Sequential innovation, naked exclusion, and upfront lump-sum payments

Abstract

We present a potentially benign naked exclusion mechanism that can be applied to sequential innovation; a non-patentable original innovation by the incumbent supplier fosters derivative innovation by rivals. In the absence of an appropriate legal framework, the original innovator’s equilibrium exclusivity contracts block subsequent efficient entry even if there is (leader–follower) competition in the contracting phase. However, the legal framework may maximize social welfare by imposing a ban on upfront lump-sum payments in exclusivity contracts (by all suppliers) combined with an outright ban on exclusivity contracts by the derivative innovator. The former ban precludes the exclusion of socially beneficial derivative innovation by causing the incumbent supplier to resort to accommodation, rather than to pure exclusion, strategies. The latter ban complements the former by preventing inefficient or excessive derivative innovation.

Appendices

Appendix

1.1 Equilibrium in the subgames of stages 3, 4, 5, 6 and 7

To solve the game we proceed by backward induction. In stage 7 a customer that is bound by a pure exclusivity contract purchases the product from its exclusive supplier (if the latter has entered the market) at the monopoly price, r. If in stage 7 only incumbent supplier I has entered the market, it sells to all free customers that have not signed exclusivity contracts at the monopoly price, r. If, on the other hand, both suppliers I and E have entered into the market, they compete in the stage-7 market for such free customers; in equilibrium a free customer buys from low-cost supplier E at a price c. Suppose now that in stage 7 a customer i has already signed I’s (finalized) partial exclusivity contract with a termination penalty $t_i$ and is now offered unit prices $p_i$ by incumbent I and ${p_{i}^{\prime }}$ by derivative innovator E. Then, customer i decides to buy from E if $p_i^{\prime }\le p_i -t_i$ and from I otherwise. Competition between I and E for such a customer drives the stage-7 price that customer i pays to E to $\min \{c,r-t_i \}$. Of course, in case $r-t_i <\underline{c}$, such a customer does not terminate its contract with I.

In stage 6 E enters into the market when first, I has entered into the market in stage 5 (which is a precondition for E’s entry), and second, E can recoup its upfront fixed cost $\underline{F}$.³³ Suppose, for example, that in stage 4 I and E finalized $n^{I}$ and $n^{E}$ pure exclusivity contracts with customers, respectively ($n^{I}+n^{E}\le N$). Then, E enters into the market in stage 6 if $n^{E}(r-\underline{c})+(N-n^{I}-n^{E})(c-\underline{c})>\underline{F}$. Suppose now that I finalized $n^{I}$ partial (rather than pure) exclusivity contracts with customers in stage 4 with $r-t_i \ge \underline{c}$, $\forall i\in \{1,\ldots ,n^{I}\}$. Then, E enters the market in stage 6 if $\sum _{i=1}^{n^{I}} {(\min \{c,r-t_i \}-\underline{c})} +n^{E}(r-\underline{c})+(N-n^{I}-n^{E})(c-\underline{c})>\underline{F}$.

Similarly, in stage 5 I enters into the market when it can recover its upfront fixed cost F.³⁴ In case I expects E not to enter in subsequent stage 6, and E has finalized $n^{E}$ exclusivity contracts with customers, I expects to monopolize the remaining $N-n^{E}$ customers and thus decides to enter if $(N-n^{E})(r-c)\ge F$. Suppose now that I expects E to enter. If I has finalized $n^{I}$ pure exclusivity contracts in stage 4, it enters into the market in stage 5 if $n^{I}(r-c)\ge F$. If, however, I has finalized $n^{I}$ partial exclusivity contracts in stage 4 with $r-t_i \ge \underline{c}$, $\forall i\in \{1,\ldots ,n^{I}\}$, I enters in stage 5 if $\sum _{i=1}^{n^{I}} {t_i } \ge F$.

