Universita Commerciale Luigi Bocconi. Facolta di Economia. Corso di Laurea: Economics and Social Sciences. Titolo Tesi: Agreements in Dynamic Games

Transcription

1 Universita Commerciale Luigi Bocconi Facolta di Economia Corso di Laurea: Economics and Social Sciences Titolo Tesi: Agreements in Dynamic Games Relatore: Prof. Pierpaolo Battigalli Controrelatore: Prof. Alfredo Di Tillio Tesi di Laurea Specialistica di Pietro Tebaldi ID: A.A

2 (back cover here) 2

3 Agreements in Dynamic Games Pietro Tebaldi Bocconi University, Milan Italy June 24, 2011 Abstract Many equilibrium concepts in game theory rely on the implicit motivation that the equilibrium can be seen as the result of a non-binding agreement between the players. From our perspective, the only consequence of a non-legally-enforceable agreement can be a restriction on the hierarchies of beliefs of the players taking part in the agreement. With an interactive epistemology approach we give a formal representation of the restrictions on players beliefs which must be generated by a given agreement. The coherency of these restrictions with rationality and common initial belief in rationality will determine whether or not the agreement is acceptable by the parties. The self-enforceability of an agreement, once accepted and transparent among the parties, is a stronger requirement that depends on what assumptions we make about the strategic sophistication of the players. 3

4 Contents Contents 4 1 Introduction 9 2 Preliminaries Dynamic games with payo uncertainty Strategic forms Systems of conditional probabilities Epistemic models Sequential rationality Belief operators rationalizability Weak -Rationalizability Strong -Rationalizability Non-binding agreements The environment Agreement structures Partial order based on completeness Topological characterization Epistemic implications Implied restrictions on rst-order beliefs Transparency of the agreement (among the parties)

5 4 Mutual Acceptability Epistemic priority of rationality Mutual acceptability Self-Enforceability Weak Self-Enforceability Backward Self-Enforceability Strong Self-Enforceability Discussion 69 7 Appendix Topological structure of Proofs of Section Proof of Lemma Proofs of Section Proof of Remark Proof of Remark Proof of Remark Proof of Remark Proof of Lemma Proof of Proposition Proofs of Section Proof of Remark Proof of Proposition Proof or Remark

6 7.5 Proofs of Section Proof of Remark Proof of Remark Proof of Lemma Proof of Proposition Proof of Conjecture Proof of Proposition Characterization results in continuous games References 107 6

7 Acknowledgments This work could have not been written without the guidance and teachings of Prof. Pierpaolo Battigalli. In the last year and a half he has taught me how to undertake original research, how to deal with analytical complexity and how to organize my energies in the pursuit of my results. His support has been fundamental in times of need. I must devote a special thanks to Prof. Marco Bonetti and Prof. Marcello Pagano, whose enthusiasm both in life and in academia will always be a driving example. The technical support provided by Prof. Alfredo Di Tillio and Prof. Fabio Maccheroni was necessary for the re nement of certain results, I want to thank them for their patience. All my friends and colleagues made this possible: thanks to Edo, Ema, Ricky, Carlo, Tommy, Babi, Maria, Ale, Fede, Maty, Franz, Ste, Gio, Anto, Skinko, Marty and all those (many) I am forgetting. A special thanks to Giovanni who gave me the very rst feedback when all I had was a mess of ideas. I must thank my parents and sister: in hard times they have been a necessary presence. The biggest thanks goes to Caitie, irreplaceable fuel for all I have done. 7

8 [...]The standard approach has the appearance of science in its ability to generate clear predictions from a small number of axioms. But only the appearance, since these predictions are mostly false. The environment actually faced by investors and economic policymakers is one in which actions do depend on beliefs and perceptions, must deal with uncertainty and are the product of a social context. John Kay, FT, April

9 1 Introduction The words of John Kay are able to capture in a few sentences what represents one of the most relevant motivations for the formal analysis of the foundations of economic behavior. "Actions do depend on beliefs and perceptions, must deal with uncertainty and are the product of a social context": this is one of the most recent missions of theoretical economics; how do agents act depending on their beliefs, how do they deal with uncertainty about external factors and the behavior of other agents, how does the context a ect their beliefs and thus their choices. What John Kay, who is certainly not an economic theorist, criticizes about the standard approach is the lack of consideration for these elements, although their relevance is transparent to all economists. Epistemic game theory and foundational economic theory try to put together well posed questions and try to propose correct answers to reduce this well known discrepancy existing between what really determines decision-makers behavior and what standard economic analysis implicitly assumes to be true. Speci cally, the use of many solution concepts (above all equilibrium concepts) in modern game theory often relies on the assumed existence of some sort of nonbinding agreements between players. One of the most common interpretations of Nash equilibrium in a static game is indeed the agreed course of action among players. Another important example is the intuitive representation of a Subgame Perfect Equilibrium as a contingent, non-binding contract. This is the case, in particular, for all those models in which we want to represent a real non-cooperative situation, such as an oligopolistic competition, when some form of coordination among the parts leads to Pareto improvements in the payo pro les. 9

