Imitation and price competition in a differentiated market

Transcription

1 Imitation and price competition in a differentiated market Abhimanyu Khan Ronald Peeters April 0, 207 Abstract We study the stable market outcome that evolves in a spatially differentiated market when price-competing firms choose actions by imitation of the most profitable firm. We compare and contrast the stable outcomes under two imitation procedures: one, where each firm immediately imitates the most profitable firm, and the other when a firm imitates another firm only if it is more profitable while being sufficiently similar (in context of the market segment it operates in) or sufficiently close. In either case, the unique symmetric pure strategy Nash equilibrium is always a stable outcome, and sometimes uniquely so. In addition, when imitation of the most profitable firm is immediate, and market differentiation is moderate, states with prices lower than the Nash equilibrium are also stable. When imitation of the most profitable firm is more gradual, and market differentiation is sufficiently low, states with prices above the Nash equilibrium are also stable. Thus, while competitive evolutionary pressure in this imitation based model does result in the standard profit-maximising Nash equilibrium always being stable, other outcomes may be stable as well. Interestingly, the states that are stable under gradual imitation give the firms a higher profit than the stable states under immediate imitation. JEL Classification: C72, C7, D2, D4, L, L. Keywords: immediate imitation; gradual imitation; differentiated markets; evolution Cambridge-INET. ak@cam.ac.uk Department of Economics, Maastricht University. r.peeters@maastrichtuniversity.nl

2 Introduction In this paper, we explore the stable outcomes of an evolutionary model of firm behaviour, where firms, primarily motivated by relative performance or relative profit rather than their absolute profit, imitate the most profitable firm. This primacy accorded to relative performance could be because firms may not be able to profit-maximise or best-respond due to lack of precise knowledge of the market structure (such as consumer preferences or the extent of market differentiation). In fact, these considerations led Alchian (50) to argue that market behaviour evolves through a dynamic process of imitation and occasional experimentation, rather than conscious profit-maximisation on the part of firms. On the other hand, Friedman (5) posits an as-if justification for more rational profit-maximising models of firm behaviour: even if firms do not consciously and objectively maximise profits, the market outcomes obtained because of competitive pressures would be as if firms rationally engaged in profit-maximising behaviour. One contribution of this paper is to formally construct an Alchian type of model to test Friedman s as-if justification. We find partial support for this view (as we elaborate on below) and emphasise on a subtler consideration: the speed/rate at which imitation occurs may have important consequences for stable market outcomes. For example, the stable market outcomes that arise when all firms immediately imitate the strategy of the most profitable firm may be different from when imitation is more gradual, in that in the latter, only some firms may imitate the most profitable firm at first, and thereafter, imitation proceeds if and only if the imitated strategy continues to be the most profitable strategy. Thus, the main question we focus on is the differential effect of immediate imitation versus more gradual imitation on stable market outcomes. We examine this question in the framework of a symmetric spatially differentiated oligopoly à la Salop (7). A finite number of price-setting firms, located equidistantly on a unit circle, compete for consumers who are uniformly distributed over the circumference of the circle. 2 Firms set prices and the profit obtained depends on the profile of prices chosen, following which they adjust/adapt by imitating the price of the most profitable firm. This imitative behaviour leads to a symmetric/monomorphic strategy profile or state (i.e. all firms choose the same price), and we use the notion of stochastic stability to determine the symmet- Adoption of industry best practices serve as an example of imitative behaviour. In the academic literature, Huck et al. (), Offerman et al. (2002) and Apesteguia et al. (2007), amongst others, find experimental evidence of imitative firm behaviour on provision of feedback on strategies and profits. Further, the first two papers report that while experimental subjects do follow a best-response process when they are provided the relevant information to do so, provision of additional information such as the profit of the competitors which should not have an effect on optimising behaviour actually pushes the observed outcome in the direction of the theoretical prediction under an imitation paradigm. 2 In this paper, we consider large finite oligopolies in the sense of the number or firms in the market (see Section 2), and indicate what happens with fewer number of firms in Footnote 6. 2

3 ric strategy profiles that describe stable market outcomes. Stability of a particular market outcome is determined by the ease of reaching it from other states on one hand, and the difficulty of reaching other states from it on the other hand. In a market state with a symmetric/monomorphic strategy profile, all firms receive the same profit, and so, only experimentation by firms with some other strategy may cause a transition to another market state. This happens when, for example, experimentation makes the experimenting firm(s) the most profitable, as a result of which it is imitated by other firms. Intuitively, a market state is stable if it can be reached relatively easily from other market states and relatively difficult to reach any other market state from it by means of experimentation. Before presenting our results, we first clarify the meaning of immediate and gradual imitation. In contrast to immediate imitation wherein all other firms instantly imitate the price of the most profitable firm, gradual imitation is more involved. Given the spatial nature of the market (described by the Salop circle), we assume that firms sufficiently close to the most profitable firm imitate it first; thereafter, imitation of the price spreads to other firms that are located farther away only if a firm sufficiently close to them is more profitable while using that particular price. This is similar in spirit to a diffusion process where a new innovation spreads from the innovating unit to other units which are closer to it, and then gradually to other units farther away. A possible explanation for gradual imitation is either that a firm is able to observe and hence imitate, or is inclined to imitate, only those firms that are located sufficiently close to it, i.e. the firms with whom it competes more intensely with. Thus, immediate imitation is similar to imitation under complete observability (all other firms are observed) while gradual imitation is similar to imitation under incomplete observability (only the firms in a neighbourhood can be observed and imitated). Hence, a critical feature in gradual imitation, which does not arise in immediate imitation, is that after a particular price has been imitated by some firms in the first wave of imitation, that price has to be profitable enough for the firms using it for it to be imitated by the other firms. We now contrast the stable market outcomes that obtain when imitation of the most profitable firm by other firms is immediate versus when it is gradual, and compare it to the unique symmetric pure Nash equilibrium market outcome (of the static game) in order to relate it to the arguments of Alchian and Friedman. In either case, the unique symmetric pure strategy Nash equilibrium is always a stable outcome, and sometimes uniquely so. In addition, when imitation of the most profitable firm is immediate, and market differentiation is moderate, a particular set of states with prices lower than the Nash equilibrium are also stable. However, when imitation of the most profitable firm is more gradual, a particular set of states with prices higher than the Nash equilibrium are also stable when market differentiation is not The notion of closeness or neighbourhood can be interpreted either in terms of physical distance or intensity of competition.

