Customer relationship management in competitive environments: The positive implications of a short-term focus

Abstract

Researchers and business thought leaders have emphasized that firms must think and act with a long-term horizon when managing customer relationships. We demonstrate that, in contrast to this widely held view, profits in competitive environments may be maximized when firms ignore the future and instead maximize period-by-period profits from customers. Intuitively, while a long-term focus yields more loyal customers, it greatly increases short-term price competition to gain and keep customers. Consequently, overall firm profits and customer lifetime value may be lower when firms directly maximize multi-period profits from customers. Specifically, we analyze a model with segment-level pricing where firms in a duopoly can choose between period-by-period and multi-period profit maximization and demonstrate that, in many cases, a symmetric focus on period-by-period profit maximization emerges as the Pareto-dominant Nash equilibrium. We extend the model in two directions. First, we demonstrate that this superiority of the short-term focus endures even when a revenue expansion effect applies—that is, when customer loyalty leads to enhanced revenues. Second, we examine the case where customers are strategic and incorporate the long-term implications of their choices into their decision-making. Here we demonstrate that it may pay for firms to be myopic even when customers are strategic. The focus on multi-period surplus makes customers less price sensitive to price variations at the early stage of the game. Consequently, the focus on maximizing period-by-period profits enables the firms to charge higher upfront prices and leverage this lower price sensitivity into higher profits. Overall, our results highlight the paradox that, when it comes to managing customer relationships in competitive environments, a short-term focus may constitute the optimal long-term strategy.

Appendices

Appendix A Revenue expansion effects

1.1 Proofs of Proposition 4 and Proposition 5

We begin with an analysis of period 2 decisions.

1.1.1 Analysis of Period 2

Depending on the decisions of customers in period 1, there are four possible cases that need to be analyzed in period 2.

• Case 1 Customers in Segment A purchase from firm 1 in period 1

  In period 2, the switching cost $\widetilde{s}$ for the marginal customer who is indifferent between buying from firm 1 or from firm 2 is determined by:

  $$ {\left( {1 + \delta } \right)}{\left( {1 + k - p^{A}_{{12}} } \right)} = 1 - \widetilde{s} - p^{A}_{{22}} $$

  (33)

  Therefore,

  $$ \widetilde{s} = {\left( {1 + \delta } \right)}p^{A}_{{12}} - p^{A}_{{22}} - k\prime $$

  (34)

  where k′=δ+(1+δ)k. We restrict k′ < 1, i.e., k<(1 − δ)/(1 + δ), to avoid corner solutions.

  The demand for firm 2 is $\widetilde{s}$ and the demand for firm 1 is ${\left( {1 + \delta } \right)}{\left( {1 - \widetilde{s}} \right)}$. Therefore, the profits that accrue to the two firms from segment A when these customers purchase from firm 1 in period 1 are:

  $$ \begin{array}{*{20}l} {{\pi ^{A}_{{12}} = p^{A}_{{12}} {\left( {1 + \delta } \right)}{\left[ {1 + k\prime - {\left( {1 + \delta } \right)}p^{A}_{{12}} + p^{A}_{{22}} } \right]}} \hfill} \\ {{\pi ^{A}_{{22}} = p^{A}_{{22}} {\left[ {{\left( {1 + \delta } \right)}p^{A}_{{12}} - p^{A}_{{22}} - k\prime } \right]}} \hfill} \\ \end{array} $$

  (35)

  Differentiating these profits with respect to own prices yields the relevant first order conditions. Solving these conditions, the following equilibrium outcomes for segment A are obtained:

  $$ p^{A}_{{12}} = \frac{{2 + k\prime }} {{3{\left( {1 + \delta } \right)}}},\;p^{A}_{{22}} = \frac{{1 - k\prime }} {3} $$

  (36)
  $$ \pi ^{A}_{{12}} = x = \frac{{{\left( {2 + k\prime } \right)}^{2} }} {9},\;\pi ^{A}_{{22}} = y = \frac{{{\left( {1 - k\prime } \right)}^{2} }} {9} $$

