Signaling in an infinitely repeated Cournot game with output restrictions

International Journal of Industrial Organization 9 (1991) 365-388. North-Holland

R. David Simpson* Resources for the Future, Washington, DC 20036, USA Final version received October 1990

When the space of actions is restricted, the conclusions of standard signaling models are softened. I model an infinitely repeated Cournot game with asymmetric costs and non-negativity constraints on prices. In equilibrium a price war of finite maximum duration occurs. A firm of the low-cost type always behaves aggressively, whi!e a firm of the high-cost type randomizes between doing so and leaving the market. An important theme is that of the equivalence between signaling models under such assumptions and reputation models of the type proposed by Kreps and Wilson (1982b) and Milgrom and Roberts (1982b).

1. Introduction Signaling models have been used by industrial economists to explain industrial entry and exit. Frominent contributions have been made by Milgrom and Roberts (1982a), Roberts (1986), and Saloner (1987). Perhaps their most successful application have been to counter the hypotheses associated with McGee (1958, 1980) and others that predation and limit pricing can never be rational. In these signaling models a firm is uncertain of a rival’s cost of production (or, completely analogously in Roberts’ model, industry demand). A firm sends a signal of its costs by choosing a quantity of production. Applying the refinements of subgame perfection [Selten (1965)], sequentiality [Kreps and Wilson (1982a)],’ and Cho and #reps’s ‘intuitive criterion’ (1987), it is generally possible to identify a unique equilibrium in which firms are separated by their actions. In other words, the strategy chosen by each firm serves to uniquely identify its type. Such models yield very strong, and arguably too strong, Uncertainty is resolved in a single period of play. If a winner e always a firm of the more efficient type. Conversely, ~~e~cie~t firms can ore e never succeed in passing themselves off as being of t *I thank seminar participants at the Economic Analysis Group of the Antitrust Division, U.S. Department of Justice, and an anonymous referee for helpful commenis on an earlier draft. uilibria also be actually use the somewhat weaker refinement that candi ayesian equilibria.

366

R.D. Sinr~s’on,Sig,x!ing in a Cournot game

The reason for these overly strong results is that existing signaling models are generally presented as two-period games.’ Solutions to such games do not always generalize to multiple periods. In the model I present in this paper, extending the number of periods and restricting the space of feasible actions may have a profound effect on resultant equilibria. The separating property (separation in pure - and hence identifiable - strategies) is lost, and there is no guarantee that high-cost types will ever be revealed as such. While it can always be argued that every repeated game is a two-period game if the periods are defined as ‘the first period’ and ‘all subsequent periods’, the logic of the separating equilibrium rests on the supposition that a firm of a relatively advantaged type can always take an action sufficiently drastic as to identify itself in the first period. I claim that this asumption is frequently unrealistic, and will show that relaxing the assumption leads to the type of results I have described above. A ‘period’ is that amount of time to which a firm can commit itself in making a decision concerning the strategic variable. If the s,trategic variable is the quantity of production, consider the eonstraints which may bind a firm which would like to initiate a price war of sufficient duration and intensity to oust its rival(s). Stockholders, who may be unaware of the firm’s strategic plans, may become upset and wish to replace the management.3 Antitrust authorities may take a dim view of firms which take extreme measures to eliminate Gompetition. Capacity constraints may limit price wars. Inasmuch as firms are unlikely to be able to pay their customers to take the product, non-negativity of prices may be a bintling constraint. While I use this last ”1ticIin d~e!o$r\-g constraint on the non-negativity of prices as a modeling L&r ~~~~~~~ my results, it should be recognized that any of the other factors cited above would lead to similar outcomes. The difficulties such constraints impose would be obviated if a would-be signaler could precommit to multiple periods of aggressive conduct. By definition, however, a period is that length of time beyond which a firm cannot commit itself to a course of action. Suppose that a firm, call it firm 1, did announce that it would behave aggressively for more than one period, and that this strategy were chosen in equilibrium. If a rival, firm 2, of a disadvantaged type observes in the first period that firm 1 has taken an action characteristic of an &nntaged type, firm 2 will exit rather than face further losses. If this tran.;pires, however, firm 1 would have no incentive to continue to behave aggressively. If the mere threat of aggressive behavior ?Saloner’s (1987) paper has three periods in the sense of ‘times at which actions take place’, but the second period involves only the announcement of a bid, not actual production. jA similar point is made in Gertner, Gibbons and Scharfstein (1988): if a firm signals simultaneously to both its rivals in the product market and its suppliers in the capital market it may mitigate the intensity of its messages to one group in order to avoid creating unfavorable impressions with the other.

