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5 i 15 A- DEWEY Massachusetts. Institute of Technoogy Department of Economics Working Paper Series BARGAINING OVER RISKY ASSETS Muhamet Yidiz, MIT Working Paper October 2001 Room E Memoria Drive Cambridge, MA This paper can be downoaded without charge from the Socia Science Research Network Paper Coection at id=xxxxx

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7 qy Bargaining over Risky Assets 1 Muhamet Yidiz 2 MIT October 17, 2001 ^his paper is based on my dissertation, submitted to Stanford Graduate Schoo of Business. I thank my advisor Robert Wison for his guidance and continuous hep. I aso thank Daron Acemogu, Chaya Bhuvaneswar, Darre Duffie, David Kreps, Michae Schwarz, Dimitri Vayanos, and especiay Bengt Homstrom for hepfu comments. Earier versions of this paper were presented at Econometric Society European Meeting, Touousse, 1997, XXth Bosphorus Workshop on Economic Design, Aegean Sea, 1997, and seminars at Bogazici, Harvard, MIT and Stanford Universities. I woud ike to thank the participants of these conferences, especiay Ahmet Akan and Roy Radner, for their comments. The usua discaimer appies. 2 MIT Department of Economics, Room: E52-391, 50 Memoria Dr., Cambridge, MA 02142, USA. Emai: myidiz@mit.edu. Te: ( + 1)

8 Abstract We anayze the subgame-perfect equiibria of a game where two agents bargain in order to share the risk in their assets that wi pay dividends once at some fixed date. The uncertainty about the size of the dividends is resoved graduay by the payment date and each agent has his own view about how the uncertainty wi be resoved. As agents become ess uncertain about the dividends, some contracts become unacceptabe to some party to such an extent that at the payment date no trade is possibe. The set of contracts is assumed to be rich enough to generate a the Pareto-optima aocations. We show that there exists a unique equiibrium aocation, and it is Pareto-optima. Immediate agreement is aways an equiibrium outcome; under certain conditions, we further show that in equiibrium there cannot be a deay. Finay, we characterize the conditions under which every Pareto-optima and individuay rationa aocation is obtainabe via some bargaining procedure as the unique equiibrium outcome. JEL No: C78, D89. Keywords: Bargaining, Risk-sharing, Deay.

9 1 Introduction Consider two risk-averse agents who want to share the risk in their assets. Assets wi pay dividends ony once, at some fixed date i. As in Wison (1968), each agent has his own beiefs about the dividends, but no agent has any private information. 1 Which contract wi these agents agree on, and when wi they reach an agreement? What are the determinants of "bargaining power"? These are the questions we ask in this paper. One might find it obvious that, given a sufficienty rich set of contracts, the agents wi immediatey agree on a Pareto optima contract. We wi show that this is true - but there are some interesting detais to it. In this environment, any aocation feasibe at the beginning remains feasibe unti the end, and the agents are indifferent about which date they agree on any given aocation. Thus, the set of payoffs does not shrink as time passes and there is no discounting to create impatience for agreement. Instead, as time passes, some uncertainty about the dividends is resoved, making some of the contracts individuay irrationa for some agent, given the agent's expectations about how events wi proceed. This reduces the scope for insurance. (Eventuay at date i agents ose a insurance opportunities; autarky is the ony individuay rationa contract eft.) This oss of contracting opportunities eads agents to agree eary on. One can imagine many bargaining probems where such ost contracting opportunities are the major costs of deay - not discounting by the agents or some materia osses. For exampe, many wage contracts are negotiated before the current contract expires. Thus, unti the current contract expires, a major part of the cost of deay is the ost insurance opportunities between the workers and the firm against fuctuations in the abor market. 2 Sometimes, agents may aso hod incompatibe beiefs about the size of the gains from trade and each agent's outside options. Our anaysis here may be hepfu in anayzing such probems. Our efficiency and immediate-agreement resuts are as foows. Assuming that a Paretooptima aocations can be generated by contracts throughout the bargaining process, under any sequentia bargaining procedure with immediatey expiring offers, the foowing are true: There exists an equiibrium where agents reach an agreement immediatey; any subgame-perfect equiibrium is payoff-equivaent to this equiibrium (Theorem 1); and the equiibrium aocation is unique and Pareto-optima when the first offer is made (Theorem 2). 3 Furthermore, except 'Under these conditions, assuming that a Pareto-optima aocation rue is empoyed, Wison (1968) characterizes the conditions under which the group can be represented by an expected-utiity maximizer. 2 Another exampe: it is common in ega practice that the settement is negotiated whie parts of the case are itigated. As each decision is made in court, some uncertainty about the eventua payment by the defendant is resoved, making certain settement agreements individuay irrationa for certain parties. (In wage bargaining, deaying agreement may aso resut in the costy deay of some decisions the parties might want to take after the agreement. In itigation, the ega fees are aso very arge.) 3 The proof of Theorem 1 is somewhat more invoved than the argument that, given a sufficienty rich set of contracts, the agents can imitate any deay by writing a contingent contract that requires at each contingency

10 for the trivia case where autarky is initiay Pareto-optima, immediate agreement is the ony equiibrium outcome, provided each agent can make an offer with positive probabiity throughout (Theorem 3). These resuts carify certain resuts in the bargaining iterature. Excessive optimism is presented as an expanation for deays by many authors, such as Hicks (1932), Landes (1971), Posner (1974), Farber and Bazerman (1989), and Yidiz (2001). They demonstrate that, if each agent has excessivey high expectations about his prospects in case of a deay, then deay may be necessary in equiibrium. Here, under the condition that agents can generate a the Paretooptima aocations throughout, we show that immediate agreement is aways an equiibrium outcome, and is in fact the ony equiibrium outcome for generic cases. Therefore, the deay resuts from the authors' restrictions on the set of contracts. Moreover, in order to satisfy our condition, a we need is risk-sharing agreements about the dividends and side-bets on the events about which our agents have distinct beiefs. Therefore, their resuts are based particuary on their impicit assumption that side-bets are not feasibe. We show that the continuation vaue of an agent at the beginning of any given date is the sum of the rents he expects to extract when he makes offers in the future, pus his expected utiity of consuming the dividend paid by his own asset. The vaue of the rents is determined by the way uncertainty is resoved. After a substantia resoution of uncertainty, a substantia group of the potentia contracts become individuay irrationa, and thus not viabe. Therefore, agents who make offers prior to such resoution of uncertainty extract substantia rents. For instance, in the canonica case anayzed in Section 5, reveaing an informative signa at some time t rather than s is equivaent to transferring the gain from trade associated with this signa from the agent who makes an offer at s 1 to the agent who makes an offer at t 1. Many axiomatic bargaining soutions have the property that, if an agent becomes more riskaverse, then he becomes worse off whie his opponent becomes better off. The same property hods in the Rubinstein-Stah mode for the instantaneous risk-aversion. In our mode, the impact of risk aversion is ambiguous. If the tota weath in society is risky, risk aversion of an agent may hurt his opponent. Hence, this property exhibited (and sometimes postuated) in axiomatic bargaining modes is not confirmed here. Finay, even though we aow our agents to hod different beiefs, this is not the main attribute of our mode. In fact, the earier versions of this paper assumed that the agents hod the same beiefs. The outine of the paper is as foows. In the next section we ay out our mode. In Section 3, we present exampes with deay and inefficiency. We present our main resuts in Section 4. In Section 5, we derive the soution in cosed form for a canonica case; we iustrate our resuts, that they write the very contract they woud have written in the case of deay. This argument impicity assumes that there exists an essentiay unique equiibrium in the continuation game, a resut that needs a proof.