4 sufficiently high. Thus, in the spatial differentiation model we consider, gradual imitation is actually profit-improving for the firms, and market outcomes that may appear to be collusive, in that prices maybe higher than the Nash equilibrium, may actually be generated by this type of imitation. In addition, we see a partial justification for Friedman s as-if hypothesis partial because while the Nash equilibrium prediction is always a stable outcome, irrespective of imitation being immediate or gradual, other outcomes may also be stable depending on the level of market differentiation and on the type of imitation. In order to firstly understand the intuition behind the stability of the Nash equilibrium state, we recall that stability of a market state is determined by the ease of reaching it from other market states on the one hand, and the difficulty of reaching other market states from it on the other hand. The stability of the Nash equilibrium state in case of both immediate and gradual imitation is because we find that: (i) from any other symmetric price strategy profile, if a firm experiments with the Nash equilibrium price, it always (weakly) improves its own profit this makes reaching the Nash equilibrium relatively easy, and (ii) the Nash equilibrium is strict, meaning that from the Nash equilibrium state, if a firm experiments with any other price, its profit is strictly lower, and this makes reaching other market states from it via experimentations relatively difficult. In case of immediate imitation, the reason for the stability of other market states (with prices lower than the Nash equilibrium state) is as follows. In any symmetric strategy profile, all firms obtain the same profit. When the price in an initial symmetric state is higher than the Nash equilibrium, and a firm experiments with the Nash equilibrium price, it not only (weakly) increases its profit (see (i) above) but is also the most profitable firm (because the other firms have a higher price, they do not attract additional demand and so their profit does not increase). With immediate imitation, this leads to the Nash equilibrium state. A similar experimentation from symmetric strategy profiles with prices lower than the Nash equilibrium also results in an increase in the experimenting firm s profit (see (i) above), but the neighbouring firms profits may increase as well (because the latter gain consumers due to the higher price of the experimenting firm) to the extent that the experimenting firm may not be the most profitable firm. However, when market differentiation is sufficiently high, the experimenting firm retains enough consumers to still emerge as the most profitable firm. Similarly, when market differentiation is sufficiently low, we find that the Nash equilibrium price is competitive enough that the experimenting firm does not lose substantial market share even when the other firms have a lower price. Consequently, in both these cases, the experimenting firm is the most profitable and immediate imitation leads to the Nash equilibrium. In contrast, when market differentiation is neither sufficiently high nor sufficiently low (i.e. it is moderate), the two forces mentioned above are relatively muted, which makes reaching the Nash equilibrium from other symmetric states with lower prices more difficult, 4

5 resulting in some symmetric states with lower prices also being stable. When imitation is gradual, the main force at play behind stability of other states (with prices higher than the Nash equilibrium price) is best explained by an example. If a firm experiments with the Nash equilibrium price from another symmetric profile with a higher price, then as explained in the previous paragraph, the experimenting firm is the most profitable firm. Importantly, here, the experimenting firm competes with other firms with a higher price, and attracts consumers from both of the closest neighbouring firms. But once the experimenting firm is imitated only by a few of the neighbouring firms, in the resulting market state, due to the nature of gradual imitation, the firms setting the Nash equilibrium price compete with some firms with the same price and other firms with a higher price. As a result, the Nash equilibrium price is not as profitable as before (i.e. when only the experimenting firm had experimented it) as the firms using it cannot attract consumers from both of its closest neighbouring firms. This makes reaching lower price market states (such as the Nash equilibrium state) from market states with higher prices relatively more difficult. This is especially true if market differentiation is low. After all, the profitability of the initial experimenting firm was based on drawing away sufficient consumers from its neighbouring firm on either side. So, when only a few of the neighbouring firms imitate this experimenting firm, they are only able to draw consumers from one neighbouring firm (since the other neighbouring firm has the same price). Then, the increase in demand does not compensate for the fall in the price, and the process of reaching the market state with a lower price is stymied. An analogous reasoning suggests that this concern does not arise when we consider the process of reaching higher price market states (such as the Nash equilibrium state) from market states with lower prices. Hence, some symmetric states with prices higher than the Nash equilibrium are also stable when imitation is gradual and market differentiation is not high enough. In related literature, Vega-Redondo (7) and Tanaka () study imitation by quantitysetting firms in a homogeneous market, and show that the Walrasian equilibrium (which is not a Nash equilibrium) is the unique stable outcome. The intuition is that a firm may emerge as the most profitable firm, and be imitated, when it takes a spiteful action which, while reducing its own profit, reduces the profit of competing firms even more. This may drive the stable outcome away from Nash equilibrium market outcomes to other market outcomes such as the Walrasian equilibrium. In context of homogeneous markets, Alós-Ferrer, Ania and Schenk-Hoppé (2000) study imitation by price-competing firms with decreasing returns to scale and find that the long-run equilibrium supports a strict subset of the Nash equilibria. On the other hand, in the context of differentiated markets, Tanaka (2000, 200) shows that stochastically stable behaviour under price or quantity imitation coincides with the unique finite-population evolutionary stable strategy in symmetrically differentiated oligopolies 5

