DEFINABLE AND CONTRACTIBLE CONTRACTS. 1. Self Referential Strategies and Reciprocity in Static Games

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1 DEFINABLE AND CONTRACTIBLE CONTRACTS MICHAEL PETERS AND BALÁZS SZENTES Abstract. Ths paper analyzes a normal form game n whch actons as well as contracts are contractble. The contracts are requred to be representable n a formal language. We prove a folk-theorem for games wth complete and ncomplete nformaton. Ths s accomplshed by constructng contracts whch are defnable functons of the Godel code of every other player s contract. We use ths to llustrate the meet the competton argument from Industral Organzaton and the prncple of recprocty from Trade and Publc Fnance. 1. Self Referental Strateges and Recprocty n Statc Games The dea that players n a game mght smultaneously commt themselves to react to ther compettors actons s heurstcally compellng. The best known expresson of ths dea s well known n the ndustral organzaton lterature (e.g. [8]) as the meet the competton clause. A smlar dea appears n trade theory as the prncple of recprocty ( [1]). Ths takes the form of trade agreements lke GATT that requre countres to match tarff cuts by other countres. Fnally, tax treates sometmes have ths flavor - for example, out of state resdents who work n Pennsylvana are exempt from Pennsylvana tax as long as they lve n a state that has a recprocal agreement that exempts out of state resdents (presumably from Pennsylvana) from state taxes. 1 One way to model recprocty s to embed t n a dynamc game. For example the tt for tat strategy n the repeated prsoners dlemma makes each player s acton depend on the acton of the other. The meet the competton argument could be supported formally by havng one frm acts as a Stackleberg leader, offerng a contract that commts t to an acton that depends explctly on the acton of the second mover. Tax recprocty could agan be accomplshed by embeddng the problem n a repeated game n whch states keep lsts of other states whom they consder to have an approprate tax treaty, deletng a state from the lst f they observe some knd of bad behavor. Our nterest here s whether ths same knd of recprocal behavor could be modelled n a completely statc game. To llustrate the problem, and our soluton, focus frst on the meet the competton argument. The Stackleberg leader, call t frm A, offers to sell at a very hgh prce provded ts compettor, frm B, also offers that hgh prce n the second round. If B n the second round offers any prce below the hghest prce, A commts tself to sell at margnal cost. If B beleves ths commtment, then hs best reply s to set the hghest prce. If the frms move smultaneously, then the logc of the argument becomes clouded. A could certanly wrte a contract that commts t to a hgh prce 1 1

2 2 MICHAEL PETERS AND BALÁZS SZENTES f B setsthesamehghprce. HoweversupposethatB s strategy s smply to set ths hgh prce and that for some reason ths s a best reply to A s contract. Then A should devate and smply undercut frm B. To support the hgh prce outcome, frm B would have to offer a contract smlar to A s n order to prevent A s devaton. A nave argument would suggest that B should smply offer the same contract as A, a hgh prce f A sets a hgh prce, and margnal cost otherwse. Casually, two outcomes seem consstent wth these contracts - both frms prce at margnal cost or both frms set the hgh prce. Ths seems to volate a farly fundamental property of game theory whch s that for each par of actons (contracts n ths case), there s a unque payoff to every player. 2 More to the pont, A s contract doesn t actually say what A would do f B offers a contract that promses to set a hgh prce unless A sets a lower prce, etc. The specfcaton of the problem tself seems to be ambguous about payoffs. The recprocal tax agreement also ncely llustrates the dffculty n a statc game. State A wants to exempt resdents of state B from state taxes provded B exempts resdents of state A from taxes. To wrte the law A exempts resdents from any state that has a recprocal agreement wth state A. The queston s what exactly s a recprocal agreement. It s clear enough what the ntenton s - create a stuaton n whch both states take the mutually benefcal acton of exemptng one another n a way that elmnates any ncentve for ether of them to devate. As mentoned above, t sn t enough to assume that state B uncondtonally exempts resdents of state A from tax because A would not longer have any ncentve to exempt state B. StateB has to have a law lke the law n state A, n other words, a recprocal agreement. It seems that to resolve ths knd of problem one needs to defne the term recprocal contract as follows: exempt f the other state offers a recprocal contract, recprocal contract don t otherwse Ths knd of defnton s famlar from the Bellman equaton n dynamc programmng where the value functon s defned n a self referental way. It s temptng to model ths n the followng nave way: start by defnng a collecton of contracts that seem economcally sensble. For example, t s reasonable that a state could wrte a contract that smply fxes any tax rate ndependent of what the other states do. Let C be the set of contracts that smply fx some uncondtonal tax rate. Append to ths set of feasble contracts the recprocal contract, call t r, defned above. Now model the set of feasble contracts as C {r}. The recprocal contract above s just r, whle otherwse means any contract wth a fxed tax rate. Defne a normal form game n whch the strateges are C {r} and declare the outcome f both states offer r to be (exempt, exempt). Vola, there s an equlbrum n whch the states mutually exempt (assumng they jontly want to). 2 One paper that allows multple payoffs to be assocated wth each array of actons s [9] who use ths approach to support equlbrum when t mght not otherwse exst.

