities we consider a very general setting where a utility is no mo re that a t yp e (a “ set”) with a preference which is a preorder , i. e ., a transitive and reflex ive r elation, and which we write  . A preor der is enough for what we w a n t to prove. W e assig n to the leav es, a utility function which a sso ciates a utility to each agent. W e can also tell how we a sso ciate a histor y with a game or a histor y and a utilit y function with a game (s e e the Coq s c ript). W e will see in the next 4 FINITE OR INFINITE STRA TEGY PROFILES No vember 15, 2018 – 10 section how to ass o ciate a utility with an agent in a game, this is done in the frame o f a stra teg y profile, which is des crib ed now. 4 Finite or infinite strategy profiles In this section we define fi nite or infinite binary str ate gy pr ofiles or Str atPr of s in shor t. They ar e based on games which are extensive (or sequential) games and in which ea ch agent has t wo c ho ic e s: ℓ (left) and r (right). 7 In addition these ga mes are infinite, we should say “can b e infinite”, as w e consider b o th finite and infinite games . W e do not give explicitly the definition of a finite or infinite extensive game s ince we do not use it in what follows, but it can be easily obtained by removing the choices from a str ategy profile. T o de fine finite or infinite stra tegy profiles, we s upp os e given a utility and a u tility function . As said, w e define directly stra tegy profiles as they are the o nly conce pt we are int er ested in. Indeed an equilibrium is a strategy pro file. The type of Str atPr of s is defined as a coinductive as follows: • a Ut ility function ma kes a Str atPr of . • an A gent , a Choic e and tw o Str atPr of s make a Str atPr of . Basically 8 an infinite str ategy profile which is not a le af is a no de with four items: an agent, a choice, tw o infinite strategy profiles. A str ategy pr ofile is a game plus a choice at each node. Strategy pro files of the first kind are wr itten ≪ f ≫ and s tr ategy pro file s of the seco nd kind are wr itten ≪ a, c, s l , s r ≫ . In other words, if b etw een the “ ≪ ” a nd the “ ≫ ” ther e is one comp onent, this comp onent is a utility function and the result is a leaf strategy profile and if there are four compo nent s, this is a co mpo und strateg y profile. In what follows, we say tha t s l and s r are strategy subprofiles o f ≪ a, c, s l , s r ≫ . F or instance, here a re the drawing of t wo stra tegy profiles ( s 0 and s 1 ): Alice Bob Alice 7→ 0 , Bob 7→ 1 Alice 7→ 1 , Bob 7→ 2 Alice 7→ 2 , Bob 7→ 0 Alice Bob Alice 7→ 0 , Bob 7→ 1 Alice 7→ 1 , Bob 7→ 2 Alice 7→ 2 , Bob 7→ 0 which c o rresp ond to the expre s sions 7 In pictures, we tak e a sub jective point of view: left and right are from the persp ective of the agent . 8 The formal definition in the Coq ve rnacular is given in appendix A. 4 FINITE OR INFINITE STRA TEGY PROFILES No vember 15, 2018 – 11 s 0 = ≪ Alice , ℓ , ≪ Bob , ℓ , ≪ A li ce 7→ 0 , Bob 7→ 1 ≫ , ≪ Alice 7→ 2 , Bob 7→ 0 ≫ ≫ , ≪ Alice 7→ 1 , Bob 7→ 2 ≫ ≫ and s 1 = ≪ Alice , r , ≪ Bob , ℓ , ≪ Alice 7→ 0 , Bob 7→ 1 ≫ , ≪ Ali ce 7→ 2 , Bob 7→ 0 ≫ , ≪ Alice 7→ 1 , Bob 7→ 2 ≫ , ≫ . T o describ e a s pec ific infinite str a tegy pr ofile one uses mo st of the time a fixp o int eq uation like: t = ≪ Alice , r , ≪ Alice 7→ 0 , Bob 7→ 0 ≫ , ≪ Bob , r , t, t ≫ ≫ which c o rresp onds to the pictures: t = A 1 A 1 7→ 0 ,A 2 7→ 0 A 2 t t Other examples of infinite stra tegy profile s ar e g iven in Section 6 . Usually an infinite g ame is defined as a cofixp oint, i.e., as the s olution of an eq uation, po ssibly a para metric