6 with linear demand. Thus, the Nash equilibrium is, in general, not supported in the long-run in non-spatial differentiated markets, while it always is stable in our paper. Hence, spatial differentiation makes a material difference to the market outcomes that are stable. In fact, the feature of market differentiation results in an interesting observation in relation to Apesteguia, Huck and Oechssler (2007) and Hedlund (205), who make the point that the spite effect of imitation, that is observed in homogeneous markets as in Vega-Redondo (7), is muted when firms may observe and imitate other firms in similar roles but in a different market. In our case, one may interpret immediate imitation as a situation where other similarly placed firms in a different market are imitated (in context of the above mentioned papers). Gradual imitation may then be interpreted as a limitation on the possibility of observing and imitating other similarly placed firms in a different market. In our paper, this leads to states with higher prices being stable, and it is the feature of market differentiation that drives this. Our paper is most closely related to Khan and Peeters (205) which analyses a similar evolutionary model of immediate imitation when firms choose both price and quantity simultaneously (in contrast to quantity being determined by demand as in this paper) to find that the Nash equilibrium state is the unique stochastically stable state. The reason for this difference is that it is more difficult to reach the Nash equilibrium state from other symmetric states with a lower price when firms choose price only (supplying the quantity demanded) rather than when they choose both price and quantity simultaneously. To elaborate, suppose that firms are in a symmetric state with prices lower than the Nash equilibrium. If a firm experiments with the Nash equilibrium price in a model with price and quantity choice, the higher price of the experimenting firm causes neighbouring firms to face excess demand on account of not being able to scale up production. As a result, these neighbouring firms receive the same profit as before the experimentation, whereas, if the firms choose quantity after observing the demand (as in the current paper), the neighbouring firms would experience a higher profit. Thus, the experimenting firm has to earn a higher profit when quantity is chosen after demand realisation (relative to when both price and quantity are chosen) for it to be imitated on grounds of being the most profitable firm, making reaching the Nash equilibrium state more difficult. Hence, perhaps counter-intuitively, the flexibility in production afforded to firms when they produce after observing demand is actually profit-deteriorating for them. The next section contains the basic model. Section presents the Nash equilibrium outcome and the long-run outcome under immediate and gradual imitation, and the intuition for these results. Formal proofs are presented in the appendix. Section 4 concludes. 6

7 2 Model Let n > 6 firms be located equidistantly on a (Salop) circle of unit circumference. 4,5 These firms choose price simultaneously and meet the demand that results from the price-profile by producing at constant marginal cost c. Consumers are distributed uniformly along the circumference. Each consumer purchases at most one unit of the good and makes the purchase decision after observing all prices. The gross utility received on purchase equals β. A linear transportation cost of per unit distance traveled is incurred. The net utility of a purchase is the gross utility (β) less the price set by the chosen firm and the relevant transportation cost. The reservation utility of abstinence is normalised to 0. A consumer maximises net utility and purchases only if the net utility from doing so is at least 0, in which case, the firm providing the highest (non-negative) net utility is chosen. 6 We follow Apesteguia and Selten (2005) and Khan and Peeters (205) in describing the demand faced by a firm. Let v [0, ) denote the circle coordinate and v = i n be the location of firm i. Let firm i choose price p i. The local price at a location v is p(v) = min {p i + δ(v, i)}, i=,...,n where δ(v, i) is the distance between location v and firm i: δ(v, i) = min{ v i n, v i n }. The demand from a particular location is zero if its local price is larger than β. If the local price is less than or equal to β, the demand from that location is one, and in case more than one firms offer the same local price, the firms share the demand equally in probabilistic terms. We define a marginal consumer to be a consumer who either (a) receives a maximum net utility exactly equal to 0, or (b) receives the same maximum non-negative utility from more than one firm, and so, is indifferent amongst these firms. Two or more firms are said to compete for the marginal consumer if they set prices such that there exists a consumer that receives the same maximum non-negative utility from the said firms. A firm is said to have a segregated market if it does not compete for the marginal consumer. In this paper we will particularly deal with situations of the type where firm i chooses price p i while all other firms choose the same price p. For this specific situation, and assuming all firms face positive demand, we use the above expression to specify the demand of firm i 4 While this assumption is guided by our motivation to study a differentiated market, it is noteworthy that Hehenkamp and Wambach (200) comment that equidistant location of firms on a Salop circle is the predicted long-run outcome in a two-stage evolutionary model where firms choose location by imitation in the first stage, and thereafter in the second stage, choose the Nash equilibrium price corresponding to the location profile. 5 We comment on smaller oligopolies in Footnote 6. 6 When the maximum net utility that a consumer receives on purchase is exactly equal to 0, we assume the good is purchased. If a consumer receives the maximum net utility (of at least zero) from more than one firm, each firm in contention has equal probability of being chosen. 7

