war of attrition

7/28/2019 War of Attrition

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Economics Department of the University of Pennsylvania

Institute of Social and Economic Research -- Osaka University

The War of Attrition in Continuous Time with Complete InformationAuthor(s): Ken Hendricks, Andrew Weiss, Charles WilsonReviewed work(s):Source: International Economic Review, Vol. 29, No. 4 (Nov., 1988), pp. 663-680Published by: Blackwell Publishing for the Economics Department of the University of Pennsylvania and

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INTERNATIONAL ECONOMIC REVIEWVol. 29, No. 4, November 1988

THE WAR OF ATTRITION IN CONTINUOUS TIMEWITH COMPLETE INFORMATION*

By KEN HENDRICKS, ANDREW WEISS, AND CHARLES WILSON'

1. INTRODUCTIONIn this paper, we present a general analysis of the War of Attrition in continu-ous time with complete information. In this game, each of two players mustchoose a time at which he plans to concede in the event that the other player has

not already conceded. The return to conceding decreases with time, but, at anytime, a player earns a higher return if the other concedes first. The game wasintroduced by Maynard Smith (1974) to study the evolutionary stability of cer-tain patterns of behavior in animal conflicts. It has subsequently been applied byeconomists to a variety of economic conflicts such as price wars (e.g., Fudenbergand Tirole 1986, Ghemawat and Nalebuff 1985, Kreps and Wilson 1982) andbargaining (Ordover and Rubinstein 1985, Osborne 1985).2Most authors have assumed specific functional forms for the payoffs. The onlygeneral analysis of the equilibria of this game in continuous time with completeinformation is by Bishop and Cannings (1978). They study a general symmetricgame, but restrict the equilibrium analysis to symmetric Nash equilibria whichsatisfy a certain stability property.3 This paper generalizes their model to allowfor asymmetric return functions and arbitrary payoffs in the event that neitherplayer ever concedes. We provide a complete characterization of the Nash equi-librium outcomes.The paper is organized as follows. In Section 2, we introduce the game andpresent the assumptions which define the War of Attrition. In Section 3, we statethe main theorems and heuristically describe the logic behind the proofs. InSection 4, we relate our analysis to some of the applications which have appearedin the economics literature. Section 5 contains the proofs of the theorems present-ed in Section 3. Section 6 concludes with a brief discussion of how the analysischanges when time is made discrete.

* Manuscript received December 1986; revised August 1987.l Support from the National Science Foundation, the Alfred P. Sloan Foundation, and the C. V.

Center for Applied Economics is gratefully acknowledged.2 Most of these models incorporate some degree of incomplete information. The method of analy-sis is similar, but the set of equilibria is sometimes substantially reduced.3 The equilibrium concept is known as evolutionary stable strategies (ESS). Following Selton

(1980), a strategy r is said to be evolutionary stable if (i) r is a best reply to itself and (ii) for anyalterniativebest reply r' to r, r is a better reply to r' than r' is to itself.

663


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664 KEN HENDRICKS, ANDREW WEISS, AND CHARLES WILSON2. THE GAME

Two players, a and b, must decide when to make a single move at some time tbetween 0 and I.4 The payoffs are determined as soon as one player moves. Inwhat follows, i refers to an arbitrary player and P to the other player. If player amoves first at some time t, he is called the leaderand earns a return of La(t).If theother player moves first at time t, then player oc s called thefollower and earns areturn of F (t). If both players move simultaneously at time t, the return to playerx is S t).In the strategic form of the game, a pure strategy for player a is a time t, E[0, 1] at which he plans to move given that neither player moves before that time.Given a strategy pair (ta, bb) E [0, 1] x [0, 1], the payoff to player ac s thendefined as follows:

P,(t. I tb) = L,(t,) if tot< t,B= S,(t) if t = tf=-F,j(t if ta > tf

2.1. Assumlptions on the Payoff Finctions. Our first assumption guaranteesthat the payoff functions are continuous everywherebut on the diagonal.(Al) La and F. are continuous functions on [0, 1].f

Games with this structure are generally called "noisy" games of timing.6Notice that we impose no differentiabilityconditions on either of these functions.Our next assumption characterizes the War of Attrition.(A-2) (i) Fj(t) > La(t) for t E (0, 1);

(ii) Fa(t) > Sj(t) for t E [0, 1);(iii) LX(t)s strictly decreasing for t E [0, 1).

Condition (i) requires that the return to following at any time t > 0 strictlyexceed the return to leading at time t. We do not rule out the possibility that

This is just a normalization. A game with an infinite horizon can be converted into this frame-work by a simple change of variable such as t = /(1 + -) where ZE [0, x). See Section 4 on theapplications of this model for more discussion oni this point.

5 It is niot essential that F,(l) and Ljl) be defined since the only return which can be realized attime 1 is S(l). Defining LX(t)o be continuous at 1 merely allows us to identify lim,1 LX(t)with La(1).Similarly for F,

6 The gaimies re called "noisy" because the payoff to the follower depends only on when the otherplayer moves. This reflects the assumption that a player who plans to wait until time t to move doesnot have to coiunnithimself to moving until he has observed the history of the game up to time t.Consequeintly,if the otlherplayer moves before time t, the first player can react optimally, indepen-dently of whcathe hcadplanned to do had the other player not moved at time t. A "silent" game oftiminig is onie in which eaclh player must commnilt himself to a time at which he will move independent-ly of the action of the other at the outset of the game. Silent games of timing have been also beenulsed n economic models (e.g. Reingainum1981a, 1981b).


