Acquisitions in a Patent Contest Model with Large and Small Firms

Abstract

Big companies and small innovation factories possess different advantages in a patent contest. While large firms typically have better access to product markets, small firms often have a superior R&D efficiency. These distinct advantages immediately lead to the question of cooperations between firms. In this paper, we model a patent contest with heterogeneous firms. In a pre-contest acquisition game large firms bid sequentially for small firms to combine respective advantages. Sequential bidding allows the first large firms to bid strategically to induce a reaction of its competitor. For high efficiencies both large firms prefer to acquire immediately leading to a symmetric market structure. For low efficiencies strategic waiting of the first large firm leads to an asymmetric market structure even though the initial situation is symmetric. We also discuss two different timing setups of the acquisition stage. In all setups, acquisitions increase the chances for a successful innovation.

Appendix

Proof of Proposition 1

To prove Proposition 1, the acquisition game is solved by backward induction, taking optimal decisions in the R&D stage as given.

At decision point 16 S ₂ has to decide whether to accept the price $p_{22}^C$ offered by L ₂ or denying the offer and getting a payoff of $\frac{\alpha V }{16}$. Thus, S ₂ accepts every offer $p_{22}^C\geq p_N$ and denies any other offer.

Expecting this decision from S ₂, L ₂ makes her offer in decision point 14. If S ₂ denies the offer, the game ends without any acquisition and L ₂ gets the payoff $\frac{\alpha V}{16}$. If S ₂ accepts, L ₂ gets the payoff of R&D game 8, the triopoly with one merged, one small, and one large firm. If L ₂ decides to acquire, it makes the lowest offer S ₂ accepts, i.e., $p_{22}^C=p_N$. Thus, L ₂ decides to make an offer if

$$ \frac{\alpha x V}{(1+2x)^2}(2x-1)^2-\frac{\alpha V}{16}\geq \frac{\alpha V}{16}. $$

This inequality holds for x ≥ 1.0333.

If S ₂ accepts the offer in decision point 12, it gets the payoff $p_{12}^C$. If not, it either gets $p_{22}^C=p_N$ if x ≥ 1.0333 resulting from an acceptable offer by L ₂ or $\frac{\alpha V}{16}$ if x < 1.0333 as an independent firm in R&D game 9. Thus, S ₂ accepts any offer by L ₁ that is greater or equal than p _N .

In decision point 9 L ₁ has to offer at least $p_{12}^C=p_N$ if it wants S ₂ to accept. If S ₂ accepts the bid, L ₁’s payoff is

$$ \frac{\alpha x V}{(1+2x)^2}(2x-1)^2-p_{12}^C. $$

If S ₂ denies the offer, L ₁ gets the payoff of an outsider in R&D game 8 when x ≥ 1.0333, i.e., in the situation where L ₂ decides to acquire in decision point 14. This payoff is

$$ \frac{\alpha V}{(1+2x)^2}. $$

If x < 1.0333, S ₁ gets $\frac{\alpha V}{16}$ as payoff. Consider first the case x ≥ 1.0333: L ₁ acquires if

$$ \frac{\alpha x V}{(1+2x)^2}(2x-1)^2-\frac{\alpha V}{16}\geq\frac{\alpha V}{(1+2x)^2}. $$

This inequality holds for x ≥ 1.10933. In the case x < 1.0333 L ₁ acquires if

$$\frac{\alpha x V}{(1+2x)^2}(2x-1)^2-\frac{\alpha V}{16}\geq\frac{\alpha V}{16}. $$

This inequality never holds since x < 1.0333. Thus, L ₁ decides to acquire in decision point 9 if x ≥ 1.10933.

In decision point 15 S ₂ gets the payoff of an outsider in R&D game 7 if it denies the offer. Therefore S ₂ accepts every offer $p_{22}^B\geq p_O.$

In decision point 13 L ₂ has already acquired S ₁ for p ₂₁. If L ₂ acquires S ₁, the situation is an asymmetric duopoly where L ₂ has acquired both small firms. L ₂’s payoff if it acquires is

$$ \frac{\alpha x^3V}{(1+x)^2}-p_{21}-p_{22}^B, $$

where $p_{22}^B=p_O$ is the lowest offer S ₂ accepts. If L ₂ does not acquire, its payoff is the payoff of the merged firm in R&D game 7 minus the price paid for S ₁:

$$ \frac{\alpha x V}{(1+2x)^2}(2x-1)^2-p_{21}. $$

Thus, L ₂ acquires if

$$ \frac{\alpha x^3V}{(1+x)^2}-p_{21}-\frac{\alpha V}{(1+2x)^2} \geq \frac{\alpha x V}{(1+2x)^2}(2x-1)^2-p_{21}. $$

This inequality holds for every x ≥ 1 and therefore L ₂ always bids $p_{22}^B= p_O$ and acquires S ₂ in decision point 13.