11 and discuss the effects of risk aversion. In Section 6, we present a monotonicity property of the equiibrium and characterize the aocations that can be the equiibrium outcome for some bargaining procedure. The appendix contains the proofs omitted in the text. 2 Mode In this section we ay out our mode. For some (possiby arge) ( N, we take T = {t G N 0 < t < t} to be the time space, where N denotes the set of a non-negative integers. At each date t G T, a signa Yt becomes pubicy observabe and remains observabe thereafter. Writing Y = {Y,...,Yi) and Y = (Y,,Y t ), we designate y = {ys } s( z T and y = {ys }s<t as the generic reaizations of Y and Y, respectivey. We have one consumption good, which wi be consumed at date t, and two agents, whose set wi be denoted by N = {1,2}. Each agent i has a stricty concave and stricty increasing Von-Neumann-Morgenstern utiity function u : R > R. We write P for the beiefs of i at y, and P x for his prior beiefs. We assume that P 1 and P are equivaent. That is, they assign zero probabiity to the same events. We write E % respect to P (- y f ) and P, respectivey. [- y*j and E1 (-{y 1 ) for the expectations with Given any i G N, for some x G R, we take [x\oo) as the set of feasibe consumption eves for agent i and write A' for the set of a (risky) consumption functions, i.e., the random variabes x : y > > x 1 (y) G [x%co) that are integrabe with respect to P. Each agent i G N has an asset that pays ony one dividend D1 at which our agents can trade. 1 G X, which wi be paid at i. There is no market By an aocation, we mean any x G A" 1 x A' 2 with x (y) + x 2 (y) < D 1 (y) + D 2 (y) at each y. Here, x z (y) denotes the consumption of agent i when the reaization of Y is y. By a contract x[y at a given y, we simpy mean the restriction of an aocation x to the set {y'\y's Vs for a s < t} of a continuations of y. Ceary, an aocation x is the mutua extension of contracts x[y*. We write X [y* for the set of a contracts at y and X for the f set of a aocations. We take some Ct [y C A [y and write C t CI for the set of aocations generated by the contracts in Ct [y (The set of feasibe contracts may be stricty smaer due to some additiona constraints.) We finay write as the set of feasibe contracts at y* C = (Co,,Cf)- We wi anayze the game G (C, p) in which the foowing is common knowedge: At each t, first the signa Y t is observed and an agent pt (y ) G N is recognized. Then, the recognized agent offers a contract x [j/. If the contract is accepted by the other agent, they sign the contract, yieding payoff E 1 \u (x [y { ) y' for each i ; otherwise, the offer expires and they wait unti the next date, except for date t, when the game ends and each i consumes D The process p = (p,...,pj) is caed the recognition process. (y).

12 . y s. We of such probabiity assessments a bargaining procedure. We assume 4 that P 1 (p t = i\y s ) = P 2 (p t = i\y s ) = p\ (y s ca any fu ist p = {p\ {y s )} s it s ) for some p\ (y ) at each t and We assume that p and D are independent and that p\ (y s ) depends ony on the history {p t, (yo,,yt')}t'< s ^ recognized agents. We finay assume that p is is affiiated: when an agent recognized at a given date, he wi be, if anything, more ikey to be recognized in the future. The set of a such bargaining procedures is denoted by V. Optimaity, side bets, risk-sharing agreements Given any y f, an aocation x is said to be Pareto- optima at y 1 iff [3i N, tfvtoy' >?*[«*(a)iy* =» [Sj 6 iv, ^'['( ) y* > E*[v?{x)\y*\, amost surey, for each aocation x X. That is, the (conditiona) probabiity that we can improve both agents' expected utiity at y J is zero. An aocation is said to be Pareto- optima iff it is Pareto-optima at the beginning. We say that C is Pareto-compete iff Ct contains a Pareto-optima aocations for each t G T. Our next emma states that C is Pareto-compete whenever a risk-sharing agreements and side-bets are avaiabe. Let us consider some objective components of uncertainty separatey, and write each Yj as (Yj,Yt) where the agents have a common prior on Yt, which contains information about some objective aspects of uncertainty. Likewise, we write each y as (y,y)- Lemma 1 Let X = {x G X\x(y,y) = x(y',y) V(y,y',y)}. If D G X, then (X,...,X) is Pareto-compete. That is, Pareto-optima contracts do not refer to Y = (Y, dividends and if agents hod common beiefs about Y...,Yj) MY does not affect the Put differenty, so ong as we can write contracts on a the events that are reevant for the dividends (so that they can share a the risk) or the events about which the agents have distinct beiefs (so that they can bet), we can generate a the Pareto-optima contracts. 3 Exampes of deay and inefficiency Two properties of the bargaining procedures in this paper are essentia: (i) they are sequentia and (ii) the offers expire immediatey. In addition, we assume that (iii) C is Pareto-compete. Under these conditions, we wi show that a equiibria are equivaent to an equiibrium where payers immediatey agree on an aocation that is Pareto-optima when the first offer is made. A the reguarity conditions we are about to assume are here to simpify the exposition. in Theorems 3 and 4 (see Yidiz, 2000). They are ony used

13 (We wi further show that this is the ony equiibrium outcome for generic cases.) Now, we wi demonstrate that these conditions are not superfuous. Our first exampe demonstrates that, when C is not Pareto-compete, we may have deay and inefficiency in (the unique) equiibrium. In this exampe, we restrict our contracts to the saes contracts. A saes contract can be represented by the amounts of equity shares Agent 1 owns in the assets, and a consumption-good transfer from Agent 2 to Agent 1. Exampe 1 (The set of contracts is Pareto-incompeteJ Take T = {0,1,2}. p = p2 = 1, and p 1 = 2. For each i, take x 1 = and u (x! ) = \fx. Take Yq a constant, and D (as a function of Y\ and Y 2 ) as in the foowing tabe. Y 2 = H >2 = L Yi = (16,9) (16,0) Yi = 2 (9,16) (0,16) We take P 1 1 (Yi = i) = 1 and P (Y 2 = H) = P (Y 2 = L) = 1/2 for each i. 5 We assume that ony saes contracts are feasibe. At date 2, each agent knows the true state, hence any trade wi be just a transfer from some agent i to the other, which is not individuay rationa for i. Hence, if they have not agreed by date 2, they wi not trade. Thus, Agent 1 wi accept an offer at t = 1 iff it gives him at east the expected utiity of consuming his own asset. Moreover, at t = 1, the optima contracts are saes contracts where any agent's equity shares in two assets are the same. Hence, at t = 1, Agent 2 wi offer the Pareto- optima saes contract that gives Agent 1 his continuation vaue. The offer wi be accepted. When Y\ = 1, the asset of 1 wi pay 16 for sure, yieding expected utiity of \/6 = 4. Then, Agent 2 wi offer 64/81 of each asset to Agent 1, yieding payoff vector (4, \/5/2). When Y\ = 2, if Agent 1 consumes his own asset, his expected utiity wi be ony \/9/2 + \/0/2 = 3/2. In that case, Agent 2 wi offer 1/9 of each asset to Agent 1, yieding payoff vector (3/2, 3%/2). At t = 0, Agent 1 is certain that Y\ = 1 and he wi get 4. Likewise, Agent 2 is certain that Y\ = 2 and she wi get 3v2- But the best saes contract at t = is no-trade, yieding expected payoff (4, 4) >C (3/2, 3\/2)- Therefore, they disagree at t = 0. Yieding payoff vector (4,3v2), the equiibrium is Pareto -inefficient. If Agent 1 owned both assets when Y\ = 1 and Agent 2 owned both assets when Y\ = 2 (the saes contracts avaiabe at each t G {1,2}/. the payoff vector woud be (9/2,9/2). Note that there is no aocation generated by feasibe contracts at t = that Paretodominates the equiibrium aocation. Therefore, the inefficiency is not caused by deay. Recognizing that the Pareto-dominating contracts wi be individuay irrationa when they 5 For simpicity, we take P (Vi = i) = I for each i, when P 1 and P 2 wi not be equivaent. For sufficienty sma > 0, P* (Vi = i) = 1 e woud aso work.