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THE WAR OF ATTRITION 665Lx(O) FJ(O) r the possibility that Fa(1)= La(1).Condition (ii), however, requiresthat the return to following strictly exceed the return to tying at all times lessthan 1. Combined with condition (iii), these conditions imply that, at any timet < 1, each player prefers to wait if the other player plans to move but, if forcedto move first, would prefer to move sooner than later. Note, however, that, sinceSa is not necessarily continuous, our assumptions impose no restrictions on therelation of S (1) to either Lat(1 or Fj(1). Consequently, Sat(1 may reflect theequilibrium payoff of any continuation game which is played whenever bothplayers wait until time 1.

Although not necessary for our analysis, it will also be convenient to ignore thespecial case in which return to leading at time 0 is exactly equal to the terminalreturn.(.A3) La(O)7&Sat1).

In Figure 1, we have illustrated three possible relations between the returnfunctions. In each case, the return functions are normalized so that Sa(1)= 0. Asrequired by Assumption A2, in each case the value of F, lies above La over theopen interval (0, 1) with La strictly decreasing throughout. Figure l(a) illustratesthe case in which both the return to leading and the return to following convergeto 0 as time approaches 1. Figure l(b) illustrates the case where the return fromleading is bounded above the return at time 1. Figure l(c) illustrates the casewhere the return to leading actually falls below the return at time 1 before time 1is reached. We will return to each of these examples again in our discussion of theapplications. Although not illustrated, Assumption A2 requires that S. must alsolie below F' over the interval (0, 1). However, it is not necessary that F. bedecreasing.

2.2. Equilibriium1. It is important for our results to permit agents to rando-mize across pure strategies. A mixed strategy for player oc s a probability distri-bution ftinction G. on [0, 1].7 If we extend the domain of the payoff functions tothe set of all pairs of mixed strategies in the obvious way, then a strategy combi-nation (G*, G*) is an equilibrium f P,(G*, G*) ? P,(G,, G*) for all mixed strate-gies G. ca= a, b and / # cx.For the remainder of the paper, (Ga, Gb) will refer to an equilibrium combi-nation, and q,(t) will denote the probability with which player cxmoves at exactlytime t. We will repeatedly use the fact that if (Ga, Gb) is a pair of equilibriumdistributions, then P (t, Gf) = supVE[0o ] P,(v, G.) for any t in the support of G .

7 By a probability distribution on [t, 1], we mean any right-continuous nondecreasing function Gfrom (-x., c(x] to [0, 1] with G(t)= 0 for t < 0 and G(1)= 1. Throughout this paper, we will adoptthe convention that

f(u) dG_(v) = lim,r,n f(v) dG,(t).That is, the integral does not include a mass point at time tP


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666 KEN HENDRICKS. ANDREW WEISS, AND CHARLES WILSON

3. THE MAIN THEOREMSIn this section we present the main theorems in our analysis and describe thelogic behind their proofs. In our analysis, we distinguish between two kinds ofequilibria. An equilibrium is degenerate if either one of the players moves at time

Return a

0

Time

ReturnL a = _

Time

Return Fa

TimeFIGURE 1

THREE PATTERNS FOR THE RETURN FUNCTIONS


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THE WAR OF ATTRITION 6670 or both wait until the terminal time. There is always at least one equilibrium ofthis type. In addition, there is sometimes a continuum of nondegenerate equilibria.These are equilibria in which both players move over an interval (0, t) accordingto a strictly increasing, continuous distribution, after which both wait until theterminal time. The indeterminacy comes at the beginning of the game where theonly constraint is that only one player may move immediately with positiveprobability.

3.1. Degenercate Equibria. Theorem 1 characterizes the possible degenerateequilibrium outcomes.THEOREM 1.

(a) Tlher-e is an equilibrium in which qa(1) = qb(l)-1 if and only if La(O)SI(1)for x = a, b.(b) There is an equilibr-iumwith q (O)= 1 and qf(O) = 0 if and only if,for someac,(i) Lx(0)> SJ(l) or(ii) Jbr soine t e (0, 1), La(O) F,(t).

At least one of the equilibrium outcomes described in Theorem 1 always exists.If the return to each player at time 1 is higher than his return from leading attime 0, then there is an equilibrium in which both players wait until time 1 withprobability 1. On the other hand, if his return to leading at time 0 exceeds histerminal return, then there is an equilibrium in which player cxmoves immedi-ately. Moreover, both types of equilibrium may exist if the terminal return ofsome player o exceeds his return to leading at time 0 which in turn exceeds hisreturn to following at some time later time t. In this case, the equilibrium out-come in which player a moves immediately is supported by a threat by player bto move at time t.8In any case, notice that only one player can move immediately with probability1. The reason is that Assumption A2 implies that the other player can do betterby waiting to obtain the return to following. However, if player cx s certain tomove immediately with probability 1, then there is generally a large class ofequilibrium strategies for player /3 which support this outcome. The only re-striction on his strategy is that it be an optimal response for player a is to moveimmediately. As we note below, however, the imposition of subgame perfectiondoes eliminate some of this indeterminacy.

3.2. Nocde generacte Equilibria. As we demonstrate in Section 4, any masspoints in the equilibrium strategies must be concentrated at either the beginningor the end of the game. Furthermore, unless the equilibrium is degenerate, anygaps in the support of the equilbriumndistributions must be the same for bothplayers and must extend to the terminal time. The condition that each player be

8 As we argue below, however, this equilibrium is not subgame perfect.


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668 KEN HENDRICKS, ANDREW WEISS, AND CHARLES WILSON

indifferent to moving at any time in the interior of the support then defines a pairof integral equations which determine the equilibrium strategies up to some "endpoint" conditions. The requirements for these endpoint conditions provide thenecessary and sufficient for the existence of a nondegenerate equilibrium.To state these conditions, we require some additional notation. Define

I(.(s, t) exp L i [dLa(t.)/(Fa(v) - La(v))]Note that Assumption (Al) implies that If(s, t) is well defined everywhere exceptpossibly at 0 and 1. Define I,(O, 0) _ limtl0 Ia(O,t) and I,(i, 1) _ lim,11 I,(t, 1).'