10 Whenever we interpret an equilibrium concept as agreement, the implicit assumption is that players agree on a course of action (or - and this is even harder to motivate - a randomized course of action) to be followed at every possible contingency. Reality proves to be di erent: even legally-binding contracts display incompleteness in what they prescribe, and there is no reason why a non-binding agreement, resulting from a pre-play cheap-talk between players would necessarily specify what actions are suppose to be taken in every possible contingency. The study of agreements in games creates a reconciliation between game theory and contract theory. In the literature (see for instance [14, 1998], [20, 1988], [30, 1992], [7, 2002], and [8, 2008]) we nd several attempts to model (possibly incomplete) non binding contracts in a dynamic game-theoretical framework. The paper by Greenberg et. al., "Mutually Acceptable Courses of Action" [19, 2009], aims at giving a formal representation of a large class of incomplete contracts and studies their mutual acceptability. The solution concept they introduce, MACA, stays however in between acceptability and self-enforceability, applying a tremblinghands approach to test the possibility of a given course of action to be mutually acceptable. The reason why it is impossible to identify the exact meaning of a MACA à la Greenberg is that it lacks any sort of epistemic justi cation, either formal or informal. In our work we model incomplete, non-binding contracts in dynamic games with payo uncertainty in a way that is fully compatible, though more general to the formalizations proposed by Greenberg et. al. [19, 2009] and earlier authors. We follow an interactive epistemology perspective representing events and agents beliefs in the canonical, universal type structure introduced by Battigalli and Siniscalchi [10, 1999]. 10

11 Our approach di ers from other works on this topic because we take the perspective that when we model a non-binding pre-game agreement between players we cannot consider any behavioral restriction, since the only possible e ect of such agreements is the formation of certain restrictions in parties beliefs and the transparency of these restrictions induced by the trasparency of the agreement. In this way we can show how mutual acceptability di ers from self-enforceability, coherently with the di erence between what people promise to each other and what they actually do. While mutual acceptability is a minimal requirement for an agreement to a ect players beliefs, and hence behavior, self-enforceability is a stronger concept, which is strictly related to the degree of strategic sophistication we ascribe to the players. We propose a series of epistemic results to characterize acceptability and selfenforceability under di erent assumptions about players belief in rationality, and we show how they are indeed connected to many of the most common equilibrium concepts in use, justifying the agreement interpretation often given to them. The study of incomplete contracts in our framework gives more exibility when we want to model situations in which there is no reason to assume that players would agree upon actions to be taken in every possible contingency, and this is the reason why with a traditional solution concept based on completeness and common certainty of the opponents strategies, the set of predictable outcomes often does not match what we observe in a reality characterized by incompleteness of contracts and discordance in beliefs. In what follows rst we describe our environment: dynamic games with payo uncertainty à la Battigalli, i.e. a dynamic setting allowing for incomplete information but not specifying players priors as in a traditional incomplete information 11

12 game à la Harsanyi. We then introduce in Section 3 our formal representation of incomplete, non-binding agreements among a set of parties (subset of players). We study the minimal epistemic consequences of an accepted agreement and, in Section 4, we de ne as mutually acceptable an agreement such that there is no con ict between its epistemic consequences and the minimal requirement of initial common belief in rationality. With mutual acceptability being the weakest requirement for an agreement to be e ective, in Section 5 we study the concept of self-enforceability under some alternative assumptions about players strategic thinking: initial common belief in rationality, common certainty of future rationality, and common strong-belief in rationality. 2 Preliminaries This section, adapted from [4, 2003], [11, 2002] and [10, 1999], describes the environment and the main analytical tools upon which what is to follow will be based: dynamic games with payo uncertainty (Section 2.1), systems of conditional probabilities (Section 2), epistemic models (Section 2.3), sequential rationality (Section 2.4), belief operators (Section 2.5) and -rationalizability (Section 2.6). 2.1 Dynamic games with payo uncertainty A dynamic game with payo uncertainty and observable actions is a structure = DI; i ; A i ; A i () ; u i i2i E given by the following elements: 12

13 I is a non empty, nite set of players. For each i 2 I, i R m i for player i and A i R n i is a non empty set of possible payo parameters is a non empty set of possible actions for player i (R k is the k-dimensional Euclidean space). Let = Q i2i i and A = Q i2i A i. Then A <N = fg [ [ t1 A t! ; that is, A <N is the set of nite sequences of action pro les, including the empty sequence 1, and A i () : A <N A i is a constraint correspondence assigning to each nite sequence of action pro les (a t ) T t=1 the set A i (a t ) T t=1 of feasible actions for player i after (a t ) T t=1. It is assumed that A i () 6=? and for all h 2 A <N, A i (h) =? if and only if 8` 2 I A` (h) =?. h T = (a t ) T t=1 2 A<N is a (feasible) history if a 1 2 A () and a t+1 2 A (h t ) for 1 t < T, where A (h) := Q i2i A i (h) for all h 2 A <N. Let H A <N [ A 1 denote the set of histories, and Z := z 2 H j z 2 A 1 or A (z) =? the set of terminal histories, or outcomes of. Let H := HnZ be the set of 1 We use the greek letter to indicate the empty sequence, i.e. the initial history. Thorough the paper we use the di erent symbol? when referring to the empty set. 13

14 ( nite) non-terminal histories. For all i 2 N, u i : Z! R is the payo function for player i. Parameter i represents player i s private information about the unknown payo relevant state of Nature. For brevity, we call i the payo -type of player i. It is assumed that is common knowledge. Chance moves and residual uncertainty about the environment can be modeled by having a pseudo-player c 2 I with a constant payo function. The payo -type c of this pseudo-player represents the residual uncertainty about the state of Nature which would remain after pooling the private information of the real players. Players common or heterogeneous beliefs about chance moves can be modeled as exogenous restrictions on beliefs (see below). Note that structures such as are not the same as games with incomplete information in the sense of Harsanyi (also called Bayesian games) because they contain no description (neither implicit nor explicit) of players hierarchies of beliefs about the state of nature. Battigalli and Siniscalchi [10, 1999] introduced hierarchies of beliefs of a much richer kind, obtaining a language that allows us to express assumptions about players rationality and interactive beliefs, deriving then behavioral implications from these assumptions. Game is static if Z =A (). Player i 2 I is active at h 2 H if ja i (h)j > 1. has perfect information if 8h 2 H 9! i 2 I s:t: ja i (h)j > 1. has complete 14