RD. Simpson, Signaling in a Cournot game

Were

Sufficient to

induce the rival’s

exit,

!kms

which

367

are

,n,ot

of

the

advantaged type would masquerade as such. Qne might envisage contracts wnich call for a signaler to maintain . . . . aggressive actions only if it taces continuing competition, b-&t such qrpe_ ments do not seem plausible. They would create opportunities for rentseeking behavior by would-be rivals, they would be difficult to enforce, and they might arouse antitrust concerns. In this paper I present a simple model of an infinitely repeated Cournot game with uncertainty concerning costs. Each firm knows its own marginal cost, but is uncertain of its rival’s. Each has the same fixed cost, and the magnitude of the costs is sufficient to prevent two firms of a high-cost type from profitably serving the market simultaneously. I demonstrate the existence of an equilibrium in which a price war continues for a certain maximum duration. Beliefs concerning the rival’s cost type are updated after every period. Low-cost firms always behave aggressively, while high-cost firms randomize between behaving aggre”,sively and leaving the market. The longer the price war co ntinues, the more likely a firm believes it is that its rival is in fact of the low-cost type. When one (or both) firm(s) leaves the market, the price war comes to an end. If neither has left before the last cf the possible price-war periods, the problem facing the firms reduces to a simple signaling situation, and uncertainty is resolved in a separating equilibrium. A firm which does not stick it out to the last possible price war period will never know whether its rival is in faci of tlic low-cost type, or if it has been duped. While market re-entry is always possible, it will only occur if both firms are of the high-cost type and both exit simultaneously. If both firms are of the high-cost type and one exits before the other, the survivor will engage in a type of limit nricing forever afterwards in order to avoid the L revelation of its type. The equilibrium I derive is not separating in the sense that observable actions uniquely identify types. That is, the level of output observed (or, more precisely, the level of output inferred from a firm’s knowledge of its own output and the market price) does not identify the cost type of the producer. The equilibrium is separating in the sense that different types pursue different mixed strategies: a high-cost type of firm randomizes over its output choices, a low-cost type does not. The randomizing device is presumed to be unobservable, however (if one firm could observe that its rival rolled dice before choosing from among its pure strategies, it would then know that the rival is of the type wl;;ich randomizes - i.e., of the cost type). The reader familiar with the literature will e ‘reputation’ models of i-reps and model presented here and (1982b) and Milgrom and results for a game with two-si

368

R.D. Simpson, Signaling in a Cournot game

Wilson in their paper. The extensive-form representations of the game modeled here and in those papers is quite similar. This essential similarity between signaling and reputation games has been noted by others [see, for example, Tirole (1988, pp. 376-377)]. The contribution of this paper is intended to be a clarification of the relationshipbetween the signaling and reputation models. In short, when restrictions on feasible actions prevent the existence of a separating equilibrium, a firm will seek to acquire a reputation for being a low-cost producer [analogous to a ‘tough’ rival in Kreps and Wilson (1982b)]. As in the KrepsWilson and Milgrom-Roberts papers, when it becomes common knowledge that one firm is of the high-cost type (and the other’s type remains uncertain), the firm whose type remains unknown to its rival will inherit the market. Also, as in Kreps and Wilson (1982b), the greater the duration of the period of fighting for dominance, the higher the probability assigned to the event that the rival is in fact of the low-cost type. The paragraph above emphasizes that signaling models become reputation modeis when restrictions are piaced on the space of feasible actions. It should also be noted that reputation models (with two-sided uncertainty) become signaling models if the fighting, or, in my model, price war, continues long enough. If both players continue aggressive actions long enough, there comes a point when both hold beliefs concerning the other’s type such that only a firm that is truly of the low-cost type stays in the market. Even when this Sinai period is reached, however, a iow-cost lirm imay hi1 have to produce a supracompetitive output in order to distinguish itself. The apparent difference between the modei deveiopd here and the simpler representations of Kreps-Wilson and Milgrom-Roberts - that firms can choose outputs from a continuum, rather than simply alloting payoffs depending upon choices of discrete actions - does not make as great a substantative difference as it might. In order to keep derivations tractable. I have restricted attention to equilibria in which output levels are constant during the price war period. I am, however, able to provide comparisons in outcomes on the basis of alternative price-war-output levels by computing numerical examples. It should be noted that my results hinge on two special features of the model. I restrict attention to only symmetric equilibria; one firm cannot colnmit itself to a course of actions which would motivate an asymmetric response from another firm with exactly the same observable characteristics and history of actions. I[ also assume that the equilibria of the full information games which are played after uncertainty is resolved are known and immutable. The effect of this assumption is to transform the infinitely reepated game into a finite one, as there will come a period by which uncertaint) must be resolve and actions determined for the rest of time. The ty of equilibria I derive are fra at they rest on mixing of