THIEOREM2. Tlere is ans1 quiilibriumel vithqa(0) + qa(l) < 1 for 0 a, b if andoni ai(a) I (O,0) = I(G? 0) = 1; a2nd(b) Onieof the followvinqcon-ditions is satisfied:

(i) LW(t*) StEl) = Lb(t*) - Sb(M)for som>e * E (0, 1];(ii) I (1? 1) =b( 1) = 0;(iii) Ia(1, 1) = 0 and L,(1) ? SJ(l)for somnea.

Before explaining these conditions, we will proceed with a characterization ofthe nondegenerate equilibria they imply.THEOREM 3. (Ga, Gb) is an1eqi.iiiibriumiwvith qa(0) + qa(1) < 1 for 2 = a, b if anid

0o71n f thle conditions Of Thleoi-eore2 are satisfied anid:(a) (qa(0), qb(O)) E [0, 1) X [0, 1) and qa(O)qb(O)= 0.(b) Th2ere s a t1 e (0, 1] stuchthat, for /B = a, b, either

(i) L13(t ) = Sl(l), 0or(ii) t1 = 1, a1nd(3.1) Gp(t) = 1- [1 - q(O)]If(O, t) for t E [0, t1), anid

ql3() = [1 - q13(0)] (O, t1)Considei first the statement of Theorem 3. Condition (a) is an "initial" con-dition which states that at most one player can move with positive probability attime 0. Sinice,by Assumption (A2), the return to following exceeds the return totying at time 0, it always pays at least one player to wait an instant. Condition (b)gives the integral equation which defines the equilibrium strategies after time 0.Both players move according to a continuous increasing probability distribution

up to some time t1 after which they wait until the end of the game. The require-ment that player -abe willing to move at any point in the interval (0, t1) implies" Note that O(O0) and If(l, 1) can only take on the values 0 and 1. Furthermore, if FJO) > LJOthen 1/J(O0) = 1, and if F (I) > Lx(1), then IQ(l11) = 1.


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equation (3.1). (If L. is differentiable,it is the solution to the differentGp(t)[l - Gp(t)] = L'(t)/[Fa(t) -G(t)].)Consider next the statement of Theorem 2. Condition (a) is nesufficient for equation (3.1) to define a strictly increasing functiosatisfied whenever the return to following at time 0 strictly exceeds tfollowing. Condition (b) gives the three possibilities for the strategieCondition (b) of Theorem 3 to be best responses. Condition (b)(i) coCondition (b)(i)of Theorem 3. In this case, there is a time at whichearn returns to leading that equal their terminal returns. Setting t, etime then makes both players indifferent between moving at anyinterval (0, t,) and waiting until the end of the game.Conditions (b)(ii) and (b)(iii) correspond to Condition (b)(ii) of Ththis case each player /Bmoves according to the distribution functionout the game. This strategy pair forms a pair of best responses if anof two conditions is satisfied. One possibility is for both players tmove with probability 1 making the terminal payoff to other irrother possibility is for only player axto eventually move with probabthis case, the terminal return to player ,B is irrelevant, but, for theplayer oeto be optimal, he must not prefer to wait until the end oConsequently, we require that L (1) ? Sj(1). Condition (b) then fobserving that equation (3.1) implies that limt,1 G,(t) = I if aI(1, 1) = 0.Notice that both Conditions (b)(i) and (b)(ii) might be satisfied simIn these cases, there are two one parameter families of equilibriaaccording to whether there is a gap in the supports of the equilibutions.

3.3. Subgame Perfection. Many of the applications of the wararise in situations in which the players cannot commit themselves towhich they plan to move at the beginning of the game. In these caseto maoveat time t is actually a decision to move first at time t, givenplayer has already moved by that time. For these games, it may berefine the equilibrium concept to incorporate the implications of sfection.To use this concept, however, we must first extend the concept ofspecify the plans of the player upon reaching any time t. As defined ithe game is in a reduced normal form in that the strategies do nospecify the plans of a player who deviates from his intended action.does not plan to move before time t with probability 1, then we marule to derive the strategy of the player at time t. Otherwise, at evewhich Bayes does not determine the plan of the player, we must sdistribution function stating the strategy of the player for the subgamthat time. In the interest of space, we will avoid the details of a prcation of this largerstrategy space and present only heuristic argume

Note first that equation (3.1) implies that, for any t < 1, there iprobability that it will be reached in any nondegenerate equilibriu

THE WAR OF ATTRITION 669


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immediately, therefore, that every nondegenerate equilibrium is subgame perfect.For the same reason, any degenerate equilibrium in which both players wait untiltime 1 is also subgame perfect. Consequently, the only cases in which subgameperfection may be restrictive is when one of the players moves immediately withprobability 1. In these cases, subgame perfection not only eliminates some equi-libria, but, as Fudenberg et al. and Ghemawat and Nalebuff have shown, it mayeven eliminate degenerate equilibrium outcomes as well.A general characterization of the subgame perfect equilibrium may be stated asfollows.

THEOREM 4. There is a subgameperfect equilibrium n which qa(O) = 1 if andonlv if:(a) either condition (b) of Theorem2 is satisfied; or(b) (i) Lo(O) So(1);and(ii) Lo(t)< So(1) mplies Lp(t)< Sp(1)forall t E [0, 1).