In stage 4 I decides to finalize its contracts that have been accepted (in stage 3) by customers if first, I expects to subsequently enter in stage 5 (i.e., if it expects to be able to recover its fixed cost F, as we explained above), and second, I also expects to recoup the stage-4 upfront lump-sum payments, $\sum _{i=1}^{n^{I}} {x_i }$, that it makes to customers when contracts are finalized; such lump-sum payments are sunk. Similarly, E decides to finalize the contracts that have been accepted (in stage 3) by customers if first, E expects to subsequently enter in stage 6 (as we explained above) and second, E also expects to recover the stage-4 (sunk) upfront lump-sum payments, $\sum _{i=1}^{n^{E}} {z_i }$.

In stage 3, given that customers can coordinate as in Bernheim et al. (1987), a sufficient number of customers accept I’s contracts so that the subsequent entry of I is ensured (if, of course, the contracts that I offered in stage 1 make such an accommodation of I’s entry by at least one group of consenting customers possible). There can be no stage-3 equilibrium in which an insufficient number of customers accept I’s contracts so that I’s entry is rendered non-viable. In such a hypothetical equilibrium an improving self-enforcing coalition of non-signing customers would deviate, signing I’s contracts and securing I’s entry; otherwise, there would be no available products (by I or E), and customer payoffs would be zero. Once the entry of I is secured, a sufficient number of customers consent to derivative innovator E’s contracts to ensure the subsequent entry of E provided that first, E’s proposed contracts are more favorable to such customers than I’s proposed contracts and second, it is indeed feasible to ensure E’s entry given the proposed contracts. If, however, E is expected to stay out of the market each customer i consents to I’s contract, subsequently earning a surplus $x_i \ge 0$ (as compared to a zero surplus if the customer does not consent).

Proof of Lemma 1

Suppose that in stage 2 E sets its lump-sum payments equal to $z_i =x_i$, $\forall i\in \{1,\ldots ,N^{I}\}$, and $z_i =0$, $\forall i\in \{N^{I}+1,\ldots ,N\}$. For E this is the minimum level of lump-sum payments that is necessary for eliciting contract acceptance from customers; if E offered a strictly smaller lump-sum payment to a customer, it would never convince the customer to consent to E’s contract (given I’s competing contracts).³⁵ Then, I’s contracts fail to block the subsequent entry of E if and only if in stage 3 there exists a subgroup of $n\in \{0,1,\ldots ,N^{I}-1\}$ customers that are able to allow the finalization of E’s contracts and the realization of E’s entry by consenting to E’s (rather than to I’s) contracts ($\sum _{i=1}^n {(r-x_i -\underline{c})} +(N-N^{I})(r-\underline{c})-\underline{F}=\sum _{i=1}^n {(r-x_i -c)} +n(c-\underline{c})+(N-N^{I})(r-\underline{c})-\underline{F}>0$), while also ensuring the finalization of I’s contracts and the realization of I’s entry ($\sum _{i=1}^{N^{I}-n} {(r-x_i -c)} -F\ge 0$).³⁶

If such a group of n customers existed and indeed signed E’s contracts, its members would have no incentive to deviate (and buy from I, rather than from E) individually or jointly since I’s entry would be ensured anyway (by the $N^{I}-n$ customers), and according to the tie-breaking convention, if a customer is indifferent between I and E, it chooses E.³⁷ If a subcoalition of those n customers deviated from buying from E, its members would be made strictly worse off. Furthermore, at least a subcoalition of the remaining $N^{I}-n$ customers would indeed sign I’s contracts so that I’s entry is ensured; otherwise, I (as well as E) would stay out of the market, and no products would be offered to customers. Thus given the above I’s and E’s contract offers in stages 1 and 2, there exists at least one coalition-proof equilibrium in the stage-3 subgame that allows E to enter into the market. Similarly, in case E’s entry was prevented (so that all the $N^{I}$ customers signed I’s contracts, and no customer signed E’s contracts, expecting E to stay out of the market), at least one possible improving self-enforcing coalition—i.e., the above group of n customers—would have an incentive to jointly deviate and sign E’s contracts, accommodating the entry of E. It follows that there exists no coalition-proof equilibrium in the stage-3 subgame that entails the exclusion of E from the market.