  (37)

  Here, $\pi ^{A}_{{12}} = x$ are the period 2 profits of firm 1 from its high preference segment if that segment purchased from firm 1 in period 1. Likewise, $\pi ^{A}_{{22}} = y$ are the period 2 profits of firm 2 from its low preference segment if that segment did not purchase from firm 2 in period 1. Note that the switching cost $\widetilde{s}$ for the marginal customer who is indifferent between buying from firm 1 or from firm 2 is:

  $$ \widetilde{s} = {\left( {1 + \delta } \right)}p^{A}_{{12}} - p^{A}_{{22}} - k\prime = \frac{{1 - k\prime }} {3} $$

  (38)

  In period 1, each customer knows the distribution of switching costs in period 2, but does not know her exact switching cost. On experiencing the product purchased in period 1, customers realize their switching costs for period 2. The expected surplus of a customer in segment A is (we calculate the surplus here but will use the calculation only when we analyze the case of the strategic customer in Appendix B):

  $$ \begin{array}{*{20}c} {S_{A} = {\int_{\widetilde{s}}^1 {{\left[ {1 + k\prime - {\left( {1 + \delta } \right)}p^{A}_{{12}} } \right]}{\text{d}}s} } + {\int_0^{\widetilde{s}} {{\left[ {1 - s - p^{A}_{{22}} } \right]}} }{\text{d}}s} \\ { = {\int_{\frac{{1 - k\prime }} {3}}^1 {{\left[ {1 + k\prime - \frac{{2 + k\prime }} {3}} \right]}} }{\text{d}}s + {\int_0^{\frac{{1 - k\prime }} {3}} {{\left[ {1 - s - \frac{{1 - k\prime }} {3}} \right]}{\text{d}}s} }} \\ \end{array} $$

  (39)

• Case 2 Customers in Segment B purchase from firm 2 in period 1

  Because the game is symmetric to Case 1 above, we obtain the following outcomes:

  $$ p^{B}_{{22}} = \frac{{2 + k\prime }} {3},\;p^{B}_{{12}} = \frac{{1 - k\prime }} {3} $$

  (40)
  $$ \pi ^{B}_{{22}} = x,\;\pi ^{B}_{{12}} = y,\;S_{B} = S_{A} $$

  (41)

• Case 3 Customers in Segment A purchase from firm 2 in period 1

  In period 2, the switching cost $\widetilde{s}$ for the marginal customer who is indifferent between buying from firm 1 or from firm 2, is determined by:

  $$ 1 + k - \widetilde{s} - p^{A}_{{12}} = {\left( {1 + \delta } \right)}{\left( {1 - p^{A}_{{22}} } \right)} $$

  (42)

  Therefore, the location of the indifferent customer is:

  $$ \widetilde{s} = {\left( {1 + \delta } \right)}p^{A}_{{22}} - p^{A}_{{12}} + k\prime \prime $$

  (43)

  where

  $$ k\prime \prime = k - \delta $$

  (44)

  The demand for firm 1 is $\widetilde{s}$ and the demand for firm 2 is ${\left( {1 + \delta } \right)}{\left( {1 - \widetilde{s}} \right)}$. The profits of the two firms are denoted below:

  $$ \begin{array}{*{20}c} {\pi ^{A}_{{12}} = p^{A}_{{12}} {\left[ {{\left( {1 + \delta } \right)}p^{A}_{{22}} - p^{A}_{{12}} + k\prime \prime } \right]}} \\ {\pi ^{A}_{{22}} = {\left( {1 + \delta } \right)}p^{A}_{{22}} {\left[ {1 - k\prime \prime - {\left( {1 + \delta } \right)}p^{A}_{{22}} + p^{A}_{{12}} } \right]}} \\ \end{array} $$

  (45)