To understand these conditions, we note that there are two ways in which adegenerate outcome might be made subgame perfect. The first is for the playersto adopt a nondegenerate equilibrium if the game reaches time t > 0. This ispossible if and only if condition (b) of Theorem 2 is satisfied.The other possibilityis for player a to plan to move with probability 1 upon reaching any time t up tothe time t* where his return to leading equals his terminal return. Thereafter,hewaits until the terminal time with probability 1. In response, player P must adopta strategy which makes this optimal. Here, two conditions must be satisfied. First,player /3 must not earn a higher return from leading after time t* than histerminal return. Otherwise, he will move with probability 1 upon reaching anytime after t* which will lead player a to wait at time t*. Inducting on thisargument then leads to an unravelling of the equilibrium. This argument yieldscondition (b)(ii). Second, since player ,Bearns a lower return from leading after t*than by waiting until time 1, he will never move after t* either. Consequently, thereturn to player cxfrom leading at time t* must be no less than his terminalreturn. Assumption (A2) then implies condition (b)(i).Notice that, in the absence of a nondegenerate equilibrium for any subgame,Theorem 4 implies that it is subgame perfect for some player to move immedi-ately only if and only if L,(O)> So(1) or some player a. Thus, in this case, the twotypes of degenerate equilibrium outcomes are mutually exclusive. Either one ofthe players moves with probability 1 at time 0 or both players wait until time 1.We should point out that in working in continuous time, there are someadditional subtleties that do not arise in the discrete time analogues. The prob-lem is that in continuous time a player may move with zero probability at anypoint in an interval, but nevertheless move with positive probability over theentire interval. Consider, for instance, a game in which, upon reaching any periodt, the unique subgame perfect outcome is for player a to move immediately withprobability 1. Since it cannot be optimal for player / to move at any time atwhich player a plans to move with probability 1, player ,Bcannot plan to move


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THE WAR OF ATTRITION 671with positive probability at any time before the terminal time. If time is discrete,this implies that he waits until the end with probability 1. If time is continuous,however, this restriction only eliminates strategies with mass points. It does notrule out a strictly increasing, continuous distribution function which rises at ratesufficiently low so that player cxalways prefers to move immediately upon reach-ing any time t. Consequently, we are left with an infinity of subgame perfectequilibria.

4. NONDEGENERATE EQUILIBRIA IN ECONOMIC APPLICATIONSIn any interesting application of this game, the return to leading at time 0 willexceed his terminal return for at least one of the players. In this case, there is

always a subgame perfect degenerate equilibrium in which one of the playersmoves immediately. The focus of most of the attention in the literature,however,has been on the nondegenerate equilibria.'0 In this section, we discuss the kindsof economic applications for which nondegenerate equilibria are likely to exist.Ignoring for the moment, the integral condition at time 0, the class of econom-ic applications for which nondegenerate equilibria exist can roughly be dividedinto those which satisfy Condition (b)(i) of Theorem 2 and those which satisfyCondition (b)(ii). One important class of applications which generally satisfy bothconditions are those which require an infinite horizon. Suppose, for instance, thatall of the return functions decline at an exponential rate 6. Then, if we transformtime according to the formula z = t/[t + 1], simple calculations reveal that notonly is F,(1) = L,(1) = So(1)= 0 as in Figure l(a), but Il(l, 1) = 0 as well. In thiscase, there is a one parameter family of equilibria in which both players moveover the entire interval according to a continuous distribution function.Condition (b)(ii) may also be satisfied for some applications with a finite hor-izon if the net return to following depends on the amount of time remaining inthe game. In their continuous time version of the "chain store" paradox, forexample, Kreps and Wilson (1982) assume that a contest for a market betweentwo firms takes place over a predetermined interval (0, 1). They also assume thatthe benefit from following over leading is proportional to 1-t and that the cost ofleading (in their case leaving the market) is proportional to t. Simple calculationsagain reveal that Il(1, 1) = 0. Consequently, under complete information, theirmodel possesses a one parameterfamily of nondegenerate equilibria in which oneof the players is certain to concede by time 1l"For most economic applications which require a finite horizon, however, the

1o There are exceptions. Kornhauser, Rubinstein, and Wilson (1987), for example, argue for select-ing the degenerate equilibria.

l This game may also possess another family of nondegenerate equilibria. If the firms were to playa sequence of n such contests (see Fudenberg and Kreps 1985), the terminal payoff to each firm at theend of the kth contest would represent its equilibrium payoff from playing the remaining tn-k ontests.If this is nonzero and returns are symmetric, the family of nondegenerate equilibria described incondition (i) of Theorem 2 also exists. Notice that in this equilibrium, the behavior of the firms in thekth contest depends on their behavior in subsequent contests through the value of the terminalpayoffs.


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return to following stays bounded above the return to leading. Consequently,neither Condition (b)(ii) nor (b)(iii) is satisfied, and the only possibility for es-tablishing the existence of an equilibrium is to satisfy Condition (b)(i).There aretwo reasons why this condition might not be satisfied. The first is that the returnto leading at any finite time may strictly exceed the terminal return. For instance,in the oil exploration example studied by Wilson (1983), if the firm has not drilledits lease by a certain time, it loses the lease. In this case, there is no nondegenerateequilibrium because truncating the horizon introduces a downward discontinuityin the payoffs at time 1 as in Figure l(b). (See Hendricks and Wilson 1985 for adetailed discussion of this issue.)The second reason nondegenerate equilibria may not exist is that, even ingames where the return to leading eventually falls below the terminal return as inFigure 1(c), the return to leading must equal the terminal return at exactly thesame time for both players. This property is likely to hold only for symmetricgames. Consequently, unless there is a special reason to assume that returns aresymmetric, such as in the biology models, we should not expect to find a nonde-generate equilibrium. Two economic applications where this result is of interest isin the patent race model of Fudenberg et al. (1983) and the exit model byGhemawat and Nalebuff (1985). In both of these models, returns are assumed tobe asymmetric and, as a result, the nondegenerate equilibria of Theorem 2 areeliminated.

