Starting Small and Commitment

Games and Economic Behavior 38, 176–199 (2002)doi:10.1006/game.2001.0857, available online at http://www.idealibrary.com on

Starting Small and Commitment

Joel Watson1

Department of Economics, University of California,San Diego, La Jolla, California 92093

Internet: http://weber.uscd.edu/∼jwatson/

Received September 1, 1999

I study a model of a long-term partnership with two-sided incomplete informa-tion. The partners jointly determine the stakes of their relationship and individuallydecide whether to cooperate with or betray each other over time. I characterizethe extremal—interim incentive efficient—equilibria. In these equilibria, the part-ners generally “start small,” and the level of interaction grows over time. Thetypes of players separate quickly. Further, cooperation between “good" types isviable regardless of how pessimistic the players are about each other initially. Thequick nature of separation in an extremal equilibrium contrasts with the outcomeselected by a strong renegotiation criterion (as studied in Watson (1999, J. Econ.Theory 85, 52–90). Journal of Economic Literature Classification Numbers: C72,C73, D74. 2002 Elsevier Science

At the beginning of a long-term relationship, prospective partners maybe dubious about each others’ motives. To optimally encourage cooperationat low risk, the parties balance the opportunity for high returns with thepossibility that one agent will take advantage of the other. Often the bestway to structure the relationship involves “starting small,” and then allowingthe stakes of the relationship to grow over time.

I study the phenomenon of gradualism, or “starting small,” by analyzing aclass of dynamic games with variable stakes. These games model a dynamicpartnership in which (a) the two partners can jointly choose from amonga variety of projects (levels of interaction) over time and (b) the partnersalso individually decide whether to cooperate at every instant. There istwo-sided incomplete information about each partner’s incentive to behaveopportunistically, an act which injures the other party. While “high” types

1 I appreciate the remarks of Vince Crawford, David Kreps, Georger Mailath, DarwinNeher, Clara Ponsati, Larry Samuelson, Joel Sobel, numerous seminar participants, and theanonymous referees and associate editor of this journal. This project was completed while Ivisited Yale University, and it was supported by National Science Foundation Grants SBR-9422196 and SBR-9630270.

1760899-8256/02 $35.00 2002 Elsevier ScienceAll rights reserved.

starting small and commitment 177

generally prefer to cooperate over time as long as their partners cooperate,“low” types have an incentive to betray their partner’s trust. Interacting ata low level is less risky than at a high level, because small stakes limit thedetrimental consequences of opportunistic behavior. The players’ incentivesto cooperate or betray depend on how the stakes change over time. Inparticular, although low types favor betrayal, they may prefer delaying suchan action if the stakes rise quickly enough.2

My analysis of this strategic setting is divided between the current paperand a companion paper (Watson 1999). Both papers characterize equilib-rium regimes in the partnership. An equilibrium regime is a specification of(a) how the stakes change over time and (b) an equilibrium in the players’decisions to cooperate or betray. Both papers demonstrate that coopera-tion between high types is viable, regardless of the players’ initial beliefsabout each other, as long as the relationship starts small enough.3 Beyondthe viability issue, the two papers focus on different genera of equilibriumregimes. Watson (1999) characterizes a regime that is selected by a strongrenegotiation condition. The present paper characterizes optimal regimeswhen the players have greater ability to commit to the way in which thestakes change over time.

Specifically, the present paper characterizes regimes that are interimincentive efficient (IIE). Such regimes are Pareto superior in the class ofequilibrium regimes, treating different player types separately. IIE regimesare shown to be “Q-regimes” (where “Q” stands for “quick”), which areidentified by two important properties. First, the relationship generallystarts small. Second, separation of the types is concentrated on two points intime: at the beginning of the relationship and right when the level reachesits maximum.

In a Q-regime, two-sided incomplete information is resolved at time zero,when the low type of player 1 betrays with probability 1 and the low typeof player 2 betrays with positive probability. (At the outset of the game,player 1 is weakly more pessimistic about player 2 than player 2 is aboutplayer 1.) Then there is a period of time when neither player betrays withpositive probability and the level of the relationship gradually rises. Just

2The model is intentionally abstract, allowing isolation of a complex strategic element thatis present in many economic relationships. For example, an employment relationship maybegin with significant informational asymmetry, whereby the worker and firm each does notknow whether the other intends to take advantage of him (by shirking on the job, assigningunpleasant tasks, etc.). Full commitment by the parties to one another may be contingent onsuccessful work at lower levels of interaction (marked by projects that give each party littlediscretion in the relationship). Joint ventures, international relations, and other settings oftenexhibit this strategic ingredient.

3This was first reported in Watson (1996). In this present paper I report a version of theanalysis on optimal regimes found in Watson (1996).

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when the level reaches its maximum, the low type of player 2 betrays withhis residual probability mass. Conditional on cooperation through this time,thereafter the players cooperate at the highest level of interaction.

I analyze two special cases. First, I examine the symmetric case, wherethe players are equally pessimistic about each other at the beginning ofthe game. In this setting, the low and high types of both players separatecompletely in the first instant of interaction. The separation is achieved bythe low types betraying with probability 1. After the high types are madeknown, the level of the relationship rises for awhile before settling at itsmaximum. Second, I examine the one-sided case, where player 1 is knownto be a high type. In this setting, the level of the relationship rises gradu-ally over time and the low type of player 2 waits until the maximum levelis reached before betraying. Under nonsymmetric, two-sided incompleteinformation, IIE regimes generally share the features of both the symmet-ric and one-sided cases.

The presentation begins in the following section with the descriptionof the model. Section 2 develops some useful properties of equilibrium.Section 3 characterizes the set of IIE regimes. Section 4 discusses the spe-cial cases noted earlier. Section 5 briefly discuss the issue of renegotiation,and Section 6 concludes. The Appendix contains the formal proofs. Somerelated literature is discussed in the companion paper (Watson, 1999).

1. DYNAMIC GAMES WITH VARIABLE STAKES

I consider a continuous-time model of a partnership between two agents.The model is the same as that of Watson (1999), except here a more narrowclass of games is studied for technical simplicity.4 There is a level function,α � �0�∞� → �0� 1�, specifying how the stakes of the partnership changeover time (which runs from zero to infinity). At time t ∈ �0�∞�, αt� is thelevel of the relationship.5 Prior to time zero, the players jointly select andcommit to the function α. I restrict attention to level functions that specifya positive level at some time. It is not important how the selection of α ismodeled, so I consider α exogenous for now. Issues regarding negotiationof, and commitment to, the level function are addressed in Section 5.