14 become avaiabe, agents agree on a Pareto-dominated contract. Note aso that, if C were Pareto-compete, the contract that gives the entire weath to the agent i with Y\ = i woud be feasibe at t = 0; and agents woud agree on this contract at t = 0. When the bargaining procedure is not sequentia, there may be mutipe equiibria. In that case, the equiibria in future subgames may depend on the pay at the beginning; and this may cause a disagreement at the beginning. This is demonstrated in our next exampe. (Whenever a subgame possesses mutipe equiibria, none of which dominates the other, we can add a prior date and construct a subgame-perfect equiibrium of the new game with deay.) Exampe 2 ("The bargaining procedure is non-sequentia. ) Take T = {0,1,2}, x = 0, and u 1 (x 1 ) = vx 1 for each i. Let Yq be a constant, Y\ and Y2 be identicay and independenty distributed, each taking vaues of and 1 with equa probabiities. Take D = A(Y\,1 Y\) so that the socia weath D 1 + D 2 = 4 is riskess. At dates and 2, we have sequentia bargaining procedures with p = p 2 = 1. At t = 1, simutaneousy, each agent offers a feasibe contract. If they offer the same contract, it becomes enforceabe; otherwise, they wait unti date 2. If they have not agreed by date 2, they do not trade, and each gets v4/2 + v0/2 = 1. There are mutipe equiibrium substrategies at t = 1. In one of them, agents agree on the riskess aocation (1,3), where agents 1 and 2 consume 1 and 3 (for sure), respectivey. In another one, they agree on the riskess aocation (3,1). One can check that the foowing is an equiibrium. 7/Yi = or Agent 1 offers (4,0) att = 0, agents agree on the riskess aocation (3,1) at t = 1; they agree on (1,3) otherwise. At t = 0, Agent 2 accepts an offer x iff E v? > y/3; and Agent 1 offers (4,0), which is rejected. [This is an inefficient equiibrium; it seects (1,3) and (3, 1) with equa probabiities. Finay, our next exampe demonstrates that, if the offers ived onger, the equiibrium outcome might necessariy be inefficient. Exampe 3 ('The offers ive onger. ) Take T = {0,1}, and assume that the offers made at t = expire at the end of date 1. Let p = 1 so that Agent 1 makes the offer at 0. Given any offer x 7^ D of Agent 1, the best response of Agent 2 is to wait unti date t = 1 and to accept the offer iff' x 2 (y) > D 2 (y). Hence, for any offer x, Agent 1 woud consume min{a; 1,Z) 1 } and Agent 2 woud consume max{x 2,D 2 }. In that case, Agent 1 woud offer x = D. Therefore, there is no trade in equiibrium.. Simiary, aowing the offers to ive one period in Rubinstein's (1982) mode, Avery and Zemsky (1994) show that the steady-state subgame-perfect equiibrium may be inefficient. many situations some information may arrive before the offer expires. This adds an option vaue for waiting, and causes inefficiency. (In an earier version of this paper, we show that, if C is Pareto-compete, and the offers ive one period, the equiibrium aocation wi be Paretooptima when the first offer expires, that is, conditiona on the signa observed.) In

15 4 Equiibrium In this section we anayze the subgame-perfect equiibria of a game G(C,p) with Paretocompete C. We first construct an equiibrium where agents reach an agreement immediatey. A the equiibria yied the same equiibrium aocation, which is Pareto-optima when the first offer is made. We show further that immediate agreement is the ony equiibrium outcome if there are some gains from trade and each agent is ikey to make an offer throughout. Finay, we derive an equation determining an agent's equiibrium payoff in terms of the dividends and the sequence of expected rents. As usua, a strategy of an agent is a compete contingent-pan that determines the offer the agent makes when he makes an offer, and whether he accepts or rejects an offer made by the other agent. We require an agent's offers at various y of a given date t to be measurabe. A strategy profie s = (s^s ) is a subgame-perfect equiibrium (henceforth simpy equiibrium) iff the strategies are best response to each other in each subgame. We first present two preiminary resuts used in our construction. Our first. Lemma states some famiiar properties of optimization with increasing functions. Lemma 2 Let C be Pareto-compete. Given any distinct i,j 6 N, any t, any y v = (is 1 u 1 (z 1 ) Ij/'I,E 2 [u 2 (z 2 ) y { ) with z = (z 1 ^ 2 ) X, the optimization probem, and any max #V(sV) y* subject to E j [u 3 (x 3 {y xy'scty 1 1 }) [if) > r } (1) has an (amost surey) unique soution x t [v, i; y, which does not depend on C'. Write m (v; y = ) E 1 1 ^ for the maximum. Then, the foowing are true. (x\[v,i\y t \) \y 1. xtfwjijy' is Pareto-optima at t; 2. the constraint is binding, i.e., E u j [xv^ytyj \y v J : and 3. m {v;y \ > v i, where the inequaity is strict whenever z is not Pareto-optima at y. Given any two dates t.s E T with t < s, consider a coection {2:[y s } v s of contracts at s, indexed by the continuation y s of y unti s. Write x\y for the mutua extension of these contracts. At each y s, x\y requires agents to write x[y s }. We ca x\y preemptive, for x\y is equivaent to offering the continuation in advance to preempt the deay. Our next Lemma states that x\y wi be a contract at y in a cases of concern. Lemma 3 Take any t T, and et v [y s = (E 1 some (z 1,^2 ) G X. Then, we can seect a famiy x s [v [y s that their mutua extension wi aso be an aocation. [u x (z 1 s ) \y,e 2 [u 2 (z 2 s ) \y \) at each y s for,i;y s ) of soutions (indexed by y s ) so

16 Construction of an Equiibrium For a given G(C,p) with Pareto-compete C, we now construct a subgame-perfect equiibrium, where the agents reach an agreement immediatey. We use backward induction. If the agents have not agreed by t, at t, agent j ^ pi (y) wi accept an offer x(y) iff x? (y) > D- 7 (y). This inequaity hods ony when x k (y) < D k (y) for k = Pf{y)- Hence, k wi offer xi{y) = D(y). There wi be no trade. Each wi consume his own asset, independent of which contracts were offered at previous dates. The continuation vaue of an agent i at the beginning of t wi be V?[y t^)=v\u (V)\y 1 -^. (2) Given any date t < t, assume that an aocation x t+ i X wi prevai at date i+1, independent of which contracts were offered (and rejected) before. Write V^+1 (y t ) = E \u (x\ +1 ) y for the continuation vaue of any agent i at the beginning of t + 1. Given that agent j ^ Pt (y*) g e *s ^+1(2/') when he rejects an offer xjfy', it is a best response for him to accept xty 1 J ) iff -E7-7' [u- (rc^ 7" [j/* ) j/* > V t+1 (y t ). Given this response, if agent k = p t (y*) offers a { contract x t [y, then his expected utiity-eve wi be E k [u k (x k t t [y )\y ^t+i(2/*)i and Vt+iiv*) if E^ [u^ (xjy 1 })^ 1 otherwise. To maximize this expected utiity eve, he wi either offer xt[vt+ i(y t ),ky t }, the best acceptabe offer, and get m k (Vf+ i(7/*); 2/*), or wi offer some unacceptabe offer and get VJ fc (y*). By Lemma 2.3, m k (Vt+i(y t );y t ) > V k + (y ), hence offering x t {V t+ i(y t ),k;y t is a best response for k. Agent k offers x^vt+iq/^fc;?/*, and the offer is accepted. Since xt+i X, by Lemma 3, we can choose contracts xt[vt+i(y { ), k;y at a sampe paths y 1 in such a way that their mutua extension forms an aocation, yieding a (preemptive) t_1 contract x t [Vt+i, k; - y at y. We choose them so. If agents do not reach an agreement by date t, aocation xt[vt+i, pt ; } (which is xt\vt+\,i; - yt_1 when pt (y { ) = i and xt[vt+i,j; - y when p t (y*) = j) prevais at t. This aocation is once again independent of which contracts were offered previousy. Since u % (a^[vt+i,i; ) = m (Vt+ \;-) and u 1 ^x\[vt+ i,j; ) = ^t+i(')> the continuation vaue of an agent i at the beginning of t is at each y t_1. KV" 1 ) = &[*? (F t+1 ;y 4 ) ^ct^y*- 1 + E1^ (y > 4 ) {p t ^)^}3/ t_1 ( 3 ) By backward induction, this procedure gives us a subgame-perfect equiibrium s of G(C, p). According to s, at any y', an agent j ^ p t (y ) accepts an offer xt[y' iff E^ [u^ (xjy 1 )^ > VJ+ 1 (y t ), and agent i = pt (y 4 ) offers x\\y t +\,i;, where the process V= {V t }tet soution of (2-3). Our next theorem states that s is essentiay the ony equiibrium. is the unique Theorem 1 Let C be Pareto-compete. Then, the strategy profie s is a subgame-perfect equiibrium of game G(C,p). Moreover, for any subgame-perfect equiibrium of G(C,p), the vector of continuation vaues at the beginning of any date t with y t_1 (before Yt is observed) is Vt (y t_1 ). After Yt is observed, the continuation vaue of an agent i is m x Pt (y ) = *> and ^1+1(2/') otherwise. (Vf+i(y i );y t ) if