Finally, consider the implications of the integral condition at time 0. In mostapplications, the return to following strictly exceeds the return to leading at time0, so that this condition is satisfied. In some cases, however, particularly wherethe war of attrition is a subgame of a larger game, this condition is less plausible.For example, Hendricks (1987) studies a model of adoption of a new technologyin which there are both first and second-mover advantages. Each firm mustchoose a time at which to adopt. The return functions are continuous and havethe property that, initially the returns to leading are increasing, and exceed thereturns to following. During this period, first-mover advantages dominate thesecond-mover advantages, and each firm has an incentive to preempt. Eventually,however, the return to leading decreases, falling below the return to following, sothat the second-mover advantage dominates. Consequently, if La is decreasing atthe point of intersection of F. and L. then Condition (a) of Theorem 2 is notsatisfied and hence there are no nondegenerateequilibria for this subgame.

5. THE TECHNICAL ANALYSISWe begin by establishing the relation between the supports of the strategies ofthe two players.LEMMA 1. Suppose G(t1) = Gj(t2) < 1 or t2 > t1. Then G,(t1) = Gf(t2 + i)forsomte6 > 0.PROOF. Suppose Gj(t1)= G(t2) < 1 for t1 < t2 and choose ? e (0, t2 - t1).Then, for any t E (t1 + E, t2], it follows from Assumption A2 that player # prefers


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THE WAR OF ATTRITION 673to move at time t1 + e than to move at t since there is no chance that player xwill move in the intervening interval:

Pp(t, G) - Po(t1 + ? Gj) = [L0(t)- L0(t1 + ?E)][1 G,(t )] < 0.Furthermore, for any r1 > 0 sufficiently small, right-continuity of GQmplies thatthere is an arbitrarily small 6 > 0 such that Ga(t2+ 6) - < c1. It thenfollows from Assumptions Al and A2 that, for t e (t2, t2 + 6),

Pp(t, Ga) -P(t + ?, G)t= {J?~ [F(v) - L(t1 + ?)] dGQ(v)+ [Sp(t) -Lp(t + e)]qx(t)

+ [Lp(t) - Lp(t1 + c)] [l - G(t)]= 0(?1) + o(el) + [LP(t) - Lp(tj + -)][1 - GJ(t2) + o(c1)] < O0

Letting - 0, we may then conclude that for any t e (t1, t2 + 6], there is anearlier time v e (t1, t) at which player / prefersto move. Q.E.D.The next Lemma rules out any mass point after time 0 but before either playermoves with probability 1.LEMMA 2. For t e (0, 1), im,?t Gp(v)< 1 implies qj(t) = 0.PROOF. Suppose, for some t e (0, 1), that qj(t) > 0. Then, for any e > 0, thereis an (arbitrarily small) 6 > 0 such that (i) L0(t - 6) - L0(t + 6) < e, and(ii) qj(t + 6) = 0 with G,(t + 6) - G,(t - 6) < qj(t) + e. It then follows from As-sumptions Al and A2 that, for e and 6 chosen sufficientlysmall,

rt+6Pp3(t+ 6, G) - P0(t, G4)= [Fp(t) - S#(t)]q,(t) + [Fp0(s) L(t)] dGQ(s)+ [Lp(t + 6) - L0(0][l - G,(t + 6)]

= [Fp(t) - SO(t)]q,(t) + o(e) + o(e) > 0.Similarly, for v e (t - 6, t),

Pp(t + 6, G) - P(v, G.) = [Fp(v) - S0(v)]q,(v) + [F0(s) - Lp(v)] dGQ(s)+ [L0(t + 6) - L0(E)][l - G,(t + 6)]

= 0(?) + [Fp(t) - L,4v)]q,(t) + O(e) + 0(e) > 0.Consequently, player /Bwill never move in the interval (t - 6, t]. This implies thatG(t - 6) = Gp(t). Then, if lim,,Tt Gp(v)< 1, Lemma 1 implies that G,(t - 6) =Gj(t),contradicting the hypothesis that qj(t) > 0. Q.E.D.

If G is strictly increasing over some interval, then we may use the fact thatplayer a must be indifferent to moving at time within the interval to explicitly


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characterize the equilibrium strategy of player fl over this interval in terms of thereturnfunctions L. and F .LEMMA 3. Suppose G. is strictly increasing over the interval [to, tl]. Then, fort0 > 0 andti E (to, t1), Gp(t)< 1 implies

(.5.1) Gfl(t)= 1-[1l- Gf(to)]Il(to, t).PROOF. Suppose that G. is strictly increasing over the interval [to, tl]. Then,since G,(t) < 1 for t < t1, it follows from Lemma 2 that Gp is continuous on

(0, t1). Therefore,for any t E [to, t1).(5.2) 0= Pj(t, Gp) - P,(to, Gp)

t= [F,(v) - L,(to)] dGp(v) + [1 - G0(t)][L,(t) - L,(tO)]Since G. and L. are both monotonic and continuous on [to, t], we may apply theformula for integration by parts (Rudin 1964, p. 122) to obtain(5.3) [l - Gp(t)][L,(t) - L,(to)]

(t rt[1 Gp(v)] dLa(v)-J [La(v) - L,(to)] dGp(v).to toSubstituting (5.3) into (5.2) and rearrangingterms then yields, for all t E [to, t1),

Ct L(1F lXFdGfi(V) dL,,(v) 1=(~.t) [L (v)-F,c(V)JL[ IJ-a()J[ - G#(v) La(v) - F(v)jBut, since [L,(U) --F()][ - G3(v)] < 0 for all v E [to, t], equation (5.4) impliesthat

t tJ dGp(v)/[1 Gp(v)]= -J dLa(v)/[Fa(v)La(v)].to to

Employing a change of variable (Rudin 1964, p. 122-124), we may then apply thefundamental theorem of calculus to obtain:log [l - G(t)] = log [1 - G0(to)] + dLa(v)/[Fa(v - La(v)].

