The Private Value of a Patent: A Cooperative Approach

By Artyom Jelnov The Leon Recanati Graduate School of Business Administration Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel E-mail: [email protected] and Yair Tauman Department of Economics, SUNY at Stony Brook, Stony Brook, NY 11794-4384, USA The Leon Recanati Graduate School of Business Administration Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel E-mail: [email protected]

November 23, 2008

We wish to thank Shigeo Muto, Dov Samet and David Wettstein for the helpful comments. We thank the associate editor and anonymous referee for very helpful and Valuable comments.

YT-The Private Value of a Patent.doc/ev

Abstract We consider a game in coalitional form played by the firms of a Cournot industry and an outside patent holder of a cost-reducing innovation. The worth of a coalition of players is the total Cournot profit of the active firms within this coalition. The number of firms that a coalition activates is determined by the Nash equilibrium of the game played by the coalition and its complement, where the strategy of each is the number of firms to be activated. Only firms in a coalition containing the patent holder are allowed to use the new technology. We prove that when the industry size increases indefinitely, the Shapley value of the patent holder approximates the payoff he obtains in a standard non-cooperative setup where he has the entire bargaining power. We also examine a partition game which considers for every coalition all structures of its complement, namely all partitions of the complement into subcoalitions. The coalition and every sub-coalition of the complement simultaneously decide how many of their firms to be activated. We prove a similar equivalence result for an extension of the Shapley value from coalitional games to partition games.

1.

Introduction The private value of a patent of cost reducing innovations was studied

extensively in the literature. Most of the literature models the interaction between the patent holder and the firms as a dynamic strategic game, using a non-cooperative approach and ignoring the fact that a patent holder can sign binding licensing agreements with one or more firms in the industry. See Kamien and Tauman (1986), Katz and Shapiro (1985 and 1986), Muto (1993), Erutco and Richelle (2007), Sen and Tauman (2007) and Kamien (1992) for a comprehensive survey. One exception to the non-cooperative approach is Tauman and Watanabe (2007) (hereafter TW) who analyze the interaction of a patent holder with the firms in a Cournot market as a cooperative game in the first stage, taking into account the Cournot competition of the second stage. The cooperative approach is a normative concept and as such it deals with questions of what should be a proper, neutral or a fair division of a cake generated when players cooperate with each other and can commit to any agreement reached. The non-cooperative approach, on the other hand, is a descriptive concept which predicts the outcome of interactions by selfish and strategic players. Thus, while the cooperative approach provides egalitarian solutions the non-cooperative approach provides utilitarian solutions. It is interesting and often insightful to compare the two, and this is what the current paper does for the private value of a patent. It is especially interesting if the two, the egalitarian and the utilitarian approaches yield the same outcome, as it is in our paper. This result provides another justification for the noncooperative more practical approach and suggests that the private value of a patent is a robust notion.

2 A key problem is how to model the interaction between the patent holder and the firms as a cooperative game. In other words we need to define the worth of any subset of firms whether or not they use the superior technology. Any subset of firms together with the patent holder has an access to the new technology. Any subset of firms which does not include the patent holder has no access to the new technology and can only use the old technology. In the model of TW any subset of firms can merge and establish a single entity. This entity then decides which of its firms to operate and which to close down. The traditional Von-Neumann-Morgenstern (1944) maxmin approach defines the worth of a coalition of players as the maximum total utility level that this coalition can attain under the assumption that the complement uses its most offensive approach against the coalition. In our context the worth of a subset of firms is the maximum sum of the profits of the active firms in this subset when the firms in the complement take a joint action against it. Any subset of firms that form a coalition with the patent holder chooses how many firms to operate. These firms use the superior technology for their production. The complement coalition (of all other firms) who has access only to the old technology chooses how many firms to activate in an attempt to minimize the total profit of the active firms with the new technology. It is assumed that all active firms in the coalition and in the complement compete á la Cournot. Similarly, when computing the worth of a coalition of firms which does not include the patent holder we use the same procedure. This coalition decides how many of its firms to operate and given this decision, the complement (which includes the patent holder and hence has an access to the superior technology) decides how many of its firms to operate as to minimize the sum of the profits of its rivals.

3 The maxmin approach represents a pessimistic attitude. The worth of a coalition is computed taking the worth case scenario. In this paper we take a more realistic approach in two directions. First, we allow the complement coalition to be partitioned into several independent and competing entities (i.e., we extend the analysis from coalitional games to partition games). Second, for every coalition S and every partition of the complement we model the interaction of S with the entities of the partition of the complement as a strategic game where every player ( S and every entity of the complement) chooses simultaneously the number of active firms and the number of firms to be shut down with the target of maximizing their own payoff. We then compute the Nash equilibrium of this game which is the number of firms every player operates. The worth of S is then defined as the total Cournot profit of the active firms in S . One of the main objects of coalitional game theory is to provide an a priori evaluation of games, i.e., to assign to each player of a game a number which represents what he would be willing to pay in order to participate. Under the superadditivity assumption which asserts that any set of players can do at least as well as in a coalition as in any sub coalition, the grand coalition (the set of all players) should form. Our game is indeed super-additive. The payoff that the grand coalition can attain, which is the monopoly profit under the new technology, is therefore allocated among the players in a way which supposed to reflect their “fair” share of the total cake. The most striking and useful solution concept of coalitional games is the Shapley value (Shapley (1953)). It is by now a central and very well established branch of game theory. The Shapley value is fully characterized by four simple and neutral axioms. It is a powerful tool for evaluating the power structure in a coalitional

4 game. It is argued that the Shapley value of a player reflects his bargaining power or his “real” contribution to the total cake generated by the grand coalition. Since the Shapley value can be interpreted in terms of "marginal worth", it is perhaps the game theoretic concept most closely related to traditional economic ideas. The Shapley value for coalitional games has been extended to partition games. First by Myerson (1977) and then Bolger (1989). The former introduced an explicit formula and the latter an implicit one. A somewhat more attractive value formula for partition games was introduced by Feldman (1996). Later on, Macho, Perez and Wettstein (2007) (hereafter MPW) provided an axiomatic justification for Feldman’s formula1. We use this formula in our analysis. We should mention the other two central solution concepts of coalitional games: the core and the nucleolus (Schmeidler (1969)). Like the Shapley value the nucleolus always exists and it is uniquely derived by an appealing set of axioms. The nucleolus, however, is often very difficult to compute (as it is in our case) and it is not defined for partition games. The core of a coalitional (or partition) game is often empty or very large and hence it is difficult or impossible to apply it as a predictive theory. These are the reasons we focus in this paper on the Shapley value for both coalitional and partition games. Our main result is that as the number of firms in the market increases, the value of the innovator approximates his payoff in the non-cooperative game traditionally studied in the literature, where the innovator enjoys full bargaining power. This asymptotically equivalent result is consistent with that of TW and it shows the robustness of the private value of a patent for a sufficiently competitive Cournot market. 1