15 information if for all i 2 I the payo function u i is constant with respect to 2. Turning to the topological properties of, we endow H with the standard discounting metric (see Appendix) and henceforth we rely on the following assumption: Assumption 0. A and are compact, A i (h) is compact for all i and for all h and, u i is continuous for every i Strategic forms A strategy for player i 2 I is a function s i : H! A i such that s i (h) 2 A i (h) for all h 2 H. The set of strategies for player i is denoted S i, S i := Y h2h A i (h) A i H. (By de nition of H, for all h 2 H, A i (h) is nonempty. Therefore S i is also nonempty.) The basic elements of our analysis are type-strategy pairs: ( i ; s i ). A generic pair for player i is denoted i and the set of such pairs for player i is the product i := i S i. The sets of pro les of pairs for all players and for the opponents of a player i are, respectively, = Q i2i i and i = Q`6=i `. Each pro le s 2 S := Q i2i S i induces a terminal history (s) 2 Z. Therefore, for each player i, we can derive 2 This de nition is coherent with the one given by Osborne and Rubinstein, Chapter 5 [23, 1994]. 15

16 the following strategic form payo function : U i :! R; U i (; s) = u i (; (s)). Furthermore, for each history h 2 H we can de ne the set of pro les of strategies consistent with h: S(h) = fs 2 S j h is a pre x of (s)g: Let (h) := S (h) denote the set of pairs consistent with history h. Clearly, () =. We let i (h) denote the projection of (h) on i, that is, the set of ( i ; s i ) such that strategy s i does not prevent history h. It can be easily checked that, for all h 2 H, (h) = Y i (h) 6=?: i2i The information of player i about the strategies of his opponents at history h is represented in strategic form by i (h), the projection of (h) on i. We endow the sets i (i 2 I) with the standard metrics derived from the metric on H, and (as well as i ) with the product metric (see the Appendix). Lemma 1 For all h 2 H, (h) is closed. We say that a history h is preterminal if, for all actions pro les a 2 Q i2i A i (h), the sequence (h; a) is a terminal history: i.e. (h; a) 2 Z. De nition 1 A game is simple if is compact and either (a) A is nite or (b) A is compact and for every h A (h) is non- nite only if h is preterminal. 16

17 Clearly, nite games and in nitely repeated games with a nite stage game are simple. Moreover, a large variety of games with a continuum of actions at the last stage are simple. Henceforth in our exposition we focus on simple games. 2.2 Systems of conditional probabilities To model the way players update and revise their beliefs as the game unfolds, it is convenient to represent beliefs by means of conditional probability systems [28, 1995]. For a given measurable space (X; X ) consider a non-empty collection C X of events such that? =2 C. The interpretation is that a player is uncertain about the "true" element x 2 X, and C is a collection of observable events - or "relevant hypotheses" - concerning x. De nition 2 A conditional probability system (CPS henceforth) on (X; X ; C) is a mapping (j) : X C! [0; 1] satisfying the following three axioms: CPS1 For all B 2 C ; (BjB) = 1. CPS2 For all B 2 C, (jb) is a probability measure on (X; X ). CPS3 For all A 2 X ; B; C 2 C ; if A B C then (AjB) (BjC) = (AjC). (X; X ; C; ) is called conditional probability space. We assume that X is a metric space and that X is the Borel -algebra on X. We will then omit X and denote the set of conditional probablity systems 17

18 on (X; C) as C (X), where (X) is the usual notation for the set of probability measures on X. The set C (X) can be regarded as a subset of [ (X)] C. Then, we endow (X) with the topology of weak convergence of measures, [ (X)] C with the product topology and C (X) with the relative topology. To represent the rst-order beliefs of a player i in a dynamic game with incomplete information is we de ne the set C i ( i ) of conditional probability systems on ( i ; S i ; C i ), where i is the set of type-strategy pro les for her opponents, S i is the Borel sigma algebra of i and C i = fb i j B = i (h) for some h 2 Hg. Note that assuming observable actions the only "relevant hypothesis" we are interested in correspond to the event that a given non-terminal history h has occurred. For this reason, given the natural correspondence between C i and H (for all i 2 I), we will simplify our notation in the following intuitive way: for every player i 2 I, C i ( i ) H ( i ) and for every history h 2 H, for every 2 H ( i ) (j i (h)) (jh). Battigalli and Prestipino (see [9, 2011]) allow for a greater level of generality, adapting the notation to embrace asymmetric information games (in which information sets, and thus conditioning events, contain more than a single partial history). By Lemma 1, C i is a collection of closed subsets and thus C i ( i ) is indeed a well-de ned space of conditional probability systems. To represent in nite hierarchies of conditional beliefs which correspond to the 18