.toTaking antilogs and rearrangingterms then yields equation (5.1). Q.E.D.

Note that, if L. is continuously differentiable, equation (5.1) is simply thesolution to the differentialequationG/X(0)E1 Gp(t)] = L0(t)/[L,(t) - F,(t)]with initial condition G0(to). n this case, G. has a continuous density function g0over the interval (to, t1).We establish next that if neither player moves with probability 1 at time 0, thenthe game cannot end with certainty until time 1.


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THE WAR OF ATTRITION 675LEMMA4. Stuppose G(O) < 1. Then G,(O) < 1 implies Gp(t) < l or t < 1.PROOF. Let t^= sup {t > 0: G,(t) < 1 for x = a, b} be the earliest time by

which one of the players plans to move with certainty. The lemma is equivalentto the requirement that t 0 (0, 1). Suppose 0 < t < 1.

We will show first that the strategy of at least one of the players must have amass point at time t. Suppose not. Then, for some player ,B, Gp s strictly increas-ing over an interval (t', t) and limtT1 G#(t) = 1. Then, for any t E (t', t), Lemma 3combined with Assumption A2 implies that GQ s strictly increasing over (t', t).Combining Lemma 3 with Assumption A2 again, we then obtain that limtTtGf(t) = 1 - [1 - Gf(t')]Ifl(t', f) < 1. A contradiction.But, if qfl([) > 0, then Lemma 2 implies that lirmtT G,(t) = 1. The definition of t

then implies that G is strictly increasing over some interval (t', f). It then followsfrom Assumption A2 and Lemma 3.3 that limtT1 G,(t) < 1. This contradictionproves the lemma. Q.E.D.Define

t* _ sup {t > 0: Ga is strictly increasing on [0, t) for x = a, b}to be the beginning of the first interval during which one of the players moveswith probability 0. Combining Lemmata 1 and 4, we can show that, unless one ofthe players moves with probability 1 at time 0, neither player ever moves in theinterval (t*, 1).

LEMMA5. Suppose GQ(O) 1. Then(i) Gfl(t)= 1 - [1 - qf(O)]f(O, t)for 0 < t < t*, and(ii) Gf3(t)= G,3(t*)for t* < t < 1.

PROOF. Suppose that G(O) < 1. Then Lemma 3 and right-continuity of G.imply that G,(t)= 1 - [1 -qfl(O)]Ifl(O, t) for 0 < t < t*. Therefore, the lemma willbe proved if we can establish part (ii). Suppose q,(O) < 1 and t* < 1. Then Lemma4 implies that Gf(t*) < 1. Lett' = sup uv> 0: G#(v) = G,3(t*)I.

Since Gf(t*) < 1, it follows that t' < 1. We need to show that t' = 1.Suppose first that t' = t*. Then the definition of t* implies that, for some

t" > t', Gj(t") = G,(t*). Since G,(t*) < 1, it then follows from Lemma 1 thatGf(t") = G(t*), contradicting our assumption that t' = t*.Suppose next that t* < t' < 1. Then, since Lemma 4 implies that GQ(t') < 1, it

follows from Lemma 2 that G,(t*) = Gf(t') < 1. But then Lemma 1 implies that,for some 8 > 0, GQ(t*) = G,(t' + 3) < 1. Applying Lemma 1 again then yieldsGf(t*) = Gfl(t' + 3), contradicting the definition of t'. Q.E.D.

Lemma 5 implies that the support of the equilibrium strategies is composed ofat most an interval [O, t*] and {1}. Furthermore, any differences among theequilibrium strategies of player /3 must occur in the values of either q,(O) or t*.


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5. 1. Equilibritum Restrictio(ns on the Strategies at Time 0. Next we establishthe restrictions imposed on the equilibrium strategies at time 0. They are basedon the argument behind Lemma 2 which, at time 0, implies only that at most oneplayer can move with positive probability.

LEMMA 6. qJ(O)qb(o) = 0PROOF. Suppose that q,(O) > 0. Then, for any E> 0, there is an (arbitrarilysmall) 8 > 0 such that (i) Lp(O) Lp(6) < , and (ii) q3(6) = 0 with G(6) - G(O) 0

which implies that qp(0) = 0. Q.E.D.

5.2. Equiilibriu-mi}>estrictions on the Value of t*. In this section, we considerthe equilibrium restrictions implied by the value of t*. These are determined bycomparing the payoff to a player from moving at or before time t* with his payofffrom waiting until time 1. Given Assumption A3, there are two cases to consideras determined by the value of t*.

LEMMA 7. Suppose t* = 0.(i) qj(O) = 0 and qp(0) < 1 implies La(O)< Sj(l).(ii) qj(O)= I inmplies ither La(O)? S,(1) or LJ(O)? Fa(t)for some t E [O, 1).

PROOF. Suppose t* = 0.(i) If qj(0) = O, then it follows from Lemma 5 that qj(l) = 1. Therefore, GQsan optimal response only if0 < P,(l, Gp) limtio Pj(t, Gp)= q0(l)[S(l) - L(O)].