  Differentiating these profits with respect to own prices yields the relevant first order conditions. Solving these conditions, the following equilibrium outcomes for period 2 are obtained:

  $$ \begin{array}{*{20}l} {{p^{A}_{{22}} = \frac{{2 - k\prime \prime }} {{3{\left( {1 + \delta } \right)}}},\;p^{A}_{{12}} = \frac{{1 + k\prime \prime }} {3}} \hfill} \\ {{\pi ^{A}_{{22}} = w = \frac{{{\left( {2 - k\prime \prime } \right)}^{2} }} {9},\;\pi ^{A}_{{12}} = w\prime = \frac{{{\left( {1 + k\prime \prime } \right)}^{2} }} {9}} \hfill} \\ \end{array} $$

  (46)

  Here, w represents the period 2 profits of firm 2 from segment A (its low preference segment), if it captures segment A in period 1, and w′ represents the period 2 profits of firm 1 from segment A (its high preference segment) if firm 2 captures segment A in period 1. After substitutions, the switching cost for the marginal customer is:

  $$ \widetilde{s} = {\left( {1 + \delta } \right)}p^{A}_{{22}} - p^{A}_{{12}} + k\prime \prime = \frac{{1 + k\prime \prime }} {3} $$

  (47)

  The expected surplus of a customer in segment A is:

  $$ \begin{array}{*{20}l} {{S^{\prime }_{A} = {\int_0^{\widetilde{s}} {{\left[ {1 + k - s - p^{A}_{{12}} } \right]}{\text{d}}s} } + {\int_{\widetilde{s}}^1 {{\left[ {{\left( {1 + \delta } \right)}{\left( {1 - p^{A}_{{22}} } \right)}} \right]}{\text{d}}s} }} \hfill} \\ {{ = {\int_0^{\frac{{1 + k\prime \prime }} {3}} {{\left[ {1 + k - s - \frac{{1 + k^{{\prime \prime }} }} {3}} \right]}} }{\text{d}}s + {\int_{\frac{{1 + k\prime \prime }} {3}}^1 {{\left[ {{\left( {1 + \delta } \right)}{\left( {1 - \frac{{2 - k\prime \prime }} {3}} \right)}} \right]}{\text{d}}s} }} \hfill} \\ \end{array} $$

  (48)

• Case 4 Customers in Segment B purchase from firm 1 in period 1

This is symmetric to Case 3 above. Therefore:

$$ \begin{array}{*{20}c} {{p^{B}_{{12}} = \frac{{2 - k\prime \prime }} {{3{\left( {1 + \delta } \right)}}},\;p^{B}_{{22}} = \frac{{1 + k\prime \prime }} {3}}} \\ {{\pi ^{B}_{{12}} = w = \frac{{{\left( {2 - k\prime \prime } \right)}^{2} }} {9},\;\pi ^{B}_{{22}} = w\prime = \frac{{{\left( {1 + k\prime \prime } \right)}^{2} }} {9}}} \\ {{S^{\prime }_{B} = S^{\prime }_{A} }} \\ \end{array} $$

(49)

This completes the analysis of period 2 under various period 1 outcomes. Applying backward induction, we now proceed to analyze firm decisions in period 1—these decisions will depend on whether the firms adopt ST or LT strategies.

1.1.2 Analysis of Period 1.

ST–ST strategy pairing

In period 1, firms engage in Bertrand competition for each segment. Therefore, in equilibrium:

$$ \begin{array}{*{20}c} {{p^{A}_{{11}} = \pi _{{11}} ^{A} = k,{\text{ }}p^{A}_{{21}} = \pi _{{21}} ^{A} = 0}} \\ {{p^{B}_{{11}} = \pi _{{11}} ^{B} = 0,{\text{ }}p^{B}_{{21}} = \pi _{{21}} ^{B} = k}} \\ {{\pi _{{11}} = \pi _{{11}} ^{A} + \pi _{{11}} ^{B} = k,{\text{ }}\pi _{{21}} = \pi _{{21}} ^{A} + \pi _{{21}} ^{B} = k}} \\ \end{array} $$