RD. Simpson, Signaling in a Cournot game

369

strategies by high-cost types. Such a result strains the logic of mixed strategy eqi~ilibria. Mxed strategy equilibria in static games can be motivated by the argument that players who do not randomize will be victimized by rivals who anticipate their actions. In many repeated games, however, players would be better off for being inflexible; if one can make a believable threat of endurance no matter what, it is at an advantage. A high-cost type of firm would like to deviate if it would be believed (and, because it’s playing a mixed strategy, it is indifferent between deviating and not deviating). It can be argued that this model places extreme informational and computational burdens on the firms. In addition to the difficulties inherent in randomizing across mixed strategies, the expansion of the space of feasible actions in this game results in the possibility of a continuum of equilibria. This is in contrast to refinements of the equilibria of most two-period signaling games, in which firms choose between only two actions in the first period, one denoting an agressive stance, the other a passive best response. I have restricted attention to strategies in which the aggressive action is the same over all periods of the game’s ‘price war’ phase. A more thorough treatment would model the selection of the level of this aggressive action as well. I have found this problem to be intractable. I can, however, make some conjectures concerning the nature of a solution. There will be some choice of aggressive action which will maximize the expected value of the game to a firm which is in fact of the low-cost type. This seems the most reasonable choice of equilibrium. I identify (approximately! such a strategy a:liong the set of constant-level-of-aggressive-action strategies in a numerical t,.::ample at the end of the paper. The construction of an argument for the uniqueness of this equilibrium is problematic, however. If there exists an equilibrium which is not optimal for a firm of the low-cost type, a firm might deviate to a strategy which, if accepted as an equilibrium, would be optimal for such a firm. Once again, however, since it is impossible to signal type in one period, this deviation could not be defended by a statement to the effect that it was taken because the deviator is of the low-cost type. This model provides some inkling of the complexity which may be involved in real-world decision making. I have restricted attention to a single extension of the two-player, two-type signaling model: the addition of additional periods (and restrictions on the action space which makes this extension meaningful). The addition of a continuum of types, a greater number of players, and uncertainty about other parameters would make the model completely unwieldy. This begs the question of whether a ode1 so difficult as to be unsolvable by a well-trained economist as any relevance to real-world decision making. I suspect that it does. The usefulness of the model the empirical plausibility of its

370

R.D. Simpson,Signaling in a Couraot game

advantages by taking aggressive actions, firms which do not in fact enjoy such advantages try to give the appearance that they do, price wars of different intensities and durations occur, firms which achieve positions of market dominance may not in fact possess absolute advantages in efficiency, and the longer aggressive behavior continues, whether masquerade or not, the less optimistic become rivals who observe it. The model is described in the next section of the paper. The case of onesided uncertainty is discussed in section 3. Section 4 begins with a brief overview of the construction of the equilibrium and then is divided into four parts. The first demonstrates that if it is common knowledge that one firm is of the high-cost type while the other’s type is not common knowledge, the uninformed firm will drop out [this is essentially the result of Kreps and Wilson (1982b) and Milgrom and Roberts (1982b)]. The probability with which a firm of the high-cost type participates in the game with two-sided 9 t. In the third part the maximum uncertainty is derived in the second p-r duration of the price war is derived. The fourth part is devoted to the derivation of the va!ue of the game to a firm of the low-cost type. Numerical examples are presented in a table and discussed in section 5. Section 6 concludes.

