Dynamics of Cournot and Bertrand Firms: Exploring Imitation and Replicator Processes

This study delves into the evolution of Cournot and Bertrand firms within the framework of imitation and replicator dynamics, encompassing diverse scenarios. In each time period, firms are randomly paired with either Bertrand or Cournot counterparts for a duopoly game. Subsequently, the populations of these firms evolve in accordance with either imitation or replicator dynamics. Within this context, the potential outcomes include the establishment of globally asymptotically stable limits for replicator dynamics, such as the coexistence of both-type firms, as well as the dominance of either all-Cournot or all-Bertrand firms. However, imitation dynamics tend to yield only the latter two equilibria, excluding the possibility of both-type coexistence. In the specific case of linear demand and cost, the stable limits of replicator dynamics hinge on factors like the uniformity of product differentiation levels among Cournot and Bertrand firms, as well as the nature of substitute or complement goods. Notably, these considerations do not exert the same influence on the long-term equilibria of imitation dynamics. Moreover, the outcomes reveal that while all evolutionary stable strategies of duopoly games featuring differentiated goods fail to serve as stable limits within our replicator dynamics, some of these strategies do attain stability when homogeneous goods are produced. Lastly, the application of our models to the analysis of firms producing goods of varying quality sheds light on intriguing disparities compared to outcomes obtained from static models.

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1. As posited within the realm of evolutionary games literature, this tournament mechanism serves to substantiate the utilization of expectations derived from a large number of players.

2. As detailed in Sect. 5, we will demonstrate that our results hold their qualitative integrity even when replicator dynamics are substituted with replicator-mutation dynamics.

3. Our findings hold valid even when the population sizes of firms vary over time.

4. There exist several real-world examples that reflect similar dynamics. For instance, Sato [26] highlights the strategies adopted by companies in Japan, such as Matsushita’s focus on sales expansion, Sanyo’s preference for undercutting and Sony’s emphasis on quality products. Consumers often perceive differentiation among the products offered by these three companies. Furthermore, the repercussions of price adjustments made by Sanyo on Sony’s customer base may vary from the impact of Sony’s price changes on Sanyo’s clientele. This discrepancy can be attributed to the divergent competing strategies pursued by these companies, which in turn shape distinct consumer perceptions and potential asymmetric brand loyalties. Comparable scenarios can also be observed within the cell phone markets.

5. In this context, \(\pi ^{CC} \) accounts for both firms’ equilibrium profits, regardless of whether homogeneous goods are being produced or not. Novshek [21] presents the expression for \( \pi ^{CC} \) within a framework involving homogeneous goods, subject to certain regularity conditions. These conditions encompass factors such as continuous inverse demand, inverse demand that is twice continuously differentiable and strictly decreasing within the relevant region, one firm’s marginal revenue decreasing as the aggregate output of other firms rises, and the cost functions of firms being lower-semi-continuous. Okuguchi [22] delves into the conditions required for the existence of Cournot equilibria in scenarios involving the production of differentiated goods. These prerequisites encompass characteristics like twice differentiable and strictly concave profit functions, as well as twice differentiable cost functions of firms. An illustrative instance of symmetric Cournot equilibria can be observed in the work of Singh and Vives [27] within the context of linear demand and cost.

6. It is widely recognized fact that \( \pi ^{BB}=0\) when identical duopoly firms engage in the production of homogeneous goods. In the case of differentiated goods, Okuguchi [22] establishes the conditions necessary for the existence of Bertrand equilibria.

7. Tremblay and Tremblay [33] determine the values of \(\pi ^{CB}_{C} \) and \(\pi ^{CB}_{B} \) in scenarios characterized by linear demand, linear cost and differentiated goods. Similarly, Chen and Liu [7] establish the existence of \(\pi ^{CB}_{C} \) and \(\pi ^{CB}_{B} \) within settings featuring linear demand, linear cost and homogeneous goods.

8. These ratios can also be interpreted as the probabilities of encountering both a Cournot and a Bertrand firm.

9. We have \( Q_{ \epsilon }(\textbf{s}, \textbf{u}) \approx \text{ constant } \cdot \epsilon ^{ U(\textbf{s}, \textbf{u})} \text{ for } \textbf{s}, \textbf{u} \in S\), where \( U(\textbf{s}, \textbf{u}) = d(r(\textbf{s}), \; \textbf{u}) = |\{ i \in \{1, \; 2, \ldots , n \}: r_{i}(\textbf{s}) \ne u_{i}\}| \) counts the total number of firm i revising its boundedly rational choice \( r_{i}(\textbf{s}).\) Additionally, \( U(\textbf{s}, \; \textbf{u}) \) can be interpreted as the cost associated with jumping from \(\textbf{s}\) to \(\textbf{u}\).

10. If \( b(t) =0, \) it must follow that \( c(t) \ne 0.\) In this case, we can apply the same method to analyze the limit of \( \frac{b(t+1)}{c(t+1)} \).

11. This scenario can occur when the products of both-type firms have varying levels of differentiation, as demonstrated in Sect. 4.1.

12. In the case where firms produce homogeneous goods and engage in price competition, we will observe \( \pi ^{BB} =0 \). Similarly, when firms produce homogeneous goods and participate in the Cournot–Bertrand competition, we will find \(\pi ^{CB}_{C} \) and \( \pi ^{CB}_{B} \), as elaborated in Sect. 4.2.