(50)

where $p^{A}_{{11}}$ is firm 1’s price in period 1 for segment A, $p^{A}_{{21}}$ is firm 2’s price in period 1 for segment A, $p^{B}_{{11}}$ is firm 1’s price in period 1 for segment B, $p^{B}_{{21}}$ is firm 2’s price in period 1 for segment B, π ₁₁ is firm 1’s profit in period 1 and π ₂₁ is firm 2’s profit in period 1. Total firm profits under ST–ST are:

$$ \pi _{{{\text{ST}} - {\text{ST}}}} = \pi _{{11}} + \pi _{{12}} = \pi _{{11}} + \pi _{{12}} ^{A} + \pi _{{12}} ^{B} = k + x + y $$

(51)

where x and y are obtained from Eq. 37 above.

LT–LT strategy pairing

On account of Bertrand competition in period 1, firm 1 will obtain segment A and firm 2 will obtain segment B in equilibrium. Further, in equilibrium, firm 1’s price for segment A will be higher than firm 2’s price by k because segment A has a differential preference of k for firm 1’s product. Firm 2’s price will not be set lower than the level that makes firm 2 indifferent between obtaining or not obtaining segment A in period 1. Therefore:

$$ p^{A}_{{11}} = p^{A}_{{21}} + k $$

(52)

We also have

$$ p^{A}_{{21}} + w = y $$

(53)

In Eq. 53, the left hand side is the total two-period profit to firm 2 (from segment A) if segment A purchased from firm 2 in period 1. Here, w represents the period 2 profits of firm 2 from segment A (its low preference segment) if it obtains segment A in period 1. The right hand side of Eq. 53 is the total two-period profit to firm 2 from segment A if that segment did not buy from firm 2 in period 1. The two-period profits to the two firms from segment A are (with x and y as defined in Eq. 37 above) are:

$$ \begin{array}{*{20}c} {\pi ^{A}_{1} = p^{A}_{{11}} + x} \\ {\pi ^{A}_{2} = y} \\ \end{array} $$

(54)

Therefore, we have:

$$ \begin{array}{*{20}c} {p^{A}_{{21}} = y - w,{\text{ }}p^{A}_{{11}} = p^{A}_{{21}} + k = y - w + k} \\ {\pi ^{A}_{1} = p^{A}_{{11}} + x = y - w + k + x,\;\pi ^{A}_{2} = y} \\ \end{array} $$

(55)

and

$$ \begin{array}{*{20}c} {{\pi _{{{\text{LT}} - {\text{LT}}}} = \pi ^{A}_{1} + \pi ^{B}_{1} = \pi ^{A}_{1} + \pi ^{A}_{2} = k + 2y - w + x}} \\ {{ < \pi _{{{\text{ST}} - {\text{ST}}}} = k + x + y}} \\ \end{array} $$

(56)

This last inequality holds because $y = \frac{{{\left( {1 - \delta - k - \delta k} \right)}^{2} }} {9} < w = \frac{{{\left( {2 - k + \delta } \right)}^{2} }} {9}$.

LT–ST strategy pairing

When firm 1 adopts LT, the equilibrium outcomes related to segment A are identical to those in ST–ST case. This is because firm 2, which adopts ST, will continue to price at zero to segment A in the ensuing Bertrand competition. For segment B, firm 2’s lowest price is zero. Therefore, firm 1 will not price below −k. In addition firm 1 will not price below $\widehat{p}^{B}_{{11}}$, the price which makes the firm indifferent between obtaining this segment or not. Therefore:

$$ p^{B}_{{11}} = \max {\left( { - k,\widehat{p}^{B}_{{11}} } \right)} $$

(57)

where,

$$ \widehat{p}^{B}_{{11}} + w = y \Rightarrow \widehat{p}^{B}_{{11}} = y - w $$

(58)