Given the level function, the players interact over time in a prisoners’dilemma-like setting called the partnership game. At every instant of time,the players individually choose whether to continue cooperating or to take

4Here I examine the class of dynamic games with variable stakes that are linear with respectto the stakes. In Watson (1999) I study a more general class of games.

5For concreteness, think of αt� as the number of projects on which the players have agreedto work at time t.


a selfish action that ends the partnership. In referring to selfish behavior,I use the terms “betray” and “quit.” A player’s strategy specifies when (ifever) the player will betray the other, ending the game. Cooperation entailsa flow payoff, ziαt�, to player i at time t. The selfish action at t bringsabout terminal payoffs. Player i’s terminal payoff at t is xiαt� if player ibetrays the other player at this time, yiαt� if player i’s opponent betraysat t, and zero if both players betray at t. The numbers zi, xi, and yi areprivate information to the players, as discussed later.

To better understand the specification of payoffs, suppose that the playerscooperate until t, at which time the game ends due to betrayal by oneor both players. Let t = ∞ for the case of perpetual cooperation. Thenplayer i’s payoff in the game is given by∫ t

0ziαs�e−rsds + giαt�e−rt � (1)

where gi ≡ xi if player i alone betrayed at t, gi ≡ yi if player j (i’s oppo-nent) alone betrayed at t, and gi ≡ 0 if both players betrayed at t. Note thatthe level determines the scale of the cooperative flow and terminal payoffsat each moment of time; a low level implies that the benefits of ongoingcooperation, as well as the values due to betrayal, are small, while a highlevel implies the opposite.

The players have private information about their flow and terminal payofffunctions. Each player is of two possible types, low (L) and high (H). Thelow type has payoff parameters z, x, y; the high type has parameters z, x,y. For i ∈ �1� 2� and K ∈ �L�H�, “iK” denotes player i of type K. Theex ante probability that player i is the high type is pi, which is commonknowledge. Assume that p1 ≥ p2. The model is otherwise symmetric, inthat players of the same type K ∈ �L�H� have the same payoff parametersx, y, and z. As stipulated later, I associate a greater incentive to cooperateover time with the high type relative to the low type.

Before specifying how the types are assumed to differ, I make someassumptions common to the two types. I wish to study a setting in which thepartners can benefit from long-term cooperation, each may have a short-term incentive to betray the other, and a player suffers if the other betrayshim. These properties are captured by:Assumption 1. x� z� x� z ≥ 0, y� y < 0.Note that games with variable stakes are similar to wars of attrition, in

that they are played in continuous time and end when one player decidesto quit or both players decide to quit.6

6There are two substantive differences between these classes of games. First, in wars ofattrition a player generally prefers that his opponent quit, while in the games studied hereplayers may wish to cooperate perpetually. Second, games with variable stakes include a levelfunction, which the players jointly determine.

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FIG. 1. An example with constant level over time.

I next turn to distinctions between the high and low types, regarding inparticular the incentive to cooperate in the partnership game. To developintuition, take the example of a level function specifying αt� = a for allt and any a ∈ 0� 1�. The matrix in Figure 1 depicts the payoffs when theplayers are constrained to choose between betraying now or cooperatingforever. Note that

∫∞0 ziae

−rsds = zia/r. As the figure indicates, perpetualcooperation can be sustained in the game only if z1/r ≥ x1 and z2/r ≥ x2.Betrayal is the dominant choice for player i if zi/r < xi.

Evaluating incentives for games with nonconstant level functions is morecomplicated than for those with constant level functions. In particular, if αrises over time, then player i may be willing to cooperate for awhile evenif zi/r < xi. Delaying betrayal can be optimal if the betrayal payoff risesfast enough over time to overcome discounting. However, since the level isbounded, this player must eventually prefer quitting. The following lemmafrom Watson (1999) makes this point formal.

Lemma 1. Suppose that xi > zi/r. Take any level function α such thatαt� > 0 for some t. There are times T� S ∈ �0�∞�, with T < S, such thatthe following holds regardless of player j’s strategy. Conditional on reachingtime T in the game, payoff-maximizing player i must quit before time S withprobability 1.

I define the low and high types so that only the high types may have theincentive to cooperate perpetually in the partnership game.

Assumption 2. x < z/r, z = 0 < x, and x+ y ≤ 0.

This assumption yields the interesting class of games in which the low-type players must eventually betray in every equilibrium. The high typeswould like to establish long-term cooperation with each other, but privateinformation makes building the relationship difficult. Cooperation entails arisk that one’s opponent may betray, since y < 0. The assumption z = 0 < xis stronger than is necessary (x > z/r would do), but it greatly simplifiessome of the analysis.7 The inequality x+ y ≤ 0, which is required for oneof the components of the analysis, means that no value is created betweenthe low types from betrayal.

7Those interested in a more general treatment of the basic model are invited to readWatson (1996, 1999).


Equilibrium

A level function and a strategy profile for the partnership game is calleda regime. If the strategy profile is an equilibrium in the partnership game,then the regime is termed an equilibrium regime. Obviously, regardless of α,there is always an equilibrium in which both types of both players betray att = 0. Thus, every level function is associated with at least one equilibriumregime. However, I shall look for more interesting cooperative equilibria,where the partnership is viable between two high types. More precisely, Irestrict attention to equilibria in which the high types cooperate perpetually.I assume that the high types adopt a strategy of never betraying, subject toestablishing later that this strategy is rational.8 The low type players willbetray at some point, but they may have an incentive to cooperate for astretch of time before betraying. An important component of the analysisinvolves demonstrating that cooperative equilibria exist.

Each low type has a strategy defined by a cumulative distribution func-tion F , where Ft� is the probability that this player betrays at or beforetime t. Since the low type must quit with probability 1 in bounded time,Fs� = 1 for some time s. Assuming that the high types cooperate perpet-ually, if player j of low type plays strategy Fj , then player i effectively facesthe strategy defined by 1 − pj�Fj . We can then associate the low type’sstrategy Fj with the function θj ≡ pj + 1 − pj�1 − Fj�, where θjt� is theprobability that player i’s opponent cooperates through time t. It will beeasiest to work with the function θj directly, with the understanding thatθj is weakly decreasing and continuous from the right, θjt� ≥ pj for all t,and θjs� = pj for some s.

2. COOPERATIVE EQUILIBRIUM REGIMES

In this section, I study cooperative equilibrium in the partnership gamefor an arbitrary level function α and initial beliefs p1 and p2. I assume thatθi is continuously differentiable, except perhaps on a closed set of measurezero. I also assume that α has well-defined right- and left-hand limits andis differentiable except on a closed set of measure zero. The assumptionson α are relaxed in subsequent sections.