3 DEFINABLE AND CONTRACTIBLE CONTRACTS 3 We would argue that ths s unsatsfactory for a number of reasons. Frst, t s undesrable to restrct the set of feasble conracts n order to support the outcome you are lookng for. The approach descrbed above amount to lttle more than sayng that r s the only feasble contract, then clamng t s an equlbrum for both states to offer r. A more satsfactory approach s to defne a set of actons that seem economcally meanngful, then to allow the broadest set of contracts possble. In the same manner that the value functon emerges endogenously from the economc envronment, the recprocal contract should be derved from economc fundamentals. One complcaton that makes ths problem conceptually more dffcult than the Bellman problem s that t sn t clear what the approprate set of feasble contracts should be. Clearly states can go further than smple uncondtonal tax rates. Exstng laws do allow them to make thngs contngent on laws n other states, as the recprocal tax agreement llustrates. One contrbuton of our approach s to provde a potental framework for thnkng about ths set of feasble contracts. As we descrbe n more detal below, defnable functons can all be wrtten as fnte sentences n a formal language. In a heurstc sense, ths seems exactly what the set of contracts should look lke. Of course, n the way t s used here, defnablty s very abstract. Yet t s abstract n the way that drect mechansms are abstract - t captures n a formal way ndrect contracts that look much more famlar. Second, the approach descrbed above msses the essence of recprocty whch s the nfnte regress nvolved n self referental objects. A contract that makes formal sense s the followng: exempt f other State exempts any State who exempts any State who exempts... C = don t otherwse where the statement n the top lne s repeated ad nfntum. Arguably, the contract C s a recprocal contract snce t would exempt any State offerng a recprocal contract. Yet t smply sn t feasble under the nave descrpton gven above. Of course, n the sprt of the ad hoc approach above, we could try to add the contract C to r and C. Ths approach breaks down once the game becomes asymmetrc. For example, f State A s supposed to exempt, whle state B s supposed to take some other acton, say partly exempt, then to support the rght outcome, the contracts should look somethng lke the followng: exempt f other State offers recprocal contract B recprocal contract A don t exempt otherwse and partally exempt f other State offers recprocal contract A recprocal contract B don t exempt otherwse Now the contracts are not drectly self referental, as s the Bellman equaton, nstead they are cross referental. A sngle self referental or recprocal contract smply doesn t go far enough. Furthermore, the contracts above use a blanket punshment for devatons. Desrable or nterestng

4 4 MICHAEL PETERS AND BALÁZS SZENTES equlbrum allocatons may not look lke ths. For example, n a Bayesan game between the states, State A mght want to do somethng dfferent for each dfferent thng that state B mght do. Ths mght arse f the acton and contract chosen by B convey some nformaton to A that affects A s most desrable acton. In ths paper, we offer a formalsm that provdes a way to thnk about self-referentalty and recprocty. Suppose there are N players n a normal form game n whch each player has a countable number of actons. Endow players wth a formal language that they can use to wrte contracts and thnk of the set of feasble contracts as the set of fnte sequences of characters n ths formal language. It s well known that there are bjectons from the set of fnte texts nto N. One such a mappng s called the Godel Codng. Provded the language ncludes all the natural numbers and the usual arthmetc operatons, t s possble for players to wrte contracts that are defnable functons from N N 1 nto that player s acton space. Snce defnable functons can be wrtten as fnte sequences of characters n the language, they have Godel codes assocated wth them. Hence we could nterpret the defnable functons as contracts that make the players acton depend on the Godel code of the other player s contract. To make the argument easer to relate to conventonal contract theory, we assume below that the contract space for each player s the set of defnable functons from N N 1 nto the subsets of the player s acton spaces. Implctly, ths approach makes t possble for players to offer any fnte text as a contract. We defer dscusson of ths pont to later n the paper. Every defnable functoncanbeassocatedwthaunquenteger, and conversely f the nteger n s assocated wth a defnable functon, then t s assocated wth a unque text. Now for each array of functons chosen by the players, compute the Godel Code of each such functon. Ft the codes of the other players strateges nto each player s strategy to determne aunque subset of actons for every player. Then, players smultaneously take actons from these subsets. Our objectve s to try to characterze the set of equlbra of ths game. To see how t works, we mght as well restrct attenton to a two player prsoner s dlemma. Call the players 1 and 2, and the actons C and D wth the usual payoff structure n whch D s a domnant strategy and both players are strctly better off f they both play C than they are f they both play D. Astrategyc for a player s a defnable functon from N to {C, D}. One obvous equlbrum of ths game occurs when both players use a strategy that chooses acton D no matter what the Godel code of the other player s strategy. Every defnable strategy has a Godel code. Let [c] denote the Godel code of the strategy c and refer to [c] as the encodng of c. Snce the Godel codng s an njecton from the set of defnable strateges to the set of ntegers. For any par of strateges c 1 and c 2, the acton (C or D) takenby player 1 s c 1 ([c 2 ]) and smlarly for player 2. Snce every par of actons determnes a payoff, ths procedure assocates a unque payoff wth every par of strateges. There are many thngs that aren t defnable strateges that also have Godel codes. We want to make use of some of these other thngs. In partcular, we want to use defnable strateges wth free

5 DEFINABLE AND CONTRACTIBLE CONTRACTS 5 varables. For example, there s a subclass of defnable strateges for player 1 defned parametrcally by C n = x, γ x (n) = D otherwse. Ths s smply a defnable strategy wth a free varable x, wherex s the target code of the other player s strategy that wll trgger the cooperatve acton. Defnable strateges wth free varables are also defnable, and so they too have Godel codes. The strategy wth free varable that we want s a slght modfcaton of the one above, n partcular h C n = hx (x), (1.1) c x (n) = D otherwse. The mappng <x> (x) s the composton of two functons. Frst, the functon hx s the nverse operaton to the Godel codng. That s, <n>s the text whose Godel code s n. Second,fφ s a text wth one free varable, then φ (n) s the same text where the value of the free varable s set to be n. Hence, f n s a Godel code of a defnable strategy h wth one free varable, then <n> (n) s tself a defnable strategy (wthout a free varable). hn (n) s just the Godel code of whatever ths defnable strategy happens to be. Notce that n ths case, [hx (x)] won t be equal to x snce a defnable strategy must have a dfferent Godel code from a defnable strategy wth one free varable because of the fact that the Godel codng s njectve. We want to defne a strategy by fxng a value for x n (1.1). In partcular, the value of x we are nterested n s [c x ]. Snce [c x ] s the Godel code of a strategy wth a free varable, the rght hand sde of (1.1) requres that we decode [c x ] to get c x,thenfx x at [c x ] to get the contract c [cx ]. Puttng all ths together gves C n = c [cx] c [cx] (n) = D otherwse So C [c 2 ]= c [cx ] c [cx ] ([c 2 ]) = D otherwse s a the recprocal or self-referental contract mentoned above. Now we smply need to verfy what happens when both players use strategy c [cx ]. If player 2 uses strategy c [cx ], then [c 2 ]= c [cx ], whch evdently trggers the cooperatve acton by player 1. The same argument apples for player 2. Player 2 can devate to any alternatve defnable strategy c 0 that she lkes. Snce every defnable strategy has a Godel code, the reacton of player 1, and consequently both players payoffs arewelldefned. As the Godel codng s njectve, c 0 6= c [cx] mples the Godel code of c 0 s not equal to c [cx], and the devaton by 2 nduces 1 to respond by swtchng from C to D. Notce that ths argument makes use of an encodng of the strategy wth free varable c x,whch sn t a defnable strategy. One mght have expected the target code number to be assocated