Sub-case 1: If $p^{B}_{{11}} = \widehat{p}^{B}_{{11}}$, that is,

$$ y - w > - k $$

(59)

$$ i.e.,k > \frac{1} {{4\delta + 2\delta ^{2} }}{\left[ {{\left( A \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} - 11 + 2\delta - 4\delta - 2\delta ^{2} } \right]} $$

(60)

where

$$ A = 121 + 68\delta + 108\delta ^{2} + 32\delta ^{3} + 4\delta ^{4} $$

(61)

then firm 1 will not compete aggressively for segment B and firm 2 will still capture that segment in period 1. In this case we have:

$$ \begin{array}{*{20}l} {{\quad \quad \quad \;\;\pi ^{B}_{{12}} = y} \hfill} \\ {{\pi ^{B}_{{21}} = \widehat{p}^{B}_{{11}} + k = y - w + k} \hfill} \\ {{\quad \quad \quad \;\;\pi ^{B}_{{22}} = x} \hfill} \\ \end{array} $$

(62)

and,

$$ \pi _{{{\text{LT}} - {\text{ST}}}} = \pi ^{A}_{{11}} + \pi ^{A}_{{12}} + \pi ^{B}_{1} = k + x + y = \pi _{{{\text{ST}} - {\text{ST}}}} $$

(63)

$$ \pi _{{{\text{ST}} - {\text{LT}}}} = \pi ^{A}_{{21}} + \pi ^{A}_{{22}} + \pi ^{B}_{{21}} + \pi ^{B}_{{22}} = y + y - w + k + x = \pi _{{{\text{LT}} - {\text{LT}}}} $$

(64)

Note that this will hold for all k that satisfy Eq. 60, i.e., when:

$$ k > k_{{{\text{threshold}}}} = \frac{{ - 11 = 2\delta {\left( {1 + \delta } \right)} + {\sqrt {121 + 4\delta {\left( {17 + \delta {\left( {27 + \delta {\left( {8 + \delta } \right)}} \right)}} \right)}} }}} {{2\delta {\left( {2 + \delta } \right)}}} $$

(65)

Thus, as is evident from Eqs. 63 and 64, the symmetric strategy pairings (ST–ST and LT–LT) and the asymmetric strategy pairings (LT–ST and ST–LT) all constitute subgame perfect Nash equilibria when condition in Eq. 65 holds. However, as shown in Eq. 56 $\pi _{{{\text{ST}} - {\text{ST}}}} > \pi _{{{\text{LT}} - {\text{LT}}}}$ and hence ST–ST is the Pareto-dominant Nash equilibrium. This proves Proposition 4. ▪

Next, we consider the case where, in Eq. 57, $p^{B}_{{11}} = - k$, that is, y - w ≤  and k. In this case, firm 1 will obtain segment B in period 1. The corresponding period 1 profits are:

$$ \pi ^{{\text{B}}}_{{\text{1}}} = - k + w,\pi ^{B}_{{21}} = 0\pi ^{B}_{{22}} = w\prime $$

(66)

and,

$$ \begin{array}{*{20}c} {\pi _{{{\text{LT}} = {\text{ST}}}} = \pi ^{A}_{{11}} + \pi ^{A}_{{12}} + \pi ^{B}_{1} = k + x - k + w > \pi _{{{\text{ST}} - {\text{ST}}}} = k + x + y} \\ {\pi _{{{\text{ST}} - {\text{LT}}}} = \pi ^{A}_{{21}} + \pi ^{A}_{{22}} + \pi ^{B}_{{21}} + \pi ^{B}_{{22}} = y + w\prime < \pi _{{{\text{LT}} - {\text{LT}}}} = k + 2y - w + x} \\ \end{array} $$

(67)

Therefore, for

$$ k < k_{{{\text{threshold}}}} = \frac{{ - 11 - 2\delta {\left( {1 + \delta } \right)} + {\sqrt {121 + 4\delta {\left( {17 + \delta {\left( {27 + \delta {\left( {8 + \delta } \right)}} \right)}} \right)}} }}} {{2\delta {\left( {2 + \delta } \right)}}} $$