13. If \( \pi ^{CC} > \pi ^{CB}_{B} \) and \(\pi ^{BB} > \pi ^{CB}_{C},\) then the stable limit of (14) assumes the form \( c(t) =0, \; b(0) x^{*} \) or 1 depending on whether \( c(0) < b(0) x^{*}, \; c(0) = b(0) x^{*} \) or \( c(0) > b(0) x^{*},\) respectively. Conversely, the stable limit of (14) is \( c(t) = b(0) x^{*} \) if \( \pi ^{CC} < \pi ^{CB}_{B} \) and \(\pi ^{BB} < \pi ^{CB}_{C}; \; c(t)= 1 \) if \( \pi ^{CC} > \pi ^{CB}_{B} \) and \(\pi ^{BB} < \pi ^{CB}_{C}; \) and \( c(t)= 0 \) if \( \pi ^{CC} < \pi ^{CB}_{B} \) and \(\pi ^{BB} > \pi ^{CB}_{C}.\)

14. It is a well-known fact that discrete-time and continuous-time replicator dynamics can lead to different limits, as illustrated by examples presented in [13].

15. For instance, if the stationary point corresponds to an all-Cournot firm scenario, then over time, each firm and its competitor will tend to adopt the Cournot equilibrium in duopoly games. Analogous conclusions can be drawn for the remaining two stationary points.

16. Under these conditions, the scenario where \( \pi ^{CC} = \pi _{B}^{CB} \) and \( \pi ^{BB} = \pi _{C}^{CB} \) will only manifest when \((a_{C}-c_{C}) = b_{1}^{\prime } = b_{1}^{*},\) which is a rare occurrence.

17. We have \( (\pi ^{CC} - \pi ^{CB}_{B}) = \frac{(a-c)^{2} f(d)}{(2+d)^{2}(4-3d^{2})^{2}} \) and \(( \pi ^{BB} - \pi ^{CB}_{C}) = \frac{(1-d^{2})(a-c)^{2} g(d) }{(1+d^{2})(2-d)^{2}(4-3d^{2})^{2}},\) where \( f(d) =[ (4-3d^{2})^{2} -(2+d)^{2}(2-d-d^{2})^{2}] \) and \( g(d) = [ (4-3d^{2})^{2} -(1+d)^{2}(2-d)^{4}].\) The graphs of f(d) and g(d) show that \( g(d)> 0 > f(d) \) for all \( d \in (-1, \; 0); \; f(d) = g(d)=0 \) at \( d = 0; \) and \( f(d)> 0 > g(d) \) for all \( d \in (0, \;1).\)

18. Further details and information can be provided upon request.

19. Through calculations, we find \( p^{*} - p^{B} = \frac{(a-c)d(d-1)(d+2) }{ 2(4-d^{2})} \ge \; (\le ) \; 0 \) off \( d \le \; (\ge ) \; 0.\)

20. We cannot apply Lemma 1(i) even in the case where \( d_{C} = d_{B} = d\) as \( d \rightarrow 1.\)

21. We cannot directly apply Proposition 2(iic) due to the requirement of \( \pi ^{BB} > 0 \). Nevertheless, we can utilize similar proofs, and these details are available upon request.

22. Applying a similar approach, we can analyze the evolution of firms characterized by high costs and low costs.

23. This is due to the fact that consumers are willing to pay more for high-quality goods compared to low-quality ones.

24. By substituting \( a_{C} = a_{H} \) and \( d_{C}=d \) into (15) and (16) and solving for the Cournot equilibrium, we obtain \( q^{HH} \) and \( \pi ^{HH}.\) Similarly, by substituting \( a_{C} = a_{L} \) and \( d_{C}=d \) into (15) and (16) and solving for the Cournot equilibrium, we derive \( q^{LL} \) and \( \pi ^{LL}.\) Moreover, substituting \( a_{C} = a_{H} \) and \( d_{C}=d \) in (15) and \(a_{C} = a_{L} \) and \( d_{C} =d \) into (16) and then solving for the Cournot equilibria will yield \( q^{HL}_{L}, \; q^{HL}_{H}, \; \pi ^{HL}_{L} \) and \( \pi ^{HL}_{H}.\)

25. When we set \( a_{B} = a_{H} \) and \( d_{B}=d \) in (19) and (20) and solve for the Bertrand equilibrium, we obtain values for \( {\hat{q}}^{HH} \) and \( {\hat{\pi }}^{HH}.\) Similarly, letting \( a_{B} = a_{L} \) and \( d_{B}=d \) in (19) and (20) and solving for the Bertrand equilibrium give us \( {\hat{q}}^{LL} \) and \( {\hat{\pi }}^{LL}.\) Further, by substituting \( a_{B} = a_{H} \) and \( d_{B}=d \) into (19) and \(a_{B} = a_{L} \) and \( d_{B} =d \) into (20) and solving for the Bertrand equilibria, we derive \( {\hat{q}}^{HL}_{L}, \; {\hat{q}}^{HL}_{H}, \; {\hat{\pi }}^{HL}_{L} \) and \( {\hat{\pi }}^{HL}_{H}.\)

26. For \( d = -0.9, \) the long-run outcomes could encompass either all-Bertrand or all-Cournot firms under the condition \( c_{B} > c_{C}.\)

27. Alternative replicator-mutation dynamics can be formulated as follows: \( c(t+1) = \frac{(1-\delta ) c(t) {\bar{\pi }}^{C}(t) }{ c(t) {\bar{\pi }}^{C}(t) + b(t) {\bar{\pi }}^{B}(t) } + \delta \) and \( b(t+1) = \frac{(1-\delta ) b(t) {\bar{\pi }}^{B}(t) }{ c(t) {\bar{\pi }}^{C}(t) + b(t) {\bar{\pi }}^{B}(t) } + \delta .\) However, due to the complexity of obtaining their analytical solutions, we choose not to delve into their exploration.

28. In the scenario where \( q_{CB} = q_{BC} =0, \) the model aligns with the one discussed in Sect. 3.1 Hence, our attention is directed toward examining the instances where \( q_{CB} \) and \( q_{BC} \in (0, \; 1).\)

29. For a substantial value of N,  we can treat \( C(t) q_{CC}, \; B(t) q_{BC}, \; B(t)q_{BB}\) and \( C(t) q_{CB}\) as positive integers.