6 6 MICHAEL PETERS AND BALÁZS SZENTES wth a strategy nstead of a strategy wth a free varable. For example, t seems that to enforce cooperaton there needs to be a defnable strategy c wth encodng [c ]=n such that c C [c 2 ]=n = D otherwse Of course, for arbtrary n t wll be false that [c n ]=n. Ths leads to a fxed pont problem that, n fact, does not have a soluton n general. More generally, one could try to construct a self-referental contract by fndng a fxed pont of the the followng problem. For each n, consder C f [c 2 ]=g(n), c n ([c 2 ]) = D otherwse, where g s a defnable functon. If there exsts an n such that [c n ]=g (n ),thenc n s obvously a self-referental contract. Indeed, what we dd above s that we chose g (n) to be [< n> (n) ] and showed that n =[c x ] s a correspondng fxed pont. Toseehowthestrategywthfreevarablec x works, recall the recprocal tax agreement exempt other State offers recprocal contract recprocal contract don t exempt otherwse and ts recursve counterpart exempt f other State exempts any State who exempts any State who exempts... C = don t otherwse The recprocal contract s c [cx] and the statement other state offers recprocal contract s [c 2 ]= c[cx]. State A wants to exempt any state whose law fullflls a condton. For example, f the condton t s lookng for s that the other state smply exempts State S, then t would compute the Godel code n 0 =[ n; c (n) =C] then use the strategy C [c 2 ]=n 0 c n0 = D otherwse If t does that, then t can t be an equlbrum as explaned above. So what t needs to do s to exempt any State whose law fullflls a condton that exempts any state whose law fullflls a condton. For example, f t wanted to exempt State B f and only f State B s law exempts state A f and only f State A uncondtonally exempts state B, then t would adopt the strategy c [cn0 ], and so on. Ths s where the partcular structure of the contract c x comes nto play. Recall that h C n = hx (x), c x (n) = D otherwse.

7 DEFINABLE AND CONTRACTIBLE CONTRACTS 7 It specfes exempton f and only f a condton s fullflled, but t doesn t seem to specfy what the condton s. However, t does requre that whatever the condton x s, f x n turn depends on a condton, then the condton that t depends on must be the same as the condton tself. To see f x depends on a condton, we frstdecodetandfnd the statement hx that the nteger x corresponds to. Then f t depends on some condton, we requre that that condton be x tself, whch s the meanng of hx (x). Sonowwecandothenfnte regress. State A adopts a law that exempts state B f and only f the Godel code of State B s law s c [cx]. Ths means that state B s law must be c [cx], orthatb exempt A f and only f the Godel code of State A s law s c [cx],.e., the same condton that A requres. 2. Lterature The approach we develop here s not the only way to sustan cooperaton wthout repetton. An alternatve logc has been developed n the theory of common agency ([6] or [7]) n whch punshments are carred out through an agent. In problems n whch prncpals nteract through many agents, ths logc can be used to prove folk theorems. The argument appears n a recent paper by [5] and more generally n the workng paper by [11]. The basc logc n the latter paper works as follows - each prncple offers a contract wth a message space consstng of all the actons that he could take. He then asks the agents to tell hm what to do. If all but one of the agents who nteract wth hm names the same acton, the prncple takes that acton, otherwse he takes some default acton. It pays the agents to agree wth everyone else, because they expect everyone to agree and accomplsh nothng by devatng. If all prncpals offer ths contract, then every agent makes the same report to any gven prncple tellng the prncpal to take some acton that, along wth the nstructons gven to the other prncpals, provdes the prncple a payoff at least as large as hs mnmax payoff. If any prncple devates and offers any alternatve mappng from messages to outcomes, the agents nstruct the other prncpals to mnmax the devator. In a very specalzed envronment, [5] makes an even smpler argument. Workng n an envronment n whch agents take actons on behalf of prncples (as n, for example, [3]), Katz mposes enough quas-lnearty and separablty such that for any fxed acton, the prncpals can offer the agent a contract such that the optmal effort for the agent under that contract mplements the desred acton and provdes the agent exactly hs reservaton payoff. Now take any collecton of actons for the prncples that provdes each prncpal at least hs or her mnmax payoff. Eachprn- cpal then offers the agent a contract wth a bnary message. If the agent sends the message 1, the prncple offers the agent a contract that mplements hs part of the collusve outcome. If the agent sends the message zero, the prncple offers a contract that mplements the acton that mnmaxes the other prncpal. Each agent sends hs or her prncpal the message 1 as long as hs prncple offers ths contract - devatons cause agents to send message zero, and lead to punshment. One objecton to contracts lke ths, and the folk theorems that they generate, s that they rely heavly on agents coordnatng ther messages. Yamashta s paper shows ths n the most strkng

8 8 MICHAEL PETERS AND BALÁZS SZENTES way. Not only must agents coordnate on the same message, but the must also coordnate on the message that the prncpals beleve s drvng the game before they offer ther contracts. Not only s such a strong relance on equlbrum selecton senstve to common knowledge assumptons, t s also very senstve to possble colluson or unmodelled communcaton between agents. All the arguments n ths lterature bascally allow agents to tell prncpals what they should do. Katz s argument deals wth a somewhat smpler envronment n whch a sngle agent s hred to carry out an acton on the part of the prncpal. The envronment satsfes the assumptons n [4], so that coordnated actons by the prncpals can be carred out by havng each prncpal offer the agent a menu of optons, as they would n common agency. Agents punsh devatons by changng the choce they make from ths menu. Snce agents must be ndfferent between the choces n the menu to make ths work, the envronment s very restrcted. Indfference agan means that prncpals must rely on agents choosng approprately among actons to whch they are ndfferent. The lterature that most closely resembles what we do uses more general commtment devces than those that can be mplemented by hrng and communcatng wth self nterested agents. The general approach s llustrated n [5] who essentally treat commtment devces as f they were messages n a mechansm desgn problem. Players n a game submt ther commtment devces to a process that acts as the prncpal would n mechansm desgn, and resolves the commtment devces to an unambguous outcome. It elmnates (by defnton) any problem wth crcularty or ambguty of contracts that we dscussed above. They llustrate that for any jont dstrbuton of actons for the players that gves each player hs mnmax payoff, the prncpal can desgn a set of commtment devces such that s s an equlbrum for players to nput a set of devces that support that jont dstrbuton as an equlbrum outcome. We focus here on a dfferent ssue, whch s to descrbe a set of commtment devces that are perhaps more natural than those descrbed n [5] and whch wll work no matter what game the players are nvolved wth. In ths sense, our paper s more closely related to the paper by [10]. He models the set of commtment devces as a set of computer programs. The programs are submtted to a central computer whch then uses them to determne an array of actons for the players. The mportant pont that he llustrates s that these computer programs can use other players programs as bts of data they can use to commt themselves to actons. He shows that jont dstrbutons of actons n whch all players actons are ndependent can be supported as equlbrum outcomes n the game n programs provded that each player receves hs mnmax payoff. Theway ths s done s to have each program mplement some acton, say the cooperatve acton n the prsoner s dllemma, provded that the other player s program s syntactcally dentcal to hs. To be syntactcally dentcal, the other program must specfy cooperaton f the other program s syntactcally dentcal, etc. Our approach s qute smlar to ths. Intutvely we replace the computer that Tennenholtz uses to mplement the outcome wth a legal system. Beyond ths, there are a couple of more mportant dfferences. Frst, the problem of desgnng commtment devces s a problem n mechansm desgn.

9 DEFINABLE AND CONTRACTIBLE CONTRACTS 9 If the mechansm desgner can compel partcpaton, then there s no real need for commtment devces at all, at least wth perfect nformaton. The mechansm desgner can smply nstruct players whch actons to use. Commtment devces only play a role n stuatons n whch the mechansm desgner s unable to force players to take actons, and nstead has to reply on the players themselves to enforce cooperaton. We accomplsh ths by allowng players to wrte contracts that specfy a subset of all ther feasble actons from whch they must ultmately make a choce. We then assume that f more than one player has specfed a contract that leads to a subset, then actons wll be selected from these subsets non-cooperatvely. Contracts unquely determne sets wthout any crcularty or ambguty, whle actual outcomes mght requre addtonal choces. Ths devce means that a player can wrte a contract that specfes that he reserves the rght to select hs acton ex post no matter what contracts the others use. Ths ensures that all players are always wllng to partcpate n the contractng process. Of course, ths approach also allows partcpants to commt themselves to the way they wll respond to non-partcpants. Ths s the sense n whch the mechansm desgner uses the players themselves to enforce cooperaton. Secondly, we deal wth randomzaton dfferently than [10]. As he does, we allow each player to use the syntax of the other player s contract to determne hs own acton. To make t possble to check ths syntax, we don t allow players to wrte contracts that nvolve nfnte expansons lke π, or to use rratonal numbers that have to be checked aganst an uncountable set of possbltes. Instead, we requre that contracts consst of a fnte set of words (n a language that may contan a countable set of words). Snce we ultmately requre players to take actons ndependently, as Tennenholtz does, ths restrcts us to contracts that always specfy pure actons. Our characterzaton theorems are restrcted to pure actons as a result. We don t vew ths as a crtcal restrcton for two reasons. Frst, we can vew the set of pure actons over whch our players contract to be a fnte approxmaton of the uncountable set of mxtures over some underlyng set of pure actons. In ths sense all of our theorems can be nterpreted as approxmaton theorems n an approprate way. Secondly, f the contracts are vewed as messages n an underlyng mechansm desgn problem, for example by vewng the mechansm desgner as the government, and the mechansm as the law, then the law could n prncple use the government to further correlate the actons of the agents usng some publc randomzng devce. Fnally, we show how to apply the approach to problems n whch the players are mperfectly nformed, or n whch they nteract wth agents who have prvate nformaton. 3. The Language and the Gödel Codng We consder a formal language, whch s suffcently rch to allow ts user to state propostons n arthmetc. Furthermore, the set of statements n ths language s closed under the fnte applcatons of the Boolean operatons: q,, and. Ths mples that one can express, for example, the followng statement: n, x, y, z {[(n 3) (x 6= 0) (y 6= 0) (z 6= 0)] (x n + y n 6= z n )]}.

10 10 MICHAEL PETERS AND BALÁZS SZENTES In addton, one can also express statements n the language that nvolve any fnte number of free varables. For example, x s a prme number s a statement n the language. The symbol x s a free varable n the statement. Another example for a predcate that has one free varable s x <4. One can substtute any nteger nto x and then the predcate s ether true or false. Ths partcular one s true f x =0, 1, 2, 3 and false otherwse. Let L be the set of all formulas of the formal language. Each of ts element s a fnte strng of symbols. It s well known that one can construct a one-to-one functon L N. Let[ϕ] be the value of ths functon at ϕ L, and call t the Gödel Code of the text ϕ. Defnton 3.1. The functon f : N k 2 N s sad to be defnable f there exsts a frst-order predcate φ n k +1free varables such that b f (a 1,..., a k ) f and only f φ (a 1,..., a k,b) s true. In the defnton, the mappng f s a correspondence from N k to N. Of course, f f (n) s a sngleton for all n N k,thenf s a functon. We llustrate the prevous defnton wth an example. Example. Consder the followng functon defned on N: ( 0 f a s an even number, f (a) = 1 f a s an odd number. We show that ths functon s defnable by constructng the correspondng predcate φ. φ (x, y) {{y =1} {y =0}} { z :2z = y + x}. Notce that φ ndeed has two free varables. (The varable z s not free because there s a quantfer front of t.) The frstpartofφ states that y s ether one or zero. The second part says that x + y sdvsblebytwo. Notcethatf (a) =0f and only f φ (a, 0) s true. To see ths, frst notce that φ (a, b) s false whenever b/ {0, 1}. (Ths s because the frst part of φ requres b to be zero or one.) If b =0then φ (a, 0) s ndeed true. If b =1, then the second part of φ becomes false because a + b s an odd number. 4. A Normal Form Contractng Game Suppose there are m players. Each player has a fnte acton space A. The payoff of Player s u (a 1,...,a m ). We use the conventonal notaton that u (a,a ) s the payoff to player f he takes acton a whle the other players take acton a. Each player smultaneously submts a contract, whch s a defnable correspondences from N m to 2 N, where defnable s to be understood n the sense of Defnton 3.1. The correspondences, as descrbe above, unquely and unambguously determne an array of sets from whch players then select actons non-cooperatvely. At stage two, players take actons smultaneously from subsets of ther actons spaces. These subsets are determned by the frst-stage contracts as follows. If at stage one player j submtted contract c j (j =1,..., m), then player can only take acton a k at stage two f k c ([c 1 ],..., [c m ]). We restrct attenton to pure-strategy subgame perfect equlbra of ths game.