equation. Whereas in the finite cas e one ca n easily a sso ciate with a stra teg y profile a utility function, i.e., a function whic h assigns a utility to an a gent, as the result of a r e cursive ev alua tio n, this is no more the ca se with infinite stra tegy profiles. O ne r e a son is that it is no more the case that the utilit y function can b e computed since the stra tegy pr ofile may run for ever. This makes the function partial 9 and it c a nnot b e defined as an inductiv e or a c oinductive. Therefore we make s 2 u (an a bbreviation for Stra te gy-pr ofile-to-Utility ) a relation b etw een a strategy profile and a utilit y function and we define it coinductiv ely; s 2 u app ea r s in expression of the form 10 ( s 2 u s a u ) wher e s is a strategy profile, a is an a gent and u is a utility . It r e ads “ u is a utility of the agent a in the stra tegy profile s ”. s 2 u is a predicate defined inductiv ely as follows: • s 2 u ≪ f ≫ a ( f ( a )) holds, • if s 2 u s l a u holds then s 2 u ≪ a ′ , ℓ, s l , s r ≫ a u holds, • if s 2 u s r a u holds then s 2 u ≪ a ′ , r , s l , s r ≫ a u holds. This means the utilit y of a for the lea f stra tegy pr ofile ≪ f ≫ is f ( a ), i.e., the v alue delivered by the function f when applied to a . The utility of a for the strategy profile ≪ a ′ , ℓ, s l , s r ≫ is u if the utility of a for the strategy pro file s l 9 Assigning arbitrarily (i. e. , not algorithmically) a utility function to an infinite “history”, as it is made sometimes in the literature, is artificial and not really handy f or formal reasoning. 10 Notice the lighter notat ion ( f x y z ) for what i s usually written f ( x )( y )( z ). 5 EQUILIBRIA Nov ember 15, 2018 – 1 2 is u . In the case of s 0 , the first ab ove strategy profile, o ne has s 2 u s 0 Alice 2 , which mea ns that, for the strategy profile s 0 , the utility o f Alice is 2. F or a game ther e are ma n y asso ciated p ossible stra tegy profiles, which hav e a similar structure, but on the o ther hand there is a function which re tur ns a game given a strategy profile. 5 Subgame p erfect and Nash equilibria 5.1 Con v ertibility An imp ortant bina r y re lation on strateg y profiles is c onvertibility . W e write ⊢ a ⊣ . the conv ertibility of ag ent a . The rela tion ⊢ a ⊣ is defined inductively a s follows: • ⊢ a ⊣ is refle x ive, i.e., for all s , s ⊢ a ⊣ s . • If the no de has the sa me age n t as the agent in ⊢ a ⊣ then the choice may change, i.e., s 1 ⊢ a ⊣ s ′ 1 s 2 ⊢ a ⊣ s ′ 2 ≪ a, c, s 1 , s 2 ≫ ⊢ a ⊣ ≪ a, c ′ , s ′ 1 , s ′ 2 ≫ • If the no de do es not hav e the same age n t a s in ⊢ a ⊣ , then the choice has to b e the same: s 1 ⊢ a ⊣ s ′ 1 s 2 ⊢ a ⊣ s ′ 2 ≪ a ′ , c, s 1 , s 2 ≫ ⊢ a ⊣ ≪ a ′ , c, s ′ 1 , s ′ 2 ≫ Roughly sp eaking tw o s trategy pr ofiles are co nv ertible fo r a if they change only for the choices for a . Since it is defined inductively , this means that tho s e changes are finitely man y . W e feels that this makes s ense since an ag ent ca n only co nceive finitely many issues. 5.2 Nash equilibria The notion of Nash equilibrium is translated from the notion in textb o oks . The concept of Nash equilibr ium is ba sed on a compa rison of utilities; this ass umes that a n actual utilit y exists and therefor e this requires c onv ertible strateg y pro- files to “le ad to a le af ” . s is a Nash e quilibrium if the following implication holds: If s “le ads to a le af ” and for all agent a and for all strateg y profile s ′ which is conv ertible to s , i.e., s ⊢ a ⊣ s ′ , and whic h “le ads to a le af ” , if u is the utilit y o f s for a and u ′ is the utility of s ’ for a , then u ′  u . Roughly sp eaking this means tha t a Nas h equilibrium is a strateg y pro file in which no agent has interest to change his choice since doing so he cannot ge t a better payoff. 