Other notable works on externality games are Myerson (1997), Clippel and Serrano (2005) and Navarro (2007)

5 In the non cooperative strategic approach the patent holder who owns a technology which reduces the per-unit cost by a certain constant has full bargaining power. He first decides on the number of licenses to be sold and then auctions them off to the highest bidders. For a sufficiently large industry the innovator extracts through the auction the entire industry profit, which falls below the monopoly profit under the new technology (the cooperative outcome). The equilibrium number of licenses that he auctions off is the minimal number which induces any non licensee firm to exit the industry. The innovator extracts a revenue equals to the magnitude of the innovation (the per unit cost decrease) times the total demand under the preinnovation competitive price. The result does not change much when the licensing schemes consist of combinations of linear royalty and up-front fee (which may be determined in an auction). In this case however, the innovator sells licenses to every firm (except perhaps one firm). But asymptotically, as the number of firms increases indefinitely, the revenue of the innovator remains the same (see Kamien and Tauman (1984) and Sen and Tauman (2007)). In the cooperative case the total industry profit is higher but the innovator extracts only a fraction of this profit. As the number of firms increases indefinitely, the revenue he attains in the cooperative game coincides with his revenue in the non cooperative game. While licensing strategies which consist of a combination of linear royalty and up-front fee are well observed in practice, other schemes may yield the innovator higher revenues. Erutco and Richell (2007) show that the innovator can actually extract the entire monopoly profit with the new technology using non-linear fees which depend on both the output level of the licensee firm and on the total industry output. For another related optimal scheme see Kamien, Tauman and Zamir (1990).

6 Finally, similar to TW we also show that the core of the game with sufficiently many firms is empty and therefore can not be used to predict the cooperative outcome. The fact that the core of a game is empty reflects more on the solution concept itself than on the game. The core of very simple games like all majority games (with no veto players) is empty. Furthermore, the core is empty for every game where the marginal contribution of at least one player to a coalition, decreases with the size of the coalition (namely where a player contributes less to larger coalitions). This is the case with our game too. As a closing remark, let us mention that the worth of the grand coalition (the one containing the innovator and very firm in the industry) is assumed to be the monopoly profit with the new technology. This is the case when the grand coalition operates one firm only. This assumption however may conflict with anti-trust laws and perhaps call for the modification of the worth of the grand coalition. Interestingly enough our equivalence result does not change even if we limit the actions of this coalition and reduce its worth (and for that matter even that of the very large or very small coalitions). Such modification will affect the Shapley value of the firms but not the Shapley value of the patent holder and hence will not have an impact on the private value of the patent. Note also that Franchise is a good example for the ability of coalitions to control the number of their firms and to operate them in a competitive environment. The paper is organized as follows: We first introduce the model for coalitional games where a coalition interacts with its complement only and prove in this case the equivalence result. We then extend this result to partition games. The core of the game is analyzed last.

7

2.

The Model Consider the set N = {1,L, n} of firms in a Cournot market.

Each firm

produces a homogeneous commodity. The production cost is c per unit. We consider a linear demand function Q = max (0 , a − p ) , where

p is a market price and

0
We consider a non-drastic a +c−ε 2

under the new

technology is higher than the competitive pre-innovation price c (see Arrow (1962)). This is equivalent to a − c − ε > 0 . Denote by qi the output level of firm i ∈ N . The profit of an efficient firm is qi ( p − c + ε ) .

The profit of an inefficient firm is qi ( p − c ) .

The innovator is

denoted by 0. Let N 0 = N U {0} be the set of players. Consider a coalition (a subset of N 0 ) and its complement. One coalition contains the innovator, and every firm in this coalition has access to the new technology.

The other coalition contains inefficient firms (who use the old

technology). Let S ⊆ N be any coalition of firms. We denote S 0 = S U {0}. The coalition S 0 can choose to operate a number m E ≤ S of firms (each has access to the new technology) and to shut down the other S − m E firms.

Similarly, the

complement coalition will choose a number l N E ≤ N \ S of firms to operate (each produces with the old technology). The m E + l N E active firms will engage in a Cournot competition (in quantities). Denote by qiE (m E , l N E ) the Cournot output of an

8 efficient firm i . Similarly we denote by q Nj E (m E , l N E ) the output of an inefficient firm j . By their symmetry, qiE (m E , l N E ) and q Nj E (m E , l N E ) do not depend on i and j respectively and hence we can omit the indices i and j .

Let S ⊆ N . The total profit of S 0 is Π S0 (m E , l N E ) = m E q E (m E , l N E ) ( p − c + ε )

And the total profit of a coalition S which does not contain the patent holder is

Π S ` (m E , l N E ) = l N E q N E (m E , l N E ) ( p − c ) . Let K = (a − c ) / ε . For simplicity we assume that K is an integer. As mentioned above an innovation is non-drastic iff K > 1 . It is easy to verify, that if the number of efficient and active firms is at least K , the inefficient firms are all driven out of the market. The number K is the smallest with this property. Let S ⊆ N . Consider the following game G (S ) where the two players are the coalitions S 0 and its complement N \ S .

In the first stage both S 0 and N \ S

simultaneously choose the number of firms to activate. In the second stage the decisions of the first stage become commonly known and the active firms in S0 and in N \ S all compete à la Cournot.

We first analyze the subgame perfect Nash

equilibrium of G (S ) for all S ⊆ N . The worth of a coalition is defined to be the total equilibrium profit of its active firms. This defines a game in a coalitional form. We will compute the Shapley value of this game and its limit when the industry size increases indefinitely.

9

3.