11 DEFINABLE AND CONTRACTIBLE CONTRACTS 11 Our objectve s to prove a folk theorem for ths contractng game. The lowest payoff for any player n any pure strategy equlbrum of the game n whch players choose actons from A s u = mn a A max u (a,a ), a A Let a j be any one of the actons that j uses to attan hs mnmax payoff. Letusfx anactona j for player, suchthat, a 1 j1,..., a m j m arg mn u j a a j,a j. That s, a j s the acton that player uses to punsh player j. In addton, defne =1for all {1,...,m}. Theorem 4.1. Let ª a 1 k 1,...,a m k m be any array of actons. These actons are supportable as an equlbrum outcome n the contractng game wth pure strategy SPNE f and only f u (a) u for each. Before we proceed wth the proof of the theorem, we recall two peces notatons from the ntroducton. Frst, f n N then <n>denotes the text whose Gödel code s n. That s, [< n>]=n. Second, for any text ϕ, letϕ (n 1,...,n k ) denote the statement where f the letter x stands for a free varable n ϕ then x s evaluated at n n ϕ for =1,...,n. For example, f ϕ s x>yand n =2,thenϕ (n) s 2 >y. Consder now the followng text n one free varable: <x> (x). One can evaluate ths statement at any nteger. Snce the Godel codng was a bjecton <n>s a text for each n N. In addton, ϕ (n) s defned for all ϕ and n. In addton, t s a well-known result n Mathematcal Logc, that f f (n) = <n> (n),thenf s a defnable functon. Proof. Frst, we prove the only f part. Fx an equlbrum n the contractng game. Let c j denote the equlbrum contract of player j (j =1,..., m) andletu denote player s equlbrum payoff. Notce, that player can always offer a contract that does not restrct hs acton space. That s, he can offer c : N m N, suchthatc (n 1,..., n m )=N for all (n 1,..., n m ) N m. The contract c s obvously defnable. 3 We show that f u <u, player can proftably devate at the frst stage by offerng c nstead of c.letec j = c j f j 6= and ec = c. LetA e n o j = a j k : k ec j ([ec 1 ],...,[ec m ]).That s, Aj e s the acton space of player j n the subgame generated by the contract profle (ec 1,..., ec m ). Also notce that A f = A. The payoff of player n any pure strategy equlbrum of ths subgame s weakly larger than mn a ea max u (a,a ) a A mn a A max u (a,a ). a A The weak nequalty follows from A e j A j for all j. Therefore, player can always acheve hs pure mnmax value by offerng the contract c. 3 For example, the predcate {x 1 = x 1 }... {x m = x m } {y = y} defnes c. Thats,forally N the predcate s true no matter how the free varables are evaluated.

12 12 MICHAEL PETERS AND BALÁZS SZENTES For the f part, consder the followng contract of Player, c x,x,nm free varables: ³n c c j o x 1,...,x m = j6= (4.1) ( k f k : <x k > (x 1,...,x m ) 6= c k ª 6= 1, f k : <x k > (x 1,...,x n ) 6=[c k ] ª = {j} j Ths contract wth free varables s a defnable functon wth free varables from N m 1 to N as long as the actons are replaced wth ther ndces. The expresson (4.1) s not a contract, but rather a contract wth free varables. Each such expresson has a Godel code, so let γ = n c x 1,...,x m. The functons c γ 1,...,γ m have no free n o o m varables, so they consttute a set of contracts. We wll now show that c γ 1,...,γ consttutes m =1 an equlbrum profle of contracts whch support the outcome ª a 1 k 1,...,a m k m. Frst observe what happens when all players use contract c γ 1,...,γ m.notcethat ³n c c j ( o k f k : <γ γ 1,...,γ m = k > (γ 1,...,γ ) m 6=[c k ] ª 6= 1, j6= j f k : <γ k > (γ 1,...,γ ) m 6=[c k ] ª = {j}. Player needs to check whether the Godel code of< γ k > (γ 1,...,γ m ) s equal to the Godel code of c k γ 1,...,γ m. The nteger γ k s the Godel code of the contract wth free varable c x 1,...,x m. Player s contract says to take ths contract wth free varable, fx thefreevarablesatγ 1,..., γ m (whch gves the contract c k γ 1,...,γ m ),thenevaluatetsgodelcode. Thsswhatstobecomparedwth the Godel code of the contract offered by k. Of course, these are the same. Snce ths s the case for all m 1 of the other players, player ends up takng acton a. So these contracts support the outcome we want f everyone uses them. Player j can devate to any defnable contract mappng N nto N. However, any such contract wll have a dfferent Godel code, and so wll nduce the punshment a j from the other players. ª6=j Snce u j (a) u j ths devaton wll be unproftable. Onemghtarguethatrestrctngthespaceofcontractstobedefnable functons of Godel codes s both arbtrary and unnatural. Indeed, there s no reason for a judge to nterpret a contract as a descrpton of a mappng from the Godel codes of the contracts offered by the other players to the actons space of the player. For that matter, the judge mght not even know about the Godel codng. It s mportant to note that the salent feature of defnable contracts s that they can be wrtten as texts that use a fnte number of words n a formal language. The set of fnte texts seems a very natural descrpton of the set of feasble contracts. In fact, from ths perspectve t seems that any reasonable descrpton of the set of feasble contracts should allow any such text. The complcaton wth such a broad descrpton of the set of contracts s that to properly defne a game, one must fully descrbe the mappngs from profles of texts nto payoffs. Many texts wll be complete nonsense and some modellng decson has to be taken about how these would translate nto actons and payoffs. The contracts that we specfy above are defnable texts that have two advantages n ths regard. Frst, snce every fnte text has a Godel code, they te down the