As a result, I’s contracts are successful in blocking the subsequent entry of E if and only if there exists no such group of n customers that are able to allow the entry of both suppliers by signing E’s contracts. In particular, given I’s contract offers, the maximum surplus that the two suppliers, I and E, together can extract from all the N customers is $Nr-\sum _{i=1}^{N^{I}} {x_i }$; as we explained above, such maximum surplus extraction is attained when E sets its lump-sum payments equal to $z_i =x_i$, $\forall i\in \{1,\ldots ,N^{I}\}$, and $z_i =0$, $\forall i\in \{N^{I}+1,\ldots ,N\}$.³⁸ Then, if even in the presence of such maximum surplus extraction on the part of the two suppliers, there exists no possible group of n customers that (by signing E’s contracts) can ensure the viability of both E’s and I’s entry, I’s contracts can successfully block the entry of E on any occasion. It follows that the entry of E is blocked if there is no group of n customers so that $\sum _{i=1}^n {(r-x_i -c)} +n(c-\underline{c})+(N-N^{I})(r-\underline{c})-\underline{F}>0$ and $\sum _{i=1}^{N^{I}-n} {(r-x_i -c)} -F=0$, or so that $V+n(c-\underline{c})+(N-N^{I})(r-\underline{c})-F-\underline{F}>0,$ where $V=\sum _{i=1}^{N^{I}} {(r-c-x_i )}$ and $\sum _{i=1}^{N^{I}-n} {(r-x_i -c)} -F=0$.

In stage 3 let $n^{L}$ be the number of customers that have been offered the lowest lump-sum payments $x_i$ by I (i.e., I’s most lucrative potential contract participants) and that can accommodate I’s entry by consenting to I’s contracts, i.e., $\sum _{i=1}^{n^{L}} {(r-c-x_i )} -F=0$. Since $\partial [V+n(c-\underline{c})+(N-N^{I})(r-\underline{c})-F-\underline{F}]/\partial n=c-\underline{c}>0$, the preferable (or most profitable) coalition of customers that E can capture in its effort to ensure the viability of its entry (without preventing the entry of I) is the group of $N-n^{L}$ customers. In particular, E maximizes its profit when I ensures the viability of its entry by selling to a small number, $n^{L}$, of lucrative customers so that E can sell to the largest possible number of remaining customers, thereby taking full advantage of its lower unit cost; thus E can earn the largest possible $(N-n^{L})(c-\underline{c})$ in addition to stealing I’s potential rent from those $N-n^{L}$ customers. It follows that I successfully blocks the entry of E if and only if $V+(N^{I}-n^{L})(c-\underline{c})+(N-N^{I})(r-\underline{c})-F-\underline{F}\le 0$.

Thus in an exclusion subgame I offers pure exclusivity contracts in stage 1 so that $V+(N^{I}-n^{L})(c-\underline{c})+(N-N^{I})(r-\underline{c})-F-\underline{F}=0$.³⁹ If in stage 3 customers rationally expect E to stay out of the market, all the $N^{I}$ customers consent to I’s contracts (as we explain in Appendix on the equilibrium of stage 3), and I’s profit is $N(r-c)-\sum _{i=1}^{N^{I}} {x_i } -F$. Suppose now that I makes uniform contract offers to all N customers, i.e., $x_i =x$, $\forall i\in \{1,\ldots ,N\}$, which implies that $n^{L}=F/(r-c-x)$. Then, $V+(N^{I}-n^{L})(c-\underline{c})+(N-N^{I})(r-\underline{c})-F-\underline{F}$, or $N(r-\underline{c}-x)-F(c-\underline{c})/(r-c-x)-F-\underline{F}$, is strictly decreasing in x (since $-N-F(c-\underline{c})/(r-c-x)^{2}<0)$. Furthermore, $N(r-\underline{c}-x)-F(c-\underline{c})/(r-c-x)-F-\underline{F}$ is strictly positive when $x=0$ and strictly negative when $x=r-c-F/N$. There thus exists a unique $x^*\in (0,r-c-F/N)$ so that $N(r-\underline{c}-x)-F(c-\underline{c})/(r-c-x)-F-\underline{F}=0$.⁴⁰ It follows that if in stage 1 I offers pure exclusivity contracts $x_i =x^*$, $\forall i\in \{1,\ldots ,N\}$, E’s entry is blocked (regardless of E’s strategy in stage 2), and all N customers sign I’s contracts in stage 3. In such an exclusion strategy, I’s profit is $N(r-c)-Nx^*-F>0$.⁴¹ Overall, such contract offers (i.e., $x_i =x^*$, $\forall i\in \{1,\ldots ,N\})$ constitute I’s optimal stage-1 pure exclusion strategy, i.e., the unique strategy that maximizes I’s profit while also preventing E’s entry.