17 This That is, any equiibrium is payoff equivaent to s. Moreover, according to s, the agents reach an agreement t = 0, so ong. as the space of feasibe contracts is Pareto-compete, e.g., so ong as a side-bets and a risk-sharing agreements are feasibe. (Therefore, the deay resuts cited in the Introduction are based on the authors' excusion of some of these contracts.) In any equiibrium, by Theorem 1, when agents earn which of them is to make the first offer, the continuation vaues are m % (V\(y );y ) and V^(y ) where = pq (j/ ) i ^ j. By Lemma 2.1, this is a Pareto-optima expected-utiity eve. Since the utiity functions are stricty quasiconcave, it is uniquey obtained by xo[v\,p (j/ ) ;y. Therefore, {xo[vi,/? (j/ ) ''V }} o is the unique equiibrium aocation. It is Pareto optima at y, when they earn which payer makes the first offer. Before they earn y, however, agents are uncertain of which xo[v\,p (y ) ;y wi prevai. When xo[v\,p (y ) ;y varies as Yq takes different vaues of yo, the uncertainty about xo[vi,p ; ( ) may hurt our risk-averse agents, causing some inefficiency. In order to avoid this ambiguity in measuring efficiency, we wi state our efficiency resut for the case that Yq is constant. Theorem 2 LetC be Pareto-compete. Then. G(C,p) admits a unique equiibrium aocation, which is Pareto-optima at y. aocation is Pareto-optima at the beginning whenever Yo is constant. We have seen in Exampe 1 that, when the space of contracts is Pareto-incompete, some deay may be necessary for reaching an agreement. When the space of the contracts is Pareto compete, there exists a (preemptive) contract at t = that imitates such deay, as the deayed equiibrium aocation is unique. Therefore, even though the unique equiibrium aocation is reachabe via contracts at t = 0, it is not cear that these contract are not preemptive, imitating a deay. Theorem 2 impies that, typicay, the contracts written in equiibrium are not preemptive. For, by the syndicate theory, Pareto-optima contracts use the minima information about the tota weath and the discrepancy between the agent's beiefs. When there is a common prior, they depend ony on the sum D + D 2. Our next exampe carifies this matter further and motivates our next theorem. Exampe 4 For any arge, even integer t, et Y = (Yq..., Y{) be identicay and independenty distributed, each Yt taking vaues of and 1 with equa probabiities. As in Exampe 2, take D = 4 (Yi, 1 Yj) and set u (x ) = vx! for each i. First consider the bargaining procedure with aternating offers where Agent 1 makes offers at even dates. Att 1, Agent 2 offers the riskess aocation (1,3), which is accepted. This aocation is aready Pareto-optima. Hence, there are mutipe equiibria yieding the same aocation: at each t < t 1, the recognized agent offers the riskess aocation (1,3), and the other agent accepts this offer ony at some t* <i- 1.

18 Now consider the recognition process according to which Agent 1 makes an offer at t iff y = 0. t Note that p\ = p\ = 1/2 for each t. Under this procedure there is a unique equiibrium. At i 1, if recognized, agents 1 and 2 wi offer (3,1) and (1,3), respectivey. The offer wi be accepted. Hence, at the beginning oft 1, the risky aocation that gives (3,1) and (1,3) with equa probabiities wi prevai. Thus, at t 2, the recognized agent wi offer the other agent his continuation vaue, ( + \/3) /2 and keep the rest, which is arger than ( + \/3) /2. Therefore, a risky aocation prevais att 2, too, yieding ((1 + \/3)/2,4 (1 + \/3)/2) and (4 (1 + v3)/2, (1 + v3)/2) with equa probabiities. Repeating this procedure, one can see that, at each date, a risky aocation prevais, eading agents to reach an agreement at the previous date, in which the randomy determined proposer gets a somewhat arger share. Therefore, there is a unique equiibrium, which yieds immediate agreement.^ Therefore, under this random recognition procedure, the immediate agreement is not merey an imitation of a deay, as deay is no onger a possibe equiibrium outcome. This is generay true in our mode: Theorem 3 Let p be a bargaining procedure with p\ (ps ) G (0,1) throughout. If D is independent ofyo and not Pareto- optima initiay, then the unique equiibrium path-of-pay is that at t = the recognized agent i offers the contract xo[vo,i; <fi and the other agent accepts the offer. The rationae behind Theorem 3 is as foows. When there are gains from trade, the recognized agent wi get some positive rent at some history y. Since either agent is ikey to be recognized at t, at the previous date there is a need for insurance against the risk associated with the aocation of this rent. When ony one of the agents can get a positive rent when he is recognized, he faces a risk of whether he wi be recognized so that he wi get the rent, a risk that can be shared. When both agents can get a positive rent when they are recognized, they face uncertainty about which agent wi get the rent; and once again they woud ike to share the risk. In either case, there is a gain to be reaized at t 1, aocating a positive rent to the agent recognized at t 1. Appying the same argument inductivey, one can concude that there are gains to be reaized at 0, in which case deaying agreement wi cause an inefficiency (when the first offer made), which cannot be an equiibrium outcome according to Theorem 2. That is, when there are gains from trade, a stricty stochastic bargaining procedure wi generate a positive risk (and thus a positive cost of deaying agreement) throughout the game, rendering deay impossibe as an equiibrium outcome. Note that, as t oo, the equiibrium aocation approaches to (2,2), whie the aternating-offer procedure aways yieds (1,3). simiar outcomes in These two procedures thus yied very different outcomes in our mode - whereas they yied Rubinstein's (1982) mode with discounting. 10

19 ps Summary In our mode the set of feasibe outcomes does not necessariy shrink as time passes. Yet, we are abe to pin down a unique (Pareto-optima) aocation as an equiibrium outcome. Here, reveation of some information makes certain contracts individuay irrationa for some parties, and therefore not viabe to society as a whoe, a phenomenon known as the Hirsheifer effect. This happens to such an extent that at date t the ony individuay rationa aocation is D itsef. This sets further restrictions on the set of viabe aocations at the previous date, through Lemmas 2 and 3. The inductive appication of this procedure gives us a unique (Pareto-optima) eve of utiities at the beginning of the game. Risk-aversion shows up at this stage and tes us that the contract must aso be unique, and further requires that, genericay, there is no deay in equiibrium. Equiibrium payoffs and rents In any equiibrium, given any y, an agent i N gets rri 1 (V t+ i(y t );y t ) if he is to make an offer at y', and gets ony V t ' +1 (y ( ) if he is not to make an offer at y. Hence, the recognized agent i = pt (y ( ) extracts a (non-informationa) rent R\ (y 4 ) = m< (Vt+ifo*);^) - V^fe,*) > 0. (4) Together with D, these rents determine the equiibrium payoffs. Substituting (4) in (3), we obtain I? [y t- 1 )=Ei this yieds [R\ (y { ) {My r )=} \y^}+e [vy +1 (</) y'" 1. Since V? = (u 1 (D 1 ),u2 (D 2 )), Vt {y*- 1 ) = # I** (y s ) i wy*)=oy t_1 + E ' W s=t Di t_1 ( ) y ( 5 ) for each t T. Equation (5) dispays the determinants of bargaining power in this environment. It states that the continuation vaue of an agent is the sum of the rents he expects to extract in the future when he is recognized, pus his expected utiity from consuming the dividend his own asset pays. These rents wi be decouped in our next section. 5 Equiibrium in a Canonica Case In order to iustrate some of the basic ideas in this paper, we anayze the equiibrium in the foowing canonica case. We assume that our agents have a common prior and the credit constraints are not binding, i.e., the set of feasibe consumption eves is E. We take u (x) = exp(-aix) for some a* > for each i e N. We et Y = {Y t : t e T be independenty distributed where each Y t takes vaues in R 2, E[Y t = at every f > 0, and Yq e M 2 is a constant. We take D = Y$ + Y\ + + Y{ so that as time passes agents earn the independent increments in the dividends. Finay, we take a purey deterministic bargaining procedure, and write Ti[t) = {s T \ agent i is to make an offer. = i, s > t for the set of a future dates at which 11