13 DEFINABLE AND CONTRACTIBLE CONTRACTS 13 acton of the player who offers such a contract even f the other players n the game offer contracts nvolvng texts that make no economc sense. Furthermore, f all players offer contracts from the set we specfy, an outcome for every player s unquely determned. So no matter how ambguous outcomes are set when players offer non-sense texts as contracts, the equlbrum outcome we descrbe above wll persst. From the perspectve of the judge who has never heard of a Godel code, Theorem 4.1 and the theorems that follow have a knd of normatve mplcaton. Ths s smply that self or cross referental contracts that to nvolve an nfnte regress can nonetheless be unambguously wrtten usng fnte texts. Generalzatons. Everythng about ths theorem nvolves pure strateges. Ths mposes lmts on ts applcaton. Next, we dscuss how to extend ourresulttothecasewhenplayerscanmx over ther restrcted acton space at the second stage of the game but cannot randomze over the contracts they offer at the frst stage. Allowng such mxng expands the set of payoff profles that can be supported by equlbra for two reasons. Frst, snce players can randomze certan convex combnatons of payoff profles can now be supported. Second, players can use mxng when punshng a devator, and hence the mnmax value of the players wll be smaller. Formally, for all S = S, S A,defne a game, G S, where the acton space of player s S, and the payoff functon of player s u restrcted S. LetE (S) denote the set of mxed equlbra n G S.Defne the mnmax value of player, u,as Z u = mn max mn u (a) dσ (a). S A S A σ E(S A) S = j6= S j The dea s that n the contractng game, players can restrct ther acton spaces arbtrarly, hence, when they punsh player they can choose S arbtrarly. On the other hand, ther second-stage actons must be best responses, and that s why we have to consder equlbrum payoffs nthe restrcted game. An argument dentcal to the proof of Theorem 1 shows that the random allocaton σ (A) can be supported as an equlbrum f () S A for all, suchthatσ E ( S ),and () R u (a) dσ (a) u for all. What happens f players are allowed to randomze over the contracts they offer? It s possble to show that part () can be completely relaxed. That s, the dstrbuton over the outcomes does not have to be an equlbrum n a G S, and t does not even have to be generated by ndependent randomzatons of the acton spaces of the players. The constructon of mxed equlbra n our contractng game that supports correlated outcomes s entrely based on Kala et.al. (2008). The authors consder a two-person game smlar to ours. Instead of takng actons, players submt commtment devces from a certan set. The devces then determne the acton profle. The authors construct a set of devces such that any ndvdually ratonal correlated outcome can be mplemented as a mxed equlbrum n the game. (That s, although the players mx ndependently

14 14 MICHAEL PETERS AND BALÁZS SZENTES over ther devces, the dstrbuton over the actons profles wll be correlated.) It s not hard to extent ther results to our model and obtan the followng theorem. Theorem 1. Suppose that σ (A), andσ (a) Q for all a A. Theσ can be supported as a mxed-strategy equlbrum outcome n the contractng game f and only f R u (a) dσ (a) u for all {1,..., m}. Another queston s why we use defnable functons as opposed to programs or Turng machnes. One mght want to requre that the contracts must be computable and assume that the set of avalable contracts s the set (or a subset) of Turng machnes. In such a model, f player ( =1, 2) chooses machne τ,thenτ runs on the descrpton of τ j, and the output wll be a subset of the acton space of player. It s well-known, that one can construct self- and cross-referental contracts (machnes) n ths space too. 4 In fact, ths constructon s essentally dentcal to our constructon of cross-referental defnable functons. Most mportantly, the equlbrum contracts we construct to support ndvdually ratonal allocatons are, n fact, recursve functons, and hence they are computable by Turng machnes. Therefore, f the reader nssts on computablty, he can restrct attenton to the space of Turng machnes. There are, however, several advantages of our approach over modellng contracts wth Turng machnes. Let us explan. 1. Turng machnes do not always halt. Therefore, t s not clear how one can defne our contractng game properly. In partcular, t s not streghforward how to defne the restrcton on the acton space of a player, f hs machne does not halt. One mght suggest that f a player submtts a Turng machne that does not halt, then defne hs second stage acton space to be the whole space (or a default subset). We thnk that such a defnton mght be arbtrary. In addton, the problem whether or not a Turng machne halts s an undecdable problem. That s, there s no Turng machne whch can determne whether a player devated or not. An alternatve way to handle the haltng problem s to restrct the space of Turng machnes to be the set of machnes that always halts. We fnd such restrctons also arbtrary. Instead of restrctng the space of recursve functons, we expanded t to be the set of defnable functons and avoded the haltng problem that way. 2. Another problem wth Turng machnes s that they can only condton on the actual descrpton of the machnes submtted by the other players but cannot condton on the functons what the machnes compute. Take the example of the prsoner dlemma. It s possble to construct a Turng machne, τ, suchthat ( C f [τ 2 ]=[τ] τ ([τ 2 ]) = D otherwse. 4 Such machnes were constructed even n the context of Game Theory, see Anderln 1990 and Cannng 1992.