19 epistemic type of a player, that is, the beliefs that this player would have, conditional on each history, about the state of Nature, his opponents strategies and his opponents epistemic types, we use the construction introduced by Battigalli and Siniscalchi [10, 1999]. Speci cally, we will consider a set of "possible worlds" = Q i, where for i2i all i 2 I i = i T i. In the following section we clarify the speci c meaning of the collection (T i ) i2i. For now, T i is a Polish space, a technical requirement that implies that in a simple game i and are Polish as well (it follows by Lemma 3.4 in [4, 2003]). This ensures that the following is a countable collection of clopen sets in the Borel sigma-algebra B ( i ): F i = fb i j B = i (h) T i for some h 2 Hg. Therefore, the set F i ( i ) of conditional probability systems on ( i ; B ( i ) ; F i ) is well de ned for all i s and, as shown by Battigalli and Siniscalchi (see [10, 1999]), the countability and the clopeness of F i guarantee that F i ( i ) is a Polish subset of the Cartesian set [ ( i )] F i. Again, we can simplify our notation in a very intuitive fashion by letting F i ( i ) H ( i ) for all i s and setting (j i (h) T i ) (jh) for all h and for all 2 H ( i ). 2.3 Epistemic models We introduce here our basic representation of hierarchical conditional beliefs. De nition 3 A type structure on (H; ; S () ; I) is a tuple T = H; ; S () ; I; ( i ; T i ; g i ) i2i such that, for every i 2 I; T i is a Polish space and for all i 2 I 19

20 1. i is a closed subset of i T i such that proj i i = i ; 2. g i : T i! H ( i ) is a continuous mapping. For any player i 2 I, the elements of the set T i are referred to as Player i s epistemic types. A type structure is compact if all the sets T i, i 2 I; are compact topological spaces. At any "possible world"! = (; s; t), we specify a state of nature 2, players dispositions to act s 2 S as well as players dispositions to believe, the latter represented through the systems of conditional probabilities g i (t i ) = (g i;h (t i )) h2h. Note that these dispositions also include what a player i would do and think at histories inconsistent with his own strategy (i.e. a history h such that s i 62 S i (h)). In the spirit of the literature introduced by Mertens and Zamir [22, 1985], Brandenburger and Dekel [15, 1993] and extended by Battigalli and Siniscalchi [10, 1999], we work in a type structure which encodes all "conceivable" hierarchical beliefs. We introduce the following: De nition 4 A belief-complete type structure on (H; ; S () ; I) is a type structure T = H; ; S () ; I; ( i ; T i ; g i ) i2i such that, for every i 2 I; i = i T i and g i : T i! H ( i ) is onto. Battigalli and Siniscalchi [10, 1999] show that for simple games a belief-complete type structure can always be constructed. Moreover, every type structure can be seen as a belief-closed subspace of the space of ini nite hierarchies of beliefs, that together with the compactness of implies that the belief-complete type structure so constucted is also compact. 20

21 To remove redundancies from this representation, one can work with a type structure for which di erent types lead to di erent hierarchies of beliefs in a bicontinuous fashion: De nition 5 A belief-complete, canonical type structure on (H; ; S () ; I) is a type structure T = H; ; S () ; I; ( i ; T i ; g i ) i2i such that, for every i 2 I; i = i T i and g i : T i! H ( i ) is a homeomorphism. Battigalli and Siniscalchi [10, 1999] show that these type structures can always be constructed. From now on in our exposition we refer to the belief-complete, canonical type structure to represent players hierarchies of beliefs. 2.4 Sequential rationality A strategy ^s i is sequentially rational for a player with payo type ^ i and conditional beliefs i if it maximizes the conditional expected utility at every history h consistent with ^s i. Note that this is a notion of rationality for plans of actions 3 rather than strategies. Let H (s i ) := fh 2 H j s i 2 S i (h)g denote the set of histories consistent with strategy s i. Given a rst order CPS i 2 H ( i ) and a history h 2 H (s i ), let U i i ; s i ; i (jh) Z = i (h) U i ( i ; s i ; i ) d i ( i jh) 3 Formally, a plan of action is a maximal set of strategies consistent with the same histories and prescribing the same actions at such histories. 21

22 denote the expected payo for player i with payo type i from playing s i given history h, provided that the integral in the right-hand side is well de ned. De nition 6 A strategy ^s i is a sequential best reply for player i with payo type ^i and rst-order beliefs i 2 H ( i ), written (^ i ; ^s i ) 2 i ( i ) or ^s i 2 r i (^ i ; i ) if, 8h 2 H (s i ) ; 8s i 2 S i (h) U i ^i ; ^s i ; i (jh) U i ^i ; s i ; i (jh) provided that the inequality is well de ned. Battigalli (see [4, 2003]) shows that i is a nonempy-valued and upper-hemicontinuous correspondence. We introduce additional notation to represent players rationality in the belief-complete type structure introduced above. De nition 7 Fix a -based type structure; for every player i 2 I, let f i = (f i;h ) h2h : T i! H ( i ) denote his rst-order belief mapping, that is 8t i 2 T i ; 8h 2 H f i;h (t i ) = marg i g i;h (t i ). It is easy to check that given the continuity of g i, also f i turns out to be continuous. Using the de nition of rst-order belief mapping we can now formalize our main behavioral axiom. De nition 8 A player i 2 I is rational at state! 2 if ( i ; s i ) 2 i (f i (t i )). The event R i := f! 2 j ( i ; s i ) 2 i (f i (t i ))g 22