20 . Notation The tota weath is denoted by W = D 1 + D 2. We define the certainty equivaent CE\[x 1 } of any x i for any agent i at any date t as CE^x* = (u 1 )' 1 (E [u 1 (x 2 ) \Y,...,Y t). When x 1 is a function of (Yt+i,.,Y hence..,yj), it is stochasticay independent of (Yo,... t ), and CE\[x ) = - \oge[exp(-aix 1 ) = M[x { ai ). (6) Oii In Figure 1, we pot the equiibrium contracts {x t } t&t in terms of their certainty equivaents computed at date 0. In this figure, the vector of certainty equivaents of any given,y aocation x t of the form x t = E[D\Yq,... E[D\Yo,...,Yt = Yo + Y\ + + Y t. Note t + f(yt+ i,...,yt_i) stays under the ine t, where that t moves inwards as t increases. At t, they consume D, which is depicted at the origin. Assume that Agent 2 makes an offer at i 1. He offers to share the unresoved risk Y^ + Y 2 optimay, and takes a the rent. The certainty equivaent of this aocation is found by going up unti we reach the ine f i- At i 2, Agent 1 now offers to share the risk Y^_ x + Y^_, as we. He now extracts a the rent of agreeing one day earier. Hence, the vector of certainty equivaents for this contract is found by going horizontay unti we reach the ine ^ - Using this agorithm unti we reach the ine Iq, we find the certainty equivaents of the contract signed at t = 0. We compute that the equiibrium aocation at any date t T is x = E[D*\Y,...,Y t } + + a-i CX\ + Ot-2 (W - E M s Tj(t) E M sst,(t) E[W\Yo,...,Y t}) OLi '(Ys+ +^s+u a.\ + «2 Oti a\ + Q2 (Y s\i+y s+) (a*) - M [Yi +1 aj)-m yd ( (a,) («i), (7) where i,j N with i ^ j. (One can easiy check the vaidity of (7) by induction.) Note that the share of the socia risk W E[W\Yo,, Y scheme is Q1+Q2 (W E[W\Yq,. his individua risk Y"/ +1, agent - ' a (W jb[w Yb,. t borne by agent i at any optima risk-sharing, Yt). At any s Ti(t), in order to insure himsef against j is offered to bear - ' a (Y^+ + Y 2 +1 ) in addition to the risk -, Ys+ i) that he woud bear if they were to agree at the next date. Since the agent i makes an offer that can be rejected ony by deaying the agreement unti s + 1, he extracts the entire vaue M ai+q2 (*?+!+ y s 2 + i) (<*j)-m Y 3 (ctj) of this transaction for agent j. This vaue is the certainty equivaent of the extra risk that agent j bears, minus the certainty equivaent of the risk that agent j avoids. Thus, (7) is read as: The consumption aocated to an agent at some t is the certainty equivaent of his own asset's dividend pus the optima risk that he shares, minus the rents that he pays to the other agent when the other agent makes an offer (in the future), pus the rent that he extracts from the other agent when he makes an offer (in the future). 12

21 ^2r^2 CE [ij Figure 1: The sequence of the equiibrium contracts in the space of certainty equivaents computed at t = 0. 13

22 Using (7), we compute 7 that CE{x=CEHD>}+ Y, G B, (8) s Ti(t) where CE^D* = E[D<\Y,...,Y t + s >t M [Y* +1 (on) and g s = m [y;vi + y s\i} (-^-) ~ \m f^i («;) + M [YJ +1 (a,). (9) Given that they wi share the risk optimay, the vaue of the increment Y* +1 + Y 2 +1 weath for the agents at date s is M [Yg +1 + Y 2 +1 ( Y 2 +1 in tota a X +a j, the certainty equivaent of Y* +1 + with respect to the group's surrogate utiity-function of Wison (1968). If they were not abe to share the risk, the vaue of the increments Y* + and Yj +1 in the socia weath woud be ony M Xs+i ( a j) + M \Y* + \ (cti). Therefore, G s is the gains from sharing the risk in Y s+ j, optimay Therefore, (8) states that, in CE terms, the consumption that our equiibrium aocates to agent i at date t consists of what he aready owns and the sum of gains from (optima) trade when he makes offers in the future. Equation (8) is simiar to (5). Moreover, in CE space we have transferabe utiity, which aows us to decoupe the gains from trade at each date from the rest of the probem. Observations strengthens the position of a bargainer. To see this, et Y= (Yo,.., Y. s,y s+ i + Z, Y s+ 2,, Yd and Y* = (Y,...Y S *,Y S Z,Y. s. +2,..,Y s,y s+1,y s+2.,..,,y {) be such that {Y x,...,yh Z} Firsty, being abe to make an off er prior to an important information-reveation is independenty distributed and ps.. ^ If we repace ps Y with Y*, then CE Ps ' {xq 3 *} increases by G[Z =M{Z + Z 2 (^j) the same amount. - [M [Z* (a,-) + M [Z J (a z )}, whie CEP-[3% decreases by Second, if we repace a bargaining procedure p with p by changing p\. = to p\» = 1 at some t* for some i, then CE [x increases by Gr- [Cf. Theorem 4 beow. Third, any Pareto- optima aocation is approximatey obtainabe via a deterministic bargaining procedure as ong as information arrives smoothy. Assume that Y\,...,Yf are identicay distributed with variance a 2 /t. Then, given any Pareto-optima and individuay rationa aocation x S X and any e > 0, there exists a t such that we can find a deterministic bargaining procedure p that yieds CE'[xq CE*[a;* < e for each i AT as ong as t > t. To do this, etting Ai,A 2 G R be such that Ai ICE 1 ^ 1 and Ai + A2 > 0, we write n for the integer part of A ^A procedure p with p\ = 1 iff t < n. Now, by (8), CE x [x) - CE [D = - CE [D }\ = A 2 \CE 2 [x 2 - CE 2 [D 2 \ t. We consider the bargaining ng = (n/t)g M [ 7 By independence, CE\[ 1^{W - E[W\Y, 8Q2 a: (qi) + M I oi ' a2 x (q 2 ' = M ) [x ( a ai...,k) = E s>, M [^j(?+i + *H-i) («) As - + a 2, ) at any x. Substitute these equaities in (7). 14

23 and CE 2 [x 2 - CE 2 [D 2 = ) } (1 - n/t) G, where G = T S G S is the tota gain from trade. Thus, C '[if) -CE [x = )\ (^Ai-n) G/t < G/i, where the right hand side goes to as i -* oo. [Cf. Theorem 5 beow. The impact of risk aversion if a bargainer gets more risk-averse, then he becomes worse off, this. Many axiomatic bargaining soutions have the property that, whie his opponent benefits from Exampes of such bargaining soutions are Nash (1950), Masher-Peres (1980) and Kaai- Smorodinsky (1975). Roth (1985) shows that Rubinstein's bargaining mode has the same property if we consider "instantaneous risk- aversion." 8 We wi now show that this property does not hod in our mode, where the bargaining is driven by risk aversion itsef. 9 In our mode, risk-aversion of an agent may hurt or benefit his opponent. To see this, note that an increase in a ; affects CE[x\ through G s for s e Xj( ). As a_j increases, on the one hand, Q M Y/ +1 («_,) decreases, which increases G s. On the other hand, M \Y} +1 + Y? +a +1 ( a ) decreases, which resuts in a decrease in G s. Either of these effects can be dominant: If D +D 1 2 is constant, the atter is identicay 0, hence, the risk aversion of j benefits i. If D 3 is a constant, the former is identicay 0, thus an increase in oj renders CEj[xJ ower, i.e.. risk-aversion of j hurts i. Intuitivey, the decrease in M \Y} + +Y \ s +i ( a ^+a ' s ) ^ e ^oss ^^ asent j' s ^skaversion causes to society, part of which is borne by agent i; and the decrease in M y/+1 (aj) is the decrease in j's reservation vaue, which strengthens Vs position in bargaining. In order to demonstrate which effect is dominant in which cases, et Y s+ \ be a bivariate norma random vector with variance-covariance matrix M [y; +1 <7 j ro-\oi ra\go o 2 Then, we have (a) -- -\cto- 2 and M [Y} + +Y 2 + (a) = -\a(o\ + o\ + 2ra x a 2 ). Substituting these equaities into (9) and differentiating with respect to a }, we find that G s (and hence CE\x\\) is increasing in q^ iff r < r = (a [i )\^ 11. IZi (10) 2 K a i, I <*i 0"i 2 a j 8 See Roth (1985) for the references to these resuts. In contrast with these resuts, White (1999) shows that adding a non-contractibe background-noise to an agent's consumption woud make him better off, eaving his opponent worse off. One can check that the certainty equivaent of an agent is decreasing with his own risk-aversion. But this hardy impies that he is in a weaker bargaining position, as the certainty equivaent of his consumption woud sti decrease even if his consumption had not changed at a. In comparing a payer's equiibrium wefare-eves as his own risk-aversion changes, we ceary perform an interpersona comparison between the two seves of the agent, each sef having a different utiity function. We can safey say that an agent gets worse off as he becomes more risk-averse, ony if each sef finds the agent's consumption worse under higher risk-aversion. the equiibrium outcome is Pareto optima, the more risk-averse sef wi find himsef worse off ony when his risk-aversion benefits his opponent. Since L5