5 EQUILIBRIA Nov ember 15, 2018 – 1 3 5.3 Subgame Perfect Equilibria In order to insure that s 2 u ha s a r esult w e define an op erato r “le ads to a le af ” that says tha t if one follows the choices shown by the strategy pr o file one reaches a lea f, i.e., one do es not go forever. The predica te “le ads to a le af ” is defined inductively a s • the s trategy pr o file ≪ f ≫ “le ads t o a le af ” , • if s l “le ads to a le af ” , then ≪ a, ℓ, s l , s r ≫ “le ads to a le af ” , • if s r “le ads to a le af ” , then ≪ a, r , s l , s r ≫ “le ads to a le af ” . This means that a s trategy pro file, which is itself a leaf, “le ads to a le af ” and if the stra tegy pro file is a no de, if the choice is ℓ and if the left strategy subprofile “le ads to a le af ” then the whole str ategy profile “le ads to a le af ” and similarly if the choice is r . If s is a strateg y profile that s atisfies the pre dicate “le ads to a le af ” then the utilit y exists and is unique, in other words: • F or all agent a and for all str a tegy pr ofile s , if s “le ads to a le af ” then there exists a utility u which “is a utility of the agent a in the strategy profile s ”. • F or all a gent a and for all str a tegy profile s , if s “le ads to a le af ” , if “ u is a utility o f the agent a in the strategy profile s ” and “ v is a utilit y of the agent a in the strategy profile s ” then u = v . This means s 2 u works like a function on strategy profiles whic h le ad to a le af . W e also c o nsider a predicate “always le ads to a le af ” which means that everywhere in the str ategy profile, if one follows the choices, one leads to a leaf. This prop er t y is defined everywhere on an infinite strateg y pr ofile a nd is therefore co inductiv e. The predicate “always le ads to a le af ” is defined coinductiv el y by saying: • the s trategy pr o file ≪ f ≫ “always le ads to a le af ” , • for all choice c , if ≪ a, c, s l , s r ≫ “le ads t o a le af ” , if s l “al- ways le ads to a le af ” , if s r “always le ads to a le af ” , then ≪ a, c, s l , s r ≫ “always le ads to a le af ” . This s ays that a strategy profile, which is a leaf, “always le ads to a le af ” and tha t a co mpo sed strategy profile inherits the predicate from its s trategy subprofiles pr ovided its e lf “le ads t o a le af ” . Let us consider now sub game p erfe ct e quilibria , which we write S GP E . S GP E is a prop erty o f strategy profiles. It r equires the strategy subprofiles to fulfill coinductively the same prop erty , na mely to b e a S GP E , and to insure that the str ategy profile with the best utility for the no de agent to be cho- sen. Since b oth the strategy profile a nd its strateg y s ubpr ofiles are p otentially infinite, it makes sens e to define S GP E coinductively . 5 EQUILIBRIA Nov ember 15, 2018 – 1 4 S GP E is defined coinductively a s follows: • S GP E ≪ f ≫ , • if ≪ a, ℓ, s l , s r ≫ “always le ads t o a le af ” , if S GP E ( s l ) a nd S GP E ( s r ), if s 2 u s l a u and s 2 u s r a v , if v  u then S GP E ≪ a, ℓ, s l , s r ≫ , • if ≪ a, r , s l , s r ≫ “always le ads to a le af ” , if S GP E ( s l ) a nd S GP E ( s r ), if s 2 u s l a u and s 2 u s r a v , if u  v then S GP E ≪ a, r , s l , s r ≫ , This means that a strategy profile, which is a leaf, is a subgame per fect equilibrium. Moreover if the strategy pro file is a no de, if the strategy profile “always le ads to a le af ” , if it has agent a and choice ℓ , if b oth stra tegy subprofile s are subga me per fect eq uilibria and if the utilit y o f the agent a for the right strategy subpr ofile is les s than this for the left strategy s ubprofile then the whole str ategy pro file is a subg a me p erfect equilibrium and vice versa. If the choice is r this works similarly . Notice that since we r equire that the utility can b e computed not only for the s trategy pro file, but for the str ategy subpro files and for the strategy sub- subprofiles a nd so on, we require these strategy profiles not only to “le ad t o a le af ” but to “always le ad to a le af ” . W e define orders (one for each a gent a ) betw een s trategy profiles which we write ≤ a . s ′ ≤ a s iff : If u (resp ectively u’) is the utility for a in s (resp. s ′ ), then u ′  u Prop ositio n 1 ≤ a is an or der (the pr o of is stra ight forwar d). Prop ositio n 2 A sub game p erfe ct e quilibrium is a Nash e quilibrium. Pro of: Supp os e that s is a strategy pro file which is a S GP E and which has to b e prov ed to b e is a Nash equilibrium. Assuming that s ′ is a strateg y profile such that s ⊢ a ⊣ s ′ , le t us prov e by induction on s ⊢ a ⊣ s ′ that s ′ ≤ a s : • Case s = s ′ , by r eflexivity , s ′ ≤ a s . • Case s = ≪ x, ℓ, s l , s r ≫ and s ′ = ≪ x, ℓ, s ′ l , s ′ r ≫ with x 6 = a . s ⊢ a ⊣ s ′ and the de finitio n of ⊢ a ⊣ imply s l ⊢ a ⊣ s ′ l and s r ⊢ a ⊣ s ′ r . s l which is a s trategy subprofile of a S GP E is a S GP E as well. Hence b y induction h yp othesis , s ′ l ≤ a s l . The utilit y of s (resp ectively of s ′ ) for a is the utility of s l (resp ectively of s ′ l ) fo r a , then s ′ ≤ a s . • The case s = ≪ x, r , s l , s r ≫ and s ′ = ≪ x, r , s ′ l , s ′ r ≫ is similar . • Case s = ≪ a, ℓ, s l , s r ≫ and s ′ = ≪ a, r , s ′ l , s ′ r ≫ , then s l ⊢ a ⊣ s ′ l and s r ⊢ a ⊣ s ′ r . Since s is a S GP E , s r ≤ a s l . 6 DOLLAR A UCTION Nov ember 15, 2018 – 1 5 Alice Bob Alice Bob v + n, n n +1 ,v + n v + n +1 ,n + 1 n +2 ,v + n +1 Figure 3: The dol lar auction ga me Moreov er , since s r is a S GP E , by induction hypothesis, s ′ r ≤ s r . Hence, by transitivity of ≤ a , s ′ r ≤ a s l . But we know that the utility of s ′ for a is this o f s ′ r and the utility of s for a is this of s l , hence s ′ ≤ a s . • The case s = ≪ a, r , s l , s r ≫ and s ′ = ≪ a, ℓ, s ′ l , s ′ r ≫ is s imilar.  The a b ove pro of is a presentation of the formal pro of written with the help of the pro of assistant Coq . Notice that it is by inductio n on ⊢ a ⊣ which is p ossible since ⊢ a ⊣ is inductively defined. Notice also that s a nd s ′ are p otentially infinite. 6 Dollar auction games and Nash equilibria The dollar auction has been pr esented by Shubik [19 71] a s the paradigm of escalation, insisting on its par adoxical asp ect. It is a seque ntial ga me pr esented as an auction in which tw o agents co mpete to a cquire an ob ject of v alue v ( v > 0) (see Gintis [2 000, Ex. 3.13]). Suppo se that b oth a gents bid $1 a t each turn. If one of them gives up, the other receives the ob ject and b oth pay the amount o f their bid. 11 F or instance, if ag ent Al ice stops immedia tely , she pays nothing and agent Bob , who acq uir es the o b ject, has a pay off v . In the genera l turn of the auction, if Alice abandons, she lo os es the auction and has a pay off − n and Bo b who has alrea dy bid − n has a payoff v − n . A t the next turn after Alice decides to contin ue, bids $ 1 for this a nd acq uires the ob ject due to Bob stopping, Alice ha s a pay off v − ( n + 1 ) and Bob ha s a pay off − n . In our formalization we ha ve consider ed the dol lar auction up to infinit y . Since we are int er ested only b y the “ asymptotic” b ehavior, we can co nsider the auction after the v alue of the ob ject has be e n pa s sed and the pay offs are neg a tive. The dollar auction game can b e summarize d by Fig. 3. Notice that we a s sume that Alice starts. W e have recognized three c la sses o f infinite strategy pr ofiles, index e d by n : 1. The strategy profile always give up , in which b oth A lice and Bob stop at each turn, in short dolAsBs n . 