In particular, suppose that although I still offers the same total amount of lump-sum payments to customers, i.e., $\sum _{i=1}^N {x_i } =Nx^*$, it does not make uniform contract offers, i.e., there exist at least two customers i and j so that $x_i \ne x_j$. This implies that $n^{L}$ is strictly smaller than under the uniform contract arrangement, i.e., $n^{L}<F/(r-c-x^*)$. The average lump-sum payment in the group of the $F/(r-c-x^*)$ customers that have been offered the smallest lump-sum payments is strictly smaller than the average, $x^*$, in the group of all N customers; I’s entry can be accommodated if strictly less then $F/(r-c-x^*)$ customers consent to I’s contracts. Thus if $\sum _{i=1}^N {x_i } =Nx^*$ and I’s contract offers are not uniform, the entry of E is not blocked. Similarly, if I does not make contract offers to all customers, i.e., if $N^{I}<N$ (although we still have $\sum _{i=1}^{N^{I}} {x_i } =Nx^*)$, E has the opportunity to relinquish its right to sell to the $N-N^{I}$ customers, effectively granting I monopoly power over them (i.e., effectively setting $x_i =0$, $\forall i\in \{1,\ldots ,N-N^{I}\})$. Then, the average lump-sum payment in the group of the $F/(r-c-x^*)$ customers that have been offered the smallest lump-sum payments (including the $N-N^{I}$ customers for which $x_i$ is effectively zero) is strictly smaller than the average, $x^*$, in the group of all N customers. As before, the entry of E is not blocked.

Suppose now that $\sum _{i=1}^N {x_i } =Nx^{\prime }>Nx^*$, and that the entry of E is successfully blocked.⁴² I’s profit will be strictly smaller than in the optimal exclusion strategy, i.e., $N(r-c)-Nx^{\prime }-F<N(r-c)-Nx^*-F$. Furthermore, suppose that $\sum _{i=1}^N {x_i } =Nx^{\prime }<Nx^*$. Then, the above analysis implies that I is unable to block the entry of E even if I makes uniform contract offers $x_i =x^{\prime }$, $\forall i\in \{1,\ldots ,N\}$ (and, of course, if I does not make uniform offers). It follows that I’s unique optimal exclusion strategy is offering $x_i =x^*$, $\forall i\in \{1,\ldots ,N\}$ in stage 1. Lemma 1 follows.

Proof of Lemma 2

(i) Suppose that in stage 1 I offers partial exclusivity contracts with $x_i =0$, $t_i =r-c$, $\forall i\in \{1,\ldots ,N\}$. Suppose also that in stage 2 E does not offer any contracts to customers. Then, in the equilibrium of the stage-3 subgame $F/(r-c)$ customers sign I’s contracts, while the remaining $N-F/(r-c)$ do not sign. In stage 4 I decides to finalize the contracts (and in stage 5 I decides to enter into the market) since $t_i F/(r-c)-F=F-F=0$. In stage 6 E also decides to enter since $N(c-\underline{c})-\underline{F}>0$. We can see that given the contract offers of I and E in stages 1 and 2, the unique stage-3 coalition-proof equilibrium indeed entails $F/(r-c)$ signing customers.