24 Here, an agent benefits from the risk aversion of his opponent when r is sma enough, the case when the socia cost of this risk-aversion is not significant. Moreover, the upper bound f increases with both cvj/a, and o~j/oi. An intuition for this observation comes from insurance: As an insured becomes more risk-averse, we expect the insurer to benefit. On the other hand, if the insurer is the one who becomes more risk-averse, we woud not expect the other party -the insured- to benefit. As Qj/ojj and <Jj/<Ji increase, agent i becomes the insurer and benefits from the risk aversion of agent j. 6 Monotonicity and Obtainabiity In this section, we present two resuts about the equiibrium. The first one is an extension of a resut in Mero and Wison (1995) and Yidiz (2001) to our mode. It states that an agent cannot ose when he becomes more ikey to make an offer in the future. This fact wi aso hep us in proving our second resut. The second resut states that any individuay rationa and Pareto optima aocation is obtainabe as an equiibrium outcome of some bargaining procedure iff "the first- mover advantage" is not too arge, i.e., the uncertainty resoved at t = 1 is not substantia. Given any pep, et us write Uq (p) for the vector Vq of equiibrium-continuation vaues at the beginning for game G(C,p) where C is Pareto-compete. Theorem 4 Given any two bargaining procedures p,p V with p\ > p\ at each t T for some i N, we have U (p) > U x {p) and U 3 {p) < U 3 (p) for j ^ i. Mero and Wison (1995) provide a counter-exampe, showing that this resut cannot be extended to more than two agents in their mode. An increase in the probabiity of recognition for an agent, say 1, decreases the continuation vaue of another agent, say 3, which may in turn increase the continuation vaue of some other agent, say 2; and this may hurt agent 1. Using an exampe in which the agents may have different beiefs about the recognition process, Yidiz (2001) shows that the concusion of this teorem woud not be vaid if the recognition process were not affiiated. Given any (Pareto-optima and individuay rationa 10 ) aocation x X, can we find a bargaining procedure p V that gives us x as its equiibrium aocation (of game G(C,p))? If such a bargaining procedure p exists, then we wi say that x is obtainabe (via p). We wi now answer this question. To this end, take any i N, and set V% = {p^p\pq = 1}. The best bargaining procedure for agent i (by Theorem 4) is p [i V\ that recognizes agent i with probabiity 1 at each history and date. Under p[i, agent i extracts a the rent, eaving agent j with his reservation utiity-eve. On the other hand, by Theorem 4, the bargaining 10 That is, E i [«' (x { ) > E i [u { (D') for each i. 16

25 /o(p[2)' U {V 2 ) Uo(p[2)) Uo(P[}) Figure 2: Obtainabe utiity eves. The disagreement utiities are at the origin. procedure minimizing C/q over V x is p[i Vi where p[i\\ is 1 when t = 0, and otherwise. Under p[i, i makes ony the first offer. (In that case, at date t = 1, agent j can extract the 1 entire rent, eaving agent i indifferent to consuming D, when both agents know y. At date 0, agent i extracts the remaining rent, which is associated with the uncertainty in y.) Theorem 5 Let C be Pareto- compete. Every Pareto- optima and individuay-rationa aocation is obtainabe iffu^ (p[i) < Uq (p[j) for some i ^ j. The condition that Uq (p[i) < Uq (p[j) for some i ^ j expresses that it is not the case that, for some agent, the worst outcome when he moves first is better than the best outcome when he does not, i.e., the first-mover advantage is not too arge. Therefore, Theorem 5 can be read as: Every Pareto-optima and individuay-rationa aocation is obtainabe so ong as the first-mover advantage is not too arge. The first-mover advantage is arge when a substantia part of uncertainty resoved at date t 1. If this is the case, then many axiomatic bargaining soutions that seect centray ocated utiity pairs, such as the Nash bargaining soution, wi not be obtainabe. When uncertainty is resoved smoothy, the first-mover advantage wi be sma, and therefore every Pareto-optima and individuay-rationa aocation wi be obtainabe. 17

26 The proof is iustrated in Figure 2. Given any i, since Uo is continuous and Vi is connected, UoiVi) is a connected set with the end-points Uq [p[i) and Uo(p{i\), which we computed above. By Theorem 2, Uo (Vi) is on the Pareto-frontier, hence it is the part of Pareto-frontier connecting these two end-points. Moreover, any bargaining procedure p G V can be written as a convex combination of two such bargaining procedures p [1 V\ and p[2) Vi, where p [j is the same as p after j is recognized at 0, but recognizes j at with certainty. Then, by inearity, L^o (V) is the convex hu of Uq (V\) U Uo (V2), as shown in the figure. Since the agents are stricty risk averse, the Pareto-frontier is stricty concave, hence the set of Paretooptima payoff vectors in Uq (V) is Uo (V\) U Uo (V2)- This set contains a the Pareto-optima and individuay rationa payoff vectors if and ony if Uo (p [2) is to the right of Uq (p [1), the characterizing condition in our Theorem. 7 Concusion Consider a two-person risk-sharing probem where the agents' beiefs may differ but there is no private information. Assuming that a Pareto-optima sharing rue is empoyed, Wison (1968), in his semina paper "The theory of syndicates", shows that the group can be represented by an expected-utiity maximizer ony if the agents hod the same beiefs or the sharing rue happens to be inear. Since there is a continuum of Pareto-optima and individuay rationa aocations, we woud expect the agents to aocate the risk through negotiation. But, if the agents hod different beiefs, the negotiation may yied Pareto-suboptima outcomes. Therefore, the Pareto-optimaity assumption of Wison (1968) may not hod. In this paper, confining oursef to sequentia bargaining procedures with immediatey expiring offers, we show that the equiibrium aocation wi be Pareto-optima and that there wi be no deay for generic cases, provided there is a sufficienty rich set of contracts to generate the Pareto-optima aocations throughout the negotiation -an impicit assumption in Wison (1968). This resut strengthens Wison's theory by predicting its Pareto-optimaity assumption. The aocation chosen by the group in equiibrium depends on the way uncertainty is to be resoved. For exampe, in Section 5, where the agents hod the same beiefs and the sharing rue is inear, reveaing an informative signa at some time t rather than s is equivaent to transferring the gain from trade associated with this signa from the agent who makes an offer at s 1 to the agent who makes an offer at t 1. Since the decision of an expected utiity maximizer is typicay independent of such detais, this suggests that we may not be abe to find an expected-utiity maximizer that represents the group even if the agents hod the same beiefs. This may strengthen Wison's negative resut. 18