8 as: d i = { 2 β p i if p i > 2β p n (p i p + n ) if p i 2β p n To explain the above, suppose all firms j i set the price equal to p β. When firm i sets the price at p i, the consumers that are indifferent between buying from firm i or one of its direct neighbours are located at distance ˆv n from firm i, where ˆv is such that p i + ˆv = p + ( n ˆv), or ˆv = p p i+ n. This is precisely the demand faced by firm i on either side in case these consumers receive a nonnegative utility (i.e., they are marginal consumers of type (b) referred to earlier), which occurs when β p i p p i+ n 0, or p i 2β p n. Otherwise, in case p i > 2β p n, the consumers farthest away from firm i that purchase from firm i are located at distance v for which β p i v = 0 (i.e., they are marginal consumers of type (a)), or v = β p i, which is then the demand on either side. Thus, firm i faces direct competition for the marginal consumers with its neighbouring firms when p i 2β p n, and faces a segregate market otherwise. Firm i receives a profit of π i = (p i c) d i. Results We analyse and contrast the stable outcome(s) under immediate and gradual imitation of the most profitable firm, and benchmark the stable outcome(s) in either case to the Nash equilibrium of the static game.. Nash equilibrium The following proposition presents the unique symmetric pure strategy Nash equilibrium of the static game (see Khan and Peeters (205) for the proof). Proposition. (i) Suppose > n (β c). In the unique symmetric pure strategy Nash equilibrium, all firms choose the monopoly price p m = β+c (ii) Suppose [ 2 n (β c), n (β c) ]. In the unique symmetric pure strategy Nash equilibrium, all firms choose the price p n = β 2n. (iii) Suppose < 2 n (β c). In the unique symmetric pure strategy Nash equilibrium, all firms choose the price p c = c + n. The thick line in Figure illustrates the Nash equilibrium price (p ) as a function of the market differentiation i.e. transportation cost parameter. observations that play an imporant role in the imitation process. 7 Observation. The Nash equilibrium described above is strict. 7 See Khan and Peeters (205) for details. 2. Here, we point out two related 8

9 p β (iii) (ii) (i) (e) (d) (c) (b) (a) () (2) () 2 β + c 2 β + 2 c 4 β + 5 c c 0 n(β c) 2 n(β c) n(β c) 2 n(β c) 6 7 n(β c) Figure : The thick line presents the Nash equilibrium price as a function of the market differentiation parameter. This line together with the light gray (dark gray) area represent the prices of the monomorphic states that form the stochastically stable set as a function of the transportation costs when imitation is immediate (more gradual). The labels on top refer to the different cases in Propositions, 2 and 4. Observation 2. The Nash equilibrium price is a weak better-reply from all other symmetric strategy profiles; i.e., if a firm deviates or experiments with the Nash equilibrium price from any strategy profile where all firms choose an identical price, its profit never decreases..2 Immediate imitation of the most profitable firm In the imitation framework, firms choose price simultaneously from a finite grid of prices, and without loss of generality, we restrict attention to prices being at least equal to marginal cost. 8 The state, defined by the profile of prices (p i ) i=,...,n, determines the demand faced and the profit received by each firm. The price and profit of each firm are observed by all other firms, and the strategy of the most profitable firm is imitated by all other firms. When there is more than one imitable action (for example, when multiple firms are in the most profitable set but with the use of disparate strategies), we assume there is an equal probability of each of the imitable actions being chosen. This describes the unperturbed process of imitation. States where all firms choose an identical price are called monomorphic. A state (minimal set of states) is said to be an absorbing state (set) when there is no possibility of transiting from it via imitation. The next observation immediately follows. 8 The finite grid is assumed to be fine enough so as to not exclude prices of any significance. A sufficient condition on the price grid is to have the prices in the set P = {c, c +, c + 2, β+c, β, c +, 2β (c + n n 2 2n n ), 2β+c, 2β+c n, 2β 2β+c n, β} contained in it. n n n