23 corresponds to the expression "player i rational". We also refer to the event R := T R i ( "all players are rational") and R i := T R j ( "all opponents of i are i2i rational"). j2infig 2.5 Belief operators Within the framework of belief-complete, canonical, -based type structures we represent (conditional) certainty, or (conditional) probability-one belief with the notation introduced below. This notation will give us the greatest level of exibility to describe epistemic events such as "player i believes in her opponents rationality" or, once pre-play agreements will be formally introduced, "party ` believes that her counterparts will honor the agreement". De nition 9 Given a -based type structure T, for every measurable event E, for every player i 2 I, for every history h 2 H, the event B i;h (E) =! 2 j! i 2 proj i E ^ g i;h (t i ) proj i E = 1 corresponds to the statement "player i is certain of E when observing that h has occurred". Note that this de nition allows a player to be certain of a given event only if this is consistent with his own strategy and epistemic type at the observed history. Battigalli and Prestipino [9, 2011] and Battigalli and Siniscalchi [13, 2007] use a di erent, yet equivalent de nition, expanding on the consistency of a player s beliefs with his own behavior (and beliefs) by introducing an additional representation of players conditional beliefs assigning probabilities to events in 23

24 B () and not just in B ( i ). For our exposition this is not necessary and there should be no confusion for the reader familiar with the literature to which we refer. Note that, for each player and each history, De nition 9 identi es a set-to-set operator B i;h : B ()! B () which satis es the usual properties of falsi able beliefs; in particular monotonicity and conjunction (see [9, 2011]). For convenience we introduce the following auxiliary notation: for every set C (or D T ), the operator [] maps the set into the corresponding rectangle in the product space T : [C] = C T (or [D] = D). A player i 2 I strongly believes that an event E 6=? is true if and only if she is certain of E at every history consistent with E itself. Formally, De nition 10 Given a, the event -based type structure T, for every measurable event E \ SB i (E) = B i;h (E) h:[(h)]\e6=? corresponds to the statement "player i strongly believes E". Note that also this de nition identi es a set-to-set operator SB i : B ()! B (), allowing us to have a well-de ned k-fold composition (see below). As showed in [11, 2002] the strong belief operator fails monotonicity and conjunction in nontrivial games. To complete this section we introduce other operators that will be used in our analysis: De nition 11 Given a -based type structure T, for every measurable event E, for every subset of players L I we de ne the following events (and corre- 24

25 sponding set-to-set operators): 4 Denition Notation Expression Initial Belief F ull Belief Initial Mutual Belief Initial M utual Belief Among L IB i (E) = B i; (E) B i (E) = T B i;h (E) h2h IB (E) = T IB i (E) i2i IB L (E) = T j2l IB j (E) Player i is certain of E at the initial history Player i is certain of E at every history All players are certain of E at the initial history All players in L are certain of E at the initial history Mutual Belief B (E) = T i2i B i (E) All players are certain of E at every history M utual Belief Among L Correct M utual Strong Belief B L (E) = T j2l B j (E) CSB (E) = E \ T i2i SB i (E) All players in L are certain of E at every history E is true and all players strongly believe E As anticipated, working with set-to-set operators (in general, O : B ()! B ()) allows us to de ne their k-fold composition in a very intuitive way: for all E 2 B (), O 0 (E) = E and for k 1 O k (E) = O O k 1 (E). 4 Note that the full belief operator is well de ned only if B i;h (E) is well de ned for all h. This requires that E \ [ i (h)] 6=? for all h. If E is epistemic (proj E = ) then the full belief operator is always well de ned. 25

26 It can be easily shown that for a given compact event E and subset L I, IB k (E) k0, B k (E) k0, IB k L (E) k0, B k L (E) k0, CSB k (E) k0 are all sequences of (compact) events in B (). 5 Therefore the following "common belief" operators are well de ned: ICB (E) := \ k0 IB k (E) ; CB (E) := \ k0 B k (E) ; ICB L (E) = \ k0 IB k L (E) ; CB L (E) := \ k0 B k L (E) ; CSB 1 := \ k0 CSB k (E). The nite intersection property applies to all the above cases when E is compact, hence a su cient (and necessary) condition for the non-emptiness of ICB (E) (respectively, CB (E) ; ICB L (E) ; CB L (E) ; CSB 1 (E)) is the non-emptiness of IB k (E) (respectively, B k (E) ; IBL k (E) ; Bk L (E) ; CSBk (E)) for all k 2 N rationalizability Weak and strong -rationalizability are two nested extensions of the rationalizability solution concept to dynamic games of incomplete information, which take as given some exogenous restrictions on players beliefs represented by compact subsets of CPSs i H ( i ), i 2 I. These solution concepts were rst introduced in the literature by Battigalli [3, 1999] and developed in Battigalli and Siniscalchi [11, 2002], [12, 2003] and [13, 2007]. In the latter the authors provide the characterization results for weak -rationalizability and a simpli ed, more operational, version of strong -rationalizability. In a recent working paper, Battigalli and Prestipino [9, 2011] provide the characterization result for the general 5 Compactness is a consequence of the Portmanteau Theorem. 26

27 case of strong -rationalizability. Recall that in the complete, canonical type structure, a state of the world describes the state of Nature 2 and players dispositions to act and believe conditioning on each observable event h 2 H. In the canonical type structure, in particular, there are no redundancies in players beliefs: every triplet ( i ; s i ; t i ) for player i is in a one-to-one (continuous) relation with the triplet ( i ; s i ; g i ), where g i 2 H ( i ) and by the continuity of the projection operator ( i ; s i ; t i ) is also homeomorphic to ( i ; s i ; i ) = ( i ; s i ; f i ), where i 2 H ( i ). Each -rationalizability solution concept characterizes the feasible type-strategy realized at states where (a) every player i 2 I is sequentially rational and has rstorder beliefs in i, and (b) the players higher order conditional beliefs satisfy conditions concerning mutual certainty of (a) and/or robustness of beliefs about (a) Weak -Rationalizability Weak -rationalizability characterizes the set of feasible type-strategy pairs realized at states of the world! 2 ICB (R\), that is, initial common belief in rationality and. De nition 12 Consider the following procedure. (Step 0) For every i 2 I; W i (0; ) = i. As usual, W i (0; ) = Q`6=i W` (0; ) and W (0; ) = Q i2i W i (0; ). (Step n 0) Suppose W (k; ) has been de ned for k = 0; 1; :::; n 1. For each i 2 I; let i 2 W i (n; ) if and only if i 2 W i (n 1; ) and 9 i 2 i such that supp i (j) W i (n 1; ) and i 2 i ( i ). 27