For a useful benchmark, consider the partnership game with αt� = 1for all t. Obviously, with this level function, a low type has no incentive todelay betraying his partner, regardless of the partner’s strategy. Therefore,every equilibrium involves the low-type players betraying at the beginning

8I therefore rule out regimes in which the high types cooperate perpetually with positiveprobability and quit with positive probability.

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of the game. Next, assess the high types’ incentive to cooperate. Supposethat player jH cooperates perpetually. If player iH is inclined to betray,then the best time to do so is at t = 0. Player iH obtains pjx by quit-ting at the beginning of the game and 1 − pj�y + pjz/r from cooperat-ing perpetually. Thus, cooperation by the high types is an equilibrium ifand only if 1 − pi�y + piz/r ≥ pix for i = 1� 2. This condition simplifiesto pi ≥ −y�z/r − x − y� ∈ 0� 1�. Observe that in this constant αt� case,incomplete information denies any hope of cooperation between high typesif either p1 or p2 is small.

Next, I derive equilibrium conditions for more general (nonconstant)level functions, where the low types may cooperate for some time beforequitting. It is helpful to start by recognizing how a player’s payoff dependson the level function, the time at which this player plans to quit, and theopponent’s quit distribution. To be precise, if player i plans to quit at timet, then her expected payoff is∫ t

0ziαs�e−rsθjs�ds + lim

τ→t−

∫ τ0yiαs�e−rs�−dθjs�� + xiαt�e−rtθjt�� (2)

This is the expectation of payoff expression (1) when player i plans toquit at t and player i’s opponent behaves according to θj . Remember thatθjs� gives the probability that player j cooperates through time s. Thefirst integral is the expected value of the first term in (1)—the payoff ofcooperation with player j. Observe that player i obtains the cooperativeflow payoff at time s < t with probability θjs�. The second and third termsof (2) give the expected value of the second term of (1). In particular,the second integral of (2) is the expected negative payoff arising becauseplayer j quits before time t; here, the limit is used to exclude any masspoint at t. Finally, the last term is player i’s expected instantaneous payofffrom quitting at t; note that player i obtains terminal payoff xiαt� withprobability θjt�, which is the probability that player j does not quit bytime t.

The functions θ1 and θ2 define an equilibrium if the strategy Fi of thelow type of player i is a best response to θj , for i = 1� 2, and the strategyof cooperating perpetually is a best response for the high-type players.

Conditions Regarding Low Types

I examine how the low types’ betrayal distribution relates to the levelfunction in equilibrium. There may be times at which θi is discontinu-ous and intervals in which θi is continuously decreasing, reflecting atomsand continuous phases in the low types’ betrayal distributions. In fact, inboth cases the level function α constrains equilibrium strategies in a pre-cise and easily identified way. Some additional notation will be helpful in


demonstrating the connection. For all t > 0, define α−t� and α+t� as thedirectional limits of α at t from the left and the right, respectively.9 I usethe convention that α−0� ≡ 0. Define θ−i t� analogously, with θ−i 0� ≡ 1for i = 1� 2.

Consider first the implications of player iL weakly preferring to quitat some time t. Let qj ≡ θjt�/θ−j t� denote the probability that player jcooperates at t, conditional on j cooperating before t. Note that qj = 1if θj is continuous at t. Conditional on reaching t, player iL’s normalizedexpected payoff from quitting at t is qjxαt�.10 If iL waits until just aftert to quit, then her expected payoff can be made arbitrarily close to 1 −qj�yαt� + qjxα+t�. Thus, given that player iL weakly prefers to quit at t,it must be that

qjxαt� ≥ 1 − qj�yαt� + qjxα+t�� (3)

Defining

Dq� ≡ qx

qx− 1 − q�y �

the inequality becomes αt� ≥ α+t�Dqj�. This condition holds as anequality if player iL is indifferent between betraying at t and at times after,but arbitrarily close to t.

If player iL quits just before time t, then her payoff (conditional onplayer j cooperating before t) is arbitrarily close to xα−t�, so the weakpreference to wait until t implies that

qjαt� ≥ α−t�� (4)

As before, this condition holds as an equality if player iL is indifferentbetween betraying at t and just before t.

Next, consider the implications of player iL quitting at a positive rateover an interval of time where α is continuous. In particular, suppose that αis continuous at some t ≥ 0 and θ′it� < 0. It must be that θ−j t� = θjt�, forotherwise the continuity of α and the payoff parameters contradict inequal-ity (4). Thus the quit distribution of player i’s opponent has no mass pointat t. Since player iL plans to betray at a positive rate in a neighborhood oft, this player must be indifferent between quitting at any time in the neigh-borhood. Using the low type’s payoff parameters in Eq. (2), and recallingthat z = 0, this player’s expected payoff from planning to betray at time wis ∫ w

0yαs�e−rs�−dθjs�� + xαw�e−rwθjw��

9For t > 0, α−t� ≡ lims→t− αs�. For t ≥ 0, α+t� ≡ lims→t+ αs�.10Recall that players earn a terminal payoff of zero if they quit at the same time.

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which must be constant in the neighborhood of t. Taking the derivative withrespect to w, evaluating it at w = t, and rearranging terms yields

xα′t� − rxαt� − θ′jt�θjt�

�y − x�αt� = 0� (5)

The left side of this equation is the value at t of player iL waiting beforequitting, relative to quitting at t. This value must be 0 if player iL is indif-ferent. The first two terms are the increase in the betrayal payoff fromchanges in the level and the decrease in the betrayal payoff due to discount-ing. (Remember that the cooperative flow payoff for the low type is 0.) Thelast term is the potential loss if the opponent quits before player i quits,times the rate that the opponent quits conditional on reaching t (the hazardrate of the opponent’s quit distribution).

In equilibrium, α must gradually rise in any interval in which a playerquits at a positive rate. To see this, note that the last term on the leftside of Eq. (5) is nonpositive. Observe that α must rise enough to offsetdiscounting for the low type to be indifferent. Furthermore, if the opponentbetrays at a positive rate, then α must rise even faster for the low type to beindifferent between quitting and waiting. That is, the “prize” of betrayal inthe future must be large enough to counter the chance that the opponentwill betray first.

Phases of Equilibrium Regimes

Other general properties of equilibrium regimes can be derived. Givenθi, define Ti ≡ min�t ≥ 0 � θit� = pi� as the time by which the low type ofplayer i betrays with probability 1.11 I pay special attention to regimes withcontinuous level functions. As the following lemma from Watson (1999)establishes, these regimes have a three-phase structure in equilibrium.