30. This is because the quantities \( ({\hat{A}} - {\hat{C}}) = q_{CC} [\pi ^{CC} - \pi _{B}^{CB} ] + q_{CB} [\pi _{C}^{CB} - \pi ^{BB} ] \) and \( ({\hat{B}} - {\hat{D}}) = q_{BC} [\pi ^{CC} - \pi _{B}^{CB} ] + q_{BB} [\pi _{C}^{CB} - \pi ^{BB} ] \) are positive (negative) under the conditions \( \pi ^{CC} > \pi _{B}^{CB} \) and \( \pi _{C}^{CB} > \pi ^{BB} \) (\( \pi ^{CC} < \pi _{B}^{CB} \) and \( \pi _{C}^{CB} < \pi ^{BB} \)), which arise due to \( q_{ij} > 0 \) for \( i, \; j = C, \; B.\)

We would like to thank the responsible editor, Dr. Georges Zaccour, and two anonymous referees for their valuable comments.

This research was financially supported by the Ministry of Science and Technology in Taiwan, under grant number MOST 106-2410-H-305-002-MY3, with Professor Hsiao-Chi Chen being the recipient of the support.

Authors and Affiliations

1. Department of Economics, National Taipei University, 151, University Road, San-Shia District, New Taipei City, 23741, Taiwan, R.O.C.

   Hsiao-Chi Chen & Shi-Miin Liu

2. Institute of Mathematics, Academia Sinica and Department of Mathematics, National Central University, Taipei, Taiwan, R.O.C.

   Yunshyong Chow

Contributions

(i) Professor H-CC is responsible for model development, participating in mathematical derivations, providing economic insights and drafting the initial manuscript. (ii) Professor YC is responsible for the mathematical derivations. (iii) Professor S-ML is responsible for conducting numerical simulations and writing.

Corresponding author

Correspondence to Hsiao-Chi Chen.

Conflict of interest

This declaration is not applicable.

Ethical Approval

This declaration is not applicable.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Proof of Proposition 1. (i) Suppose \( \pi ^{CC} = \pi ^{CB}_{B}.\) We then have

If \( \pi ^{CB}_{C} > \pi ^{BB}\) and \( b(t) >0, \) then \( {\bar{\pi }}^{C}(\textbf{s}) > {\bar{\pi }}^{B}(\textbf{s}) \) for all \( \textbf{s} \in S,\) which implies \( r(\textbf{s}) = \textbf{C} \) for all \( \textbf{s} \ne \textbf{B}.\) Thus, all mixed states are in the basin of \( \textbf{C},\) and \( S_{0} = \{\textbf{B}, \; \textbf{C} \}.\) Given state \( \textbf{B},\) we only need a single firm deviating to leave \( \textbf{B} \) for \( \textbf{C}.\) In contrast, we need N firms deviating to leave \( \textbf{C} \) for \( \textbf{B}.\) Hence \( S_{*} = \{ \textbf{C} \}.\) Similarly, we have \( S_{*} = \{ \textbf{B} \}\) if \( \pi ^{CB}_{C} < \pi ^{BB}.\) However, if \( \pi ^{CB}_{C} = \pi ^{BB},\) then \( {\bar{\pi }}^{C}(\textbf{s}) = {\bar{\pi }}^{B}(\textbf{s}), \) and \( r(\textbf{s} ) = \textbf{s} \) for all \( \textbf{s} \in S.\) Thus, \( S_{0} = S,\) and any two states in \(S_{0} \) can communicate with each other by a sequence of transitions as shown in (6) such that the minimum outgoing cost of each state is one. Hence \( S_{*} = S.\) These results prove Proposition 1(i).

(iia) Suppose \( \pi ^{CB}_{C} = \pi ^{BB}.\) Then,

By adopting the arguments used in part (i), we can get \(S_{*} = \{ \textbf{C} \} \) if \( \pi ^{CC} > \pi ^{CB}_{B}, \) and \(S_{*} = \{ \textbf{B} \} \) if \( \pi ^{CC} < \pi ^{CB}_{B}.\) These prove Proposition 1(iia).

(iib) Suppose \( \pi ^{CB}_{C} \ne \pi ^{BB}, \; \pi ^{CC} > \pi ^{CB}_{B}\) and \( \pi ^{BB} > \pi ^{CB}_{C}.\) We then have

where \( x^{*} = \frac{\pi ^{BB} -\pi ^{CB}_{C} }{\pi ^{CC} -\pi ^{CB}_{B}} >0.\) Two cases based on the values of \( \frac{x^{*}}{1+ x^{*}}\) exist. First, if \( \frac{x^{*}}{1+ x^{*}} \notin \{ \frac{1}{N}, \; \frac{2}{N}, \ldots , \frac{N}{N} \};\) then for all \( \textbf{s} \notin \{ \textbf{B}, \; \textbf{C} \}, \) we have \( r(\textbf{s}) = \textbf{C} \) if \( C(\textbf{s}) \ge \lceil \frac{Nx^{*}}{1+ x^{*}} \rceil , \) and \( r(\textbf{s}) = \textbf{B} \) if \( C(\textbf{s}) < \lceil \frac{Nx^{*}}{1+ x^{*}} \rceil \) by (43), where \( C(\textbf{s}) \) is the number of C at state \( \textbf{s}.\) These imply that each mixed stationary state is either in the basin of \( \textbf{C} \) or in the basin of \( \textbf{B}.\) Thus, \( S_{0} = \{ \textbf{C}, \; \textbf{B} \}, \) and the minimum outgoing cost of the basin of \( \textbf{C} \; (\textbf{B}) \) is \( N - \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor \; (\lceil \frac{Nx^{*}}{1+ x^{*}} \rceil ).\) Hence, \( S_{*} = \{ \textbf{ C} \} \) if \( \lceil \frac{Nx^{*}}{1+ x^{*}} \rceil < N - \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor , \; S_{*} = \{ \textbf{B}, \; \textbf{ C} \} \) if \( \lceil \frac{Nx^{*}}{1+ x^{*}} \rceil = N - \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor , \) and \( S_{*} = \{ \textbf{B} \} \) if \( \lceil \frac{Nx^{*}}{1+ x^{*}} \rceil > N - \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor .\)