11 In a v ariant, eac h bidder, when he bi ds, puts a dollar bill in a hat or in a piggy bank and their is no return at the end of the auction. The last bidder gets the ob ject. 6 DOLLAR A UCTION Nov ember 15, 2018 – 1 6 2. The stra teg y profile Alice st ops always and Bob c ontinues always , in short dolAsBc n . 3. The stra teg y profile Alice c ontinu es always and Bob stops always , in short dolAcBs n . The three kinds of strategy profiles a re presented in Fig. 4. Alice Bob Alice Bob v + n, n n +1 ,v + n v + n +1 ,n + 1 n +2 ,v + n +1 dolAsBs n ak a Alwa ys give up Alice Bob Alice Bob v + n, n n +1 ,v + n v + n +1 ,n + 1 n +2 ,v + n +1 dolAsBc n ak a Alice abandons alw ay s and Bob contin ue s alwa ys Alice Bob Alice Bob v + n, n n +1 ,v + n v + n +1 ,n + 1 n +2 ,v + n +1 dolAcBs n ak a Alice contin ues alwa ys and Bob abandons alw a ys Figure 4: Three strategy profiles W e hav e s hown 12 that the se cond and third kinds of strategy profiles , in which one of the ag ent s always stops and the other contin ues, are subgame per fect e q uilibria. F or insta nce, consider the strategy pr ofile dolAsBc n . Assume SGPE ( dolAsBc n +1 ). It works a s follows: if do lAsBc n +1 is a subg a me p er fect equilibrium co rresp onding to the payoff − ( v + n + 1) , − ( n + 1), then ≪ Bob , ℓ, dolAsBc n +1 , ≪ Alice 7→ n + 1 , Bob 7→ v + n ≫ ≫ is again a subgame perfect equilibrium (since v + n ≥ n + 1) and ther efore dolAsBc n is a subga me per fect equilibrium, since ag ain v + n ≥ n + 1. 13 W e 12 The pro ofs are typical uses of the Coq cofix tactic. 13 Since the cofix tactic has b een used on a strict strategy subprofile, the r easoning is correct. 8 ANOTHER EXAMPLE: THE INFINIPEDE Nov ember 15, 2018 – 1 7 can conclude that fo r a ll n , dolAsBc n is a sub game p erfe ct e quilibrium . In other words, we ha ve assumed tha t SGPE ( dolAsBc n ) is an invariant a ll alo ng the game and that this inv ariant is pr eserved as we pro c eed backw ard, through time, into the game. With the co nditio n v > 1, we ca n pr ov e that dolAsBs 0 is not a Nash eq ui- librium, then as a co nsequence not a subgame p erfect equilibrium. Ther e fore, the strategy profile that consists in sto pping fro m the b eginning and fo r ever is not a Nas h equilibrium, this co n tra dicts what is said in the literature [Shu bik, 1971, O’Neill, 19 8 6, Leininger, 198 9, Gintis, 2000]. 7 Wh y escalation is rational? Many authors a gree (see how ever [Halp ern, 2001, Stalnaker, 19 98]) that choo sing a subg ame p er fect equilibrium is rational [Aumann, 1995]. Le t us show that this can lea d to a n esca la tion. Supp ose I am Al ice in the middle of the auction, I hav e tw o o ptions that are r a tional: one option is to sto p right aw ay , since I assume that Bo b will contin ue alwa ys . But the se c o nd option says that it could be the case that from now on Bob will s top alwa ys (stra tegy pr ofile do lAcBs n ) and I will always con tinue which is a subg ame p er fect e quilibrium hence rational. If Bob acts similar ly this is the es calation. So at eac h step an a gent can stop and be rationa l, a s well a s at each step an a