28 The set of weakly -rationalizable type-strategy pairs is de ned as W (1; ) := \ k0 W (k; ) : In [10, 1999] and [13, 2007] (sketch only) it is proved that, for every k, which implies k\ 1 W (k; ) = proj n=0 W (1; ) = proj ICB (R\). IB n (R\) (1) Note that this solution concept is silent about how the players would change their beliefs if they observed a history h which they believed impossible at the beginning of the game, even if h is consistent with rationality or mutual certainty of rationality of any order. In this it di ers from strong -rationalizability. For notational convenience, when no restrictions on players beliefs are imposed (i.e. when i = H ( i ) for all i 2 I) we omit from the notation Strong -Rationalizability Strong -rationalizability characterizes the feasible type-strategy pairs realized at states of the world! 2 CSB 1 (R\CB ()), that is common strong belief in rationality and transparency of. 6 De nition 13 Consider the following procedure. (Step 0) For every i 2 I; ^i (0; ) = i. As usual, ^ i (0; ) = Q`6=i ^ i (0; ) and 6 An event E is transparent if E B (E). 28

29 ^ (0; ) = Q i2i ^ i (0; ). (Step n 0) Suppose ^ (k; ) have been de ned for all i 2 I, for all k = 0; 1; :::; n 1. For each i 2 I; and for all i 2 i let i 2 ^ i (n; ) if and only if 9 i 2 i such that 1. 8m 2 f1; :::; n 1g, 8h 2 H, ^ i (m; )\ i (h) 6=? )supp i (jh) ^ i (m; ); 2. i 2 i ( i ). The set of strongly -rationalizable type-strategy pairs is de ned as ^ (1; ) := \ k0 ^ (k; ) : In [9, 2011] it is proven that, for every k, ^ (k + 1; ) = proj CSB k (R\CB ()) = proj CSB k (R\) and, consequentely, ^ (1; ) = proj CSB 1 (R\CB ()) = proj CSB 1 (R\). Again, when no (transparent) restrictions are imposed, we simplify the notation omitting the symbol. According to strong rationalizability each player believes that his opponent is rational as long as this is consistent with his observed behavior. More generally, each player bestows on his opponent the highest degree of strategic sophistication 29

30 consistent with his observed behavior. This best rationalization principle is a form of forward induction reasoning; it also induces the backward induction path in games of perfect and complete information. For more on this solution concept and for a formal description of the underlying epistemic assumption we refer to [3, 1999], [12, 2003], [13, 2007] and [9, 2011]. 3 Non-binding agreements A contract help parties to expect what other parties will do in future contingencies. A non-binding contract is an agreement not intended by the parties to be legally enforceable, but yet it is expected to be performed or followed The environment To represent (possibly incomplete) non-binding agreements among players (or a group of them) we introduce a structure that allows for a high level of generality, without neglecting any of the main, necessary elements that need to be considered when studying incomplete contracts in a game-theoretic framework. A non-binding (incomplete) contract, however the way it is de ned, cannot have direct e ects on the behavior of players, neither explicit nor implicit in the analysis. Instead, we focus on the epistemic consequences of a pre-play agreement between players. The exogenous restrictions on players beliefs structures is the only conceivable consequence of a non-binding contract. These restrictions could then a ect behavior 7 Adapted from Hendrikse, G. and Hu, Y., Incompleteness in Chinese Fruit and Vegetable Contracts; and YourDictionary.com 30

31 when embedded in together with the working assumptions of rationality and common certainty of rationality Agreement structures To model agreements within the framework of games with incomplete information we use the following formalization. Henceforth x a simple game. E De nition 14 An agreement A for is a structure A := DP; (A j ) j2p. P I is the set of parties, i.e. players aware of A; (A j ) j2p ; A j Q A j (h) = S j, is a list of correspondences describing what is prescribed by A in every contingency. 8 h2h The interpretation is that at every history h 2 H each one of the parties j 2 P is "expected" by the other parties k in P n fjg to play an action in A j (h). This de nition allows us to pin down all the elements required for the analysis. In particular, because we focus on the epistemic implications of these agreements, it is noteworthy that the set P represents the set of players informed about A: for instance we could have a player j in P without any prescription (i.e. 8h 2 H ; A j (h) A j (h)), but because j 2 P, her beliefs will be subjected to the restrictions we are going to study in the following sections. Moreover, because we talk about non-binding agreements, these implications of A on players beliefs is actually all that really matter for the analysis: every player is still free to act as described in by (A i ()) i2i, the only "concrete" consequence of A is its impact on the beliefs of the players who know that A was drawn. "Behaviorally" speaking, there is no real di erence between a player j 2 P with no prescriptions and an 8 We do not use the term pro le because this will always denote lists of elements, one for every player i 2 I. Henceforth we will consistently use the term list whenever we refer to collections of elements regarding only a subset of players. 31