Second, if there exists some positive integer k with \( 1 \le k \le N \) such that \( \frac{x^{*}}{1+ x^{*}} = \frac{k}{N},\) then the states in set \( M \equiv \{ \textbf{s} \in S \; | \; C( \textbf{s} ) = k \} \) will belong to \( S_{0} \) in addition to \( \textbf{B} \) and \( \textbf{C}.\) That is because (43) implies that \( r(\textbf{s} ) = \textbf{s} \) by \( {\bar{\pi }}^{C}(\textbf{s}) = {\bar{\pi }}^{B}(\textbf{s}) \) for all \( \textbf{s} \in M,\) each state with more than k’s C is in the basin of \( \textbf{C},\) and each state with less than k’s C is in the basin of \( \textbf{B}.\) Thus, the minimum outgoing cost of each state in M to the basin of \(\textbf{C} \) or to the basin of \( \textbf{B} \) is 1, the minimum outgoing cost from the basin of \( \textbf{C} \) to M is \( (N-k),\) and the minimum outgoing cost from the basin of \( \textbf{B} \) to M is k as shown by (7). Since \( N \ge k \ge 1, \; S_{*} \) is determined by relative sizes of k and \( (N-k).\) Thus, \( S_{*} = \{ \textbf{ C} \} \) if \( k < \frac{N}{2}, \; S_{*} = \{ \textbf{B}, \; \textbf{ C} \} \) if \( k = \frac{N}{2}, \) and \( S_{*} = \{ \textbf{B} \} \) if \( k > \frac{N}{2}.\) These results prove Proposition 1(iib).

(iic) Suppose \( \pi ^{CB}_{C} \ne \pi ^{BB}, \; \pi ^{CC} < \pi ^{CB}_{B}\) and \( \pi ^{BB} < \pi ^{CB}_{C}.\) We then have

Since \( x^{*} > 0, \) we have \( \frac{x^{*}}{1+ x^{*}} \in (0, \; 1).\) Again, there are two cases. First, if \( \frac{x^{*}}{1+ x^{*}} \notin \{ \frac{1}{N}, \; \frac{2}{N}, \ldots , \frac{N-1}{N} \},\) then \( r(\textbf{s}) = \textbf{C} \) as \( C(\textbf{s}) \le \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor \) and \( r(\textbf{s}) = \textbf{B} \) as \( C(\textbf{s}) > \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor \) for all \( \textbf{s} \notin \{ \textbf{B}, \; \textbf{C} \} \) by (44). These imply that all states with more than \( \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor \)’s C are in the basin of \( \textbf{B},\) and the rest states with no more than \( \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor \)’s C are in the basin of \( \textbf{C}.\) Hence, \( S_{0} = \{ \textbf{B}, \; \textbf{C} \},\) and the values of \( \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor \) decide the LREs. If \( \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor =0,\) then all mixed states are in the basin of \( \textbf{B}.\) The minimum cost of leaving the basin of \( \textbf{C} \) for \( \textbf{B} \) is N,  while the minimum cost of leaving \( \textbf{C} \) for the basin of \(\textbf{B}\) is 1. Thus, \( S_{*} = \{ \textbf{B} \}.\) In contrast, if \( \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor =(N-1),\) then all mixed states are in the basin of \( \textbf{C}.\) The minimum cost of leaving \( \textbf{B} \) for the basin of \( \textbf{C} \) is N,  while the minimum cost of leaving \( \textbf{B} \) for the basin of \( \textbf{C} \) is 1. Thus, \( S_{*} = \{ \textbf{C} \}.\) However, if \( 1 \le \lfloor \frac{Nx^{*}}{1+ x^{*}} \rfloor \le (N-2),\) then the minimum outgoing cost of both basins of \( \textbf{C} \) and \( \textbf{B}\) are 1, and hence \( S_{*} = \{ \textbf{B}, \; \textbf{C} \}. \)

Second, if there exists some positive integer k with \( 1 \le k \le (N-1) \) such that \( \frac{x^{*}}{1+ x^{*}} = \frac{k}{N},\) then we have \( S_{0} = \{ \textbf{B}, \; \textbf{C} \} \cup M, \) where set M is defined as in part (iib). As in part (iib), (44) implies that states with less than k’s C are in the basin of \( \textbf{C},\) states with more than k’s C are in the basin of \( \textbf{B},\) and states with exactly k’s C are in set M. Thus, the minimum outgoing cost of all states in M to the basin of \(\textbf{C} \) or \( \textbf{B} \) is 1. However, the minimum outgoing cost of the basin of \( \textbf{C} \) to M is k,  while the minimum outgoing cost of the basin of \( \textbf{B} \) to M is \( (N-k).\) Since \( (N-k) \ge k \ge 1, \; S_{*} \) is determined by relative sizes of k and \( (N-k).\) Thus, \( S_{*} = \{ \textbf{ C} \} \) as \( k > \frac{N}{2}; \; S_{*} = \{ \textbf{B}, \; \textbf{ C} \} \) as \( k = \frac{N}{2}; \) and \( S_{*} = \{ \textbf{B} \} \) as \( k < \frac{N}{2}.\) These results prove Proposition 1(iic).