In particular, in stage 3 each of the $F/(r-c)$ customers that consent to I’s contracts earns a zero surplus, while each of the $N-F/(r-c)$ customers that do not consent earns a strictly positive surplus ($r-c>0$). Thus no subcoalition of the group of the $N-F/(r-c)$ non-signing customers would have an incentive to deviate (and consent to I’s contract) since its members would be made strictly worse off, earning a zero surplus. Furthermore, even if one of the $F/(r-c)$ signing customers deviated (refusing to consent to I’s contract), I would be unable to recover its fixed cost F and would stay out of the market; no products would be offered to customers. Thus no subcoalition of the $F/(r-c)$ signing customers would have an incentive to deviate. Similarly, in case strictly less (strictly more) than $F/(r-c)$ customers consented to I’s contracts, an improving self-enforcing coalition of customers would have an incentive to deviate by signing (not signing) I’s contracts so that there are exactly $F/(r-c)$ signing customers.

We can also see that given I’s contract offers in stage 1, E’s optimal stage-2 strategy is to offer no contracts to customers; E is unable to earn a strictly larger profit by choosing any other stage-2 strategy. In particular, to elicit contract acceptance by one of the $N-F/(r-c)$ customers that are expected to decline I’s contracts and buy in the stage-7 market (at a price c), E would have to offer a lump-sum payment $z_i \ge r-c$, which leads to a weakly smaller profit for E than the strategy above. Overall, it follows that if I offers partial exclusivity contracts with $x_i =0$, $t_i =r-c$, $\forall i\in \{1,\ldots ,N\}$ in stage 1, the entry of both I and E is always accommodated in the equilibrium of the subgame. Thus there exists at least one feasible accommodation strategy for I (i.e., the above strategy) in stage 1. Lemma 2(i) follows.

(ii) Suppose that given I’s offers of partial exclusivity contracts in stage 1 and E’s offers of pure exclusivity contracts in stage 2, n customers consent to I’s contracts in stage 3, all contracts are finalized in stage 4, and both I and E decide to enter into the market in stages 5 and 6. Furthermore, suppose that $\sum _{i=1}^n {(t_i -x_i )} >F$, i.e., I earns a strictly positive profit. In such a subgame equilibrium, although each of the n customers is not individually pivotal for the finalization of I’s contracts and the realization of I’s entry (since $\sum _{i=1}^n {(t_i -x_i )} >F)$, it is apparently better off buying from I under the terms of I’s offered contract than buying from E under the terms of E’s offered contract or buying from E without a contract at a price c in the stage-7 market.

Suppose now that in stage 2 E deviates by offering a contract with a lump-sum payment $z_i =\max \{r-c-t_i +x_i, x_i \}$ to one of the above n customers. In stage 3 such a customer would choose to consent to E’s contract since according to the tie-breaking convention, if a customer is indifferent between I and E, it chooses to buy from E, and I’s entry is ensured anyway by the remaining $n-1$ customers (given that $\sum _{i=1}^n {(t_i -x_i )} >F)$. Furthermore, E’s profit is increased by capturing such a customer (i.e., $r-z_i -\underline{c}=\min \{c+t_i -x_i, r-x_i \}-\underline{c}\ge 0)$ as long as $t_i \ge x_i$, i.e., as long as the customer’s contract with I was not loss-making for I. Thus E indeed has an incentive to make such a deviation in stage 2 to “steal” I’s non-loss-making customer, which implies that initial player strategies after stage 1 did not constitute an equilibrium. We can also see that since all other contracts offers by I and E are unchanged, the remaining $n-1$ customers indeed continue to consent to I’s contracts as before (since I’s contracts are still preferable to E’s contracts or to no contracts at all), ensuring the entry of I. Overall, it follows that there exists no possible accommodation strategy for I in stage 1 that leads to a strictly positive profit for I in the equilibrium of the subgame. Lemma 2(ii) follows.