Lemma 2. Consider any level function α that is continuous at every t > 0.Suppose that θ1 and θ2 define a cooperative equilibrium in the partnershipgame. Then T1 ≤ T2 and θ2T1� ≤ θ1T1�. Furthermore, θ1t� = θ2t� for allt < T1 and these functions are continuous on 0� T1�.

To interpret Lemma 2, remember that p1 ≥ p2. The three phases of acooperative equilibrium regime are �0� T1�, �T1� T2�, and �T2�∞�. In the firstphase, the low types behave symmetrically, betraying with the same proba-bility over time. This phase ends when the probability mass on the low typeof player 1 has “run out” (when θ1t� = p1) and, conditional on coopera-tion, the updated probability that player 1 is the high type equals 1. In thesecond phase, only the low type of player 2 quits with positive probability.

11This is well-defined since θi is right-continuous.


Just after T2, the updated probability that player 2 is the high type also hasreached 1. The high types cooperate through the third phase, where onlythey remain.

The proof of Lemma 2 builds on the following intuition about the incen-tives of the low types. Suppose that there is a time interval (perhaps apoint) in which player 1 betrays with higher probability than does player 2,and also suppose that player 2 betrays later in the game with positive prob-ability. Note that player 2 is willing to quit later even though there is achance that player 1 will quit in the interval. Since player 1 is at less riskin the interval, he should strictly prefer to wait, which contradicts that hequits with positive probability in the interval.

Examples and Existence

It is not difficult to construct a variety of different equilibrium regimes.For example, consider a setting with one-sided incomplete information,where p1 = 1. For every T > 0, define αT to be the continuous function sat-isfying αT t� = 1 for all t ≥ T and α′T t�/αT t� = r for all t ∈ 0� T �. Thisdifferential condition is Eq. (5) for the case in which θ′j = 0. If the hightypes cooperate perpetually, then such a level function makes player 2Lindifferent between betraying at all times from 0 to T . Any quit distributionsatisfying θ2T � = p2 characterizes an equilibrium, as long as player 1Hweakly prefers to cooperate perpetually. (Obviously, player 2H prefers tocooperate.) In fact, take any θ2 such that θ2T ′� = p2 for some T ′. Onecan show that there is a number T ′′ ≥ T ′ such that for every T ≥ T ′′, theregime defined by θ2 and αT is an equilibrium. Figure 2(a) illustrates thisclass of regimes.

Equilibria of the same flavor can be constructed for games with two-sidedincomplete information. For example, consider a symmetric game, wherep1 = p2. Look for a symmetric equilibrium by taking any differentiableθ1 = θ2 and, for T such that θ1T � = p1, selecting α to satisfy Eq. (5) forall t ∈ 0� T �. As in the one-sided case, one can prove that this is an equi-librium if the level starts small enough (so that T is large enough). Thereare also symmetric equilibria with atoms in the low types’ quit distribution.Such equilibria generally involve discontinuities in α, because inequalities(3) and (4) must hold where θ1 = θ2 is discontinuous. In addition, α maybe nonmonotone before or after the low types betray. Figure 2(b) showsan example.

As the examples confirm, a cooperative equilibrium regime always exists.

Theorem 1. Regardless of p1 > 0 and p2 > 0, there is a cooperativeequilibrium regime.

Given the example with constant α discussed at the beginning of thissection, Theorem 1 reveals the value of starting small. In the fixed α case,

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FIG. 2. Equilibrium regimes.

the partnership is not viable when the players are initially very pessimisticabout one another. However, cooperation between two high type players canalways be sustained if the players start small enough in their relationship.

3. INCENTIVE EFFICIENT REGIMES

The examples indicate that there is a plethora of equilibrium regimes.However, to the extent that the players jointly determine the regime, itseems reasonable to expect them to settle on one that is interim incentiveefficient (IIE). An equilibrium regime is IIE if, within the class of equilib-rium regimes, there is no way of making one type of one player better offwithout making another type or other player worse off. In this section Icharacterize the IIE regimes. I constrain attention to cooperative regimeswhose level functions are continuous at positive times.

I prove that IIE regimes have two noteworthy properties. First, theyentail starting small, which should come as no surprise given the discussionfollowing Theorem 1. Second, and more significantly, IIE regimes involvebetrayal by low types that is concentrated on two times: at the beginning ofthe relationship and just as the level reaches its maximum. In other words,betrayal occurs quickly and is lumpy over time.

To state the result, a bit more notation and terminology is useful. Given anequilibrium regime defined by α, θ1, and θ2, let the function viH � R+ → R


associate with every time in the game the equilibrium continuation payofffor player iH, for i = 1� 2. That is, viHt� is the continuation payoff forplayer iH from time t, conditional on reaching this time in the game (withneither player betraying before t). I use the convention that the continua-tion value at t is evaluated before the parties interact at this time; thus viHreflects any mass point in the betrayal distribution of player i’s opponentat t. Let viL be defined analogously.12 Further, define by viHt� the payoffthat player iH would obtain at time t if he were to betray at this time; wehave viHt� = xαt�θjt�/θ−j t�.

For any interval of time M ⊂ R+, I say that a regime has maximal delayon M if for t� t ′ ∈M , θ2t� < θ−2 t ′� implies that v1Hs� = v1Hs� for everys ∈M with s > t. In words, if player 2 betrays in interval M by time t, thenthe high type of player 1 is indifferent between quitting and cooperatingafter t until the right endpoint of M . In this case, the mass of player 2L’squit distribution on M is pushed as close to the right endpoint of M as ispossible while still maintaining player 1H’s incentive to cooperate. I featurethe following type of equilibrium.

Definition 1. A Q-regime is a cooperative equilibrium regime in which(a) θ10� = p1 (so T1 = 0); (b) αt� = min�α0�ert/Dp1�� 1� for all t > 0;and (c) with T ≡ inf�t � αt� = 1�, the regime has maximal delay on 0� T �.

Here, Q stands for “quick” (referring to the nature of betrayal by thelow types). A Q-regime is pictured in Figure 3. Note that the entire massof quitting by player 1 is concentrated at time zero, along with at least thesame mass of quitting by player 2 (who has a weakly greater chance ofbeing a low type). The remainder of player 2L’s betrayal mass occurs atT . Thus, if θ20� > p2, then T2 = T ; otherwise, T2 = T1 = 0. The level ofthe relationship jumps up just after t = 0 and then gradually rises until T ,after which it stays at its maximum. The discontinuity at t = 0 makes thelow types weakly prefer to quit at this time. The gradual rise after timezero obeys α′/α = r, which coincides with Eq. (5) for θ′j = 0. Observe thatthe class of Q-regimes is parameterized by the numbers α0� ∈ �0� 1� andθ20� ∈ �p2� p1�.

Here is the main result of this paper.