But if q:(O) < 1, then Lemma 5 implies that qp(l) = 1 - q0(0) > 0 which inturn implies that S,(1) > L(0).(ii) If La(O) Sj(l) and La(O) Fa(t) for all t E [0, 1). Then Assumption A2implies thatPJ(1, Gfl) P(0, G) = [F(O) - S(O)]qp(0)

+ [F,(t) - L(O)] dGfp(t) + [Sa() - L,(0)]q#(l) > 0.Consequently, q,(O) > 0 cannot be an optimal response. Q.E.D.


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THE WAR OF ATTRITION 677

The implications of Lemma 7 are summarized in Table 1. Typical elementsrepresent (qX(O),,(O)).The value of x is any number in the interval [O, 1].When t* > 0, there is an additional restriction to consider besides the tradeoffbetween moving at time t* and waiting until time 1. It must also be possible toconstruct a strategy which makes the other player indifferent between moving atany time near 0. This requires that the "integral" condition at time 0 be satisfied.Recall that I,(O,0) _ limto If(O,t).LEMMA 8. Sulppose 0 < t* < 1. TlheniIf(O, 0) = I and LX(t*)= Sj(1).PROOF. The requirement that I,(O, 0) = 1 follows from Lemma 5 and therequirement that G. be right-continuous. To establish that LX(t*)= 0, note thatLemmata 4 and 5 imply that qx(1)> 0. Therefore, G, is an optimal response onlyif

(5.5) 0 = PJ(1, G13) limtt P,(t*, G13) [SX(1) - L,(t*)]qfl(l)But since Lemmata 4 and 5 also imply that qf(i) > 0, it follows that LX(t*) 0.Q.E.D.

Recall that I,(1, 1) lim,t1 II(t, 1). If t* = 1, and I,(1, 1) = 0, then Lemma 5implies that player / moves with probability 1 before time 1. In this case, thevalue of Sx(1) is irrelevant. Consequently, when t*= 1, there are no additionalrestrictions at time t* unless 1f(1, 1) > 0 for some player ,B.LEMMA 9. Suppose t* = 1. Theni

(i) Ifl(O,0) = 1 7and(ii) I,(1, 1) > 0 imiipliesLX(1)? SX(1)acnd J(1, 1)[Lx(1) - Sx(1)] = 0.PROOF. The proof of part (i) follows again from Lemma 5 and the requirment

that GJbe right-continuous. To establish (ii), note that G, is an optimal responseonly if(5.6) 0 ? P (1, Gal) limt 1 Pj(t, Gal)= [S,(1) - L,(1)]q/3(l)Lemma 5 implies that q13(l)> 0 whenever I(1, 1) > 0. Therefore, (5.6) implies that

TABLE 1POSSIBLE INITIAL MASS POINTS WHEN t* = 0

Lb(O) > Sb(l) Lb(O) < Sb(l)L/(0) > S,(1) (1, 0) (1, 0)

(0, 1) (0, 1)b(0, I)b

LX,(O) Svl(1) (0, 1) (0, 0)(1, 0O) (1, 0).

If LJ(O)? F,(t) for some t E [O, 1).h If L,(O) ? F,(t) for some t E [0, 1).


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Lx(l) ? Sj(l). Furthermore, if Ix(l, 1) > 0, then it again follows from Lemma 5that qx(1)> 0, in which case equation (5.5) must be satisfied for t* = 1. Q.E.D.5.3. Proof of Theorems. Using the restrictions derived in Lemmata 1 to 9,we can now prove the theorems stated in Sections 3.1 and 3.2.PROOF OF THEOREM 1. The necessity of these conditions follow from Lemmata1 and 7. All that remains is to show that they are sufficient.

(i) If Lx(O) FX(t)for some t E [0, 1), then choose the strategies so thatqx(O) 1 and qx(t)= 1. Then, for any v E (0, 1],

P,(v,G) -

Pf(O,GQ) [Fa(O) - Sf(O)] > 0which implies that G. is an optimal response. And

[[Lx(j) -L(O)] < 0 forj < tPj(j, Gl) - PJ(0, Gf) = [Sx(t)- L(O)] < [Fx(t) - La(O)] < 0 for j = t

[Fx(t)- LJ()] < 0 forj > twhich implies that G. is an optimal response.If Lx(O) S,(1), then a similar argument establishes that q,(O)= 1 andqf(t) = 1 form a pair of best responses.(ii) If Lx(O) S.(1) and q,(1) = 1 for a = a, b, then, for any t < 1,

Pfl(1, G) - Pf(t, Gj) = Sa(1) - Lf(t) ? 0which implies that q,(1) = 1 is an optimal response. Q.E.D.

PROOF OF THEOREM2. The necessity of Condition (a) follows from Lemmata8(i) and 9(i). The necessity of Condition (b) follows from Parts (ii) of Lemmata 8and 9. The sufficiencyof these conditions then follows upon verifying that one ofthe strategy pairs defined in Theorem 3 are equilibria. Q.E.D.PROOF OF THEOREM3. The necessity of Condition (a) follows from Lemma 6.The necessity of Condition (b) follows from Parts (ii) of Lemmata 8 and 9. Thesufficiencyof these conditions then follow upon verifying that one of the strategypairs defined in Theorem 3 are equilibria. Q.E.D.

6. DISCRETEVERSUS CONTINUOUS TIMEHendricks and Wilson (1985a, 1985b) have studied the equilibrium propertiesof Wars of Attrition in discrete time and investigated the relation between the

equilibria of these games to the equilibria of the continuous time analogues. Inthis paper, we confine ourselves to some remarks about the differences betweenthe two formulations as they pertain to our characterizationtheorems.In both formulations, the same degenerate equilibrium outcomes obtain underroughly the same conditions, although as noted in Section 3.3, the number of


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THE WAR OF ATTRITION 679subgame perfect equilibria tends to be much larger when time is continuous.Moreover, when the horizon is infinite, the class of nondegenerate equilibria inthe continuous time model essentially coincides with the discrete time model. Inboth cases, there is an equilibrium corresponding to any initial condition inwhich only one of the players moves with positive probability at time 0.