The Coalitional Game and its Value Let V (S0 ) and V (S ) be the worth of the coalitions S 0 and S , respectively.

Namely, V (S0 ) is the total Cournot profit of the active firms in S in the game G (S ) whereas V (S ) is the total Cournot profit of the active firms in S in the game G(N\S). In the next proposition we compute the value of V (S ) and V (S0 ) .

Proposition 1 Consider a non-drastic innovation. Let S ⊆ N and S 0 = S U {0} and let

K≤

n −1 . Then 2

⎧ S ⎛ a − c + ( S + 2 )ε ⎞ 2 ⎟ , S
⎧ (a − c − (n − S ) ε )2 ⎪ ⎪ 4 (n − S + 1) ⎪ ⎪ V (S ) = ⎨0 ⎪ ⎪ 2 ⎪ S ⎛⎜ a − c − S ε ⎞⎟ ⎪ 4 ⎜ S +1 ⎟ ⎠ ⎩ ⎝

S >n−K

,

Proposition 2 Suppose that K >

min(n − S , S + 1) ≥ K

,

, K −1≤ S ≤ n − K

n −1 . Then 2

10

⎧ S ⎛ a − c + ( S + 2 )ε ⎞ 2 ⎟ ⎪ ⎜ , ⎟ S +1 ⎪ 4 ⎜⎝ ⎠ ⎪ ⎪ 2 ⎪ ⎛ ⎪ S ⎜ a − c +( S + 1)ε ⎞⎟ , ⎟ ⎪ ⎜ 2 S +1 ⎝ ⎠ ⎪ V (S 0 ) = ⎨ ⎪ ⎪ n − S + 1 ⎛ a − c + (n − S + 1)ε ⎜ ⎪ ⎜ n − S +1 4 ⎪ ⎝ ⎪ ⎪ ⎪ ⎪ε (a − c ) ⎩

⎛n −1 ⎞ S ≤ min ⎜ , K − 1⎟ 2 ⎠ ⎝ n −1 ⎛n +1 ⎞ < S < min ⎜ , K⎟ 2 ⎝ 2 ⎠ ⎞ ⎟ ⎟ ⎠

2

,

⎛n +1 ⎞ ,n + 1 − K ⎟ S ≥ max ⎜ ⎝ 2 ⎠

, K ≤ S ≤ n +1− K

⎧ S ⎛ a − c − ( S + 1) ε ⎞ 2 n −1 ⎛ n + 1⎞ ⎟ ⎪ ⎜ n−K< S < , ⎜ possible only if K > ⎟ ⎜ ⎟ S + 2 ⎠ 2 4 1 ⎝ ⎪ ⎝ ⎠ ⎪ ⎪ 2 n +1 ⎪ ⎛⎜ a − c − ( S + 1)ε ⎞⎟ ⎛ n −1 ⎞ ,n − K⎟< S < , min ⎜ ⎪S ⎜ ⎟ 2 2 S +1 ⎝ 2 ⎠ ⎠ ⎪ ⎝ ⎪ 2 ⎪ V (S ) = ⎨ (n − S + 1) ⎛ a − c − (n − S ) ε ⎞ ⎛n +1 ⎞ ⎜ ⎟ , n − K + 1⎟ S , max ≥ ⎜ ⎪ ⎜ ⎟ 2 n S 4 1 − + ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎪0 , n − K ≥ S ≥ K −1 ⎪ ⎪ 2 ⎪ ⎛ ⎪ S ⎜ a − c − S ε ⎞⎟ , S ≤ min (n − K , K − 2 ) ⎪ 4 ⎜⎝ S + 1 ⎟⎠ ⎩ The proof of Propositions 1 and 2 appear in the Appendix.

The Shapley Value of V

11 Let ℜ = ( j0 , j1 , L, jn ) be an order of the players in N 0 . Let Ρ ℜj be the set of players in N 0 who precede player j ∈ N 0 in the order ℜ . Let (Shi (V ))i∈N 0 be the Shapley value of V . It is defined by

Shi (V ) =

[

]

1 Pi ℜ U {} i − V (Pi ℜ ) ∑ (n + 1)! ℜ

The Shapley value of a player i ∈ N 0 is a simple average of his marginal contribution to a coalition which is formed by a random order ℜ of N 0 , where all orders of N 0 are equally likely. Fix an order ℜ ( S + 1) where the patent holder is located at ( S + 1) . Then ℜ ( S + 1)

Ρ0

is the set of firms that precede the patent holder 0 in ℜ ( S + 1) . Since V (S )

is a function of S we can write Sh0 (V ) =

[(

)]

) (

1 n ℜ ( S + 1) ℜ ( S + 1) V Ρ0 U {0} − V Ρ0 . ∑ n + 1 S =0

(1)

By the efficiency and symmetry axioms defining the Shapley value, for each firm i Sh0 (V ) + n Shi (V ) =V (N 0 )

(2)

By Propositions 1 and 2 2

⎛a −c+ε ⎞ V (N 0 ) = ⎜ ⎟ , 2 ⎝ ⎠

(3)

which is the monopoly profit under the new technology. Let us state our main result.

Proposition 3 Consider a non drastic innovation. Then, lim Sh0 (V ) = ε (a − c ) and lim

n→∞

n→∞

∑

i∈N

⎛a −c −ε ⎞ Shi (V ) = ⎜ ⎟ 2 ⎠ ⎝

2

12 Proof See Appendix Proposition 3 asserts that for a non drastic innovation the Shapley value of the innovator approaches ε (a − c ) when the industry size increases indefinitely. This result coincides with the one obtained by the non-cooperative approach where the innovator auctions off an optimal number k of licenses. The k highest bidders pay the innovator their bids up front and in return they obtain a license to use the new technology. Then the licensee and non-licensee firms compete in the market à la Cournot. It is shown in Kamien, Oren and Tauman (1992) that if the market size n is sufficiently large then the innovator offers K =

a−c

ε

licenses, the Cournot price falls

to c and the non-licensee firms are driven out of the market. The licensee firms produce a total of a − c units and make a per unit profit of ε . The innovator extracts their entire profit, ε (a − c ) , since their opportunity cost is zero. In contrast in the cooperative approach the total pie is bigger and it is equal to the monopoly profit under the new technology.