Theorem 2. Among the class of cooperative regimes whose level functionsare continuous on 0�∞�, every IIE regime is a Q-regime. Furthermore, thereexists an IIE regime.

12Note that since the high types cooperate perpetually, viH is continuous wherever θj iscontinuous; the function is everywhere left continuous and has well-defined right-hand limits.Where α is continuous, viL has well-defined right- and left-hand limits.

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FIG. 3. Q-regime.

The proof of this theorem has several complicated components, which Iattempt to summarize in more intuitive form here. The general idea is tostart with an arbitrary regime and show how it can be transformed into a Q-regime, improving the payoffs of all player types in the process. The prooffirst establishes that efficiency requires α to eventually reach the maximumlevel and remain there forever. Further, defining S ≡ min�s � αs� = 1�,it must be that player 2L is indifferent between quitting at every timebetween 0 and S. Otherwise, one can raise α at times where player 2Lstrictly prefers not to quit, forming a new equilibrium that all player typesprefer (player 1H strictly so).13 I show that the indifference conditionimplies that Eq. (5) holds on the interval 0� T1�. The proof then addressesseparately the first two phases of an equilibrium regime (i.e., �0� T1� and�T1� T2�).

On the first phase, I show that it is optimal to shift toward time zero theprobability mass of quitting by the low types. The following heuristic argu-ment provides the intuition behind this claim. Suppose that one starts withan equilibrium regime in which a section of the first phase of interaction isdescribed by the solid lines in Figure 4. In this region, the low types quitwith positive probability between s′ and s′ + ". Note that θ1 = θ2 ≡ θ inthe first phase of the equilibrium. Dividing Eq. (5) by α and integrating

13This step is simplified by the assumption z = 0. Under this assumption, the incentives ofthe low types are preserved when the level function is modified at times where the low typesstrictly prefer not to quit.


FIG. 4. Shifting probability mass in the first phase.

reveals that αt� = ertθt�−bc for an open set containing �s� s′ + "�, whereb ≡ x− y�/x and c is a constant.

Consider another equilibrium regime constructed by shifting the betrayalmass on s′� s′ + "� to instead occur on s� s + "�. The rest of the regimeremains the same. The new regime is represented by the lines with dashedhighlights. It is not difficult to verify that the low types are indifferentbetween the old and new regimes. However, the high types strictly pre-fer the new regime. To see this, first observe that the component of thehigh types’ payoff due to betrayal by the low types is the same under thetwo regimes. This is because although betrayal occurs earlier in the newregime, it occurs at a lower level which exactly offsets the change in dis-counting between s and s′.14 The difference between the two regimes, interms of the high types’ expected payoff, thus amounts to the difference onthe interval s + "� s′�.

Cooperation occurs on the interval s+"� s′� in both the new and the oldregimes. The old regime has a higher probability of cooperation here (sincethe betrayal mass under consideration occurs after this interval), but coop-eration occurs at a lower level. Relative to the old regime, the level underthe new regime is higher by the multiplicative factor θs′ + "�−b/θs′�−b,where θ is given by the old regime. The probability of cooperation overthe interval is lower in the new regime than in the old regime by the mul-tiplicative factor θs′ + "�/θs′�. Putting these factors together, the ratioof high types’ payoff from the interval s + "� s′� in the two regimes is

14This follows from the αt� = ertθt�−bc identity.

190 joel watson

θs′ + "�1−b/θs′�1−b. Since 1 − b = y/x < 0 and θs′ + "� < θs′�, thehigh types like the new regime better.

In short, shifting probability mass forward in the first phase of interactionis preferred, because the reduced probability of cooperation in the interimis more than offset by the gain of cooperation at a higher level. Thus, inIIE regimes, the first phase of interaction must occur arbitrarily quickly.Of course, there is a difference between “arbitrarily quickly” associatedwith the limiting payoff of a sequence of continuous θk and the payoffof the limiting θ, which has a mass point at time zero. I show that, givenx + y ≤ 0, the latter is preferred by the players.15 The intuition behindthis last claim has to do with whether it is better to have signaling occurin several small stages or in one big stage. Times of quitting by low typescan be interpreted as signaling stages, where high types send a signal oftheir type by not quitting. Multiple stages of signaling may be valued if thesignaling technology creates value. This would be the case if x + y > 0,meaning that betrayal between low types yields a surplus.

The final component of the proof involves showing that the betrayal massof player 2L in the second phase of interaction is optimally shifted later intime. Note that α′/α = r between T1 and the time S at which α reaches 1.This follows from the facts that player 1L does not quit in this interval oftime and player 2L is indifferent here. Delaying the betrayal of player 2Lin this interval does not alter the payoff of this player type. However, itdoes increase the payoff of player 1H. Player 1H obtains the same negativepayoff from the cheating event, since changes in the timing are exactlyoffset by changes in the level. But this player enjoys a higher probability ofcooperation in the interim.

4. TWO SPECIAL CASES

Recall that Q-regimes are parameterized by the numbers α0 = α0� ∈�0� 1� and θ0

2 = θ20� ∈ �p2� p1�. Theorem 2 establishes that IIE regimesare Q-regimes, but it does not indicate the values α0 and θ0

2 precisely or howthey relate to the payoffs of the player types. To further characterize theset of IIE regimes and generate simple comparative statics conclusions, it ishelpful to examine two important special cases of the model: the symmetriccase and the one-sided case.

First, consider the symmetric case, distinguished by p1 = p2 = p. Undersymmetry, we obviously have θ10� = θ20� in any Q-regime. Therefore,θ0

2 = p2, and so T2 = T1 = 0. IIE regimes are thus symmetric and are

15This is why I allow the level function to be discontinuous at t = 0; efficiency demands it.


characterized by the number α0. To learn how α0 relates to the payoffs ofthe players, consider maximizing the welfare function wsymm� ≡ muH +1 −m�uL, where uH is the equilibrium expected payoff of the high typesand uL is the same for the low types. Let α0

symm�p� denote the initial levelcharacterizing the regime that maximizes wsym over all equilibrium regimes.

Lemma 3. α0symm�p� is weakly decreasing in m and weakly increasing

in p. Also, m = 1 implies that T > 0.

Equilibria that favor the high types have lower initial levels; the initiallevel is also lower when players are more pessimistic about each other.Remember that T defines the time in the regime at which α first reaches 1.Note that T > 0 means the level of the relationship rises gradually evenafter time zero, where both low type players quit with probability 1.

Next, consider the one-sided case, distinguished by p1 = 1. In this setting,there is maximum delay from the beginning of the game, so player 2L quitsat t = 0 only if α0 is large.

Lemma 4. In the one-sided case, every IIE regime has maximum delay on�0� T �, where T ≡ inf�t � αt� = 1�.