Two firms play an infinitely repeated Cournot game against one another. Price in each period is given by the linear inverse demand function P=max(A-q,-q,,O),

_a

(1)

where qi is firm i’s output choice. The restriction of the non-negativity of price will be important in the sequel. The firms are assumed to be unable to observe each other’s choice of output directly. Hence any further increases in outgut ;vhen q, +q2 > A will result in no further drop in price, and, therefore, no further transmission of information. The firms can produce any level of output at constant marginal cost. arginal cost can take on one of two values. I can assume without any great loss of generality that the lower of the two values is zero. The greater of the possible values will be denoted by c. Each firm faces a fixed cost, F. Thus the net one-period payoff to a firm is zi=max {(A -qt

-q2

where cI is either zero or c, or high-cost type, respectively.

- Ci)qi, - Ciqi) - F,

i = 1,2,

(2)

on whether the firm is of the low-

RD. Simpson, Signaling in a Cournot game

371

Firms seek to maximize the expected discounted present value of all future profits. The value of a dollar of profit one period in the future is denoted by 6; this discount factor is assumed to be constant over time. The expectation of net discounted profits is bounded below by zero; a firm always has the option of not participating and, by not participating, incurring neither fixed nor variable costs. In order to focus attention on issues relating to exclusion and exit, I will assume that no collusive behavior, either tacit or explicit, ever occurs. If a firm is going to produce at all in any period it must incur the fixed cost of F. When the firm makes the decision of whether or not to participate it also decides what quantity to produce. Both firms decide at the same time whether to participate in the market and how much to produce. Neither decision can be reversed in the period in which it is made. Firms cannot directly observe the quantity which the rival has chosen. Information regarding quantities comes only through prices. It is worth emphasizing again that no additional information can be inferred by the production of quantities in excess of those which drive price to zero. I shall assume that if it becomes common knowledge that both firms are of the low-cost type both firms will choose their one-shot Cournot-Nash equilibrium strategies in all subsequent periods. I assume that both high-cost firms cannot profitably participate in the same market period if their types are common knowledge, and that a high-cost firm cannot survive if it is common knowledge that it is of ihe high-cost type and its rival is of the lowcost type. It is easily shown that the one-period Cournot-Nash equilibrium outputs of a high- or a low-cost firm are if both are of the low-cost type and this fact is common knowledge; (A -c)/3 if each is of the high-cost type and this fact is common knowledge; (A +c)/3 for the low-cost type and (A - 2c)/3 for the high-cost type if one is of the low-cost type, the other isI of the high-cost type, and this fact is common knowledge. 43

The. assumptions concerning the sustainability of Cournot outputs can be summarized as

A high-cost firm would like to become it must masquerade as a monopolist of t had the monopoly in this market it would pr implies a price of A/2; hence and receives this price will ear

equilibrium

R.D. Simpson, Signaling in a Cow-not game

372

(A* - 2Ac)/4.

A su,fficient condition for (4) to exceed the fixed cost given the assumptions embodied in (3) is that A 2 18/5c. I will model the full-information game between two firms of the high-cost type as a war of attrition with a mixed-strategy equilibrium.4 One could, as is always the case in such situations, envisage scenarios in which one firm claims that it will always participate and produce its monopoly output. In such a case the other, if it believes the threat, would not produce. The credibility of such a threat is doubtful, however, and inasmuch as I am imposing symmetry of actions as a requirement for a plausible equilibrium, such a strategy certainly cannot constitute a symmetric equilibrium. Instead, I will suppose that the full-information game is a variant on the war-of-attrition model. In such a case each firm will randomize between participation and nonparticipation. Let the probability with which the firms decide to participate be cc. Two conditions must be satisfied for a sensible mixed-strategy equilibrium to the ftt!!-information game. First, each firm must be indifferent between participating and not participating, given its assumptions of the rival’s probability of participating; i.e., the expected payoff to participation must equal zero, the payoff to nonparticipation. Second, each firm must play a best response given its assumptions concerning the rival’s probability of participating. Let qw(H) be the quantity chosen by a high-cost type of firm in this game. As I am only concerned with the high-cost type of firm here, I will drop the argument H. The first condition above may be stated as (5) The second condition requires that each firm’s choice of qw is a best response to the other’s choice of qw, i.e., q”=argmax 4

[p(A-c-qW-q)q+(l-p)(A--c-q)q].