Proof of Proposition 2

Suppose that in stage 1 I offers pure exclusivity contracts with $x_i =0$, $\forall i\in \{1,\ldots ,N^{I}\}$, $N^{I}\le N$ (since $x_i >0$ are banned). By following the same procedure as the proof of Lemma 1, we can see that E is always able to secure its entry. There exist feasible stage-2 strategies for E (e.g., offering pure exclusivity contracts with $z_i =x_i =0$, $\forall i\in \{1,\ldots ,N\})$ so that a self-enforcing coalition of $n\le N-F/(r-c)$ customers always ensures E’s entry by accepting E’s contracts in stage 3. Similar to the proof of Lemma 2(i) $F/(r-c)$ customers consent to I’s contracts in stage 3, while the remaining $N-F/(r-c)$ buy from E (either by signing E’s contracts or by waiting to buy from E without a contract in stage 7). It follows that if lump-sum payments are banned, there are no feasible exclusion strategies for I in stage 1.⁴³ Furthermore, along the lines of the proof of Lemma 2(ii), since the entry of E is accommodated, the equilibrium profit of I is zero.⁴⁴ In this subgame social welfare—i.e., the sum of I’s profit ($F(r-c)/(r-c)-F=0$), E’s profit ($n^{E}(r-\underline{c})+[N-F/(r-c)-n^{E}](c-\underline{c})-\underline{F}$, where $n^{E}$ customers have finalized contracts with E) and customer surplus ($[N-F/(r-c)-n^{E}](r-c)$)—is $[N-F/(r-c)](r-\underline{c})-\underline{F}$.

As the proof of Lemma 2(i) shows, even after a prohibition on upfront lump-sum payments, there exists at least one feasible accommodation strategy (i.e., the strategy outlined in the proof of Lemma 2(i)) for I that entails the use of partial exclusivity contracts with $t_i \le r-\underline{c}$, $\forall i\in \{1,\ldots ,N\}$ (see note 19) and $N^{I}\le N$, in which all N customers buy from E in equilibrium. As Lemma 2(ii) also shows, in all such accommodation subgames the equilibrium profit of I is zero. Suppose that in such a subgame I and E finalize contracts with $n^{I}$ and $n^{E}$ customers, respectively. Then, social welfare—i.e., the sum of I’s profit ($\sum _{i=1}^{n^{I}} {t_i } -F$), E’s profit ($\sum _{i=1}^{n^{I}} {[\min \{c,r-t_i \}-\underline{c}]} +\sum _{i=1}^{n^{E}} {(r-\underline{c})} +(N-n^{I}-n^{E})(c-\underline{c})-\underline{F}$ and customer surplus ($\sum _{i=1}^{n^{I}} {[r-\min \{c+t_i, r\}]} +(N-n^{I}-n^{E})(r-c)$—is $N(r-\underline{c})-F-\underline{F}$, which is strictly larger than social welfare in the subgame where I offers pure exclusivity contracts ($N(r-\underline{c})-F-\underline{F}>[N-F/(r-c)](r-\underline{c})-\underline{F}$). Since a ban on upfront lump-sum payments always leads to a zero equilibrium profit for I anyway, I chooses to offer partial exclusivity contracts in stage 1 so that E sells to all N customers in equilibrium (given that according to the tie-breaking convention, if I is indifferent between two strategies, it chooses the strategy with the larger social welfare). Proposition 2 follows.

Proof of Proposition 3

(i) As the appendix on the equilibrium of stage 7 implies, if in stage 6 E enters into the market without having any exclusivity contracts with customers, in stage 7 it obtains a price c from each free customer without any contract and a price $\min \{c,r-t_i \}$ from a customer i that has a partial exclusivity contract with I. Thus E’s profit is always weakly smaller than $N(c-\underline{c})$. It follows that if E is banned from offering exclusivity contracts, it never decides to enter into the market in stage 6 if $N(c-\underline{c})-\underline{F}\le 0$.

(ii) Similar to the proof of Proposition 2.