10 Observation. Each monomorphic state is an absorbing state, and the imitation process causes a transition from any non-monomorphic state to a monomorphic state. The unperturbed imitation dynamic is now augmented with the possibility of firms independently experimenting with their choice of price with probability ε. As the above observation states, without experimentation, the unperturbed imitation process would converge to (and would stay locked into) an absorbing state that depends on the initial profile of prices. Now, experimentation makes the transition across absorbing states of the unperturbed process possible. For example, starting from a monomorphic state, if a firm experiments with a particular price and on account of this turns out to be the most profitable firm, the other firms imitate its strategy, thereby causing a transition from the initial monomorphic state. The resulting experimentation-augmented imitation dynamic (or the perturbed process) has no absorbing states. 0 Stochastic stability is determined by the relative ease of transiting into and transiting out of absorbing states. Absorbing states that are comparatively easier to reach from the other absorbing states (i.e. needs fewer experimentations) and more difficult to move out (i.e. needs more experimentations) belong to the stochastically stable set. The next proposition presents the stochastically stable set, Ω, when imitation of the most profitable firm is immediate. Proposition 2. (a) Suppose > n (β c). In the stochastically stable state, all firms set the monopoly price p m = β+c 2. (b) Suppose ( 6 7 n (β c), n (β c) ]. In the stochastically stable state, all firms set the price p n = β 2n. (c) Suppose [ 2 n (β c), 6 7 n (β c) ]. The stochastically stable set consists of the monomorphic states where all firms set the same price p [c + n, pn ], where p n = β 2n. (d) Suppose [ 2 n (β c), 2 n (β c)]. The stochastically stable set consists of the monomorphic states where all firms set the same price p [ 2β+c n, p c ], where p c = c + n. (e) Suppose < 2 n (β c). In the stochastically stable state, all firms set the price pc = c+ n. A comparison of the stochastically stable set, Ω, with the Nash equilibrium in Proposition (also see Figure ) reveals that: () when the level of differentiation is sufficiently high (i.e. > 6 7 n (β c) or regions (a) and (b) in Figure ), Ω uniquely contains the Nash equilibrium state, (2) for moderate levels of differentiation (i.e. [ 2 n (β c), 6 7 n (β c)] or regions (c) and (d) in Figure ), Ω contains the Nash equilibrium state, along with a set of other monomorphic states with lower price, and () for sufficiently low levels of differentiation (i.e. < 2 n (β c) or region (e) in Figure ), Ω uniquely contains the Nash equilibrium state. The transition from an absorbing state of the unperturbed process may require more than one firm to experiment. 0 Henceforth, by absorbing states, we mean the absorbing states of the unperturbed process. For more details, see Young () or Kandori et al. (). 0

11 We use the following two lemmata to convey the intuition behind the result stated in the proposition. Let p represent the monomorphic state with price p, where each firm receives a profit of π( p). The profit of a firm which chooses price p when its two neighbouring firms choose p 2 and p is denoted by π(p, p 2, p ). We anchor the discussion on the long-run stability of the Nash equilibrium. Lemma. Consider a monomorphic state p, and suppose that a firm experiments with the Nash equilibrium price p. Then: (i) if p > p, the experimenting firm is (one of) the most profitable firm(s); imitation of p leads to the Nash equilibrium state p. (ii) if p < p, one experimentation is needed for the transition to p when 6 7 n (β c) and two experimentations when < 6 7 n (β c). Proof sketch: From the initial monomorphic state p, let a firm experiment with p. It receives π(p, p, p). The non-experimenting firms obtain either π(p, p, p) or π(p, p, p ). By Observation 2, π(p, p, p) π(p, p, p). So, if π(p, p, p) π(p, p, p ), imitation leads to p. When p > p, π(p, p, p) π(p, p, p ) holds, and so, π(p, p, p) π(p, p, p ) as well; imitation of the experimenting firm leads to p. 2 However, when p < p, the higher priced experimenting firm loses consumers to the lower priced neighbouring firms. Now, π(p, p, p) π(p, p, p ) holds, and π(p, p, p) < π(p, p, p ) may be possible, in which case the experimenting firm is not the most profitable firm (and is not imitated). The proof in the appendix shows that the level of market differentiation plays a pivotal role in this calculus. When market differentiation is sufficiently high (i.e. 6 7 n (β c)), the market power held by firms results in lower prices of the neighbouring firms not drawing away a substantial mass of consumers from the experimenting firm; here, the single firm experimentation with the Nash equilibrium price makes the experimenting firm the most profitable i.e. π(p, p, p) > π(p, p, p ) holds. However, for differentiation lower than that (i.e. 6 7 n (β c)), π(p, p, p) < π(p, p, p ) holds; here, the experimenting firm is not the most profitable firm (and is not imitated) and it now takes two experimentations to transit to p (details in the appendix). Lemma 2. It is not possible to exit the Nash equilibrium state when only one firm experiments. In particular, it takes two experimentations to transit from the Nash equilibrium state when [ 2 n (β c), 6 7 n (β c) ] and more than two experimentations when < 2 n (β c). Proof sketch: Suppose that the initial state is p, and a firm experiments with p. As the Nash equilibrium is strict (recall Observation ), π(p, p, p ) < π( p ). Now, importantly, there 2 A similar result holds when p > 2β p. Here, the experimenting firm does not compete for the marginal n consumer with its neighbouring firms, and hence, does not affect their profit (i.e. π(p, p, p) = π(p, p, p )). So, a single experimentation also leads to p in this case as well. This merits separate mention only when 2β p n < p ; the complementary case is subsumed under p > p.