Nevertheless, the Shapley value of the innovator is

asymptotically the same for these two different approaches. The intuition behind the proof is simple. We first show that a coalition S0 = S

U {0}

exactly K =

containing the innovator and satisfies K ≤ S ≤ n − K , will activate

a−c

ε

firms which is the minimal number of efficient firms required to

reduce the market price to c , the pre-innovation marginal cost. Hence all firms in

N \ S are driven out of the market. The total industry profit in this case is ε (a − c ) . Consequently V (S 0 ) = ε (a − c ) and this is true for all coalitions S 0 such that K ≤ S ≤ n − K . Since K does not depend on n most of the coalitions S 0 have

V (S 0 ) = ε (a − c ) provided that n is sufficiently large. Also every such coalition will

13 obtain zero if the innovator deviates to N \ S . The marginal contribution of the innovator to most of the coalitions is therefore ε (a − c ) . The marginal contribution of the innovator to any other coalition is bounded above, say, by the monopoly profit. Since the Shapley value of a player is his marginal contribution to a random coalition, the Shapley value of the innovator is asymptotically ε (a − c ) , as claimed.

4.

The Partition Game and its Value We next introduce the family of games in partition form. These are games of

the form w (S , π ) where π is a partition of the set of players into disjoint coalitions and S is a coalition in π . Namely, the worth of a coalition S depends not only on its players but also on the way the complement is partitioned into coalitions. Once a coalition is formed, one has to consider all possible structures of its complement. The complement can form a single entity (like in coalitional games) but also can be broken down into smaller coalitions. Feldman (1996) suggested a formula for the value of every player i in a partition game w (S , π ) , S ∈ π . It is given by

φi

∏ (T − 1)! (w ) = ∑ β (S ) w (S , π ) ( π ) (n − S )! T ∈π \ S

i

S, S∈π

where2

⎧ ( S − 1)! (n − S )! ⎪ n! ⎪ β i (S ) = ⎨ ⎪ S (n − S − 1)! ⎪− n! ⎩

2

For short we write

π\S

instead of

π \ {S }.

i∈S

i∉S

14 Remark Suppose that w depends on S only, namely for every S and every two partitions π and π ′ such that S ∈ π I π ′

w (S , π ) = w (S , π ′)

Define V (S ) = w (S , π ) . Then it can be verified that for every player i

ϕ i V = φi w That is, the value φi is an extension of the Shapley value from coalitional games to partition games. Let us apply the above to our context. Consider a partition π = (T1 , T2 , L , Tk ) of N 0 where Ti I T j = φ for i ≠ j and

k

UT

i

= N 0 . In the first stage, every coalition

i =1

Ti of firms chooses how many of their firms to operate and how many to shut down.

The choices of the k coalitions are made independently and simultaneously. Let mi k

be the number of firms which Ti activates, 1 ≤ i ≤ k . Let m = ∑ mi . In the second i =1

stage the m active firms compete a la Cournot. Suppose that 0 ∈ T j then the m j firms

of T j use the efficient technology and all other active firms use the old technology. The payoff of Ti is the total Cournot profit of the active firms in Ti . Let G (T1 , L , Tk ) be the two-stage game in a strategic form described above played by the k players T1 , L , Tk . Then w (Ti , π ) is defined to be the equilibrium payoff of Ti in

the strategic game G (T1 , L , Tk ) . Our next goal is to show that the value φ0 of the innovator in the partition game w (S , π ) approximate ε (a − c ) when n increases indefinitely.

Proposition 4

lim φ0 w = ε (a − c )

n →∞

15 Proof See Appendix

Proposition 4 strengthens the equivalence result stated in Proposition 3. Namely, whether we use coalitional games or partition games, the cooperative and the non-cooperative approaches yield the innovator the same payoff if the market is sufficiently competitive. This value is the private value of the patent.

5.

The Core of the Coalitional Game

Another well known solution concept is the core. The core of a game may be empty and unfortunately this is the case with our game V in a coalitional form, when the number of firms is large. Definition

Let u be a coalitional game on the set N = {1, 2, L , n } of players. The

core of u is the set of all payoff vectors ( x1 , K , x n ) such that for every S ⊆ N

∑x i∈S

i

≥ u (S ) and

∑x

i∈N

i

= u (N ) .

Proposition 5 Consider a non-drastic innovation and suppose that n ≥ 2 . Then the core of the game V is empty. Proof See Appendix

Conclusions We extended the result of Tauman and Watanabe (2007) to the case where the characteristic function is defined by the Nash equilibrium concept rather than by the minmax or the maxmin concept. We prove that the equivalence between the Shapley

16 value egalitarian solution and the non-cooperative utilitarian solution still holds for large markets whether the cooperative game is a coalitional game or a partition game.

References Arrow K.J.: Economic Welfare and the Allocation of Resources for Invention, in: The Rate and Direction of Inventive Activity: Economic and Social Factors, Nelson R.R. ed. Princeton University Press, pp. 609-625 (1962). Bolger, E.M.: A Set of Axioms for a value for partition function games. International Journal of Game Theory, 18, 37-44 (1989). De Clippel, G., Serrano, R.: Marginal Contributions and Externalities in the Value, Brown University, Working Paper, 2005-11 (2005). Erutco C., Richelle Y.: Optimal Licensing Contracts and the Value of a Patent. Journal of Economics and Management Strategy, 16, 407-436 (2007). Feldman, B.E.: Bargaining, coalition formation and value, Ph.D. Thesis, State University of New York at Stony Brook (1996). Kamien M.I.: Patent Licensing, in: Handbook of Game Theory, vol. 1, Aumann R.J., Hart S. eds. North-Holland, pp. 332-254 (1992). Kamien M.I., Tauman Y.: Fees versus Royalties and the Private Value of a Patent. Quarterly Journal of Economics, 101, 471-491 (1986). Kamien M.I., Tauman Y. and Zamir S.: The Value of Information in a Strategic Conflict. Games and Economic Behavior, 2, 129-153 (1990).

Katz M.L., Shapiro C.: On the Licensing of Innovation. Rand Journal of Economics, 16, 504-520 (1985).