In decision point 11 S ₂ accepts any offer $p_{12}^B \geq p_O.$

In decision point 8 L ₁ has to offer at least $p_{12}^B=p_O$ if it wants S ₂ to accept. If S ₂ accepts, the resulting situation is R&D game 4, a symmetric duopoly. If S ₂ denies the offer, L ₁ gets the profit of an outsider in an asymmetric duopoly in R&D game 6 since L ₂ always makes an offer that S ₂ accepts in decision point 13. L ₁ therefore decides to offer $p_{12}^B=p_O$ and to acquire S ₂ if

$$ \frac{\alpha x V }{4}-\frac{\alpha V}{(1+2x)^2} \geq \frac{\alpha V}{(1+x)^2}. $$

This inequality holds for x ≥ 1.18731.

If S ₁ denies the offer in decision point 6, the acquisition game ends in an asymmetric triopoly (where S ₁ is an outsider) for x ≥ 1.0333 and in R&D game 9 for x < 1.0333. Thus, S ₁ accepts every offer p ₂₁ ≥ p _O for x ≥ 1.0333 and every offer p ₂₁ ≥ p _N for x < 1.0333.

If L ₂ decides to acquire S ₁ in decision point 4, the acquisition game ends in R&D game 4 for x ≥ 1.18731 and in R&D game 6 for x < 1.18731. If L ₂ does not acquire S ₁, the acquisition game ends in R&D game 5 for x ≥ 1.10933, in R&D game 8 for 1.10933 > x ≥ 1.0333, and in R&D game 9 for x < 1.0333. Consider first the case x ≥ 1.18731: The payoff for L ₂ in R&D game 4 is $\frac{\alpha x V }{4}-p_{21};$ in R&D game 5 it is $\frac{\alpha V}{(1+2x)^2}.$ The payoff in R&D game 4 is larger for the lowest accepted offer p ₂₁. For x ≥ 1.18731 L ₂ therefore bids p ₂₁ = p _O and acquires S ₁. The next case to consider is 1.18731 > x ≥ 1.10933: The payoff for L ₂ if it acquires S ₁ is given by R&D game 6 and is thus $\frac{\alpha x^3V}{(1+x)^2}-p_{21}-p_{22}^B,$ where $p_{22}^B=p_O$ and the lowest accepted offer is p ₂₁ = p _O . If L ₂ does not acquire, its payoff is given by R&D game 5 and it is $\frac{\alpha V}{(1+2x)^2}.$ Thus, L ₂ acquires if

$$ \frac{\alpha x^3V}{(1+x)^2}-2\cdot \frac{\alpha V}{(1+2x)^2} \geq \frac{\alpha V}{(1+2x)^2}. $$

This inequality holds for every 1.18731 > x ≥ 1.10933. The third case is 1.10933 > x ≥ 1.0333: The payoff for L ₂ if it acquires is the same as in the previous case. If L ₂ does not acquire, its payoff is given by R&D game 8 and it is $\frac{\alpha x V}{(1+2x)^2}(2x-1)^2-p_{22}^C$ with $p_{22}^C=p_N$. Thus, L ₂ acquires if

$$ \frac{\alpha x^3V}{(1+x)^2}-2\cdot \frac{\alpha V}{(1+2x)^2} \geq \frac{\alpha x V}{(1+2x)^2}(2x-1)^2-\frac{\alpha V}{16}. $$

This inequality holds for x ≥ 1.05678. The last case to consider is x < 1.0333. The payoff for L ₂ if it acquires is again given by R&D game 6, but L ₂ only has to pay p ₂₁ = p _N to make S ₁ accept the offer. If L ₂ does not acquire, its payoff is given by R&D game 9 and it is $\frac{\alpha V}{16}$. Thus, L ₂ acquires if

$$ \frac{\alpha x^3V}{(1+x)^2}- \frac{\alpha V}{(1+2x)^2}-\frac{\alpha V}{16}\geq \frac{\alpha V}{16}. $$

This inequality holds for every x < 1.0333. Thus, in decision point 4 L ₂ bids p ₂₁ = p _O and acquires S ₁ for x ≥ 1.05678, does not acquire S ₁ for 1.05678 > x ≥ 1.0333, and bids p ₂₁ = p _N and acquires S ₁ for x < 1.0333.

In decision point 10 S ₂ accepts every offer that gives it a higher payoff than the payoff as an outsider in R&D game 3. Thus, S ₂ accepts every offer $p_{12}^A \geq p_O.$

In decision point 7 L ₁ has already acquired S ₁ for p ₁₁. If L ₁ acquires S ₂, L ₁ is the merged firm in R&D game 2 and has a payoff $\frac{\alpha x^3V}{(1+x)^2}-p_{12}^A-p_{11},$ where the lowest accepted offer is $p_{12}^A=p_O.$ If it decides not to acquire, its payoff is the one of the merged firm in a triopoly, i.e, $\frac{\alpha x V}{(1+2x)^2}(2x-1)^2-p_{11}.$ Thus, L ₁ acquires if

$$ \frac{\alpha x^3V}{(1+x)^2}-\frac{\alpha V}{(1+2x)^2} \geq \frac{\alpha x V}{(1+2x)^2}(2x-1)^2. $$

This inequality always holds and thus L ₁ always bids $p_{12}^A =p_O$ and acquires S ₂ in decision point 7.