(iid) Suppose \( \pi ^{CB}_{C} \ne \pi ^{BB}, \; \pi ^{CC} > \pi ^{CB}_{B}\) and \( \pi ^{BB} < \pi ^{CB}_{C}.\) We then have

for all \( b(t) \ge 0, \; c(t) \ge 0, \; t \ge 0\) and all \( \textbf{s} \notin \{ \textbf{B}, \; \textbf{C} \}.\) Thus, (45) implies \( r(\textbf{s} )= \textbf{C} \) for all \( \textbf{s} \notin \{ \textbf{B}, \; \textbf{C} \},\) and all mixed states are in the basin of \( \textbf{C}.\) Hence, \( S_{0} = \{ \textbf{B}, \; \textbf{C} \}.\) While a single firm deviating is enough to leave \( \textbf{B} \) for \( \textbf{C},\) N firms deviating are needed to leave \( \textbf{C} \) for \( \textbf{B}.\) Thus, \( S_{*} = \{ \textbf{C} \}.\) These prove Proposition 1(iid).

(iie) Suppose \( \pi ^{CB}_{C} \ne \pi ^{BB}, \; \pi ^{CC} < \pi ^{CB}_{B}\) and \( \pi ^{BB} > \pi ^{CB}_{C}.\) We then have

for all \( b(t) \ge 0, \; c(t) \ge 0, \; t \ge 0\) and all \( \textbf{s} \notin \{ \textbf{B}, \; \textbf{C} \}.\) Thus, (46) implies \( r(\textbf{s} )= \textbf{B} \) for all \( \textbf{s} \notin \{ \textbf{B}, \; \textbf{C} \},\) and all mixed states are in the basin of \( \textbf{B}.\) Hence, \( S_{0} = \{ \textbf{B}, \; \textbf{C} \}.\) While a single firm deviating is enough to leave \( \textbf{C} \) for \( \textbf{B},\) N firms deviating are needed to leave \( \textbf{B} \) for \( \textbf{C}.\) Thus, \( S_{*} = \{ \textbf{B} \}.\) These prove Proposition 1(iie).

Proof of Proposition 2. To simplify our analyses, denote \( x_{t} \equiv \frac{c(t)}{b(t)} \) for \( t =0, \; 1, \ldots , \; A \equiv \pi ^{CC}, \; B \equiv \pi ^{CB}_{C}, \; C \equiv \pi _{B}^{CB}\) and \( D \equiv \pi ^{BB} \) with \( A, \; B, C \) and \( D > 0.\) The dynamic system in (13) is then reduced to

Some calculations yield \( f^{\prime } (x_{t}) = \frac{AC x_{t}^{2} + 2 AD x_{t} + BD}{(Cx_{t} + D)^{2} } > 0 \) for all \( x_{t} \ge 0, \; f^{\prime }(0) = \frac{B}{D} \) and \( f^{\prime \prime } (x) = \frac{ 2D(AD - BC) }{(Cx_{t} + D)^{3} }.\) The stationary points of the dynamic in (47) are \(0, \; \infty \) and \( x_{1}^{*} = \frac{D-B}{A-C} \) if \( A \ne C.\) According to relative sizes of \( A, \; B, \; C \) and D,  we have two cases below.

Case 1: Suppose \( A =C. \) There are three sub-cases.

Case 1a: If \( B =D, \) then \( f(x_{t}) = x_{t} \) and \( x_{t+1} = f( x_{t} )= x_{t} = f( x_{t-1} ) = \ldots = x_{0} \) for all \( t \ge 1.\) Therefore, Proposition 2(ia) is proved.

Case 1b: If \( B > D, \) then \( f(x_{t}) =\frac{Ax_{t}^{2} + B x_{t}}{ A x_{t} + D} = x_{t} + \frac{(B-D)x_{t}}{Ax_{t}+ D} > x_{t}, \; f(0) =0, \; \lim _{x_{t} \rightarrow \infty } \) \( f(x_{t}) = \infty , \; f^{\prime }(0) = \frac{B}{D} > 1 \) and \( f^{\prime \prime } (x_{t}) = \frac{2AD(D-B)}{(Ax_{t} + D)^{3}} < 0 \) for all \( x_{t} \ge 0.\) These imply that \( f(x_{t}) \) is strictly concave and lies above the 45-degree line for all \( x_{t} > 0.\) Thus, \( x_{t+1} = f(x_{t} ) > x_{t} \) for all \( t \ge 0, \) and \( \lim _{t \rightarrow \infty } x_{t} = \infty . \) Hence, Proposition 2(ib) is proved.

Case 1c: If \( B < D, \) then \( f(x_{t})< x_{t}, \; f(0) =0, \; f^{\prime }(0) = \frac{B}{D} < 1 \) and \( f^{\prime \prime } (x_{t}) = \frac{2AD(D-B)}{(Ax_{t} + D)^{3}} > 0 \) for all \( x_{t} \ge 0.\) These imply that \( f(x_{t}) \) is strictly convex and lies below the 45-degree line for all \( x_{t} >0.\) Thus, \( x_{t+1} = f(x_{t} ) < x_{t} \) for all \( t \ge 0,\) and hence \( \lim _{t \rightarrow \infty } x_{t} = 0. \) Hence, Proposition 2(ic) is proved.

Case 2: Suppose \( A \ne C. \) There are three sub-cases.