32 opponent k 2 P with ja k (h)j = 1 8h 2 H; nevertheless there is a substantial di erence between a player j 2 P and an opponent ` 2 InP "unaware" of A. For the sake of simplicity, we shall eventually refer to an agreement as A = (A j ) j2p, but it is important to bear in mind the information on the "disclosure" of A brought by the set P. The class of agreements A for is denoted AG, with a slight abuse of notation, AG can be seen as a class of subsets of the cartesian product I S, i.e. AG 2 IS. It is also noteworthy that the agreements we consider do not allow information sharing between players regarding the state of Nature. A player i can make "promises" about her future behavior but she cannot disclose to her opponents the private information contained in her parameter value i (we expand on this in the Discussion) Partial order based on completeness One relevant aspect in the analysis of incomplete agreements, as emphasized in Greenberg et al. [19, 2009], is the concept of completeness. Not to be confused with completeness in metric spaces, nor with completeness of type spaces, the term completeness here simply refers to the degree at which the agreement dictates how players should behave as the game unfolds. The following de nition of completeness is consistent with the literature on incomplete contracts, see for instance [14, 1998], [20, 1988], [30, 1992], [7, 2002], and [8, 2008]. De nition 15 Fix a game. For any two agreements A; A 0 for, we say that A is less complete than A 0, written A 0 if (a) P P 0, (b) 8j 2 P A j A 0 j 32

33 and (c) A 6= A 0. A v A 0 if (a) and (b) hold. If A v A 0 ) A 0 w A. It can be checked that AG ; w is a partially ordered set. Intuitively, an agreement is more complete if it speci es more about the behavior that players are expected to follow or, ceteris paribus, if more agents are involved in the agreement itself. Remark 1 The set of maximal elements of AG is naturally homeomorphic to the set of strategies S, we will eventually refer to a strategy pro le s 2 S as to a complete agreement. The minimum of AG exists and it is homeomorphic to the game itself. Intuituvely, there is nothing more complete than a strategy pro le, specifying for each player a singleton in the action set at every history. Conversely, if nothing is said, and no exogenous restrictions are imposed through an incomplete agreement, all is left is just the game (and the common knowledge of ). We will say that A is public whenever P = I. To our knowledge, this is the standard in all the literature on incomplete contracts, in which no explicit assumption regards the degree at which players are informed about the settlement. Example 1 We refer to Greenberg et al. introductory example, just to make the reader familiar with the notation. A graphic representation of the extensive form of is reported in Figure 1. What the authors point out is that while the only Subgame Perfect Equilibria of the game are the two complete agreements bs = (L1; R2; L3; R4) es = (L1; R2; R3; L4) 33

34 both inducing the terminal node z = L1, the incomplete agreement A = hf1; 2g ; ((R1; L1) ; (L2))i "might" lead to the non-spe outcome z = (R1; L2). We put emphasis on "might" because we have not yet introduced any behavioral implication of agreement structures. We shall be able to tell much more about this simple example once the epistemic implications will be introduced (Section 3.2) and the working assumptions exploited in this restricted epistemic environment (Section 4). Figure 1. From J. Greenberg et al. [19, 2009] Topological characterization Throughout the paper, we rely on this assumption on AG : 34

35 Assumption 1. For every A 2 AG, for every history h 2 H, A (h) is compact. Notice that this is not merely a technical assumption, there is no real meaning in allowing players to agree on open subsets of A (h). Looking at AG as a subset of 2 IS = 2 I 2 S, using the topological structure of (see Appendix) we can de ne a topology on AG induced by the metric d :AG AG! R constructed as follows: Let S A := fs 2 S j 8h 2 H; s 2 A (h) S P g. S A is the set of strategies consistent with A. We can identify A with the pair P; S A without loss of generality. It is necessary, however, to specify P, so to have (A j ) j2p = proj SP S A. The nite set I is endowed with the discrete metric topology, and its power set 2 I with the corresponding Hausdor metric topology, say d I : 2 I! R: 9 8P; Q I d I (P; Q) = 1 fp 6=Qg, where 1 fg is the indicator function. Let K S denote the class of compact sets in 2 S. The metric on S introduced in the Appendix induces the Hausdor metric on K S, say d S. 9 Let X; Y be non-empty subsets of a metric space (M; d). The Haudor distance between X and Y is de ned as d H (X; Y ) = max sup inf x2x y2y d (x; y) ; sup y2y inf x2x d (x; y). 35

36 We endow AG with the product metric d: d (A; A 0 ) = d I (P; P 0 ) + d S S A ; S A0. Note that for a sequence of agreements fa n g n0 AG, A n! A implies that there exists N P such that P n = P 8n N P. Thus 8n N P ; d (A n ; A) = d S S An ; S A. Therefore A n! A only if S An! S A, that is only if 8h 2 H; 8j 2 P; A n j (h)! A j (h). Intuitively, convergence of a sequence of agreements requires that eventually the set of parties is the same and that for every history the prescriptions converge in the Hausdor metric to a (compact) subset of S. 3.2 Epistemic implications An (incomplete) non-binding agreement A is not part of the game structure for which the agreement is designed. Players are all facing the same game, and they are free to act independently of what A prescribes. Before looking at whether or not an agreement A is "reasonable", that is, before imposing rationality assumptions, it is necessary to describe what restrictions on parties beliefs are induced by A Implied restrictions on rst-order beliefs Once A is drawn, every party j 2 P believes that A is going to be honored. Let us denote with A j () : H j the correspondence mapping every non-terminal history h in the set of strategies of player j which are consistent with h and "honor" A in future histories: 8h 2 H; A j (h) := j fs j 2 S j (h) j 8h 0 h; s j (h 0 ) 2 A j (h 0 )g. 36