(68)

LT–LT constitutes the sole subgame-perfect Nash equilibrium. This proves Proposition 5. ▪

Appendix B Strategic Customers

1.1 Proofs of Proposition 6 and Proposition 7

A strategic customer is forward looking and assesses the total expected surplus across periods when choosing between firms in period 1. Effectively, customers expecting a higher surplus in period 2 from staying with a firm would require to be compensated by even lower prices during period 1 by the competitor in order to get them to switch. We analyze the possible strategy pairings below.

ST–ST strategy pairing

In period 1, a customer in segment A will buy from firm 1 if and only if

$$ 1 + k - p^{A}_{{11}} + S_{A} \geqslant 1 - p^{A}_{{21}} S\prime _{A} $$

(69)

Here, S _A (S′ _A ) is the customer’s period 2 surplus obtained by going with firm 1 (firm 2) in period 1. The remaining terms in the expression relate to period 1 surplus. Firms engage in Bertrand competition for segment A. Denote:

$$ \Delta = S_{A} - S\prime _{A} $$

(70)

where S _A and S′ _A can be obtained from Eqs. 39 and 48, by substituting δ = 0. Specifically,

$$ \Delta = S_{A} - S\prime _{A} = \frac{1} {{18}}{\left( {k^{2} + 10k - 11} \right)} + 1 - {\left[ {\frac{1} {{18}}{\left( {k^{2} + 8k - 11} \right)} + 1} \right]} = \frac{k} {9} $$

(71)

Because customers are forward looking, firms need to accommodate this period 2 surplus when setting period 1 prices. We have in equilibrium:

$$ \begin{array}{*{20}c} {{p^{A}_{{11}} = k + \Delta ,p^{A}_{{21}} = 0}} \\ {{p^{B}_{{11}} = 0,p^{B}_{{21}} = k + \Delta }} \\ {{\pi _{{11}} = \pi _{{21}} = k + \Delta }} \\ \end{array} $$

(72)

Thus the total profit of each firm in the ST–ST case is:

$$ \pi = _{{{\text{ST}} - {\text{ST}}}} = \pi _{{11}} + \pi _{{12}} = k + \Delta + x + y $$

(73)

LT–LT strategy pairing

In equilibrium, firm 1 will capture segment A in period 1 and firm 2 will capture segment B in period 1 in the ensuing Bertrand competition. In equilibrium, firm 1’s price will be higher than firm 2’s price by k + Δ, and firm 2’s price will be the one that makes it indifferent between obtaining or not obtaining this segment in period 1. Therefore, we have:

$$ \begin{array}{*{20}c} {{p^{A}_{{11}} = p^{A}_{{21}} + k + \Delta }} \\ {{p^{A}_{{21}} + w = y}} \\ \end{array} $$

(74)

and,

$$ \begin{array}{*{20}c} {{\pi ^{A}_{1} = p^{A}_{{11}} + x}} \\ {{\pi ^{A}_{2} = y}} \\ \end{array} $$

(75)

where $\pi ^{A}_{i}$ is firm i’s total profit from segment A in both periods. Therefore, we have:

$$ \begin{array}{*{20}c} {p^{A}_{{21}} = y - w,\;p^{A}_{{11}} = p^{A}_{{21}} + k + \Delta = y - w + k + \Delta } \\ {\pi ^{A}_{1} = p^{A}_{{11}} + x = y - w + k + \Delta + x,\;\pi ^{A}_{2} = y} \\ \end{array} $$

(76)

and,

$$ \pi _{{{\text{LT}} - {\text{LT}}}} = \pi ^{A}_{1} + \pi ^{B}_{1} = \pi ^{A}_{1} + \pi ^{A}_{2} = k + \Delta + 2y - w + x < \pi _{{{\text{ST}} - {\text{ST}}}} = k + \Delta + x + y $$