gent can cont inue and b e ra tional; b oth options make per fect sense. W e claim tha t human agents reason coinductively unknowingly . Ther efore, for them, e s calation is one of their rational options at least if one considers strictly the rules of the dolla r auction game, in par ticula r with no limit on the bankroll. Ma ny exp eriences [Colman, 1999] have shown that h uman a re inclined to escalate or at least to go very far in the auction when playing the do lla r auction g a me. W e prop ose the fo llowing explanation: the finiteness of the game was not explicit for the par ticipants and for them the game was na turally infinite. Ther efore they adopted a form of r easoning similar to the o ne w e develope d he r e, probably in an intuit ive form and they co nclude it was equa lly rational to co nt inue o r to leav e acco rding to their feeling on the threat of their o pp o nent, hence their attitude. Actua lly our theo retical work reconciles e xpe r iences with log ic, 14 and h uman with rationality . 8 Another example: the infinip ede An often studied extensive g ame is the so-ca lled centipede 15 int ro duced by Rosenthal [1981] (see also Binmor e [1987], Co lman [1998], Osb o r ne and Rubinstein [1994]). Whereas centipedes ar e finite ex tensive games, we have studied games with infinitely many “legs” , which we pr o p o se to call infi nip e des . Infinip e des are g eneralizatio n to infinity of cent ip edes. In infinipedes, we hav e identified 14 A logic which includes coinduction. 15 A cen tip ede has hundred l egs, whereas a m illip ede has thousand. Al l b elong to the group of myriapo ds which m eans “ten thousand legs”. REFERENCES Nov ember 15, 2018 – 1 8 only one subgame per fect equilibrium, namely this where bo th a gents aba ndon at each turn. T his sho ws that even in the infinite generalizatio n, a gents are rational if they do not start the game and abandon fro m the b eginning . Hence the pa radox discussed by the a utho r s still re ma ins, namely the age nts do not get the somewha t better payoff, they would get if they would be more flexible with res pec t to r ationality . The pr oblem for the ag ent s in the infinip ede ga me is that when they start an infinite game, they do not know when to stop. W e notice the sp ecific status of the stra tegy profile ac in which all age nts contin ue for ever. Since ac cannot attribute payoffs to the ag en ts, it cannot b e compared w ith any other strategy pro file and lies isolated in its own attractor (in term o f eq uilibrium). The headlong run ac is somewhat rational despite it do es no t deliver any reward. 9 Conclusion W e have shown that coinduction is the right to ol to s tudy infinite structur es, e. g., the infinite dol lar auction game . This w ay we get results which c o nt r adict forty years of claims that escalatio n is irrational. W e can show where the failure comes fro m, namely from the fact that authors have extr ap olated o n infinite structures results obtained on finite ones. Actually in a strategy profile in which one of the age nts threatens credibly the o ther to contin ue in every cas e , common sense says that the other ag ent sho uld a bandon a t each step (taking seriously the threat), this is a subgame p erfect eq uilibrium. If the thr e a t to contin ue is not credible, the o ther agent ma y think that his opp onent bluffs and will abandon at every step from now on, hence a rationa l a ttitude fo r him is to contin ue. As a matter of fact, coinductio n meets common sense. 