15 DEFINABLE AND CONTRACTIBLE CONTRACTS 15 The problem s that f player 2 submtts a machne, say τ 0, whch s computatonally equvalent wth τ, but has a dfferent descrpton, then player 1 would defect. In fact, t s not possble to construct a machne whch does not suffer from ths problem. That s, the equlbrum contract s senstve to the way t s wrtten. A player does not only requre the other player to have the rght ntentons, but also requres hm to express hmself n a unque way. Ths feature makes us doubt whether machne contracts s the rght way of modelng contractble contracts. Weavod suchproblemswthdefnable functons. Indeed, t s possble to express contracts that do not condton on the actual way the other contract s wrtten, but on the functon tself that the other contract descrbes. Consder ( C f c c 1 ([c 2 ]) = 2 c 2, D otherwse. The contract c 1 s obvously defnable, but does not condton on the actual form of c 2.Aslongas c 2 represents the same functon as c 2, cooperaton s prescrbed. 5. Contractng n a Bayesan Envronment Ths secton shows how to extent the result of the prevous secton to games wth ncomplete nformaton. We shall show that the set of allocatons that can be supported by contractble contracts s lmted because the contracts reveal publc nformaton about the players and ths makes t harder to punsh devatons. The model s the same as n the prevous secton, wth the addton of player types. There are n players. Player s actons space s a fnte set denoted by A.Eachplayer has a type t drawn from a fnte set T. The jont dstrbuton types s common knowledge. The payoff of player s u (a,a,t) where t T 1 T n. Our goal s to characterze the set of equlbra of the contractng game (to be defned formally later) n ths physcal envronment. Our strategy s to defne a mechansm desgn problem, and then to show that the set of allocatons that can be mplemented by these mechansms are dentcal to the set of equlbrum outcomes n the contractng game. To ths end, we frst descrbe a class of mechansms. Publc Message Mechansms. Consder the followng class of three-stage mechansms. Frst, players smultaneously decde whether or not to partcpate n the mechansm. Those players who decde to partcpate send publc messages to the mechansm desgner (MD). Player s message space s a countable set. At the same tme, players who do not partcpate publcly submt a functon whch wll mpose a restrcton on ther acton spaces as a functon of the publc messages sent by the partcpatng players. (It s mportant to note that these functons must be submtted before players can observe the publc messages.) At the second stage, the MD can arbtrarly restrct the acton spaces of those players who have decded to partcpate at the frst stage, as a functon of the messages. These restrctons, however, cannot depend on the functons submtted

16 16 MICHAEL PETERS AND BALÁZS SZENTES at the frst stage by the nonpartcpants. Fnally, at the thrd stage, players take actons from ther restrcted actons spaces smultaneously. We restrct attenton to determnstc mechansms and pure-strategy Weak Perfect Bayesan Nash Equlbra. By standard arguments n mechansm desgn, wthout loss of generalty, one can restrct attenton to mechansm-equlbrum pars n whch () each player partcpates, () the messages space of player s the set of elements of a certan partton of hs type space, and () each player reports the element of hs partton whch contans hs true type at stage one. In what follows we characterze the allocatons that can be mplemented by these mechansms by constrants. There are three sets of constrants. The frst one s the partcpaton constrants whch guarantees that each player prefers to partcpate n the mechansm ndependently of hs type. The second one s the ncentve compatblty constrants whch guarantees that each player reports the element of the partton of hs type space whch contans hs type. Fnally, the thrd sets of constrants guarantee that each player takes an acton at stage three whch s a best response aganst the strateges of the others. Before we proceed wth characterzng these constrants, we ntroduce some notatons. Let τ : T 2 T denote a partton of player s type space. That s, t τ t for all t,and f t τ t 0 then τ t = τ t 0. Let τ, τ,andτ j denote n =1 τ, j6= τ j,and k6=,j τ k respectvely. Let r (t) A denote the restrcted acton space of player f each player partcpates, and the message sent by player j s τ j t j. That s, r must be measurable wth respect to τ. Ths means that r (t) =r (t 0 ) whenever τ t = τ t 0 for all. Smlarly, r j t j denotes the restrcton on the actons space of player f all players but player j partcpate, and the message sent by player q s τ q (t q ).Thefunctonr j t j s measurable wth respect to τ j. Fnally, let n F τ = f f : τ T 2 Ao, where τ T = τ t : t T ª. The set F τ s the acton space of player at stage one f he does not partcpate n the mechansm. If player submts f ( F τ ) and player j reports τ j t for j 6=, thenplayer s restrcted acton space at stage three s f τ t. Let s denote the strategy of player at stage three f each player partcpates, that s, s : T T A,suchthats t,t r (t) for all t, ands s measurable wth respect to τ. That s, s t,t = s t,t 0 f τ t = τ t 0. Smlarly, s j denotes the strategy of player at stage three f all players but player j partcpates. That s, s j : T T j F j A such that s t,t j,f j r j t j,ands j s measurable wth respect to τ j. Best Response. In the last stage, players optmally choose ther acton gven the others strateges, and truthful reports at stage two. That s, for all, t T : (5.1) s (t) = arg max a r (t) E t u a, s (t),t : t,τ t.

17 DEFINABLE AND CONTRACTIBLE CONTRACTS 17 Ths constrant ensures that s (t) ª s a Bayesan Equlbrum n the game where player, wth type t,observesτ (t). Incentve Compatblty. At stage two, each player reports the element of the partton of hs type space truthfully, gven the strateges n the last stage and that everybody else reports truthfully. That s, for all, t, and t 0 T : (5.2) E t u (s (t, τ),t):t µ E t max E t u a, s t 0,t,t : t,τ t : t. a r (t 0,t ) Partcpaton Constrant. At stage one, players prefer to partcpate to optng out, gven that everybody else partcpates, everybody reports truthfully, and the strateges at the fnal stage. To characterze ths constrant, we frst compute the payoff of player wth type t f he does not partcpate. For all, t consder µ max E t max E t f F u a, s a f (τ (t t,f,t : t,τ t : t. )) Let us denote the value of ths problem by u t. Then, player wth type t prefers to partcpate f and only f (5.3) E t u (s (t, τ),t):t u t. Let α : T A be an allocaton. Ths allocaton can be mplemented by the mechansm f there are parttons of the type spaces, {τ },restrctonsofthemd, ª r,rjª,andthestrateges s,s j such that s (t) =α (t) and the three sets of constrants (5.1), (5.2), and (5.3) are satsfed. The Contractng Game. The contractng game s the same as n the prevous secton. The game has two stages. In the frst stage, players offer contracts smultaneously. A contract of player s a defnable functon from N n to subsets of A.Theth coordnate of the doman s the Godel code of the defnable functon offered by the player. At stage two, players take actons smultaneously from the subsets of ther actons spaces whch s specfed by the contracts. If player q offers a defnable functon c q,thenc c 1,..., [c n ] A s the subset of the acton space of player pnned down by the contracts. Of course, at the second stage, players make nferences about the types of the other players from the contracts they have submtted. We restrct attenton to pure-strategy Weak Perfect Bayesan Equlbra of ths game. Theorem 2. An allocaton s mplementable wth the publc-message mechansm f and only f t s mplementable as an equlbrum n the contractng game.