(77)

This last inequality holds because, as can be seen from Eqs. 9 and 15, y < w

LT–ST strategy pairing

The equilibrium for segment A is the same as in the ST–ST case when firm 1 adopts LT. This is because firm 2 will continue to price at zero in this Bertrand competition. For segment B, firm 2’s lowest price is zero. Therefore firm 1 will not price below (−k − Δ). In addition, firm 1 will not price below $\widehat{p}^{B}_{{11}}$, which makes it indifferent between obtaining this segment or not. Therefore:

$$ p^{B}_{{11}} = \max {\left( { - k - \Delta ,\widehat{p}^{B}_{{11}} } \right)} $$

(78)

where,

$$ \widehat{p}^{B}_{{11}} = w = y \Rightarrow \widehat{p}^{B}_{{11}} = y - w $$

(79)

If $p^{B}_{{11}} = \widehat{p}^{B}_{{11}}$, that is, y − w  > − k − Δ, firm 2 will still obtain segment B. Substituting from Eqs. 9, 15, and 71 for y, w and Δ respectively, the condition y − w  > − k − Δ simplifies to k > (1/4). When this condition holds, we have:

$$ \begin{array}{*{20}c} {{\pi ^{B}_{1} = y}} \\ {{\pi ^{B}_{{21}} = \widehat{p}^{B}_{{11}} + k + \Delta }} \\ {{\pi ^{B}_{{22}} = x}} \\ \end{array} $$

(80)

and

$$ \pi _{{{\text{LT}} - {\text{ST}}}} = \pi ^{A}_{{11}} + \pi ^{A}_{{12}} + \pi ^{B}_{1} = k + \Delta + x + y = \pi _{{{\text{ST}} - {\text{ST}}}} $$

(81)

$$ \pi _{{{\text{ST}} - {\text{LT}}}} = \pi ^{A}_{{21}} + \pi ^{A}_{{22}} + \pi ^{B}_{{21}} + \pi ^{B}_{{22}} = y + y - w + k + \Delta + x = \pi _{{{\text{LT}} - {\text{LT}}}} $$

(82)

Therefore, when k > (1/4), the symmetric strategic pairings (ST–ST and LT–LT) and the asymmetric strategy pairings (LT–ST and ST–LT) all constitute subgame-perfect Nash equilibria. However, as can be seen by comparing Eqs. 81 and 82, given that $y = \frac{{{\left( {1 - k} \right)}^{2} }} {9} < w = \frac{{{\left( {2 - k} \right)}^{2} }} {9}$, ST–ST is the Pareto-dominant Nash equilibrium. This proves Proposition 6. ▪

Next, we consider the case where, in Eq. 78, $p^{B}_{{11}} = - k - \Delta$. This holds when y − w ≤ − k − Δ or k < (1/4). In this case, firm 1 will obtain segment B in period 1. We have:

$$ \begin{array}{*{20}c} {\pi ^{B}_{1} = - k - \Delta + w} \\ {\pi ^{B}_{{21}} = 0,\pi ^{B}_{{22}} = w\prime } \\ \end{array} $$

(83)

and:

$$ \pi _{{{\text{LT}} - {\text{ST}}}} = \pi ^{A}_{{11}} + \pi ^{A}_{{12}} + \pi ^{B}_{1} = k + x - k - \Delta + w > \pi _{{{\text{ST}} - {\text{ST}}}} = k + x + y $$

(84)

$$ \pi _{{{\text{ST}} - {\text{LT}}}} = \pi ^{A}_{{21}} + \pi ^{A}_{{22}} + \pi ^{B}_{{21}} + \pi ^{B}_{{22}} = y + w\prime < \pi _{{{\text{LT}} - {\text{LT}}}} = k + \Delta + 2y - w + x $$

(85)

Therefore, for k < (1/4), LT–LT constitutes the sole subgame–perfect Nash equilibrium. This proves Proposition 7. ▪