17 Katz M.L., Shapiro C.: How to License Intangible Property. Quarterly Journal of Economics, 101, 567-589 (1986). Macho-Stadler, I., Perez-Castrillo, D. and Wettstein, D.: Sharing the Surplus: An Extension of the Shapley Value for Environments with Externalities. Journal of Economic Theory, 135, 339-356 (2007). Muto,S.:On Licensing Policiesin in Bertrand Competition, Games and Economic Behavior,5, 257-267 (1993). Myerson, R.B.: Values of Games in Partition Function Form. International Journal of Game Theory, 6, 23-31 (1977). Myerson, R.B.: Graphs and Cooperation in Games, Math of Operation Research, 2, 225-229 (1997). Navarro, N.: Fair Allocation in Networks with Externalities, Games and Economic Behavior, 58, 354-364 (2007). Schmeidler, D.: The Nucleolus of a Characteristic Function Game. SIAM Journal of Applied Mathematics, 17, 1163-1170 (1969). Shapley L.S.: A Value for n-person Games, in: Contributions to the Theory of Games II, Kuhn H.W., Tucker A.W. eds.

Princeton University Press,

pp. 307-317 (1953). Sen D., Tauman Y.: General Licensing Schemes for a Cost-reducing Innovation. Games and Economic Behavior, 59, 163-186 (2007). Tauman Y., Watanabe N.: The Shapley Value of a Patent Licensing Game: The Asymptotic Equivalence to Non-Cooperative Results. Economic Theory, 30, 135-149 (2007). Von Neumann J., Morgenstern O.:

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Princeton University Press, 1944 (2nd ed. 1947).

18

19

Appendix Proof of Propositons 1 and 2 Consider the game G (S ) and assume that every firm in S has access to the new technology while every firm in N \ S uses the old technology. The following is the equilibrium outcome of G (S ) .

Lemma 1

Suppose that K ≤

n −1 2

(i)

If S < K then m E = S and l N E = S + 1 .

(ii)

If n − S + 1 < K ≤ S then m E = n − S + 1 and l N E = n − S .

(iii)

If K ≤ min ( S , n − S + 1) then m E = K and either l N E = K − 1 or l N E =K . In both these two cases the inefficient firms are driven out of the market.

Lemma 2

Suppose that K >

n −1 2

⎛ n −1 ⎞ (i) If S ≤ min ⎜ , K − 1⎟ then m E = S and l N E = S + 1 . ⎝ 2 ⎠ (ii)

If

n −1 ⎛n +1 ⎞ < S ≤ min ⎜ , K ⎟ then m E = S and l N E = n − S . In this case n 2 2 ⎝ ⎠

is even and S = N \ S =

n . 2

n +1 ≤ S < K then m E = n − S + 1 and l N E = n − S . 2

(iii)

If

(iv)

If n − S + 1 < K ≤ S then m E = n − S + 1 and l N E = n − S .

(v)

If K ≤ min ( S , n − S + 1) then m E = K and either l N E = K − 1 or l N E = K .

20 By Lemmas 1 and 2 if min ( S , n − S + 1) ≥ K then S 0 activates K efficient firms. Even though any K efficient firms in a Cournot equilibrium, drives every inefficient firm out of the market l N E = K − 1 or l N E = K . This means that N \ S will activate either K − 1 or K of its inefficient firms but they will produce no output. If N \ S activates no firm, then S 0 is better off deviating from m E = K to m E = 1 .

Corollary For every S ⊆ N the equilibrium number of firms that S 0 and N \ S each activate is bounded above by K .

Proof of Lemmas 1 and 2 The following three lemmas are essential for the proof.

Lemma 3

Consider a market with m E + l N E firms where the m E operate with the new technology and the l N E firms operate with the old technology. Then, the Cournot outcome is ⎧ a − c + (l N E + 1)ε , ⎪ E NE +1 ⎪ m +l ⎪ q E (m E , l N E ) = ⎨ ⎪ ⎪a − c + ε , ⎪ E ⎩ m +1

mE < K

mE ≥ K

21

qN E

⎧ a − c − mE ε , ⎪ E NE +1 ⎪m + l (m E , l N E ) = ⎪⎨ ⎪ ⎪ 0 , ⎪ ⎩

mE < K

mE ≥ K

The Cournot profit of an efficient firm is (q E ) , and the profit of an inefficient firm is 2

(q )

NE 2

The proof is straightforward. The next lemma characterizes the best strategy of a coalition S 0 of efficient firms as a function of the number of firms that is activated by its complement.

Lemma 4 (TW)

(1)

Let S 0 ⊆ N , S ≥ 1 , be the set of efficient firms. If the coalition N \ S

operates l N E inefficient firms, where 1 ≤ l N E ≤ n − S , then the optimal number m E (S0 , l N E ) that S 0 should operate is

⎧min ( S , l N E + 1) , 1 ≤ S < K ⎪ ⎪⎪ m E (S , l N E ) = ⎨l N E + 1 , S ≥ K > lNE +1 ⎪ ⎪ otherwise ⎩⎪ K , (2)

[

m E < K → Π S0 (m E , l N E ) > Π S0 (K , l N E ) ⇔ m E K > (l N E + 1) The proof of Lemma 4 appears in TW.

2

]

22 The next lemma characterizes the best reply strategy of a coalition of inefficient firms as a function of the number of efficient active firms.

Lemma 5

Let S 0 ⊆ N , n − S ≥ 1 . If the coalition S 0 operates m E efficient firms, where 1 ≤ m E ≤ S , and m E < K , then the optimal number l N E (S , m E ) of inefficient firms

that N \ S operates is l N E (S , m E ) = min (n − S , m E + 1) . Proof By Lemma 3, the payoff of N \ S is Π N \ S (m , l E

NE

)= l

2

NE

⎛ a − c − mE ε ⎞ ⎟⎟ , ⎜⎜ E NE ⎠ ⎝ m +l

which is maximized for l N E = m E + 1 , if m E + 1 ≤ n − S , and for l N E = n − S , otherwise.

■

We now use Lemmas 4 and 5 to characterize the Nash equilibrium of G (S ) for all S ⊆ N .

We are now ready to prove the proposition Let S 0 = S U {0}, N 0 = N U {0}. Consider first the case: S < K : By Lemmas 4 and 5, the case m E < S , l NE < n − S is not possible in equilibrium (because every coalition can be better off operating one more firm). Therefore either m E = S or l N E = n − S . Let us consider three subcases.