27 I 1. Part A Proofs This section contains the proofs omitted in the text. 4 C is Pareto-compete iff, given any x X and any y E i [u i (x i )\y t > E i [u i {x i )\y t at each i N. Towards proving Lemma 1, write Throughout these proofs, we wi use the fact that, there exists a feasibe aocation x Ct such that each P 1 aspxp", where P is the common prior associated with 1 Y. Given any (i,x\?/), we define the conditiona marginas E [x' i/ (y) f-x (y,y) dp (-\y ) and E i [x i \y t (y) = f-f.x i (y,y)dp(-\y t ). ft [D \y Proof. (Lemma 1) Take any y 1 and any x X. Since.r 1 + x 2 < D 1 + D 2, ft [x^y* + ft [x 2 \y f + ft {D'\y 1 }. Moreover, when D X, ft [ >V = D for each i N, hence ft [x^y 1 + ft [x 2 \y'} < D 1 + D 2. Thus, &\E [u«(x! ) y* y ( is Pareto-compete. < (ft [x 1 ^/,ft [x 2 j/) X. But, by Jensen's inequaity, E i < [^(x 1 )^} = #[«*( [iy*)^ = *[«<( [*»*) y* for each i JV. Therefore, (X,...,J?) Proof. (Lemma 2) Since (u 1, u 2 ) is increasing, the optimization probem (1) corresponds to the optimization probem max w 1 subject to w 3 > v J (11) w U(C t ;y<) in the utiity space where U(C i ;y t ) = {w R 2 3x[j/ Cife* s.t. w; 1 < ' [u I (.r I [j/ 1 )?/' Vi JV}. Firsty, [/(Cj;?/) = f/(^y;j/*) for each CV, 11 hence, if exists, the soution nf(v,y') to (11) does not depend on C (. Moreover, U {Xrf) = {(E^u^x 1 )^ 1^ 2^! 2 )^* ) s X) is compact. 12 Hence, a soution m (i';y*) to (11) (and thus a soution xt^ijy* to (1)) exist. As we wi see, it[u,i;y* is Pareto optima at y*, hence xt[v,2;j/ ( is unique (amost surey), and is in Ct [y 1 In the foowing, without oss of generaity, we wi take Ct = X. Towards proving part 1, suppose that xj[i;,i;y' is not Pareto optima at y. Then, E i [u i (x i t t 1 [y })\y > EV(aj[i;,t;y*)iy* = rrffoy ) and E 3 [u 3 4 (x»v) y > there exists some x[y' -X[j/'such that - SV^^/W. and one of the inequaities is strict. Since xt[i>,z;y* is a soution to the probem (1), we aso have E 3 \u (x 3 t [v,i\y t )\y < > v 3, and hence we have E 3 [vp (x J [j/') y* > u J, i.e., x[y' is feasibe to the optimization probem. Thus, E*[u (x^y') \y < Tn (v;y ). Therefore, E^u 1 (x! [y') y at east one of the inequaities was strict, this impies that E 3 \u 3 (x J f [j/ ) \y > ( = m (v;y t ). Since v 3. Since v? is increasing and continuous, it foows that there exists some random variabe 6 > such that x J [y* 6 > x 1 and E 3 [u 3' m, ( %* - 6) \y \ > v 3. But then, for (x [y'+m-v -<5) X[y*,we have j^v ( sv + 6 ) y* > (v;y t ), which is a contradiction. Therefore, it[v,i;y t is Pareto-optima at j/ f 2 of the emma can be proven by simiar arguments to the ast one. As for Part 3, since z X, m'^ujy') > v 2. Moreover, since t[v,i;y* is Pareto-optima at y and E 3 [u 3 (x^[v, i; y 1 })} = E 3 [u 3 (z 3 t ) \y then z wi be Pareto-optima at y, showing the contrapositive of the ast statement. ), \im 1 (v;y t ) = v\ n To see this, take any w G U(X;y ) so that w < (E 1 [u 1 (x 1 t )\y,e 2 [u 2 {x 2 t )\y ) for some x 6 X. Since C is Pareto-compete, there exists some x Ct such that '[u (x') i/! < E'\u % (x')\y \ for each i N. Thus,,1 1 iu < ( [«( 1 ) y',e 2 [u 2 (i 2 1 ) 2/ ), showing that w c7(c t ;j/'). Therefore, U(X;y ) C L/(C,;y'). On the other hand, C, C X, thus U(C t \y ) C U{,X;y ). Therefore, [7(C,;y') = t/(x;y'). '"Since u 1 and u 2 are continuous and ^[y' is cosed, {/ (X;?/ J ) Hence, i < x huge = (D 1 +D is cosed. Moreover, given any x X, x > x. 2 -i 2, D + D 2 -x}). Since u 1 is increasing, it foows that u' (x*) < E i [u i (x i )\y EL \p\xh* g *)\V t - Therefore, U (X,y ) is bounded. t < )

28 \ + Proof. (Lemma 3) Let us write x[y s = x s [v,i;y s for each y s. We want to seect x[y s in such a way that x X. By construction, x > x and x 1 + x 2 < D 1 + D 2. Hence, we ony need to guarantee that x k is integrabe for each k s N. Now, by Lemma 2.1, x[y s is Pareto-optima. Thus, by the syndicate theory, there exists some Ao > such that the equations x(y;a) = arg max u 1 (x )P x (y\y s ) + \u 2 (x 2 )P 2 (y\y s ) (12) x 1 +x 2 <W(y) x k >x k (at each continuation y of y s ) and *(A) = /"u'"(s»'(y; A))dP>(y y s ) = >' (") y s (13) hod when A = Ao- We set x(y) = x(y;ao). [We choose x(y;a) to satisfy Equation (12) at each y rather than amost surey. Hence the phrase "we can choose" in the statement of Lemma. Ceary, x(y;x) is continuous in A. Hence, so is $. Thus, Ao that soves equation (13) is upper semi-continuous in v 3 = E 3 [u 3 (z- 7 s ) \y. Since x(y) = x(y;ao) is continuous in Ao, it wi aso be continuous in v = 3 E 3 [u 3 (z- 7 ) \y s \. Since E 3 [u 3 (z- 7 3 ) \y \ is measurabe, we concude that x is measurabe. Moreover, x k < E k [\D + D 2 -x'\ < oo; and this D 1 + D 2 - x X k for I N\{k} and hence E k [\x k \ < \x k competes the proof. Proof. (Theorem 1) We have aready shown in the text that s is an equiibrium. Now, given any equiibrium s*, we wi show (via mathematica induction) that the vector of continuation vaues at the beginning of any t is V t. This is triviay true at t. Assume that it is true at some t + 1. Since s* is an equiibrium, an agent j must accept an offer x t [y* if E 3 [u 3 (xwy 1 )^ 1 3 t } > V t +1 (y ), and reject it if E 3 [u J (xjy 1 )\y t < V/ +1 (y*). Thus, j's action in s* can differ from his action in s ony by rejecting an offer x t [y' when E 3 [u 3 (x^y 1 })^} = V/ +1 (y'). The ony such distinction that can make a difference is when j rejects x t [Vt+\(y t ),i;y t. Let us consider this case. If VJ+i is not Pareto-optima at y', then by Lemma 2.3, rn (V t+ i(y t )\y t ) > V7 +1 (y*) where i = pt (y t ). Since x t [V t+1 (y t ),i;y t is rejected, the best response correspondence of i at y* is empty This contradicts the fact that s* is an equiibrium. Therefore, when V t+ i is not Pareto-optima at y', Xt[V t+ i(y t ),i-y t must be accepted, and hence s* and s yied the same outcome at y*. On the other hand, when V t+ i is aready Pareto-optima at y', t ' t m*(v t+ i(y ); y*) > V t + (y ). Hence, if j is to reject x t [V t+ i(y t ),i;y t, the best response of i is to make an offer that is to be rejected, yieding m (Vt+i(y t );y t ) for i and V'/ +1 (y t ) for j - as in s. Therefore, in either case, s* and s yied the same continuation vaues at y* (after the agent is recognized). Therefore, at the beginning of t, they yied the same vector of continuation vaues, which is V t (y t_1 ). We now prove Theorem 3. Using the notation of Lemma 1, we take Y = p, and write each y* as (p t,y ) and each y as (p,y), where pt is the history of the recognized agents. (Y consists of a the information except for the recognition process.) The recognition process and the dividends are stochasticay independent, hence D X, i.e., D(p,y) = D(p',y) for each (p,p',y). The aocations generated by contracts beow wi a be in X, hence the irreevant argument p wi be omitted. We first prove the foowing Lemma. Lemma 4 i = Pt(y*) J s Let p 6 V be such that p\ (y ) (0, 1) for each (i,t,y s ). If R\ (y ) = for some y* with then for each j,k N. p 3 (R k + (yt+ )\ Pt^) = k = *\y t ) = ^ 20