(6)

It is a well-known artifact of the linear-demands, constant-costs Cournot game that expected variable profits are the square of the best-response output. Thus it is immediate that qW=,/F

and

p=(A--c)/JF-2.

(71

@The game I discuss here differs slightly from what is usually meant by a war of attrition, as I admit the possibility of t-e-entry. Rather than coin a more complex phrase to describe this game, I will simply call it a war of attrition.

R.D. Simpson, Signaling in a Cow-not game

373

Note that, for (A-c)~/~~FzZ(A-C)~/~, 05~5 1; firms will randomize for any value of the fixed cost at least as great as Cournot profits but not greater than monopoly profits. 3. Conditions for a s

arating equilibriu

The types of results derived for the two-period case analyzed by Milgrom and Roberts (1982a), Roberts (1986), and Saloner (1987) may be valid if the discount factor is relatively low, the length of time commitment is relatively long, the feasible action space is relatively large, or (all of these conditions are in effect equivalent) the probability with which the rival is believed to be of the low-cost type is relatively large. In this section I derive conditions sufficient for the existence of a separating equilibrium in one period. This result is an important building block for constructing the equilibrium when separation cannot occur in a single period. On the one hand, when the conditions derived in this section do not obtain, the analysis conducted in section 4 of the multi-stage price war becomes relevant. On the other hand, when the conditions derived in this section are satisfied, that price war comes to an end. I denote by 8 the probability with which one firm believes its rival to be of the low-cost type. In this section I derive an expression relating 8 to the invariant parameters of the model: A, c, F, and 6, the present value of a dollar of profits to be received one period in the future. I find that there is a critical value of 0 such that for all higher values separating signaling can occur in one period. Heuristically, when it appears very likely that the rival is in fact of the low-cost type, it is very risky for a high-cost type to masquerade as a low-cost type. Doing so affords it only a very small probability of attaining monopoly while exposing it to a very high risk that its rival will responc1 in kind, causing it further losses. When 8 is relatively large, then, it may f le possible for a low-cost firm to take an action in one period which would not be profitable for a high-cost firm, even if by doing so the high-cost firm were perceived to be of the low-cost type. In the equilibrium I construct to the overall game in the following section, the high-cost type will always randomize during price wars. This being the case, the probability with which a firm has participated is perceived as being of the low-cost type is always rising as it continues to participate. Eventually, then - unless the Ii!-ocess is terminated by the exit of one (or both) of the firms - @ reaches otae critical value, the analysis conducted in this section becomes relevant, a ,:onvincing signal is, sent, and the price war e Suppose that a separating equilibrium exists in the infinit Cournot game as described above in which a firm makes a choice of whether i-kt or not to participate in the first period, ch ose t and as a result of its choices, uniquely i

374

R.D. Simpson, Signaling in a Cow-not game

tities at or in excess of some level qL as chosen that qn~ntity is of t L will result in a beli If this is to be a separatin type of firm must choose to produce qL (or more), and a ust decide to produce less thn qL. 1 turn out that low-cost firms will, in general, n wish to deviate 7 of choosing qL in favor of earning hi er profits in the ficing monopoly profits later on.5 Consider the incentives the high-cost type. If it takes the action expected of it in of a firm which is will become known after the first period. Thus, in any bsequent period(s) of the game the best it can hope for is that the other m will be revealed to be of the high-cost type, in which case both will earn . On the other hand, if the other firm reveals itself to be e, a firm of the high-cost type will prefer not to participate period, and hence, again, the value of the game will be zero. A h“+-cost firm will have no incentive to participate at all in the first period. hat would have to be true if the high-cost type of firm were to be willing viate from the equilibrium strategy of not participa.ting at all in the first riod? Suppose it were to participate and produce q’ (clearly, it would not choose to participate and produce any other quantity, since if it did so, it would lose money in the present and reveal its type for the future). With robability I-8 its rival would not participate, in which case the firm could earn immediate profits of (A -c-qL)qL - F followed by profits of (it must duplicate the low-cost firm’s monopoly output osing its own most preferred level, or else it will reveal its in subsequent periods) in perpetuity. On the other hand, with probability 8 the rival is in fact of the low-cost type, in which case. since both L, the firm would earn one-period profits (actually, net losses) -cqL) -F in the first period, and woLtld never choose Thus a high-cost type of firm would choose to riod if