23 Subcase 1 S >

n +1 . In this case S > n − S + 1 . Using Lemmas 4 and 5 it must be 2

that m E = l N E + 1 and l N E = n − S .

Hence m E =n − S + 1 .

By Lemma 3

⎛ a − c + (n − S + 1) ⎞ ⎟ . Π S0 (n − S + 1, n − S ) = (n − S + 1) ⎜⎜ ⎟ ( ) − + 2 n S 2 ⎝ ⎠ 2

Subcase 2

S ≤

n −1 . In this case n − S ≥ S + 1 . Again by Lemmas 4 and 5, s

m E = S , l NE = S + 1 , and by Lemma 3

⎛ S ⎞ ⎛ a − c + ( S + 2 )ε ⎞ ⎟ . Π S0 ( S , S + 1) = ⎜⎜ ⎟⎟ ⎜⎜ ⎟ S +1 ⎝ 4 ⎠⎝ ⎠ 2

Subcase 3 S =

n and n is even. By Lemmas 4 and 5 2

S = l NE = N \ S

and by Lemma 3 ⎛ a − c + ( S + 1)ε ⎞ ⎟ . Π S ( S , S ) = S ⎜⎜ ⎟ + 2 S 1 ⎝ ⎠ 2

This completes the proof of the case where S < K . Consider next the case: S ≥ K . If l NE + 1 ≥ K by Lemma 4 the optimal strategy for S 0 is m E = K (note, that q j (m E , l NE ) = 0 in this case).

By Lemma 4, if 0 ≤ l N E < K − 1 , the optimal m E for S 0 is m E = l NE + 1 . Thus if n − S + 1 < K , we apply Lemma 5 to the case l NE ≤ n − S , and obtain that the unique equilibrium outcome is m E = n − S + 1, l NE = n − S . By Lemma 3

(n − S +1, n − S ) = (a − c + ε (n − S + 1)) 2 (n − S + 1)

2

Π S0

.

24 If on the other hand n − S + 1 ≥ K , there is no equilibrium where m E < K (by Lemmas 4 and 5). Therefore the only equilibrium outcomes are m E = K and either 2

⎛a −c +ε ⎞ ⎟⎟ = ε (a − c ) l N E = K − 1 or l NE = K . In this case Π S0 (K , l NE ) = K ⎜⎜ ⎝ K +1 ⎠ for both l N E = K − 1 and l N E = K . Let us next compute the equilibrium profit of

S where 0 ∉ S . Similarly to the

previous two cases we also consider the case S > n − K

and S ≤ n − K . We start

with the case S > n − K . By

arguments

similar

to

the

previous

cases

the

two

inequalities

l NE < S , m E < n − S are not possible in equilibrium. By Lemmas 3, 4 and 5.

If

S ≥

n +1 , then 2

m E = n − S , l NE = n − S + 1 . Thus

⎛ a −c − (n − S )ε ⎞ ⎟ . Π S (n − S + 1, n − S ) = (n − S + 1) ⎜⎜ ⎟ ⎝ 2 (n − S + 2 ) ⎠ 2

If

S ≤

n −1 then l NE = S , m E = S + 1 and 2

⎛ S ⎞ ⎛ a − c − ( S + 1)ε ⎞ ⎟ . Π S ( S , S + 1) = ⎜⎜ ⎟⎟ ⎜⎜ ⎟ S 4 ⎝ ⎠⎝ ⎠ 2

If

S =

n and if n is even then l NE = S = m E = n − S , 2

and ⎛ a − c − ( S + 1)ε ⎞ ⎟ . Π S ( S , S ) = S ⎜⎜ ⎟ 2 S +1 ⎝ ⎠ 2

25 Finally consider the case:

S ≤ n − K . It can be easily verified using the above

arguments that: if

S + 1 < K then the equilibrium is l NE = S , m E = S + 1 , and

⎛a −c − S ε ⎞ ⎟ Π S ( S , S + 1) = S ⎜⎜ ⎟ ⎝ 2 ( S + 1) ⎠ If

2

S + 1 ≥ K then m E = K and either l NE = K − 1 or l NE = K . By Lemma 3,

Π S (l NE , K ) = 0 . This completes the proof of both, Lemma 1 and Lemma 2.

■

The proof of Proposition 1 and 2 are now immediate consequence of Lemmas 1-5. Proof of Proposition 3

n − 1⎞ ⎛ Suppose that n > 2 K + 1 ⎜ or K < ⎟ . Then K < n − K . By Lemma 4 (part (ii) 2 ⎠ ⎝ and by Proposition 1, 2 2 ⎧ ⎛ a − c − S ε ⎞ ⎫⎪ ⎛ 1 K − 2 ( S ) ⎪⎛ a − c + ( S + 2 ) ε ⎞ ⎜ ⎟ −⎜ ⎟ ⎬ + 1 (K − 1)⎜ a − c + (K + 1)ε Sh0 (V ) = ⎨ ∑ ⎜ ⎟ ⎜ 2K n + 1 S = 1 4 ⎪⎜⎝ S +1 S + 1 ⎟⎠ ⎪ n + 1 ⎝ ⎠ ⎝ ⎩ ⎭ 2 ( a − c − (K + 1)ε ) 1 1 ⎛ n−K ⎜ + ∑ ε (a − c ) + n + 1 ⎜ ε (a − c ) − n + 1 S =K 4 (K + 2 ) ⎝

+

1 n −1 ∑ n + 1 S =n−K +2

(

⎞ ⎟+ ⎟ ⎠

)

⎧⎪ a − c + (n − S ε )2 (a − c − (n − S ) ε )2 ⎫⎪ − ⎨ ⎬ ⋅ IK ≥3 + 4 (n − S + 1) 4 (n − S + 1) ⎪ ⎪⎩ ⎭

2 2 1 ⎧⎪ ⎛ a − c + ε ⎞ ⎛ a − c ⎞ ⎫⎪ + ⎟ −⎜ ⎟ ⎬ → ε (a − c ), as n → ∞, ⎨⎜ n + 1 ⎪⎩ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎪⎭

⎧1 K ≥ 3 where I K ≥ 3 = ⎨ ⎩0 K < 3 2 2 ⎞ 1 ⎛ ⎛a −c +ε ⎞ ⎛a −c+ε ⎞ Since V (N 0 ) = ⎜ ⎟ − Sh0 (V )⎟⎟ . ⎟ , by (2) we have Shi (V ) = ⎜⎜ ⎜ n⎝⎝ 2 2 ⎠ ⎝ ⎠ ⎠