In decision point 5 S ₂ accepts every offer $p_{22}^A \geq p_O.$

In decision point 3 L ₂ has to decide whether to acquire S ₂ and create a symmetric duopoly in R&D game 1 or not to acquire S ₂ and become the outsider in the asymmetric duopoly in R&D game 2. If L ₂ acquires, its payoff is $\frac{\alpha x V }{4}-p_{22}^A$, where the lowest accepted offer is $p_{22}^A = p_O.$ If L ₂ does not acquire, its payoff is $\frac{\alpha V}{(1+x)^2}$. Thus, L ₂ acquires if

$$ \frac{\alpha x V }{4}-\frac{\alpha V}{(1+2x)^2} \geq \frac{\alpha V}{(1+x)^2}. $$

This inequality holds for x ≥ 1.18731 and therefore L ₂ acquires S ₂ for x ≥ 1.18731 in decision point 3.

If S ₁ denies the offer in decision point 2, the acquisition game ends with a payoff of $\frac{\alpha V}{(1+2x)^2}$ for S ₁.²² Thus, S ₁ accepts every offer p ₁₁ ≥ p _O .

If L ₁ decides to acquire in decision point 1, the acquisition game ends with R&D game 1 for x ≥ 1.18731 and with R&D game 2 if x < 1.18731. If L ₁ decides not to acquire, the acquisition game ends with R&D game 4 for x ≥ 1.18731, with R&D game 6 for 1.18731 > x ≥ 1.05678 or x < 1.0333 and with R&D game 8 for 1.05678 > x ≥ 1.0333. Consider first the case x ≥ 1.18731: L ₁’s payoff if it acquires is then given by R&D game 1 minus the price p ₁₁ paid for S ₁. The lowest price accepted by S ₁ is p _O . If L ₁ decides not to acquire in decision point 1, the game ends in R&D game 4 where L ₁’s expected profit is the same and the price paid to acquire S ₂ is p _O . Thus, for x ≥ 1.18731 L ₁ is indifferent between acquiring now or later and due to our assumption given in footnote 14 L ₁ acquires S ₁ for p ₁₁ = p _O . For all other cases L ₁’s payoff if it acquires is given by R&D game 2 and L ₁ has to pay $p_{11}=p_{12}^A =p_O.$ Thus, L ₁’s payoff if it acquires is

$$\label{payoffacquire}\frac{\alpha x^3 V}{(1+x)^2}-2 \frac{\alpha V}{(1+2x)^2}. $$

(7)

L ₁’s payoff if it decides not to acquire depends on x. If 1.18731 > x ≥ 1.05678 or x < 1.0333, L1’s payoff is given by R&D game 6 and it is

$$ \frac{\alpha V}{(1+x)^2}. $$

This is larger than the payoff given by Eq. 7 for x < 1.22364. If 1.05678 > x ≥ 1.0333, L ₁’s payoff is given by R&D game 8 and it is

$$ \frac{\alpha V}{(1+2x)^2}. $$

This is larger than the payoff given by Eq. 7 for x < 1.09021. Thus, in decision point 1 L ₁ bids p ₁₁ = p _O and acquires S ₁ for x ≥ 1.18731 and does not acquire S ₁ for x < 1.18731. □

Proof of Proposition 3

If all firms remain active in the innovation market (x ≤ 2), the probability of a successful innovation is

$$ \begin{array}{lll} p_{\rm ges}^5&=&2\left( \alpha \cdot \frac{\frac{3 \alpha V x (2x-1)}{4 (1+x)^2}}{2 \left( \frac{3 \alpha V x (2x-1)}{4 (1+x)^2}+\frac{3 \alpha V x (2-x)}{4 (1+x)^2}\right)} + \frac{\alpha}{x}\cdot \frac{\frac{3 \alpha V x (2-x)}{4 (1+x)^2}}{2\left(\frac{3 \alpha V x (2x-1)}{4 (1+x)^2}+ \frac{3 \alpha V x (2-x)}{4 (1+x)^2}\right)} \right)\\ &=&\frac{2 \alpha (1-x+x^2)}{x(1+x)}. \end{array} $$

Comparing this to the initial situation leads to

$$ p_{\rm ges}^5-p_{\rm ges}^4=\frac{2 \alpha (1-x+x^2)}{x(1+x)}-\frac{\alpha (1+x)}{2x}=\frac{3 \alpha (x-1)^2}{2x (1+x)}>0. $$

If large firms quit the innovation market, the probability of a successful innovation is the same as in the symmetric duopoly and $p_{\rm ges}^1>p_{\rm ges}^4$ (see Proposition 2). □