12 always exists one non-experimenting firm which receives π( p ). Since π(p, p, p ) < π( p ), the experimenting firm is not the most profitable firm and is not imitated; thus, irrespective of the level of differentiation, it is not possible to move out of p with one experimentation. It follows that a transition from the Nash equilibrium state requires at least two experimenting firms. In the appendix, we show that when market differentiation is moderate, it is indeed possible to exit the Nash equilibrium state when two firms experiment, and involves a transition to a monomorphic state with lower price. Importantly, when market differentiation is below a certain threshold (i.e. 2 n (β c)), the Nash equilibrium price pc (equal to c+ n ) decreases as market differentiation decreases. For sufficiently low market differentiation, it becomes competitive enough to withstand the experimentations that, in case of moderate differentiation, lead to monomorphic states with a lower price; here, a transition from the Nash equilibrium requires more than two experimenting firms (details in the appendix). It can be gathered from the above that: () for sufficiently high differentiation (i.e. > 6 7 n (β c)), it takes a single experimentation to transit into the Nash equilibrium state, but more than one experimentation to move out of it, making it the unique stochastically stable state; (2) for moderate levels of differentiation (i.e. [ 2 n (β c), 6 7 n (β c) ]), two experimentations are needed to cause both a transition into the Nash equilibrium state 4, and a transition from this state to other monomorphic states with lower prices; this suggests why a set comprising of the Nash equilibrium and other monomorphic states with lower prices is stochastically stable; and () for low-enough differentiation (i.e. < 2 n (β c)), it takes at most two experimentations to transit into the Nash equilibrium state, but more than two experimentations to move out of it, making it the unique stochastically stable state. 5 We conclude this sub-section by commenting on the non-monotonicity of the stochastically stable set (Ω ) when market differentiation is moderate (i.e. [ 2 n (β c), 6 7 n (β c)]). Firstly, consider [ 2 n (β c), 6 7 n (β c)]; here, as decreases, p (equal to β 2n ) increases, and the set of prices of the monomorphic states that form Ω is larger and includes higher prices (also see Figure ). The reason for this is that the least-resistant transition Suppose a firm experiments with the lowest feasible price of marginal cost from the state p. Because β p β c, the experimenting firm can at most attract the consumers up to the location of the n neighbouring firm. Hence, the non-neighbouring firms (the existence of which is established by the assumption on the number of firms) are not affected by the experimentation. 4 Recall Lemma to note that two experimentations are needed to transit into the Nash equilibrium state from monomorphic states with lower price; one experimentation is sufficient for a transition into the Nash equilibrium state from monomorphic states with higher price. 5 Apesteguia and Selten (2005) conduct an experimental study of price competition on the Salop circle, with the parameters corresponding to what we call sufficiently low differentiation. They find that behaviour differs from the Nash equilibrium. We suggest this difference may be due to two factors: (i) there are at most five firms in the experiment, while we assume the presence of at least seven firms; (ii) each firm imitates only its closest neighbouring rather than the most profitable firm in the entire population. 2

13 from the Nash equilibrium state involves a transition to monomorphic states with a lower price (say, p ), and this is made possible by one of the two experimenting firms experimenting with this lower price p and also being the most profitable firm. Now, this experimenting firm draws even more consumers from its non-experimenting neighbours as decreases (since this causes p to increase, and because lower market differentiation makes consumers more willing to travel to a firm with a lower price). Thus, not only does the experimenting firm continue to be the most profitable firm as decreases, but it would also be the most profitable firm if it were to experiment with another price that is suitably higher than p, implying a similar transition from p to another monomorphic state with price higher than p. Hence, as decreases, the most profitable experimenting firm may experiment with higher prices and remain the most profitable firm. This causes the set of prices of the monomorphic states that form Ω to be larger and to include higher prices. Secondly, the reason for the set of prices of the monomorphic states that form Ω when [ 2 n (β c), 2 n (β c)] being smaller and including lower prices is similar, once we take into consideration that p decreases as decreases. 6. Gradual imitation of the most profitable firm In the previous subsection, we assumed that the most profitable firm is immediately imitated by all other firms. However, in general, it may be likely that the most profitable firm is imitated, at first, by only those firms than are more proximate to it, and that compete with it more intensely in the sense of offering products that are relatively more substitutable compared to the products of the other firms. Consequently, imitation may not unfold in the same way as with immediate imitation. For example, an experimentation which is successful (and hence imitated by all other firms) under immediate imitation, is now only imitated by a subset of firms. Following this initial wave of imitation, it is possible that no other firm observes the price experimented with to yield the highest profit, resulting in it not being adopted by the remaining firms. This motivates us to ask how stable outcomes do compare under gradual and immediate imitation. 6 We have have assumed the number of firms n > 6. To see what happens if n 6, we first note that the transition into the Nash equilibrium state require the same number of experimentations as before. Recall the explanation of Lemma to note that the transitions from higher price states do not depend on the number of firms; for transitions from lower price states, the crucial criterion is that experimenting firm has to be more profitable than its neighbours, irrespective of the number of firms. So, in this respect, decreasing n has no effect. However, transitions out of the Nash equilibrium become relatively easier as a non-experimenting firm that is unaffected by experimentations may not exist when n 6 (recall explanation of Lemma 2). For example, if n {2, } (n {4, 5, 6}), then the Nash equilibrium state is relatively easily destabilised by spiteful experimentation by a firm (by two firms). Thus, when n 6, there would be an expansion in the scope of spiteful experimentations, resulting in an increased appearance of monomorphic states with prices lower than the Nash equilibrium in the stochastically stable set.