18 18 MICHAEL PETERS AND BALÁZS SZENTES Proofofthe f part. Frst, let us fx a mechansm and an equlbrum of t. Let us denote the parttons of the type spaces generated by the mechansm by {τ }, the restrctons by ª r,rj,,j and the strateges of the players by ª s,s j. Consder now the followng contract n T free,j varables: c t ³ x t j c 1,..., c n j j,t j rj (t) f k!τ k τ T k s.t. <x t k k > (x) = c k f t k τ k, = r j t j f k k τ T k s.t. <x t k k > (x) = c k ª f t k τ k = j, A otherwse and f k +1>kf k H (t ), ³ where x denotes x t j j and H t = j : τ t j = τ t ª. The last statement s n the thrd lne j,t j s always true. Such a statement, however, makes t possble that a player wth two dfferent types offers two dfferent but computatonally equvalent contracts. Let γ t denote the Godel Code of ths contract and let γ = γ t. The equlbrum contract offered by player wth type t wll,t be: c t γ.then c 1,..., c n = c t γ Notce that <γ tq q (5.4) = rj (t) t j f k!τ k τ T k s.t. <γ t k k > (γ) = c k f t k τ k, f k k τ T k s.t. <γ t k k > (γ) = c k ª f t k τ k = j, A otherwse and f k +1>kf k H (t ), r j > (γ) = c tq γ. Therefore, the prevous contract can be rewrtten as c 1,..., c n c t γ rj (t) t j f k!τ k τ T k s.t. c t k γ = c k f t k τ k, f k k τ T k s.t. ª c t k γ = c k f t k τ k = j, A otherwse and f k +1>kf k H (t ), r j Next, we specfy the strateges of the players n at the second stage. If for all j there s a t j T j such that player j offeres a contract c t j γ,thenplayer takes acton s (t). Suppose now that one player devated, say player k, and he offered a contract c k, and player j offered c tj γ for all j 6= k. Defne f c k : τ as follows: (5.5) f c k τ k t k ³ = c k c k, hc t k γ, h where c t k γ denotes the vector of the Godel codes of players other than k. Then player í s strategy s s k t,t k,f k. Notce that by (5.4) these second-stage strateges are consstent wth the restrctons mposed by the contracts. We shall argue that the strateges descrbed above consttute an equlbrum n the contractng game. Frst, the strateges s ª are optmal n the second stage by (5.1). Hence, we only have to

19 DEFINABLE AND CONTRACTIBLE CONTRACTS 19 show that players do not have ncentve to devate at the contractng stage. Notce that f a player does not devate then her payoff s the same as n the mechansm. Suppose now that player wth type t offers a contract c whch s dfferent from c t γ. We shall consder two cases. Case 1: c = c t0 γ but τ t 6= τ t 0. Then, by the frst lne of (5.4) and by the defnton of s jª, the devator s j payoff cannot exceed the payoff of player wth type t n the mechansm f she decded to report τ t 0 nstead of τ t n the frst stage. Snce (5.2) holds, such a devaton s not proftable. Case 2: c 6= c t0 γ for all t 0. (That s, player offers an off-equlbrum contract.) Then, by the second lne of (5.4), the restrcton on player j s acton space s r j t j (j 6= ). The restrcton on ³ h player s acton space s c [c], c t γ = f c τ t. Thats,therestrctonsarethesameas f n the mechansm player wth would have not partcpated and submtted f c at the frst stage. In addton, the actons of players at stage two are also the same as n the mechansm at stage three. Hence, by (5.3), such a devaton s not proftable. The dffculty of provng the only f part of the theorem s the followng. In the mechansm, even f a player does not partcpate, the MD can restrct the acton space of the partcpatng players only as a functon of ther messages. But these restrctons cannot depend on the nonpartcpant player s functon he submts at the frst stage. Ths lmts the severty of the punshment that players can mpose on a devator. Snce contracts can explctly depend on other contracts, one mght thnk that a devator n the contractng game can be punshed as a functon of hs contract. That s, the restrctons on the acton spaces of the players can depend on the devator s contract. Ths observaton suggests that the punshment for a devaton can be more severe n the contractng game than n the mechansm and, hence, a larger set of allocatons cen be mplemented by the contractng game. We show that, surprsngly, ths s not true. More precsely, we show below that when player devates from hs equlbrum contracts, players cannot punsh hm more than by offerng contracts whch pn down a subset of A whch does not depend on the devator s contract. Consder an equlbrum n the contractng game. Denote the equlbrum contract of player wth type t by ec t. Let us defne a partton, τ,ofplayer s type space as follows: τ t = ª t 0 T : ec t = ec t 0. We shall also use the notaton cτ (t ) for c t. Denote the vector of equlbrum contracts of players wth type profle t by ec τ (t ). (That s, ec t s a vector of defnable functons whose coordnates are ª ³ ec τ j (t j ),andec j6= τ (t ) : N A.) For all A τ (t ) t t A τ (T ) defne S ³³A τ (t ) as follows: n³ A τ (t ) t : Aτ (t ) A, c s. t. c ec τ (t ) = A τ (t ), ec τ (t ) ([c]) = A τ (t ) µ µ Let us explan what t means that A τ (t t ³ ) S A τ (t). By the defnton t t of S, there exsts a contract, c, avalable for player suchthatfthetypeprofle of the other players s t,thenfplayer offers c then hs restrcted acton space wll be A τ ) (t and players o.