Case 2a: Suppose \( B =D \) and \( A > C. \) We then have \( f(x_{t}) =\frac{Ax_{t}^{2} + B x_{t}}{ C x_{t} + B} =x_{t} + \frac{(A-C)x_{t}^{2}}{Cx_{t}+ B} > x_{t}, \; f(0) =0, \; \lim _{x_{t} \rightarrow \infty } f(x_{t}) = \infty , \; f^{\prime }(0) = \frac{B}{D} = 1 \) and \( f^{\prime \prime } (x_{t}) = \frac{2D^{2} (A-C)}{(Cx_{t} + D)^{3}} > 0 \) for all \( x_{t} \ge 0.\) These imply that \( f(x_{t}) \) is strictly convex and lies above the 45-degree line for all \( x_{t} >0.\) Now, \( x_{t+1} = f(x_{t} ) > x_{t} \) for all \( t \ge 0, \) and hence \( \lim _{t \rightarrow \infty } x_{t} = \infty \) for all \( x_{0}.\) In contrast, if \( B =D \) and \( A < C, \) then \( f(0) =0, \; f(x_{t}) < x_{t}, \; f^{\prime }(0) = \frac{B}{D} = 1 \) and \( f^{\prime \prime } (x_{t}) < 0. \) These imply that \( f(x_{t}) \) is strictly concave and lies below the 45-degree line for all \( x_{t} > 0.\) Thus, \( x_{t+1} = f( x_{t} ) < x_{t} \) for all \( t \ge 0, \) and hence, \( \lim _{t \rightarrow \infty } x_{t} = 0. \) Hence, Proposition 2(iia) is proved.

Case 2b: Suppose \( B \ne D \) and \( x^{*} = \frac{ D - B}{A -C} > 0.\) Under the circumstance, two possibilities exist. First, we have \( A > C \) and \( D > B,\) which imply \( AD > BC, \; f(0) =0, \; f^{\prime }(0) = \frac{B}{D} < 1 \) and \( f^{\prime \prime } (x_{t}) = \frac{2D (AD-BC)}{(Cx_{t} + D)^{3}} > 0 \) for all \( x_{t} \ge 0.\) Thus, \( f(x_{t}) \) is strictly convex, lies below (above) the 45-degree line for \( x_{t} < \; (>) \; x^{*},\) and intersects with the 45-degree line at \( x_{t} = x^{*}.\) Accordingly, we have \( \lim _{t \rightarrow \infty } x_{t} = \infty \) if \( x_{0} > x^{*}, \; \lim _{t \rightarrow \infty } x_{t} = x^{*} \) if \( x_{0}= x^{*}, \) and \( \lim _{t \rightarrow \infty } x_{t} = 0 \) if \( x_{0} < x^{*}. \) That is because \( x_{t+1} = f(x_{t} ) < \; (>) \; x_{t} \) for \( x_{t} < \; ( > ) \; x^{*}. \) Hence, Proposition 2(iib) is proved. Second, if \( A < C \) and \( D < B, \) then \( AD < BC, \; f(0) =0, \; f^{\prime }(0) = \frac{B}{D} > 1 \) and \( f^{\prime \prime } (x_{t}) = \frac{2D (AD-BC)}{(Cx_{t} + D)^{3}} < 0 \) for all \( x_{t}.\) These imply that \( f(x_{t}) \) is strictly concave, lies above (below) the 45-degree line for \( x_{t} < \; (>) \; x^{*}, \) and intersects with the 45-degree line at \( x_{t} = x^{*}.\) Thus, \( \lim _{t \rightarrow \infty } x_{t} = x^{*} \) due to \( x_{t+1} = f(x_{t} ) > \; (<) \; x_{t} \) for \( x_{t} < \; (>) \; x^{*}.\) Hence, Proposition 2(iic) is proved.

Case 2c: Suppose \( B \ne D \) and \( x^{*} = \frac{ D - B}{A -C} < 0.\) Under the circumstance, two possibilities exist. First, we have \( A > C \) and \( D< B,\) which imply \( f(x_{t}) = \frac{Ax_{t}^{2} + B x_{t} }{C x_{t} +D} > x_{t} \) and \( x_{t+1} = f(x_{t}) > x_{t} \) for all \( t \ge 0.\) Since \( x_{0} \ge 0,\) we get \( \lim _{t \rightarrow \infty } x_{t} = \infty \) even though \( x^{*} < 0.\) These prove Proposition 2(iid). Second, if \( A < C \) and \( D >B, \) then \( f(x_{t}) = \frac{Ax_{t}^{2} + B x_{t} }{C x_{t} +D} < x_{t}, \) and hence \( x_{t+1} = f(x_{t}) < x_{t}.\) Thus, \( \lim _{t \rightarrow \infty } x_{t} = 0,\) and Proposition 2(iie) is shown. \( \Box \)

Proof of Lemma 1. Denote \( b_{1} \equiv (a_{C} - c_{C})>0 \) and \( b_{2} \equiv (a_{B} - c_{B})>0. \) We first explore the relationship between \( \pi ^{CB}_{C} \) and \( \pi ^{BB}\) for \( b_{1} > \frac{d_{C} b_{2}}{2}.\) Under the circumstance, we have \( (\pi ^{CB}_{C} - \pi ^{BB}) = \frac{(1-d_{C} d_{B} ) [2 b_{1} - d_{C} b_{2}]^{2} }{(4-3 d_{C} d_{B})^{2}} - \frac{(1-d_{B}^{2} )b_{2}^{2} }{(1+d_{B})^{2} (2-d_{B})^{2}} \) with \( \frac{\partial (\pi ^{CB}_{C} - \pi ^{BB})}{\partial b_{1} } = \frac{4(1-d_{C} d_{B})(2b_{1} - d_{C} b_{2})}{(4-3 d_{C} d_{B})^{2}} > 0,\) which implies that \( (\pi ^{CB}_{C} - \pi ^{BB}) \) increases with rising \( b_{1} \). Moreover, we have \( (\pi ^{CB}_{C} - \pi ^{BB}) = \frac{-(1-d_{B})b_{2}^{2} }{(1+d_{B})(2-d_{B})^{2}} <0 \) at \( b_{1} = \frac{d_{C} b_{2}}{2}.\) These suggest that \( (\pi ^{CB}_{C} - \pi ^{BB}) \) can be positive or negative for \( b_{1} > \frac{d_{C} b_{2}}{2},\) and \( b_{1}^{*} \) exists with \( b_{1}^{*} > \frac{d_{C} b_{2}}{2} \) and \( (\pi ^{CB}_{C} - \pi ^{BB}) =0 \) at \( b_{1} = b_{1}^{*}.\) Hence, \( \pi ^{CB}_{C} > \pi ^{BB} \) for \( b_{1} > b_{1}^{*}, \) and \( \pi ^{CB}_{C} < \pi ^{BB} \) for \( b_{1} < b_{1}^{*}. \) In contrast, for \( b_{1} \le \frac{d_{C} b_{2}}{2},\) we have \( \pi ^{CB}_{C} =0 \) and \( \pi ^{BB} > 0 \) for all values of \( a_{B}, c_{B} \) and \( d_{B}.\) Thus, \( \pi ^{CB}_{C} < \pi ^{BB} \) for \( b_{1} \le \frac{d_{C} b_{2}}{2}.\) In summary, we have \( \pi ^{CB}_{C} > \pi ^{BB} \) for \( b_{1} > b_{1}^{*}, \; \pi ^{CB}_{C} = \pi ^{BB} \) at \( b_{1} = b_{1}^{*}, \) and \( \pi ^{CB}_{C} < \pi ^{BB} \) for \( b_{1} < b_{1}^{*}. \)