aq=/aa =

if

12, or

Q

The condition on (4) above assures tha numerator of (%i;a) positive. Solving ’ I find that it i adratic with roots sy about (A - c)/p( ittPe sense as a so! s the lesser mot this incentive compatibility constraint (the high-cost firm should have to produce more, not less, output if it wishes to masquerade as a low-cost ty the denominator of (lOa) should also be positive. Expression (1 positive: the numerator is the same as that Zn (lOa) while the den positive under the assumption qLz A/2. In either case, t en, aqL/asis positive. ) if 1 Since the greatest verifiable value qL can take on in a replace qL by A in (9) I find that the (9) is satis~e

or equivalently, that

RD. Simpson, Signaling in a Cournot game

e situation, I will assume that if a se

erive an equi~ib~~

ill occur for a maximum of T + 1 ill be a quantity, which I denote

the same in all Ihe

Tth I\f bIa ,,GA,-AA plav~ ~1

tbm n.xmm 8~ ;n U~LIII~U rlnGnm4 &II- B~L”I~

tention to only those riods in which the price war

tn LV ka us+ tbc. wcl fi33!

’ which the i3XkX! if” high-cost type of firm is willing to choose a quantity qF with positive probability (if it has part.;.;m$ated in all previo-us periods). In every period following one in which a quantity of qF has been overserved 0, the hich a firm is believed to be of the low-cost type is ng 8 to denote its value in the indicated period, if the price war continues for the full T periods, Or, I 2 8* and separating signaling will occur in the T + first period. Either both firms will exit for the T+first a zero-expected-profit war of attrition will follow, both will T + first period and the full-information Cournot-Nash ~q~i~ib~urn will ensue, or one will stay and the other leave, in which case t successor inherits a onopoly. be following strategies describe the equilibrium:

1. \S a firm ever fails to participate or to produce a T, the separating equilibrium

amity of at least qF antity # implicitly

period T -I-1, and the if only it produced in

r will produce A/2

i

following steps: 1. Suppose that it is co high-cost type. If the : its rival% costs, regardless of how low the probability assigned by the uuiufo~~d the event that the informed firm is of the low-cost type, tbe unin firm will always exit. 2. In the case of two-sided symmetric u probabihty, which I denote by /?,6 wi other will choose to participate and probability is tbat w equilibrium involves mixed strategies, t the high-cost type of firm indifferent between maintaining its chance of high cost type) and exitin is constant. 3. From the probability derived in ste the signaling game, T, and the probabilit firm will participate and choose yI. These results are derived by probability assessment that t i osition of the condition uncertainty is resolve

R.D. Simpson, Signaling in a Cournot game

378

one firm, however small is the measure of that uncertainty,

This result is important

will lead

to

the

in deriving other results, since it assures that

this fact is common knowledge. It is reasonable to suppose that in such a on the firm whose cost type is common knowledge will choose its oneSt-response output in every period: because its type is common edge, it has nothing to gain by d.nGrg zrrything else. he fact that one of A_. firms is of the high-cost type is common ere is any doubt as to the cost type of the other firm, there is a perfect Bayesian equilibrium in which the informed firm produces the monopoly output and the uninformed firm drops out of the market. The following strategies and beliefs describe this equilibrium: 1. Both types of the informed firm produce quantity A/2. 2. So long as the informed firm has produced a quantity A/2 or higher (equivalently, the price observed is A/2 or lower) in every previous period, the uninformed firm incurs no fixed cost and produces no output; it never participates. 3. If the informed firm produces a quantity A/2 or higher (the uninformed of A/2 or lower), the uninfo

ed

is of the Iow-cost type as zero and the war of attrition

monopoly

profits, an

The remaining possibility is that q firm 2 cannot distinguish firm I’s realization of ql. Let 8, be the probability that firm 1 is in fact of the low-cost t on farm I’s choice of output in per

that fi

event that its rival is of the low-cost type. Thus, b the case that 0,26,_ 1 if the quantity chosen in perio a high- and a low-cost type could play. If e, is always increasing if firm 1 plays a strategy consistent wit type, there are two possibilities: either it eventually reaches unity, i nverges to s case firm 2 would wish to drop out, or abominator unity. The latter instance requires that t ayes’ Rule converges to unity, however, i.e., that both ty to play the lo -cost type’s strategy. If this is the case, prefer to exit than to contest the market. To summarize, there are two pos in ihe iimit, implying that firm 2 qf, and hence that firm 2 will prefer not to updated, and there will come a time after q4 in the past, it will choose to play qf again