29 E Proof. We wi prove the contrapositive. That is, assuming that P J (/?f+1 (y' +1 )i {pt+i=fc} 7^0 y')>0 (14) for some j, k TV, we wi show that R\ (y > 0, where i = ) p t (y ( ). Since there are no rents at t, our assumption hods ony for t < t 1. Considering any such t with (14), we take / / k. We take y and +1 + its continuations y and y\ as generic reaizations where 1 pt+1 (y^ = ) k and pt+1 (y' +1 = ) I. We +1 wi simpy write x t+ [y^ for x t+ [V +1,fc,y^+1 t+2 (y[ ). By Lemma 1, x t+ <= [y }.Y [y[. We +1 define Xt+\ [y( > simiary. Define x X by setting fc + x(y) =p t (y<) z t+1 [y[ } (y) + (-p k (y*)) x t+1 [y,' +1 (y) at each continuation {p,y) of y*. By concavity, we have +1 E? [u* (5») y* > ^ [pf (j/*) a 1 (i +1 [y[ ) ( - p* (y<)) + u' (x +1 [y,* 1 ) y*! = p* < (y ) [«(xj +1 [y^ 1 ) j/ + (1 - pf (y<)) [ (i + [y^) y< = V t\ Y (y*) for each i TV. We caim that E 3 [u 3 (x 3 ) y' > V/ +1 (y*). In that case, V t+ i (y*) is not a Paretooptima utiity pair at y t 1, thus by Lemma 2.3, m (Vj+i (y*) ;y ( ( > V? ) +1 (y ), showing that 7?J = (y ) * (V't+1 (y ( ) -y ) - V t\, (y f ) > for each i = pt (y<). Now we prove our caim. Since p is affiiated, by Theorem 4, we have V t k +2 (?/fc +1 where we use the convention that Vf+1 = 0. ) > ^t+2 (y' +1 )' +1 Hence, whenever -fi +1 (yfc ) 7^ 0, we have m k +1 (V t+ 2 (j/ +1 K+2 (J/1 ) + «f+i 00 > ^+2 K ) > Vft a (y? ). showing that x t+1 [y + x +1 t+ [yj. This inequaity hods for each component, as x} + + x 2 +1 = D + D 2. Since pj - ' (y 4 ) G (0, 1) and u J is stricty concave, we therefore have u 3 {x 3 k ( {y)) > p (y ) u (x 3 +1 t+1 [y[ (y)) + { - k 1 +1 p (y )) u J" (.^+1 [yj: (y)) at each continuation (p,y) of y'. Thus, by (14), E 3 [u 3 (x 3 ) \y k f > p (y (1 - p k (y*)) E 3 [u 3 (x> + [y, i+1 ) y* = V*, (y<). - ) E j u 1 (x{ +1 [y +1 ) y'j + Proof. ( Theorem 3) Suppose that there exists an equiibrium in which agents do not agree at some history y. Then, by Theorem 2, Vi(y ) is a Pareto-optima utiity eve, thus Rq = 0. Thus, by the previous Lemma, for each j,k TV, and each s > 0, P 3 [R k (y s { = ) Pa= t} 0 y ) = 1, yieding E 3 [R k \ {f>3=k) \y = 0. Therefore, by (5), Vi(y ) = {E [u (D 2 )\y, {u2 (D 2 )\y }), showing that D 1 is Pareto-optima at y. Since D is independent of Yq it must aso be Pareto-optima at the beginning - a contradiction. Proof. {Theorem 4) The proof is very simiar to that of Proposition 3 in Yidiz (2001), and omitted here. We wi now prove Theorem 5. To this end, given any set 5 C R 2, et us write Hu[S) for the convex hu of 5, the smaest covex set that incudes S. Note that Hu{Uo(V\ U Vo)} = {XUo(p [1) + (i - \)u (p[2})\\ e [o,i, P [i e Pi, P [2 g v 2 ). +1 ) iyfc ) Lemma 5 U (V) = M[(/o(PiUP 2 ). Proof. Given any p&v, starting at history ho,i, the subgame of G(C,p) that starts at some y before the recognized agent makes his offer coincides with that of and G(C,p[io\), where %o = p (y ) and p [io e Vi such that p [io = Pt for each t > and i 6 TV, whie p [ioo = 1- Therefore, the utiity eve 21

30 Conversey, y for an agent i at equiibrium consumption can be expressed as a mixture of [/q(p[1) and Uq(p[2\) with 2 probabiities p\ and p = 1, -pj. Therefore, U {p) = Po^(p[) + (1 -Po)t/ (p[ 2 ) e Hu[U {PiL>P 2 )}. On the other hand, given any v = XU (p [1) + (1 - X)U {p[2}) Hu[U a {P\ U P 2 )}, we consider p P such that Pq = A and, at any p s agent recognized at according to y s. Once v = U (p)eu (P). m and any t > 0, p\{y s s ) = p[io\ iy ) f r eacn i e N where i'o is the again, Uq(p) = Po^o(p[) + (1 Po)Uo(p[2), showing that Lemma 6 rn J (i;;y') is a continuous function of v. Proof. Given any y and any v U (X, ), m^vip') = xasx^r)om (v) ', where Dom(v) = { = ( 1 [M 1 ox%* y t, ' t 2 [u 2 ox 2 [2/ 2/) x[2/ t G C t [y',^ > v}. Note that Dom(v) is aways non-empty and compact. Moreover, since u 1 is continuous, (v) => Dom(v) is ceary upper-semi continuous, and hence by the Maximum Theorem m (u; y ) is a continuous function of v. Lemma 7 Uq is continuous in p. Proof. Using mathematica induction on t, we wi show that Vi[p(p t_1 ) is continuous in p at each t and y t ~ 1 Firsty, Vj[p(y t_1 ~ ) = u' (D (y t 1 )) is triviay continuous in p. Assume that V t+ i[p(y 4 ) is 1 t continuous in p for each p* for some t < t. Since m (-;y is continuous, it foows that Fj (y = ) Tn 1 (V 4+ i[p(p t );p*) V t I +1 [p(y*) is continuous in p for each p* and each i N. Thus, ) 4 " = 1 pkp )^ [/$ (y*) pt =i y*- 1? + * [V^jW^JIy 1-1 is continuous in p. The payoff vectors from Pareto-optima aocations form a connected curve PF in R 2, which we ca the Pareto-frontier. Given any two v,v PF, we write [v, v}pf = {v PFIminv 1,?) 1 } < v 1 < maxt) 1,!) 1 }} for the part of the Pareto-frontier connecting uto v. Since Uo is continuous and Pi is connected, Uo{Pi) is aso connected. But, by Theorem 2, we have U (Pi) C PF. Thus, U (Pi) = [Uo(2[i\),U (p\i)pf- (15) Using Lemma 5, we obtain the foowing emma. LemmaS U (P) n PF = U (P 1 U P 2 ) = [[/ (p [1), U (p[1)pf U [I7 (g[), /o (PW)}pf- Proof. We have C/ (P) n PF = Hu[U {Pi U P 2 ) n PF = t/ (Pi U P 2 ), where the first equaity is due to Lemma 5, and the second equaity is due to the fact that we have stricty concave utiity functions yieding stricty convex set U[X of materiay-feasibe utiity eves. Uo(Pi U P 2 ) = [U (P [1), Uo (p [1)pf U [U (p [1), U Q (p [1)PF. m By (15), we further have Proof. (Theorem 5) (See Figure 2 for iustration.) Firsty, the set of a Pareto-optima and individuay rationa utiity eves is \Uo{p [),Uo (P[2)pf- Thus, Every Pareto-optima and individuayrationa aocation is obtainabe iff [Uo(p[),Uo(p[2) PF C U (P). This incusion hods iff U {P\) n U (P 2 ) ^ 0- [If U {Pi) n U (P 2 ) / 0, then by Lemma 8 we have U (P) n PF = U (Pi) U U {P 2 ) = [C/ (P [1), t/ (p [2) PF. when U (Pi)nU (P 2 ) = 0, by Lemma 8, [U (p [\}),U (p [2) PF \[/ (P) [t/o (p[),[/o(p[2) PF /0. But U (Pi)DUo{P 2 ) ^0iffL/ 1 o (p[) <C#(p[2), and this inequaity hods ifff/ 2 (p[2)<c/ 2 (p[).. 22

31 "The References [1 Avery, C. and P. Zemsky (1994): "Option vaues and bargaining deays," Games and Economic Behavior, 7-2, [2 Farber, H. and M. Bazerman (1989): "Divergent expectations as a cause of disagreement in bargaining: Evidence from a comparison of arbitration schemes," Quartery Journa of Economics, 104, [3 Hicks, J. (1932): The Theory of Wages, New York: Macmian Pubishing Co. [4 Hirsheifer, J. (1971): "The private and socia vaue of information and the reward to inventive activity," The American Economic Review, Vo. 61, No. 4, pp [5 Kaai, E. and M. Smorodinsky (1975): "Other soutions to Nash's bargaining probem," Econometrica, 43-3, [6 Mero, A. and C. Wison (1995): "A stochastic mode of sequentia bargaining with compete information," Econometrica, 63-2, [7 Nash, J. F. (1950): "The bargaining probem," Econometrica, 18, [8 Pedes, M. A. and M. Mascher (1980): "The super-additive soution for the Nash bargaining game," Report No. 1/80, The institute for Advanced Studies, The Hebrew University of Jerusaem. [9 Posner (1974): "An economic approach to ega procedure and judicia administration," Journa of Lega Studies, 2, 399. [10 Roth, A. E. (1985): "A note on risk aversion in a perfect equiibrium mode of bargaining," Econometrica, 53-1, [11 Rubinstein, A. (1982): "Perfect equiibrium in a bargaining mode," Econometrica, 50-1, [12 Stah, I. (1972): Bargaining Theory, Stockhom Schoo of Economics. [13 White, L. (1999) : "Prudence in Bargaining: The effect of uncertainty in Rubinstein bargaining outcomes," Nuffied Coage, Oxford, mimeo. [14 Wison, R. (1968) : theory of syndicates," Econometrica, 36-1, [15 Yidiz, M. (2001): "Sequentia bargaining without a common prior on the recognition process," mimeo. [16 Yidiz, M. (2000): Essays on Sequentia Bargaining, Ph.D. Thesis, Graduate Schoo of Business, Stanford University. 23

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