2

⎞ ⎟⎟ + ⎠

26

∑

i∈N

2

2

⎛a −c+ε ⎞ ⎛a −c+ε ⎞ ⎛a −c −ε ⎞ Shi (V ) = ⎜ ⎟ − Sh0 (V ) → ⎜ ⎟ − ε (a − c ) = ⎜ ⎟ 2 2 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Proof of Proposition 4 Lemma 6

2

■

For the proof we use the following two lemmas

Let π be a partition of No and let S ∈ π , T ∈ π and 0 ∉ S U T . Then w (S , π ) ≤

Proof Suppose

that

(a − c − ε )2 min ( S , T

π = (S , T0 , T1 , K , Tk )

) 0 ∈ T0

where

and

T1 = T .

Let

ms , mT0 , K , mTk be the number of firms that each coalition in π operate. Clearly m s maximizes the total Cournot profit of S which is

⎛ ⎞ ⎜ ⎟ ⎜ a − c − mT0 ⋅ ε ⎟ ms ⎜ ⎟ k ⎜ m + ⎛⎜ m ⎞⎟ + 1 ⎟ Tj ⎟ ⎜ s ⎜∑ ⎟ ⎝ j =0 ⎠ ⎝ ⎠

2

⎛ k ⎞ (see Lemma 3). This profit is increasing in ms as long as ms ≤ ⎜⎜ ∑ mT j ⎟⎟ + 1 . Hence ⎝ j =0 ⎠ the optimal m s satisfies mT + 1 ≤ m s ≤ S .

Similarly, the optimal mT satisfies

m s + 1 ≤ mT ≤ S . The two inequalities may hold only if either m s = S or mT = T .

If ms = S then 2

⎛ ⎞ ⎜ ⎟ a − c − m ⋅ ε T0 ⎜ ⎟ ≤ 1 (a − c − ε )2 . w (S , π ) = ms k ⎜ ⎟ S ⎜ ms + ∑ mT j + 1 ⎟ j =0 ⎝ ⎠

If mT = T , then again either ms = S or ms ≥ T + 1 . In both cases

27

⎛ (a − c − ε ) 2 (a − c − ε )2 ⎞ (a − c − ε )2 ⎟= , w (S , π ) ≤ max ⎜⎜ ⎟ min ( S , T ) S T ⎝ ⎠ ■

Let r be a positive integer and let ℘nr +1 be the set of partitions π of N 0 into non empty coalitions T such that T ≤ r for every T ∈ π . Lemma 7

For every η > 0 and for any integer r > 0 there exists n0 such that for

all n > n0

Pr ob (℘nr + 1 ) < η Proof Let π ∈℘nr + 1 and suppose that π = {T1 , K , TK } where Ti = ri ≤ r . Then by definition k

Pr ob (π ) =

∏ (r − 1)! i =1

i

(n + 1)!

Let m be the number of partitions of N 0 containing k elements of size r1 , K, rk . Consider all the different components of (r1 , r2 ,K, rk ) . Without loss of generality assume that these are (r1 , r2 ,K, rt ) , t ≤ k . Namely, ri ≠ rj for all i ≠ j , 1 ≤ i, j ≤ t and for every l > t there is i ≤ t such that rl = ri . Let k i , 1 ≤ i ≤ t , be the number of times t

that ri appears in (r1 , K , rk ) . Then n + 1 = ∑ k i ri . i =1

Claim There exists i , 1 ≤ i ≤ t , such that

ki ≥

Proof Suppose to the contrary that k i <

2 (n + 1) r (r + 1) 2 (n + 1) for all i = 1, 2, K, t . Then r (r + 1)

28 t

n + 1 = ∑ ki ri < i =1

2 (n + 1) t ∑ ri r (r + 1) i =1

But the ri s are all different from each other and ri ≤ r . Hence

t

∑r ≤ i =1

i

(1 + r ) r , 2

contradicting the previous inequality, and the proof of the claim is complete ■ We conclude that ⎞ ⎛ ⎞ ⎞⎛ ⎛ ⎜ n + 1⎟ ⎜ n + 1 − r1 ⎟ K ⎜⎜ rk ⎟⎟ r r2 ⎠ ⎝ rk ⎠ m≤ ⎝ 1 ⎠⎝ δ (n, r )! where

δ ( n, r ) = ⎜ 2 ( n + 1) ⎟ ⎜ ⎟ ⎛

⎞

⎝ r ( r + 1) ⎠

Thus the probability P of all partitions with k elements of size r1 , K , rk such that 1 ≤ ri ≤ r satisfies k

P≤

∏ (r i =1

i

− 1)!

(n + 1)!

⋅

(n + 1)! 1 = ri ! K rk ! δ (n, r )! ri K rk δ (n, r )!

We next find an upper bound to the number of all possible integers r1 , r2 , K , rk and k k

such that

∑r i =1

i

= n + 1 and 1 ≤ ri ≤ r . Let A (n, r ) be the set of all such combinations

and let a (n, r ) = A (n, r ) . Then

a (n, r + 1) ≤ a (n, r ) + (n + 1) a (n, r ) = (n + 2 ) a (n, r ) The reason is as follows: Every combination in A (n, r + 1) can be obtained from A (n, r ) in two ways. Once by adding 1 as a new element to each combination in

29

A (n, r ) and secondly by adding 1 to every number in every combination of A (n, r ) (by doing this, repetitions occur and we have inequality rather than equality). Since the number of elements in every combination is at most n + 1 , the above inequality is established. The last inequality together with a (n, 1) = 1 imply that a (n, r ) ≤ (n + 2 )

r −1

We therefore conclude that Pr ob (℘

n +1 r

n+2 ) ≤ (δ ( n, r) )!