R.5. Simpson, Signaling in a Cow-not game?

ction has argued that a high-cost

type

of

firm

w

eriods after not

lue to it of participating

is equal a0 the value to it of not

lue of the game to a high-cost type of firm in any period in

/AnnntnA h,, R hnllrwl “, pt -rvn, \urrlvrbu -

caF)

-

F.

2(H)

l-S_,Q h;nh_cr\ot t,,mc GI t Ft ul “, L1111611-bvJc LJdvb L1llll. ,e, ;i !#a, =max

= (A -

c-

qF)qF -

{(A-c-2qF)qF,

F, and n”( If) = (A - 2Ac)/4 - F. Let j?, be

the probability with which the firm expects its rival to participate a quan?ity qF in period 1. Then I can write

and choose

V’(H) = &cF(H)i- ( 1 - /!?,)[7?(W)+ (thy H)/( 1 - S))]

+sg,v+l(H).

(16)

Hn words, the value to a high-cost type of firm of choosing to participate in the game and producing a quantity qF in period a consists of three co ONXltS. rst term on the right-bard side of the equal sign is the ex ted pay eriod t if the rival chooses to produce qF in thal period, we~gb~~~.~ by the probability of this event. The second term is the discounted esen% siellue of monopoly profits which will accrue in all subsequent periods ~,(1,,:rival choose not to participate in period t, plus the one-time

From which I can derive

Since n?(H) is negative by assumption, /?, is necessarily less than one. Note that fi depends on the time period, t, only if qf (and by i nS(Z-I) and #(Z-I)) does; if the same conduct is followed in each which the market is contested, j3 will be a constant. I e interests of tractability I will restrict attention to cases in which qF does not vary over time. Thus I will drop the subscript on /I in the sequel.

4.3.

Ikrivarion

of T

In deriving the value of /? above I[ mentioned that in any one in which both firms have chosen to participate there are for a firm of the high-cost type. The first is that it will choose to ran over its choice of participation or nonparticipation in the su reveale The second is that information which has been to it in the previous period will dissuade it In every period in of its rival’s t rm is uncertai

RD. Simpson, Signaling in a Cournot game

382

nfiniterepetitions

of the

ot Coumot

game.

ok

call for a constant t in every period. the low-cost type in riod t. Then, since /I is the unc e quantity qF will be choke, it must be true that (23) ayes’ ruk relates the posterior probability tv~e _ _ to the prior and yl: 8t + 1

Eli

-

-=

&AR + Cl - w;~,l.

that the rival is of the low-cost

(24)

sting yr from (23) and (24) yields

6+

1 =

at

0a+

II

w.

(25)

have denoted the initial value of 0 by 0,. Thus by iteration on

1=4/B”.

(26) tion of the constancy of 8, it will be true in any period for

t

can

t

This expression may be solved for T in ter as

More precisely, T is the smallest integer satisfying this condition. These results provide the elements interest. The probability with whit monopolist in the market can be most CQ the probability with which it fir probability that the first period is just 1 -yl. It will exit in t chooses to exit in the second not exit in the first in the first period, high-cost firm exit minus the sum of with which a high-cost typ

RD. Simpson, Signaling in a Cournot game

38’4

ecause

T

is also

a

functio of /!I and

81, it

is

derivation of this value is straightforward, although once again it is difficult to amalyze its dete inants. C&C ca9 thiah of deriving the value of the game to a low-cost type of firm robability fl the rival fights and the firm receives O>-F, which I will denote by xF(L). If 8, is still P, the game goes on to t e next period. If 8, is not less than 8*, and separating sig~a~li~g 11OCCLK. In this period a IOW-CGSt type of f&m ufiyvrr* GavrurPtrU ‘1 FU,“,L R9x,n\ffnf ,R~I\-~~~ !rA_(z-fl \“~-I#-& n\ _ F, \?&CrC “1 #I. \Y, - l‘PI.bA ,L,’ “T+lIY JY yVI the symbols are agp.in as defined above. If neither rival has backed down in the first T+ 1 periods, they arc: sure of each other’s type. Thus each expects payoffs of nD(L)= AZ/3 - F in perpetuity. On the other hand, if the other firm dcies not participate (an event which robability 1 --/I), the surviving firm will earn payoffs of hich I will denote by xS(L)) in the period in which the other firm first absent, followed by payoffs of A2/4-F (which I will denote by discounted present value of the game to a low-cost type