Second, we explore the relationship between \( \pi ^{CB}_{B} \) and \( \pi ^{CC} \) for \( b_{1} < \frac{b_{2}(2-d_{C} d_{B})}{d_{B}} \equiv {\bar{b}}_{1}. \) Under the circumstance, we have \( (\pi ^{CC} - \pi ^{CB}_{B}) = \frac{b_{1}^{2} }{(2+d_{C})^{2} } - \frac{[ b_{2} (2-d_{C} d_{B}) - d_{B} b_{1}]^{2} }{(4-3 d_{C} d_{B})^{2}} \) with \( \frac{\partial ^{2} (\pi ^{CC} - \pi ^{CB}_{B})}{\partial b_{1}^{2} } = \frac{8\,L }{(2+d_{C})^{2} (4-3 d_{C} d_{B})^{2} } \), where \( L = 4-6 d_{C} d_{B} - d_{C} d_{B}^{2} + 2d_{C}^{2} d_{B}^{2} - d_{B}^{2}.\) Some calculations yield \( \frac{\partial L }{\partial d_{C}} = d_{B} (-6-d_{B} + 4d_{C} d_{B}) > \; (<) \; 0 \) iff \( d_{B} < \; (>) \; 0, \; L = (4+ 6d_{B} + 2 d_{B}^{2} )>0 \) at \( d_{C} =-1, \; L = (4 - 6d_{B}) > \; (<) \; 0 \) iff \( d_{B} < \; (>) \; \frac{2}{3} \) at \( d_{C}=1, \) and \( L =0 \) at \( d_{C} = \frac{(2-d_{B})}{2d_{B}}.\) Based on the values of \( d_{C} \) and \( d_{B}, \) we have four cases below.

Case 1: Suppose \( -1< d_{C} < 1\) and \( -1< d_{B} < 0. \) We then have \( \frac{\partial L }{\partial d_{C}} > 0 \) by \( d_{B} <0, \; L> 4+ 6 d_{B} + 2d_{B}^{2} >0\) and \( \frac{\partial ^{2} (\pi ^{CC} - \pi ^{CB}_{B})}{\partial b_{1}^{2} } >0.\) Moreover, we have \( (\pi ^{CC} - \pi ^{CB}_{B})=0 \) at \( b_{1} = b_{1}^{\prime } \) and \( b_{1} = b_{1}^{\prime \prime } \) with \( b_{1}^{\prime } = \frac{b_{2} (2+d_{C})(2-d_{C} d_{B}) }{ 2(2+d_{B} - d_{C} d_{B})}> 0 \) and \( b_{1}^{\prime \prime } = \frac{ b_{2} (2+d_{C})(2- d_{C} d_{B})}{2(d_{B} + 2d_{C} d_{B} -2)} <0. \) Since \( \pi ^{CB}_{B} > 0 \) for \( b_{1} < {\bar{b}}_{1}, \) the values of \( b_{1}^{\prime } \) and \( b_{1}^{\prime \prime } \) are meaningful only when \( b_{1}^{\prime } < {\bar{b}}_{1}, \) which will hold because \( ({\bar{b}}_{1}- b_{1}^{\prime }) = \frac{b_{2} ( 4-3d_{C} d_{B})( 2- d_{C} d_{B}) }{2 d_{B} (2+d_{B} - d_{C} d_{B} ) } > 0. \) Thus, \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{\prime }\) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} > b_{1}^{\prime }.\)

Case 2: Suppose \( -1< d_{C} < 1\) and \( 0< d_{B} < \frac{2}{4}. \) Then, \( \frac{\partial L }{\partial d_{C}} < 0 \) by \( d_{B} >0 \) and \( L> 4- 6 d_{B} >0. \) Moreover, \( \frac{\partial ^{2} (\pi ^{CC} - \pi ^{CB}_{B})}{\partial b_{1}^{2} } >0 \) with \( b_{1}^{\prime }> 0, \; b_{1}^{\prime \prime } < 0 \) and \( b_{1}^{\prime } < {\bar{b}}_{1}\) as defined in Case 1. Thus, \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{\prime }\) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} > b_{1}^{\prime }.\)