Regarding the players’ preferences over IIE regimes, note that play-ers 1L, 2H, and 2L fancy α0 as large as possible; player 1H is morecautious and generally prefers starting small. Consider a welfare functionwonem� ≡ mu1H + 1−m�u2H , where uiH is the equilibrium expected pay-off of player iH. Let α0

onem�p2� denote the initial level characterizing theregime that maximizes wone over all equilibrium regimes.

Lemma 5. α0onem�p� is weakly decreasing in m and weakly increasing in

p. Also, m = 1 implies that T > 0.

5. NEGOTIATION AND COMMITMENT

The level function α is interpreted as jointly selected by the players. How-ever, in the analysis thus far I have treated α as exogenous. In this section,I address whether IIE regimes would result from negotiation between theplayers before the partnership game begins. I also comment on the issue ofrenegotiation by the players in the course of play.

Initial Selection of the Level Function

Consider the joint selection of α before play of the partnership game.There are three natural settings in which negotiation may take place: (a) the

192 joel watson

players determine the level function before obtaining their private informa-tion (ex ante), (b) the players negotiate the level function with the knowl-edge of their own types (interim), and (c) the players select α ex ante butthen can alter α after learning their own types. Some of the relevant issuesin contracting with asymmetric information have been studied for generalmechanism design problems.16 The notion of “durability”—advanced byHolmstrom and Myerson (1983) and Crawford (1985)—provides a way ofassessing whether a mechanism survives renegotiation once players obtainprivate information [case (c)] and provides a consistency criterion for amechanism to arise from initial interim negotiation [case (b)]. Crawford(1985) proves that IIE mechanisms are durable in that there is a reasonablemodel of negotiation that supports these regimes as contracted in equilib-rium. The regimes identified in the previous section here are IIE. Thus, theyare supported by a reasonable model of negotiation between the players.

Renegotiation in the Course of Play

In this paper I have focused on commitment to the level function ini-tially selected by the players. In earlier work (Watson, 1999), I concentratedon renegotiation and studied a particular strong renegotiation criterion. Iwish to point out, though, that there is a sense in which the regimes iden-tified in this paper are also renegotiation proof. Consider the followingnotion of renegotiation proofness, which is a dynamic version of Maskinand Tirole’s (1992) “weak renegotiation proofness.” Suppose that some αhas been set by contract and the partnership game begins. At some time s(s may equal 0), play is temporarily suspended and player i has the optionof suggesting an alternative level function α that will be in effect for theremainder of the game. If α �= α, then player j decides whether to accept orreject this new level function. If player j accepts α, then play continues withthis level function; otherwise, play resumes with α. Call this renegotiationphase and the ensuing partnership game from s the “renegotiation game.”The original level function α is weakly renegotiation proof if there exists anequilibrium in the renegotiation game in which both types of player i sug-gest level function α = α. It is not difficult to show that optimal Q-regimesare weakly renegotiation proof in this way.17

16Standard references include Holmstrom and Myerson (1983), Myerson (1983), andCrawford (1985); more recent work includes Maskin and Tirole (1990, 1992) and Cramtonand Palfrey (1995).

17This is because player j’s belief about player i’s type can be updated contingent onplayer i’s suggested level function at time s. Suppose that when player i suggests α, player j’sbelief does not change. However, if player i suggests any other level function, then player j’sconditional belief puts probability 1 on the low type of player i, which implies that both play-ers quit immediately when the partnership game resumes. (If one player is known to be the


6. CONCLUSION

The class of games with variable stakes permits a tractable treatment ofthe partnership problem, leading to intuitive results on the nature of start-ing small and the manner in which the players learn about one another.Possible extensions of the model include (a) incorporating a multidimen-sional “level” parameter and (b) enlarging the scope for signaling betweenthe players by including other signaling technologies. Both extensions areworthwhile pursuits, and both present some challenging technical prob-lems. Extension (b) is attempted in a preliminary and limited way inWatson (1996). Regarding (a), I simply note that the present analysis reliesheavily on symmetries between the types of different players and with theway in which the level affects the payoffs of the players. At the very least,the present paper and Watson (1999) together provide a reasonably com-plete analysis of a class of partnership problems that highlights the virtueof starting small.18

APPENDIX: PROOFS

Proof of Lemma 1

This is proved in Watson (1999), where the same lemma appears asLemma 1. Although z > 0 is assumed in Watson (1999), none of the anal-ysis is altered when z = 0.

Proof of Lemma 2

This is proved in Watson (1999), where the same lemma also appears asLemma 2.

Proof of Theorem 1

This is an immediate consequence of Lemma 3 in Watson (1999).

low type, then equilibrium requires both players to quit as soon as αs� > 0.) Given thesebeliefs, player i has no incentive to suggest a level function other than α. The comparisonbetween this renegotiation criterion and the one in Watson (1999) hinges on the whetherbeliefs about types can be altered contingent on messages sent in the renegotiation game. InWatson (1999), beliefs are not allowed to change directly due to the messages. Without com-mitment, the G-regime is selected, featuring a gradual rise of the level and gradual separationof types throughout the first two phases of equilibrium.

18Rauch and Watson (1999) offer a complementary analysis of starting small in an environ-ment with symmetric information and joint uncertainty.

194 joel watson

Proof of Theorem 2

Note that by Lemma 2, any equilibrium regime satisfying (b) of the defi-nition of a Q-regime is fully characterized by α� θ2�, with the understandingthat θ1 ≡ max�θ2� p1�. I begin by proving the following result.

Lemma 6. Take as given a cooperative equilibrium regime α� θ2�, forwhich α is continuous on 0�∞�. Then there exists a Q-regime α� θ2� inwhich the expected payoff of every player type weakly exceeds that of α� θ2�.

I prove the lemma in four steps. I focus on the case in which T2 > 0. Thecase of T2 = 0 is simpler, requiring just part of steps 1 and 4.

Step 1. Claim: For the regime α� θ2�, θ1 is continuous on 0� T1�. Toprove this claim, suppose that it does not hold and let t ∈ 0� T1� be suchthat θ−1 t� > θ1t�. (Remember that θ is right continuous.) From Lemma 2,we know that θ−2 t� > θ2t� as well, so player 2L quits with positive prob-ability at t. But then inequality (4) must hold, implying that α is discontin-uous at t, which is a contradiction.