14 For this purpose, we first define an imitable set for each firm: this is the set of other firms that a firm may imitate. The most profitable firm in the imitable set of a particular firm is the imitable firm if the former is more profitable than the latter; this is the firm that it imitates. We begin by considering a situation of almost immediate imitation defined as a situation when for each firm, there exists at most one corresponding firm it does not imitate even when the latter is the most profitable firm in the entire market. An example of this is when a firm never imitates a firm that it is most distant to, possibly because the firm considers the actions of the most distant firm as not very relevant for the particular market segment it serves. Thus, under almost immediate imitation, the imitable set of each firm contains at least n 2 firms (we exclude a firm from its own imitable set). The next proposition shows that in this case, the stable outcome(s) is identical to the stable outcome(s) under immediate imitation. Proposition. Under almost immediate imitation, the stable market states are the stable states under immediate imitation (described in Proposition 2). Proof: First, consider transitions into Ω. The type of experimentation in Proposition 2 that was successful in causing a transition into Ω from a state outside it is now imitated by at least n 2 other firms. In the following state, all firms but one choose a price that is supported in Ω. This state is equivalent to a situation where a single firm experiments from a state p Ω. One experimentation is not sufficient to effect a transition from any state in Ω under immediate imitation; so, it must be that this lone firm that posts a different price is not the most profitable firm. It follows that this firm observes at least one other firm in the imitable set being more profitable with the use of the price supported in Ω, and imitates it. Thus, the number of experimentations needed to move into Ω remains unchanged. Secondly, consider the transition out of Ω. Take any experimentation type that is unsuccessful in causing a transition from a state in Ω under immediate imitation, implying that there exists a non-experimenting firm that is the most profitable firm. Now, under almost immediate imitation, this non-experimenting firm is imitated by at least n 2 other firms. Thus, at most one firm imitates the experimenting price as the most profitable firm may not be in the former s imitable set, and it may instead imitate an experimenting firm. Again, this resulting state is akin to a situation where a single firm experiments while all other firms choose a price supported in Ω. As it takes at least two experimentations to move out of the states in Ω, this single firm is not the most profitable. In addition, its imitable set contains a non-experimenting firm that is more profitable and so, this causes a reversion to the initial state in Ω. It follows that the number of experimentations needed to transit from Ω does not decrease. In conclusion, in comparison to immediate imitation, the number of experimentations to 4

15 transit into Ω does not change, and the number of experimentations to transit out of Ω does not decrease. This establishes the proposition. We now examine a more pronounced case of gradual imitation. Gradual Imitation Assumption. max{2, n 4 4 Each firm s imitable set contains at least the closest firms on either side. 7 } firms on either side and at most the closest n 2 We allow for firms to have different numbers of firms in its imitable set subject to the thresholds above. First, we address the issue of the composition of the absorbing sets of the unperturbed imitation process with gradual imitation. Under immediate imitation, all non-monomorphic states were transient and led to a monomorphic state. Now, with gradual imitation, the price of the most profitable firm in the market may only be imitated by a subset of the firms. Following this, it is no longer obvious that the remaining firms will subsequently imitate this price as well. In fact, in the new state where some firms have imitated the most profitable firm s price, it is possible that a firm with a different price emerges as the most profitable firm. This raises the possibility of absorbing sets being formed by non-monomorphic states. However, Lemma (proof in appendix) states that even if such absorbing sets (comprising of non-monomorphic states) exist, they are very fragile, and one experimentation suffices to effect a transition from such an absorbing set. Furthermore, a series of such single firm experimentations leads from any absorbing set comprised of non-monomorphic states to a monomorphic state. We later also show that there exists a set of monomorphic states that are more resilient to experimentations and so, absorbing sets consisting of non-monomorphic states never appear in the stochastically stable set. Lemma. Under the gradual imitation assumption, a series of single experimentations causes a transition from an absorbing set comprising of non-monomorphic states to a monomorphic state. We first present the proposition characterising the stochastically stable set (proof in appendix), and explain the result with the lemmata that follow. Proposition 4. Under the gradual imitation assumption, () Suppose > 2 n (β c). In the stochastically stable state, all firms set the Nash equilibrium price p. (2) Suppose [ n (β c), 2 n (β c) ]. The stochastically stable set consists of the monomorphic states where all firms set the same price p [p, 2 β + c]. 7 x ( x ) represents the lowest integer greater than or equal (greatest integer lower than or equal) to x. The gradual imitation implies that each firm does not imitate at least one firm on each side. 5

16 () Suppose < n (β c). The stochastically stable set consists of the monomorphic states where all firms set the same price p [p, c + n ]. The above proposition (also see Figure ) shows firstly, that the unique symmetric pure strategy Nash equilibrium state is always supported as a long-run outcome. Secondly, the prices in the states that form the support of the stochastically stable set are higher (or less competitive) when imitation is gradual. The following lemmata, which contrast the transition possibilities with the immediate imitation case, provide more insight into the result. Lemma 4. If a single-firm experimentation does not induce a transition from a monomorphic state under immediate imitation, then it cannot induce a transition under gradual imitation either. Proof: From the state p, let a firm experiment with p. Now, the experimenting firm obtains π(p, p, p) while the non-experimenting firms obtain π(p, p, p) or π(p, p, p). As the experimentation is unsuccessful under immediate imitation, π(p, p, p) < max{π(p, p, p), π(p, p, p)}. Under gradual imitation, each firm s imitable set has at least two firms on each side. Now, any firm (including the experimenting firm) that has a firm receiving π(p, p, p) in its imitable set also a firm receiving π(p, p, p) and π(p, p, p) in its imitable set. Hence, the experimenting firm is never imitated by any firm; the experimenting firm itself also chooses p by imitation. The lemma follows. For the next two lemmata, we use Figure 2 as a visual aid: in the figure, there are ten firms, and in case of each firm, the imitable set contains only the two closest firms on each side. The experimenting firm(s) are depicted in grey, the firms that imitate to a different price are in black and the firms which persist with the price of the initial monomorphic state are in white. Lemma 5. If a single-firm experimentation with a higher price is successful in inducing a transition from a monomorphic state under immediate imitation, then it also induces the same transition under gradual imitation. Proof: Consider the state p, and let a firm experiment with p > p. Suppose this experimentation is successful under immediate imitation, and without loss of generality, that p Ω and p Ω. The experimenting firm receives π(p, p, p) and the non-experimenting firms receive π(p, p, p) or π(p, p, p). Since the experimentation is successful under immediate imitation, π(p, p, p) > max{π(p, p, p), π(p, p, p)}. Now, with gradual imitation, the experimenting firm, is imitated by a strict subset of the firms. After the first wave of imitation, firms that choose p obtain either π(p, p, p ) (firms, 2 and 0) or π(p, p, p) (firms and ). The other firms obtain either π(p, p, p) 6