When the horizon is finite, however, the set of nondegenerate equilibria maydiverge substantially. First, in discrete time, there is generally at most one nonde-generate equilibrium, corresponding to the continuous time equilibrium in whichneither player moves with positive probability at time 0. In contrast, when time iscontinuous, the existence of a single nondegenerate equilibrium implies the exis-tence of a continuum of nondegenerate equlibria."2

Second, there is always an equilibrium in discrete time whenever the terminalreturn lies below the return to leading at any time for both players. In contrast,there is no degenerate equilibrium in continuous time when the terminal returnlies strictly below the return to leading as time approaches the terminal date. Thisfailure of upper semi-continuity in the equilibrium correspondence is the result ofthe fact that the payoffs are not continuous in the norm topology on the space ofdistribution strategies.These differences in the set of equilibria are reflected in the rich local structureof the equilibria in discrete time games which is totally lacking in the continuoustime analogues. This local structure is particularly sensitive to the treatment ofties, which, in continuous time, occur with probability 0. We also note that thereis no analogue to the integral conditions at time 0 in the discrete time game.

State University of New York,U.S.A.Bell CommunicationsResearch, U.S.A.New York University,U.S.A.

REFERENCESBISHOP, . T. AND C. CANNINGS,"A Generalized War of Attrition," Journal of Theoretical Biology 70

(1978), 85-124., J. CANNINGS, AND J. MAYNARD SMITH, "The War of Attrition with Random Rewards,"JolurnaIlof Theooreticailiology 74 (1978), 377-388.

BLISS,C., AND B. NALEBUFF,"Dragonslaying and Ballroom Dancing: The Private Supply of a PublicGood," Journal of Public Economics25 (1984), 1-12.

DASGUPTA, P. AND E. MASKIN,"The Existence of Equilibrium in Discontinuous Games, 1: Theory,"Reviewof EconomicStudies,Vol. LIII, (1) 172 (1986), 1-26.

AND , "The Existence of Equilibrium in Discontinuous Games, 2: Applications,"Reviewof Econonmic tuidies,Vol LIII (1) 172 (1986), 27-41.

FUDENBERG, D., R. GILBERT,J. STIGLITZ,AND J. TIROLE,"Preemption, Leapfrogging, and Compe-tition in Patent Races,"European Economic Review 22 (1983), 3-31.

AND D. KREPS,"Reputation and Multiple Opponents", mimeo. (1985).AND J. TIROLE,"A Theory of Exit in Duopoly," Econometrica54 (4) (July 1986),943-960.

12This lack of lower semi-continuity in the equilibrium correspondence can be repaired if we use aconcept of s-equilibria in the discrete time games.


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680 KEN HENDRICKS, ANDREW WEISS, AND CHARLES WILSONGHEMAWAT, R. AND B. NALEBUFF, "Exit," The RanidJournialof Economics 16 (2) (1985), 184-194.HAMMERSTEIN, P. AND G. A. PARKER, "The Asymmetric War of Attrition," Journal of Theoretical

Biolovjy96 (1982), 647-682.HENDRICKS, K., "Reputation in the Adoption of New Technology", mimeo.

AND C. WILSON, "The War of Attrition in Discrete Time," C. V. Starr Working Paper, R. R.85-32 (1985).

AND , 'Discrete Versus Continuous Time in Games of Timing," SUNY-Stony BrookWorking Paper 281 (1985).

KORNHAUSER, L., A. RUBINSTEIN, AND C. WILSON, "Reputation and Patience in the 'War of At-trition',"C. V. Starr Discussion Paper, R. R. 86-3 1, New York University (1987).

KREPS, D. AND R. WILSON, "Reputation and Imperfect Information," Journal of Economic Theory 27(1982), 253-279.

AND R. WILSON "Sequential Equilibria,"Economietrica 0 (1982), 863-894.MAYNARD SMITH, J., "lThe Theory of Games and the Evolution of Animal Conflicts," Journal of

Theoretical Biology 47 (1974), 209-221.NALEBUFF, B. AND J. RILEY, "Asymmetric Equilibria in the War of Attrition," Journal of TheoreticalBiology (1985).

ORDOVER, J. A. AND A. RUBINSTEIN, "A Sequential Concession Game with Asymmetric Information,"QuarterlyJoturna1tlf Economo2ic's07 (November 1986).OSBORNE, M. J., "The Role of Risk Aversion in a Simple Bargaining Model," in A. E. Roth, ed.,

Gam-ie-Theoretic Models of Bargaining (Cambridge: Cambridge University Press, 1985).PITCHIK, C., "Equilibria of a Two-Person Non-Zerosum Noisy Game of Timing," International

Jorls-s1s7f Gamiie heory 10 (1982),207-221.REINCiANUM, J., "'On the Diffusion of New Technology: A Game-Theoretic Approach," Review of

Econonmic tludies153(1981), 395-406."Market Structure and the Diffusion of New Technology," Bell Journal of Economics 12 (2)(1981), 618-624.

RUDIN, W., Principlesof MathematicalAnalysis (New York: McGraw-Hill, 1964).SELTEN, R., "Reexaminationiof the Perfectness Concept for Equilibrium Points in Extensive Games,"Initerjiiationialourna-ll f GalmeTheory 4 (1974), 25-55.

, "A Note on Evolutionarily Stable Strategies in Asymmetric Animal Conflicts," Journal ofTheoretical Biology 84 (1980),93-101.WILSON, C. A., "Notes on Games of Timing with Incomplete Information,"mimeo (1983).

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