Case 3: Suppose \( -1< d_{C} < \frac{(2-d_{B})}{ 2d_{B}}\) and \( \frac{2}{3}< d_{B} < 1. \) We then have \( \frac{\partial L }{\partial d_{C}} < 0 \) by \( d_{B}>0, \; L >0\) at \(d_{C} =-1, \) and \( L =0 \) at \( d_{C} = \frac{(2-d_{B})}{ 2d_{B}}.\) These imply \( \frac{\partial ^{2} (\pi ^{CC} - \pi ^{CB}_{B})}{\partial b_{1}^{2} } >0\) with \( b_{1}^{\prime }> 0, \; b_{1}^{\prime \prime } < 0 \) and \( b_{1}^{\prime } < {\bar{b}}_{1}\) as defined in Case 1. Again, \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{\prime }\) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} > b_{1}^{\prime }.\)

Case 4: Suppose \( \frac{(2-d_{B})}{ 2d_{B}}< d_{C} < 1\) and \( \frac{2}{3}< d_{B} < 1. \) Then, \( \frac{\partial L }{\partial d_{C}} < 0 \) by \( d_{B} >0, \; L =0 \) at \( d_{C} = \frac{(2-d_{B})}{ 2d_{B}}, \) and \( L <0\) at \(d_{C} =1.\) These imply \( \frac{\partial ^{2} (\pi ^{CC} - \pi ^{CB}_{B})}{\partial b_{1}^{2} } <0.\) Moreover, we have \((\pi ^{CC} - \pi ^{CB}_{B}) =0 \) at \( b_{1} = b_{1}^{\prime } \) and at \( b_{1} = b_{1}^{\prime \prime } \) with \( b_{1}^{\prime }> 0, \; b_{1}^{\prime \prime } > 0 \) and \( b_{1}^{\prime } < b_{1}^{\prime \prime } \) by \( (4 -3d_{C} d_{B} ) >0, \) and \( {\bar{b}}_{1} < b_{1}^{\prime \prime } \) by \( \frac{(2-d_{B})}{ 2d_{B}}< d_{C} < 1.\) Thus, \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{\prime }\) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} \in (b_{1}^{\prime }, \; {\bar{b}}_{1}).\)

Cases 1-4 imply \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{\prime }\) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} \in (b_{1}^{\prime }, \; {\bar{b}}_{1}).\) For \( b_{1} \ge {\bar{b}}_{1}, \) we have \( \pi ^{CC} > 0 \) and \( \pi ^{CB}_{B} =0 \) regardless of the values of \( d_{C} \) and \( d_{B}.\) In summary, we have \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{\prime }, \; \pi ^{CC} = \pi ^{CB}_{B} \) for \( b_{1} = b_{1}^{\prime },\) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} > b_{1}^{\prime }.\)

Third, it remains to compare \( (\pi ^{CC} - \pi ^{CB}_{B}) \) and \( (\pi ^{CB}_{C} - \pi ^{BB}). \) Evaluating \( (\pi ^{CB}_{C} - \pi ^{BB}) \) at \( b_{1} = b_{1}^{\prime } \) yields \( (\pi ^{CB}_{C} - \pi ^{BB}) = \frac{(d_{B} b_{2} )^{2} (2+d_{C} - d_{C} d_{B})(d_{B} - d_{C} )}{(1+d_{B})(2-d_{B})^{2} (2+ d_{B} - d_{C} d_{B})^{2}} \le \; (>) \; 0 \) iff \( d_{B} \le (>) \; d_{C}.\) Since \( \pi ^{CB}_{C} > \pi ^{BB} \) for \( b_{1} > b_{1}^{*}, \; \pi ^{CB}_{C} = \pi ^{BB} \) for \( b_{1} = b_{1}^{*}, \) and \(\pi ^{CB}_{C} < \pi ^{BB} \) for \( b_{1} < b_{1}^{*}; \) we must have \( b_{1}^{\prime } < b_{1}^{*} \) as \( d_{C} > d_{B}, \; b_{1}^{\prime } = b_{1}^{*} \) as \( d_{C} = d_{B}, \) and \( b_{1}^{\prime } > b_{1}^{*} \) as \( d_{C} < d_{B}.\) Thus, we have three cases below.

Case A: Suppose \( d_{C} = d_{B}.\) Then \( b_{1}^{\prime } = b_{1}^{*}. \) Accordingly, \( \pi ^{CB}_{C} > \pi ^{BB} \) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} > b_{1}^{*}, \) and \( \pi ^{CB}_{C} < \pi ^{BB} \) and \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{*}. \) These prove Lemma 1(i).

Case B: Suppose \( d_{C} > d_{B}.\) Then \( b_{1}^{\prime } < b_{1}^{*}. \) Thus, we have \( \pi ^{CB}_{C} < \pi ^{BB} \) and \( \pi ^{CC}< \pi ^{CB}_{B} \) for \( b_{1}< b_{1}^{\prime }, \; \pi ^{CB}_{C} < \pi ^{BB} \) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} \in (b_{1}^{\prime }, \; b_{1}^{*}), \) and \( \pi ^{CB}_{C} > \pi ^{BB} \) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} > b_{1}^{*}.\) These prove Lemma 1(ii).

Case C: Suppose \( d_{C} < d_{B}.\) Then \( b_{1}^{\prime } > b_{1}^{*}. \) Thus, we have \( \pi ^{CB}_{C} < \pi ^{BB} \) and \( \pi ^{CC}< \pi ^{CB}_{B} \) for \( b_{1} < b_{1}^{*}, \; \pi ^{CB}_{C} > \pi ^{BB} \) and \( \pi ^{CC} < \pi ^{CB}_{B} \) for \( b_{1} \in ( b_{1}^{*}, \; b_{1}^{\prime }), \) and \( \pi ^{CB}_{C} > \pi ^{BB} \) and \( \pi ^{CC} > \pi ^{CB}_{B} \) for \( b_{1} > b_{1}^{\prime }.\) These prove Lemma 1(iii). \( \Box \)

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