17 imitation imitation (p < p) 2 0 imitation (p > p) imitation 4 8 imitation Figure 2: Figure supporting Lemma 5 and 6. (firms 4 and 8) or π(p, p, p) (firms 5, 6 and 7). Then, a firm with price p that either has another firm with profit π(p, p, p ) in its imitable set, or itself receives π(p, p, p) (such as firms 4 and 8), must have firms that receive a profit of π(p, p, p), π(p, p, p) and π(p, p, p) in its imitable set as well. Similarly, a firm with price p that either has another firm with profit π(p, p, p) in its imitable set, or itself receives π(p, p, p) (such as firms and ), must have other firms receiving a profit of π(p, p, p), π(p, p, p) and π(p, p, p ) in its imitable set as well. Since p > p, π(p, p, p) < π(p, p, p ). The firms that still choose price p receive a profit of π(p, p, p) or π(p, p, p), and some of these firms (such as firms 4 and 8) have another firm with profit π(p, p, p) in its imitable set. As π(p, p, p) max{π(p, p, p), π(p, p, p)}, these firms imitate the price p. On the other hand, the firms with price p are never led to imitate p. By repeated iteration of this argument, this single experimentation results in a transition to p. Lemma 6. Suppose that a transition from p to p, where p < p, is possible under immediate imitation when a single firm experiments with p. Then, the same experimentation by a firm with p from the state p leads to the same transition to p under gradual imitation: (i) if and only if π(p, p, p) π(p, p, p) or (ii) if the experimenting firm does not compete for the marginal consumer with its neighbours. Proof: (i) Consider the initial state p, and let a firm experiments with p < p. Without loss of generality suppose p Ω and p Ω. The experimenting firm receives π(p, p, p) and the non-experimenting firms receive π(p, p, p) or π(p, p, p). As the experimentation is successful 7

18 under immediate imitation, π(p, p, p) > max{π(p, p, p), π(p, p, p)}. However, under gradual imitation, p is imitated by a subset of the firms (such as firms 2,, and 0). The firms with price p receive either π(p, p, p) or π(p, p, p ). Since p < p, we have π(p, p, p) π(p, p, p ). The firms that still choose price p receive a profit of either π(p, p, p) or π(p, p, p) with π(p, p, p) < π(p, p, p). Other firms (such as firm 4) imitate p if and only if π(p, p, p) π(p, p, p); this is the only circumstance under which a firm with price p is imitable. Thus if this holds, then, by iteration of the same reasoning, the single firm experimentation is successful with gradual imitation. On the other hand, if π(p, p, p) < π(p, p, p), the firms with price p (starting first with firms and, followed in the next round by the firms and 2) revert to price p. (ii) Now, suppose that the experimenting firm does not compete with the neighbouring firms. 8 Then, all firms with price p are equally profitable, i.e. π(p, p, p) = π(p, p, p ), and similarly, all firms with price p are equally profitable, i.e. π(p, p, p) = π(p, p, p). Hence the condition π(p, p, p) π(p, p, p) in (i) above is equivalent to π(p, p, p) max{π(p, p, p), π(p, p, p)}; this last inequality holds because the single firm experimentation is successful with immediate imitation. So, p is imitated repeatedly till it is imitated by all firms. The above lemmata indicate that (in comparison to immediate imitation) transitions from p to p (where p < p ) is possible with the same number of experimentations (Lemma 5), but the reverse transition is relatively more difficult (Lemma 6). As a result, the monomorphic states where the firms price is below the Nash equilibrium are not stable now that imitation is gradual. At the same time, the increased resistance of a transition from a particular set of monomorphic states where the firms prices are higher than the Nash equilibrium price results in such states being stable, but only when market differentiation is below a threshold. Footnote suggests the reason behind similar monomorphic states with price higher than the Nash equilibrium not being stable when market differentiation is sufficiently high. addition, for low market differentiation ( 2 n (β c)), the set of monomorphic states that forms the stochastically stable set contracts as decreases, states with higher prices are progressively excluded from the stable set. The reason for this is related to the fact that the Nash equilibrium price decreases as decreases; so, as decreases, it becomes easier (more difficult) to transit into the Nash equilibrium states from the monomorphic states with higher prices (from the Nash equilibrium states to monomorphic states with higher prices). 8 While the lemma only supposes that the experimenting firm does not compete for the marginal consumers with its neighbouring firms, since the non-experimenting firm post a higher price, the same holds for them; i.e., the non-experimenting firms do not compete amongst themselves for the marginal consumer either. Importantly, this condition is more likely to be satisfied when market differentiation is sufficiently high. For sufficiently high market differentiation, all firms have a segregated market in the state p. So, if the initial monomorphic state has a price even higher than p, and a firm experiments with p, all firms continue to have a segregated market, fulfilling condition (ii). In 8