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Mathematische Annalen , 100:29 5–300 , 1928 . J. von Neumann and O. Mo rgenstern. The ory of Games and Ec onomic Beha vior . Princeton Univ. Pr ess, P rinceton, 19 44. K. W eie r strass. ¨ Uber contin uirliche F unktionen eines reellen Arguments, die f¨ ur keinen W er th des letzteren einen b estimmten Differentialquotienten b esitzen, 1872. in Karl Weiertr ass Mathematische Werke (1967 ), Abha nd lungen II . Gelesen in der K¨ onigl. Ak ademie der Wiss eschaften am 18 Juli 187 2 . A EXCERPTS OF COQ SCRIP TS Nov ember 15, 2018 – 2 2 A Excerpts of Co q scripts Infinite binary trees CoInductive LBintr e e : Set := — LbtNil : LBintr e e — LbtNo de : LBintr e e → LBintr e e → LBintr e e . CoInductive InfiniteLBT : LBintr e e → Pr op := — IBTL eft : ∀ bl br , InfiniteLBT bl → InfiniteLBT ( LbtNo de bl br ) — IBTRight : ∀ bl br , Infi niteLBT br → InfiniteLBT ( LbtNo de bl br ). CoFixp oint Zig : LBintr e e := LbtNo de Zag LbtNil with Zag : LBintre e := LbtNo de LbtNil Zig . Infinite strategy profiles CoInductive Str atPr of : Set := — sL e af : Utility fun → Str atPr of — sNo de : A gent → Choic e → Str atPr of → Str atPr of → Stra tPr of . Inductive s 2u : Str atPr of → A gent → U tility → Pr op := — s2uL e af : ∀ a f , s2u ( ≪ f ≫ ) a ( f a ) — s2uL eft : ∀ ( a a’ : A gent ) ( u : Ut ility ) ( sl sr : Str atPr of ), s2u sl a u → s2u ( ≪ a’ , l , sl , sr ≫ ) a u — s2uRight : ∀ ( a a’ : Age nt ) ( u : Utility ) ( sl sr : Stra tPr of ), s2u sr a u → s2u ( ≪ a’ , r , sl , sr ≫ ) a u . Inductive Le adsT oL e af : S tr atPr of → Pr op := — LtLL e af : ∀ f , L e adsT oL e af ( ≪ f ≫ ) — LtLL eft : ∀ ( a : A gent )( sl : Str atPr of ) ( sr : St ra tPr of ), L e adsT oL e af sl → L e adsT oL e af ( ≪ a , l , sl , sr ≫ ) — LtLRight : ∀ ( a : A gent )( sl : S tr atPr of ) ( sr : Str atPr of ), L e adsT oL e af sr → L e adsT oL e af ( ≪ a , r , sl , sr ≫ ). CoInductive Al wL e adsT oL e af : Str atPr of → Pr op := — ALtL e af : ∀ ( f : Utility fun ), AlwL e adsT oL e af ( ≪ f ≫ ) — ALtL : ∀ ( a : A gent )( c : Choic e )( sl sr : Stra tPr of ), L e adsT oL e af ( ≪ a , c , sl , sr ≫ ) → Al wL e adsT oL e af sl → AlwL e adsT oL e af sr → AlwL e adsT oL e af ( ≪ a , c , sl , sr ≫ ). SGPE CoInductive SGPE : Str atPr of → Pr op := — SGPE le af : ∀ f : Utility fun , SGPE ( ≪ f ≫ ) A EXCERPTS OF COQ SCRIP TS Nov ember 15, 2018 – 2 3 — SGPE left : ∀ ( a : A gent )( u v : U tility ) ( sl sr : Str atPr of ), AlwL e adsT oL e af ( ≪ a , l , sl , sr ≫ ) → SGPE sl → SGPE sr → s2u sl a u → s2u sr a v → ( v  u ) → SGPE ( ≪ a , l , sl , sr ≫ ) — SGPE right : ∀ ( a : A gent ) ( u v : Utility ) ( sl sr : Str atPr of ), AlwL e adsT oL e af ( ≪ a , r , sl , sr ≫ ) → SGPE sl → SGPE sr → s2u sl a u → s2u sr a v → ( u  v ) → SGPE ( ≪ a , r , sl , sr ≫ ). Nash equilibri um Definition NashEq ( s : Str atPr of ): Pr op := ∀ a s’ u u’ , s’ ⊢ a ⊣ s → L e adsT oL e af s’ → ( s2u s’ a u’ ) → L e adsT oL e af s → ( s2u s a u ) → ( u’  u ). Alice stops alwa ys and Bob contin ue s alwa ys Definition add Ali c e Bob dol ( cA cB : Choic e ) ( n : nat ) ( s : St r at ) := ≪ Alic e , cA , ≪ Bob , cB , s ,[ n +1, v + n ] ≫ ,[ v + n , n ] ≫ . CoFixp oint dolA cBs ( n : n at ): Str at := add Ali c e Bob dol l r n ( dolA cBs ( n +1)). Theorem SGPE dol A c Bs : ∀ ( n : nat ), SGPE ge ( dolA cBs n ). Alice con tinues alw ay s and Bob sto ps alw ay s CoFixp oint dolAsBc ( n : nat ): St r at := add Ali c e Bob dol r l n ( dolAsBc ( n +1 )). Theorem SGPE dol As Bc : ∀ ( n : nat ), SGPE ge ( dolAsBc n ). Alw a ys giv e up CoFixp oint dolAsBs ( n : nat ): St r at := add Ali c e Bob dol r r n ( dolAsBs ( n +1)). Theorem NotSGPE dolAsBs : ( v > 1) → ˜( NashEq ge ( dolAsBs 0)).