T-l

t-1

This expression can be rearra fences of in~~ite sums and si

Expression (35) is somew a number of numerical exa

I lacking in elegance, but will es in the next section.

While I have attempted to eep this model as complex to proceed much further analytically. _ _.^.._.. numerical examples to illustrate my findings, h-WGVGI.

demand intercept,

e calculated for

RD. Simpson, Signaling

386

Tabie A nwnkwical Parameters: A= 10

in a Cournol game

1

example (see text for variable definitions). C=2

F=8

8, =0.2

6 = 0.9

q+ = 2.828

p = 0.828

0* = 5.556

6’ = 0.833

6’

B

T

4

qL

VL

3.33 3.67 4.00 4.33 4.67 5.00 5.33 5.67 6.0-O 6.33 6.67 7.00 7.33 7.67 8.00 8.33 8.67 9.00 9.33 9.67 10.00

0.95? 0.927 0.899 0.867 0.832 0.795 0.788 0.779 0.770 0.760 0.750 0.738 0.725 0.712 0.696 0.679 0.660 0.639 0.615 0.587 0.556

21 i4 IO 8 6 5 5 5 4 4 4 4 4 4 ? 3 3 3 3 2 2

0.371 0.366 0.358 0.35 I 0.338 0.327 0.326 0.324 0.314 0.313 0.311 0.308 0.364 0.300 0.289 0.284 0.278 0.269 0.258 0.244 0.23 1

9.940 9.800 9.728 9.165 9.495 9.156 8.612 7.997 9.849 9.536 9.083 8.430 7.415 5.626 9.604 9.008 8.025 6.111 4.393 9.748 8.862

82.115 94.460 104.90 111.708 119.055 123.225 124.696 126.101 131.800 132.863 133.852 134.760 136.288 139.276 141.410 141.736 141.945 144.888 145.679 147.086 146.500

Values of variables:

sling quantity, and the expected net present value of the game to type of firm, all for the value of qF listed in the first column. pi the example I have chosen, I compare the properties of equilibria for

euristically, the low-cost

ty

higher values of 6~~ma is co~~~rrned in the

attains a monopoly. Finally, note the cycling in the one-period signaling quantity. general, be higher or lower than qF. It may so increase or depending on the value of qF. This is because T t es on only inte Sometimes the end of the game is reached when uncertainty is treme signals, sometimes w comparisons across values of 4” for qL deciines as $ increases. If qF increases without increasing sufficiently shorten the duration of the maximum price war, the probability cost type of firm has hung on falls. The comparative statics of one&tit signaling are straightforward [see expression (9) above]: when 8 is lower, less extreme signals are called for.

I have presented and devel

d a model which is

contributions in that it expands the number of peri the space of possible actions. These changes resu changes in the nature of equilibria. The separating possible that firms of the high-cost type attain mon perfect Bayesian equilibria may survive refineme

B. Simpson. Signaling in 0 Cournot game

38

~edictions

tamty

might enhance the realism of tlC3 ty

s. 1987. Signaling

of the

of model, it seems unlikely that

games and stable equilibria,

Quarterly

Journal

nd Robert Wilson. 1982a, Sequenrial equilibrium. Econometrica 50, 863-894. Ison, 11982b. Reputation and imperfect information, Journal s signalhng aynard

Smith, John, 1974, The theory of games and the evolution atory

price cutting:

The Standard

of animal

of

to the

conflicts, Journal

Oil (NJ) case, Journal

of Law and

atory pricing revisited. Journ *I of Law and Economics 27.289-330. a, Limit pricing and entry under incomplete information, tion. repnation

s, and andlung wissenschaft

incomplete

and entry deterrence,

information,

eines ~~igo~o~rnod~~~s 12. 301-324. F Press, ~~b~dge).

and

Journal

Journal

of

of