Step 2. I next define a new equilibrium regime, based on α� θ2�. Letα be defined by: (i) α0� = α0�, (ii) αt�x ≡ v2Lt� for all t ∈ 0� T2�(where v2L is the continuation value function associated with α� θ2�), and(iii) αt� ≡ min�1� αT2�ert−T2�� for all t > T2. We have αt� ∈ �0� 1� forpart (ii), since v2Lt� ∈ �0� x�. Furthermore, from step 1 we know v2L iscontinuous on 0� T1�, which implies that α is continuous on this interval.In fact, one can easily verify that α is continuous on 0�∞�.

Note that α is defined so that first, αt� = αt� where the low type quitwith positive probability in the original regime, and second, α is set to makeplayer 2L indifferent everywhere else (until α = 1). It should be obviousthat because z = 0, the incentives of players 1L and 2L are the sameunder α� θ2� as they are under α� θ2�. That is, for the low types, α� θ2� isincentive compatible. It is also not difficult to show that players 1H and 2Hhave no incentive to quit under α� θ2�. At times when low types quit, thehigh types’ value of cooperating is weakly greater relative to the value ofquitting; at other times, the high types strictly prefer to wait. Therefore,α� θ2� is a cooperative equilibrium regime. By construction, every playertype fares weakly better under α� θ2� than under α� θ2�.Step 3. I define yet another regime by modifying α� θ2�. This modifi-

cation involves shifting the probability mass for which the low types quit in0� T1� to a mass point at time zero. The new regime is denoted by α� θ2�.

I begin the construction with some general analysis. Take any cooperativeequilibrium regime β�µ� and suppose that these functions are continuouson �0� s�, with µs� ≥ p1. Here µ directly defines the cooperation distribu-tion for player 2L, and max�µ�p1� defines the same for player 1L. Further


suppose that the low types are indifferent between quitting at every time in�0� s�. Then, using Eq. (5), we have

xβ′t� − rxβt� − µ′jt�µjt�

�y − x�βt� = 0�

Integrating, we obtain

βt� = ertµt�−bc (6)

for all t ∈ 0� s�, where b ≡ x − y�/x and c = βs�e−rsµ−s�−1. Further,using inequality (3) and low-type indifference, we have

β0� = q−bDq�c� (7)

where q ≡ µ0�. Equations (6) and (7) define β as a function of µ; theyrepresent conditions from the incentives of the low types.

I next use the relationship of Eqs. (6) and (7) to write α as a function ofθ2 for the regime α� θ2�. The expected payoff of both players 1H and 2Hon the interval �0� T1� (i.e., not including the payoff from time T1) is

y1 − q�α0� +∫ T1

0zαt�e−rtθt�dt +

∫ T1

0yαt�e−rt−θ′t��dt�

where θ ≡ θ1 = θ2 on �0� T1� and q ≡ θ0�. Substituting for α using Eqs. (6)and (7), and using c = αT1�e−rT1µ−T1�−1, we obtain

c

{y1 − q�Dq�q−b +

∫ T1

0zθt�1−bdt −

∫ T1

0yθt�−bθ′t�dt

}�

Evaluating the second integral and combining the terms containing q, thisbecomes

c

{y�1 − q�Dq�q−b + qy/xx/y� +

∫ T1

0zθt�y/xdt − yx/y�θ−T1�y/x

}� (8)

Here I write b where it simplifies the expression; otherwise, I have substi-tuted its value x− y�/x. Note that 1 − b = y/x.

Next I examine H’s payoff on �0� T1� for modified versions of regimeα� θ2�. Specifically, I look at a regime β�µ� constructed to satisfy Eqs. (6)and (7) on �0� T1� and also to be equivalent to α� θ2� on �T1�∞�. Sucha regime is consistent with rationality of the low types, as the foregoinganalysis establishes. Let β�µ� be defined by a parameter m ∈ �0� 1�, suchthat µt� ≡ mθ2t�+ 1−m�θ−2 T1� for every t ∈ �0� T1�. The level functionβ is then defined by Eqs. (6) and (7). Observe that the regime is exactlyα� θ2� when m = 1; as m decreases, “quitting” probability mass is shiftedfrom the interval 0� T1� to a mass point at time zero.

196 joel watson

Rewriting Eq. (8) for regime β�µ�, we see that the expected payoff ofthe high types in the interval of time �0� T1� is cA+ B + C�, where

A = y1 − q�Dq�q−b + yqy/xx/y�

B =∫ T1

0z�mθ2t� + 1 −m�θ−2 T1��y/xdt�

C = −yx/y�θ−T1�y/x�c = βT1�e−rT1µ−T1�−1 = αT1�e−rT1θ−2 T1�−1�

and

q = µ0� = mθ20� + 1 −m�θ−2 T1��I demonstrate that the regime specified by m = 0 is an equilibrium andthat each player type prefers it to the one specified by m = 1, in terms ofexpected payoffs.

To prove that the regime with m = 0 is an equilibrium, I only need todemonstrate that the high types do not wish to quit at t = 0. (The prefer-ences of the low types were addressed earlier; the high types’ preferencesfrom T1 are captured in the original regime α� θ2�, and by construction,the high types do not wish to quit on 0� T1�.) We know that α� θ2� isan equilibrium and that the component of the high types’ payoffs from T1are independent of m. Also note that xq1−bDq�c is the high types’ pay-off from quitting at the beginning of the game. It is sufficient to showthat E ≡ cA + B + C� − xq1−bDq�c is decreasing as a function of m.Let F ≡ A− xq1−bDq�. We have dE/dm = cdF/dq�dq/dm+ cdB/dm+cdC/dm. Note that c is not a function of m, dq/dm > 0, dC/dm = 0, anddB/dm ≤ 0, since y ≤ 0. Thus, I must establish that dF/dq ≤ 0. Taking thederivative and performing some algebraic manipulation, we find that dF/dqhas the same sign as x+ y��Dq� − �Dq��, where

�Dq� ≡ qx

qx− 1 − q�y �

Recall that x + y ≤ 0 is assumed. Thus, we obtain the desired conclusionunder the condition Dq� ≥ �Dq�, which simplifies to −y/x ≥ −y/x.

In fact, the regime with m = 0 is also an equilibrium in the case in whichDq� < �Dq�. To see this, recall that Dq� is defined so that the low typeis indifferent between quitting at t = 0 and just after this time; Dq� isa scaling factor on the relation between the level at 0 and just after 0.We can compute the corresponding scaling factor for the high types; it is�Dq�. Thus, if the level is scaled according to �Dq�, then the high-typeplayer is indifferent between quitting at 0 and just after this time. When


Dq� < �Dq� and Dq� is used to scale the level, the high type strictlyprefers waiting to quitting at t = 0.