r −1

(n + 2) ⎛ 2 ( n + 1) ⎞ ⎜ ⎟ ⎝ r ( r + 1) ⎠ r −1

=

⎛ ⎜⎜ ⎝

⎞ ⎟⎟ ! ⎠

For a fixed r and n sufficiently large we have Pr ob (℘nr + 1 ) < η , as claimed ■ We now proceed to the proof of Proposition 4. Let η > 0 be an arbitrary small number ⎛ 1 2⎞ and let r0 be such that r0 > max ⎜⎜ K , (a − c − ε ) ⎟⎟ . Let n > r0 and S ⊆ N such that ⎝ η ⎠

r0 ≤ S . Consider any partition π with S 0 ∈π . For large n , S 0 will operate K firms (see Proposition 1) and every other firm in the market will produce zero. Hence the 2

⎛a − c +ε ⎞ ⎟⎟ = ε 2 and the total Cournot profit Cournot profit of every licensee firm is ⎜⎜ ⎝ K +1 ⎠ of S 0 is

w(S 0 , π ) = K ⋅ ε 2 = ε (a − c ) Let L = {1, 2, K , l }. By Lemma 7 there is an integer to such that if l > t0 the probability that a random partition of L contains an element A ⊆ L with A ≥ r0 is at least 1 − η . Let S ⊆ N be such that r0 ≤ S ≤ n − t0 where n > r0 + t0 . Then, with a

(4)

30 probability of at least 1 − η , a random partition π of N 0 \ S contains an element T ∈ π with T ≥ r0 .

By Lemma 6 and by the choice of r0

(a − c − ε ) w (S , π ) ≤

2

r0

<η

(5)

Since β 0 (S ) < 0 whenever S ⊆ N L1 ≡

∑ α (π S ) β (S ) w (S , π ) ≤ (∑π )α (π S ) β (S ) w (S , π ) ≤ 0

(s, π )

0

s, s ≤ r0 0∈s

s ≤ r0

⎛a − c +ε ⎞ ≤⎜ ⎟ 2 ⎝ ⎠

2

⎛a − c +ε ⎞ α (π S ) β 0 (S ) = ⎜ ⎟ ∑ 2 ⎝ ⎠ (s , π )

2

s ≤ r0 0∈s

r0

∑ (∑π ) j =1 s , s= j 0∈s

∏ (T

T ∈π \ s

− 1)!( j − 1)!

(n + 1)!

Since x ! y ! ≤ ( x + y )! we have

⎛a − c +ε ⎞ L1 ≤ ⎜ ⎟ 2 ⎝ ⎠

2

(n − j )! ( j − 1)! = ⎛ a − c + ε ⎞ 2 r ⎛⎜ n ⎞⎟ (n − j )! ( j − 1)! ⎜ ⎟ ∑⎜ ⎟ ∑ ∑ (n + 1)! (n + 1)! 2 ⎝ ⎠ j =1⎝ j ⎠ j = 1 (s , π ) r0

0

s = j 0∈s

(6) 2

⎛ a − c + ε 2 ⎞ r0 r ⎛a − c +ε ⎞ 1 ⎟⎟ ∑ = ⎜⎜ ≤ 0 ⎜ ⎟ → 0, as n → ∞ 2 2 ⎠ ⎝ ⎠ j =1 j (n + 1) n + 1 ⎝ Similarly, it can be shown that the probability that a random partition π of N 0 contains a coalition S with S ≥ n − t0 decreases to zero as n → ∞ . Indeed, in this case π \ {S } is a partition of N 0 \S into subsets of size of at most to. By Lemma 7

(

)

Pr ob ℘tn0 + 1 → 0 , as n → ∞ . Hence

L2 ≡

∑ α (π S ) β (S ) w (S , π ) → 0,

(s , π )

s ≥ n − t0

Finally,

0

as n → ∞

(7)

31

φ0 (w ) = L1 + L2 +

∑ α (π S ) β (S ) w (S , π ) 0

(s, π )

r0 ≤ s ≤ n − t0

where

∑ α (π S ) β (S ) w (S , π ) =

(s , π )

0

r0 ≤ s ≤ n − t0 0∈s

r0 ≤ s ≤ n − t0

+

∑ α (π S ) β (S ) w (S , π ) +

0

(s , π )

∑ α (π S ) β (S ) w (S , π ) 0

(s , π )

r0 ≤ s ≤ n − t0 0∉s

For sufficiently large n and by (4) the first summond is ε (a − c ) .By (5) the second one is bounded below by (− η ) . Hence for sufficiently large n

ε (a − c ) − η ≤

∑

(s, π )

α (π S ) β 0 (S ) w (S , π ) ≤ ε (a − c )

(8)

r0 ≤ s ≤ n − t0

By (6), (7) and (8)

φ (w ) → ε (a − c ),

as n → ∞

as claimed. ■

The proof is similar to the proof of Proposition 5 of

Proof of Proposition 5

TW. Since n is sufficiently large consider n such that n ≥ K 3 . Suppose to the contrary that C V ≠ φ . Then, there is a vector payoff in C V such that all firms in N obtain the same payoff (the equal treatment property of the core). Suppose that

(x 0 , x, L , x )∈ C V . Then

x 0 + n x = V (N 0 ) . Let S ⊆ N , then S x + x 0 ≥ V (S 0 ) .

Hence V (N 0 ) − V (S 0 ) ≥ (n − S ) x

Since V (N ) ≤ n x we have

32

V (N 0 ) − V (S 0 ) ≥

(n − S ) n

V (N )

(9)

Suppose that S ≥ K and n − S + 1 ≥ K . Then by Proposition 2 V (S 0 ) = ∈ (a − c ) , 2

2

⎛a − c⎞ ⎛ a − c + ∈⎞ V (N 0 ) = ⎜ ⎟ . Hence, by (9) ⎟ and V (N ) = ⎜ 2 ⎝ ⎠ ⎝ 2 ⎠ 2

n−K ⎛ a − c + ∈⎞ ⎜ ⎟ − ∈ (a − c ) ≥ n 2 ⎝ ⎠

⎛a −c⎞ ⎜ ⎟ ⎝ 2 ⎠

2

This is equivalent to 2

2

n − K ⎛a − c⎞ ⎛ a − c − ∈⎞ ⎜ ⎟ ≥ ⎜ ⎟ . 2 n ⎝ 2 ⎠ ⎝ ⎠ The last inequality is equivalent to

n≤

Since

a−c

ε

ε 2 (2 a − 2 c − ε )

(a − c )3

≥1

n≤2− Hence, for any n ≥ 2 the core is empty

ε a−c ■