37 Note that A j () is well de ned only for parties j 2 P : for each player not included in the agreement, i 62 P, we let A i () = i (). As usual, then, A j () := Q Q À (), A P () := A j () and A () := Q A i (). j2p i2i `6=j Remark 2 If A (h) is compact or all h 2 H and A n! A, then 8h 2 H; An (h)! A (h). The analysis of the behavioral consequences of incomplete agreements starts from the description of the restrictions on parties rst-order beliefs generated by the agreement itself. These restrictions are usually implicit in most interpretations of solution concepts as "agreements", above all Nash equilibrium and Subgame Perfect Equilibrium (see Discussion). We give the following explicit representation: De nition 16 The rst order restrictions on party s j 2 P beliefs induced by an agreement A for, denoted A j are A j;h := t 2 T j f j;h (t j ) A j (h) = 1. We let then A j := \ h2h A j;h and A = \ j2p A j. Notice that A = \ \ B j;h A j (h). j2p h2h Clearly, the restrictions regard only the players aware of A: 8i 2 InP; proj Ti A = T i and proj i A = i From the closure of the belief operators introduced in Section 2.5 it follows that A is closed in. Thanks to the non-redundancy of the canonical type structure, 37

38 the subset of the class of rst order CPSs homeomorphic to A, Q S f i (t i ) i2i t i 2proj Ti Q A H ( i ), is closed in the corresponding product topology. We will denote this i2i set as f A, which can be rewritten as f A := ( i i2i 2 Y i2i H ( i ) j 8j 2 P; 8h 2 H; j A j (h) jh = 1 ). The following is veri ed and ensures that our de nition is well posed. 10 Remark 3 Every ( i ) i2i 2 f A is a well de ned pro le of CPSs Transparency of the agreement (among the parties) Once the parties have settled on A to be played, and everyone in P is aware of this fact, the rst order restrictions described by A are object of common belief among parties j 2 P. This idea completely describes the way an agreement A 2 AG a ects the hierarchies of beliefs of players: common certainty of A among the parties is the event in the complete, canonical type structure that results when A is drawn. Recall that here we do not argue that A "makes sense", 10 We do not discuss on whether or not our de nition of the rst-order beliefs restrictions induced on the parties by A is "operationally" correct. It could be argued that imposing parties in P to strongly believe in the agreement, i.e. requiring only that they assign probability one to the set of strategies consistent with A if none deviated from the agreement, would be of easier interpretation. In this case we would have: ( ) i i2i 2 Y i2i H ( i ) j 8j 2 P; 8h : h 0 h ) s (h 0 ) 2 A (h 0 ) ; j A j (h 0 ) jh 0 = 1. Although we believe that this alternative is worth to be explored in future research, this paper aims at modelling the epistemic restrictions implicit in the interpretation of many solution concepts as "agreements", and the reader can immediately verify that this is the case, for instance, for Subgame Perfection (backward induction solutions are motivated by the fact that any player assigns probability one to the equilibrium strategy even after observing that a deviation has occurred). 38

39 but assuming that P have agreed to play according to A, without imposing any behavioral assumption, the epistemic structure in which is played is the one described in what follows. As in Battigalli and Prestipino [9, 2011] we say that an event E is transparent (among L I) at state! 2 if! 2 CB (E) (CB P (E)). It is straightforward to impose the transparency of the agreement (exclusively) among the parties, i.e. among the set P I: assuming that an agreement A has been drawn we restrict our analysis to the subset of states CB P A. This is the main di erence between our formalization and the one given by Greenberg et. al. [19, 2009] in which the epistemic structure deriving from a CA is not described. De nition 17 The epistemic restrictions induced by an agreement A 2 AG are described by the correspondence E : AG ; E (A) = CB P A. In the canonical type structure, the subset of CPSs on corresponding to E (A) (homeomorphic to proj T CB P A ) is the mapping E : AG Y H ( i ) ; E (A) = Y i2i i2i [ g i (t i ). t i 2proj Ti CB P ([ A ]) Note that that 8i 2 InP proj H ( i )E (A) = H ( i ), that is, no restrictions are imposed on players who do not take part in A. Notice also that since A is an epistemic event that cannot be contradicted by any observable event h, B j;h A is well de ned for every history h. 11 If we 11 An event E is epistemic if proj E =. If E is epistemic, for all h 2 H, E is consistent 39

40 were imposing behavioral restrictions (that would be wrong, since we are talking about non-binding agreements), this would not be the case: it is only for epistemic events that belief at every history and strong belief coincide. Remark 4 E (A) is self-evident among P : i.e. E (A) B P (E (A)). Lemma 2 E : AG ; w (; )and E : AG ; w Y i2i H ( i ) ;! are monotonically decreasing. Proposition 1 E and E are upper hemi-continuous. The results above show the regularity of the transparent (among parties) restrictions generated by the agreement A. The next step is to determine whether this restrictions are compatible with those that can be derived from the working assumptions of rationality and common initial (strong) belief in rationality. We can analyze the concept of acceptability imposing those assumptions and checking for the consistency with E (A), the only "concrete" consequence of A being "signed" by the parties. 4 Mutual Acceptability Acceptability is a feature that can be either true or false for an agreement A 2 AG at the initial history. When considering pre-play agreements, as we do here, it is intended as the possibility for A to be settled among players that are rational and commonly believe in the opponents rationality (at least) at the initial history. with (h). 40