Next, I prove that all player types fair weakly better under the regimewith m = 0 than they do if m = 1. To demonstrate that the high types preferthe m = 0 regime, I need to show that dA/dq ≤ 0. Taking the derivativeand performing some algebraic manipulation, we find that dA/dq has thesame sign as x+ y, which is nonpositive given assumption 2. To see that thelow types prefer the m = 0 regime, note that the payoff of the low types asa function of q is

qy/xDq�cx�We require that the derivative of this expression with respect to q be non-positive. Evaluating the derivative, we find that it has the same sign as x+ y,which is indeed nonpositive.

Let α� θ2� denote the regime specified by m = 0. To be precise,θ2t� = θ−2 T1� for every t ∈ �0� T1�; θ2t� = θ2t� for every t ∈ �T1�∞�;αt� = ertθ−2 T1�−bc for all t ∈ 0� T1�; αt� = αt� for all t ∈ �T1�∞�; andα0� = Dθ−2 T1��θ−2 T1�−bc. Recall that this proof formally covers thecase in which T1 > 0. If T1 = 0, then this entire step is not required.

Step 4. Define T ≡ min�t � αt� = 1�. The final step involves construct-ing a new regime by modifying α� θ2� to delay as much as possible the quitmass of player 2L in the interval �T1� T �. Let �T1 and �T2 be the times thatdefine the phases of regime α� θ2�. We have �T1 = 0 and �T2 = T2. By theconstruction of α� θ2�, we know that player 2L is indifferent between quit-ting at all times in �0� T �. It is not difficult to see that there is a uniqueregime α� θ2� satisfying (i) α ≡ α (for all t), (ii) θ20� = θ20�, and (iii)the regime has maximal delay on T1� T � = 0� T �.

The regime α� θ2� is an equilibrium by construction. In addition, it isthe case that θ2 ≥ θ2, meaning that player 2L cooperates for a longer timein this regime than with α� θ2�. The payoffs of players 1L, 2L, and 2H areidentical between regimes α� θ2� and α� θ2�. The payoff of player 1H isweakly greater under α� θ2�. To see this, note that delaying when player 2Lquits does not change the discounted (terminal) payoff of the quit event toplayer 1H, since α′/α = r on 0� T �; however, the delay gives player 1H anadditional payoff due to cooperation by player 2L.

To summarize the steps, I have identified an equilibrium regime α� θ2�under which the payoff of every player type is weakly greater than withthe original regime α� θ2�. By construction, α� θ2� is a Q-regime. Thisproves the first assertion of the theorem. To prove that an IIE regimeexists, recall that the class of Q-regimes is parameterized by the numbersα0 = α0� ∈ �0� 1� and θ0

2 = θ20� ∈ �p2� p1�. It is obvious that payoffs arecontinuous in these parameters. Furthermore, the set of parameter values

198 joel watson

yielding equilibrium regimes (a subset of �0� 1�2) is closed; this follows fromcontinuity of the relevant continuation payoff expressions. Thus, for eachwelfare function wv1H0�� v1L0�� v2H0�� v2L0�� that is strictly increasing,there is a Q-regime that maximizes w over all equilibrium regimes with levelfunctions continuous on 0�∞�. This proves the second assertion.

Proof of Lemma 3

Consider the region of m�p� for which α0symm�p� is “interior,” meaning

that α0sym is less than Dp� so T > 0. (The other case is easy to handle.) It

is not difficult to show that an interior solution is unique and continuousin m and p. To prove that α0

sym is weakly increasing in p, by the implicitfunction theorem it is enough to demonstrate that ∂2wsym/∂α0∂p ≥ 0. Wethus need to show that ∂2uL/∂α

0∂p ≥ 0 and ∂2uH/∂α0∂p ≥ 0. Equivalently,

we need ∂2uL/∂T∂p ≤ 0 and ∂2uH/∂T∂p ≤ 0, where T and α0 are relatedby the identity α0 = Dp�e−rT in an interior solution. Note that

uL = p2x2

px+ 1 − p�y e−rT

and

uH = yDp�1 − p�e−rT + pe−rT T z + pe−rT z/r� (9)

Differentiating these expressions, we obtain the desired cross-partial con-ditions. Regarding uH , the calculations are simplified by first noting that

∂uH/∂T = −ruH + pe−rT z� (10)

To prove that α0sym is weakly decreasing in m, note that

mu′Hα0symm�p�� + 1 −m�u′Lα0

symm�p�� = 0

in an interior solution. Clearly, u′Lα0symm�p�� > 0, so it must be that

u′Hα0symm�p�� < 0. The result then follows. To prove that T > 0 in the

case of m = 1, first note that if T = 0, then the high types’ payoff ismaximized with α0 = Dp�. Then it is enough to demonstrate that, treat-ing the expression for uH in the previous paragraph as a function of T ,duH0�/dT > 0. Using Eqs. (9) and (10), we have

duH0�/dT = −ryDp�1 − p� − pz + pz > 0�

which completes the proof.

Proof of Lemma 4

This follows from the argument used in the last step of the proof ofTheorem 2.


Proof of Lemma 5

As in the proof of Lemma 3, we can write the payoffs as a function ofT . Note that

u1H = y1 − p�e−rT + ρe−rT T z + pe−rT z/rand

u2H = e−rT T z + e−rT z/r�where ρ ∈ 0� 1� captures the fact that it may be necessary to have player 2Lbetray in a region s� T � for some s < T . If T = 0 defines an equilibriumregime, then it must be the case that ρ = 1. One can then follow the stepsused in the proof of Lemma 3 to complete this proof.

REFERENCES

Cramton, P., and Palfrey, T. (1995). “Ratifiable Mechanisms: Learning from Disagreement,”Games Econ. Behav. 10, 255–283.

Crawford, V. (1985). “Efficient and Durable Decision Rules: A Reformulation, Econometrica53, 817–835.

Holmstrom, B., and Myerson, R. (1983). “Efficient and Durable Decision Rules with Incom-plete Information,” Econometrica 51, 1799–1819.

Maskin, E., and Tirole, J. (1990). “The Principal-Agent Relationship with an Informed Prin-cipal, I: The Case of Private Values,” Econometrica 58, 379–410.

Maskin, E., and Tirole, J. (1992). “The Principal-Agent Relationship with an Informed Prin-cipal, II: Common Values,” Econometrica 60, 1–42.

Myerson, R. (1983). “Mechanism Design by an Informed Principal,” Econometrica 51,1767–1797.

Rauch, J., and Watson, J. (1999). “Starting Small in an Unfamiliar Environment,” WorkingPaper 96-28r, University of California San Diego, Department of Economics.

Watson, J. (1996). “Building a Relationship,” Unpublished Manuscript.Watson, J. (1999). “Starting Small and Renegotiation,” J. Econ. Theory 85, 52–90.

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Starting Small and Commitment