Equilibrium if only incumbent supplier I is able to offer exclusivity contracts

Suppose that in stage 1 I offers pure exclusivity contracts to $N^{I}\le N$ customers. Along the lines of the proof of Lemma 1, let $n^{L}$ be the number of customers that have been offered the lowest lump-sum payments $x_i$ by I and that can accommodate I’s entry by consenting to I’s contracts, i.e., $\sum _{i=1}^{n^{L}} {(r-c-x_i )} -F=0$. I can successfully block the entry of E if and only if $(N-n^{L})(c-\underline{c})-\underline{F}\le 0$; in this case, I sells to all N customers and earns a profit that is equal to $N(r-c)-\sum _{i=1}^{N^{I}} {x_i } -F=(N-n^{L})(r-c)-\sum _{i=1}^{N^{I}-n^{L}} {x_i }$. Then, in an exclusion subgame I maximizes its profit if $n^{L}$ is minimized subject to the constraint $(N-n^{L})(c-\underline{c})-\underline{F}=0$, and also $N^{I}=n^{L}$, i.e., if $n^{L}=N-\underline{F}/(c-\underline{c})=N^{I}$. Therefore, I’s optimal pure exclusion strategy in stage 1 entails the offer of pure exclusivity contracts to $N-\underline{F}/(c-\underline{c})$ customers so that $\sum _{i=1}^{N-\underline{F}/(c-\underline{c})} {x_i } =[N-\underline{F}/(c-\underline{c})](r-c)-F$ and no offers to the remaining $\underline{F}/(c-\underline{c})$ customers.⁴⁵ Along the lines of the proof of Lemma 1, we can see that in stage 3 each of the $N-\underline{F}/(c-\underline{c})$ customers consents to I’s contracts. I’s profit is $\Pi _{EX}^I {}^{**}=\underline{F}(r-c)/(c-\underline{c})>0$.

Suppose now that in stage 1 Ioffers partial exclusivity contracts to customers, and in stage 3 customers expect that both I and E will subsequently enter into the market.⁴⁶ $n^{1}$ customers are offered $t_i \le r-c$, while $N-n^{1}$ customers are offered $t_i \in (r-c,r-\underline{c}]$ (and effectively $t_i =0$ for customers that are not offered contracts). Furthermore, $x_i =\min \{t_i, r-c\}$ since for strictly smaller $x_i$, a customer i would refuse to accept I’s contract in stage 3 (buying instead without a contract at a price c in stage 7). As the appendix on the equilibrium of stage 7 shows, the stage-7 price that customer i pays to E is $\min \{c,r-t_i \}$; E’s profit is $\Pi ^{E}=\sum _{i=1}^N {[\min \{c,r-t_i \}-\underline{c}]} -\underline{F}$. Given that all I’s contracts are accepted by customers and also I’s entry is secured, the entry of E is accommodated ($\Pi ^{E}\ge 0$) if and only if $n^{1}(r-c)+\sum _{i=1}^{N-n^{1}} {t_i } \le N(r-\underline{c})-\underline{F}$. In such a subgame all customers accept I’s partial exclusivity contracts. In particular, along the lines of previous proofs, a sufficient number of customers always consent to I’s contracts to secure I’s entry. In addition, given that the entry of both I and E is secured, each customer consents to I’s partial exclusivity contract since $x_i =\min \{t_i, r-c\}$ (and in case of a tie a customer decides to consent to a contract). The profit of I is $\sum _{i=1}^N {(t_i -x_i )} -F=\sum _{i=1}^{N-n^{1}} {t_i } -(N-n^{1})(r-c)-F$.

It follows that the maximum profit of I in an accommodation subgame is $\Pi _{AC}^I {}^{**}=N(c-\underline{c})-\underline{F}-F$ and is attained when I offers partial exclusivity contracts with $t_i \le r-\underline{c}$ and $x_i =\min \{t_i, r-c\}$ so that $n^{1}(r-c)+\sum _{i=1}^{N-n^{1}} {t_i } =N(r-\underline{c})-\underline{F}$, or so that E’s entry is exactly accommodated. Such a strategy allows I to extract the entire surplus from E’s derivative innovation through the collection of termination penalties after customers breach their contracts with I.⁴⁷ I chooses to follow the optimal pure exclusion (accommodation) strategy when $\Pi _{EX}^I {}^{**}-\Pi _{AC}^I {}^{**}>0$ ($\Pi _{EX}^I{}^{**}-\Pi _{AC}^I {}^{**}\le 0$).