Firms' R&D decisions under incomplete information

European Journal of Operational Research 129 (2001) 414±433 www.elsevier.com/locate/dsw Theory and Methodology Firms' R&D decisions under incomplete information V.A. Grishagin a, Ya.D. Sergeyev a,b,* , D.B. Silipo c a c Software Department, University of Nizhi Novgorod, pr. Gagarina, 23, Nizhi Novgorod, Russian Federation b ISI-CNR, c/o University of Calabria, 87036 Rende (CS), Italy Dipartimento di Economia Politica, Universit a della Calabria, Cubo 19A, 87036 Arcavacata di Rende (CS), Italy Received 29 September 1998; accepted 7 September 1999 Abstract The paper considers a patent race in which ®rms do not know their relative positions. In this setting, ®rms that start in the same position proceed at the highest possible speed; and if one ®rm has an initial advantage it preempts the rival, but at the cost of dissipating a signi®cant part of its monopoly rent. So the paper shows that incomplete information in a patent race leads to rent dissipation. The latter is higher, the higher the value of the prize and the lower the cost of R&D. Thus, for innovations that provide relatively high pro®ts the time to discovery is shortened, but the social losses are likely to be high, due to duplication of e€ort. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Patent races; Incomplete information; Rent dissipation 1. Introduction Very often the R&D activity that leads to the production of knowledge assumes the characteristics of a race between competing ®rms. If there is a perfect patent system, the winner takes all and the losers get nothing. On the other hand, if patent protection is imperfect, losers too may bene®t from the innovation. Among the authors who have analyzed models of patent race are Loury (1979), Dasgupta and Stiglitz (1980), Reinganum (1981), Fudenberg et al. (1983), Harris and Vickers (1985, 1987), Beath et al. (1989) and Nti (1997). All assume that ®rms interact strategically and posit winner-takes-all games. Although they di€er in the characterization of the R&D game (Reinganum (1981) assumes that each ®rm chooses a time path of R&D expenditure at the outset, whereas Fudenberg et al. (1983) and Harris and Vickers (1985,1987) assume ®rms revise their decisions according to their relative position in the race), all these models posit that each ®rm knows its relative position in the race, in terms of acquired knowledge. Looking at the real world, however, it is very hard to maintain such an assumption. In competitive R&D markets, research programs are conducted secretly, and competitors know very little about the research * Corresponding author. Tel.: +39-0984-839047; fax: +39-0984-839054. E-mail address: yaro@si.deis.unical.it (Ya.D. Sergeyev). 0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 4 2 7 - 0 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 415 progress of rivals until someone gets the patent. On the other hand, the imperfect information condition in which R&D activity takes place motivates many actions, ranging from industrial espionage to blung (overstating their successes in order to induce competitors to drop out of the race). The aim of the present paper is to analyze a duopoly R&D game when ®rms do not know their relative positions. More precisely, in a framework with no uncertainty in R&D activity, we consider a duopoly model of patent race, in which the two ®rms compete for a prize of known value. Although they know their starting positions, they cannot monitor their rivalÕs progress. In this framework we compute the Nash equilibria of the patent race in relation to the entire parametric space of the game and the initial positions of the ®rms. The main conclusion is that both ®rms will engage in R&D if they are in the same position at the outset. In this case the ®rms dissipate the rent arising from the patent in the attempt to win the race. On the other hand, if they start in di€erent positions only one ®rm engages in R&D, although the winner of the race dissipates a signi®cant part of the monopoly rent in order to keep its rival from entering. Thus, the main implication is that in patent races incomplete information leads to rent dissipation. By contrast, in the models cited if one ®rm gets a lead on its rival the latter drops out; the race turns out into a monopoly. On the other hand, one feature of the above models is their inability to explain simultaneous discovery. Even ``when ®rms begin with equal experience there is a burst of R&D followed by the eventual emergence of a monopolist'' (Fudenberg et al., 1983, p. 15). As a matter of fact, in many circumstances several ®rms make the discovery simultaneously (see Jewkes et al., 1969), as a result of research conducted in parallel and pursuing the same end. Simultaneous discovery arises in a natural way when ®rms involved in a deterministic patent race have incomplete information about the position of their rivals. This is due to the fact that competitive ®rms conducting R&D activity in secret pursue research programs right up to the end of the race, so that if they pursue the same aim, ®rms that start in the same position get to the end discovery at the same time. The paper is organized as follows. Section 2 presents the main features of the model, Section 3 computes the Nash equilibrium of the symmetric game, and Section 4 extends the analysis to an initial asymmetric position. Section 5 summarizes the main results. 2. The model We consider a model in which two ®rms compete in a multistage patent race for the acquisition of a prize of positive value, V, common to both ®rms. Like Fudenberg et al. (1983), we assume that the competition is staged in discrete time t ˆ 0; 1; . . . ; and the discovery occurs with certainty when a given number of ``units of knowledge'', N, are accumulated. The patent is awarded to the ®rm that ®rst achieves level N. If both ®rms achieve the discovery simultaneously, the prize goes to the ®rm with the highest level of knowledge. If they tie, they have equal probability of getting the prize. Again like Fudenberg et al. (1983), we assume that the R&D process is deterministic. 1 That is, there is a deterministic relationship between the amount of expenditure on the project and the 1 Of course, the deterministic model of patent race is a simpli®cation of reality. In the real world, the degree of uncertainty related to R&D activity goes from true uncertainty, which characterizes basic research and fundamental inventions and innovations, to very little uncertainty, which is the case of modi®cation of the existing product or process, product di€erentiation and minor technical improvements (see Freeman and Soete, 1997). One example of the kind of races we are considering here is the race between United States and Germany during the second world war to produce the atomic bomb. Many other examples involve the use of computers through productive processes or in products in many industries. However, the implications of a stochastic R&D process will be discussed in Section 5. 416 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 knowledge accumulated. 2 However, Fudenberg et al. (1983) assume that ®rms know their relative position, so that in every stage each ®rm can decide whether to continue or to drop out, taking account of the other ®rmÕs position. Unlike these authors, we assume that research programs are secret. Although the ®rms know their relative position at the beginning of the race, 3 they are not able to monitor the progress of the rival until the game is over. Let us describe the R&D game. At time t ˆ 1, both ®rms know their relative position in the race and decide their e€ort levels for each stage of the game. They then accumulate independently units of knowledge, and the game ends when one ®rmÕs accumulated knowledge equals or exceeds a given amount, N (the discovery is supposed to occur with certainty at this moment). Di€erent levels of e€ort provide di€erent ``amounts of knowledge'' and have di€erent costs (R&D expenditure). To simplify the analysis, we assume that in each period ®rms can exert zero-e€ort, which provides no knowledge and costs nothing, c0 ˆ 0; low level, which generates one ``unit of knowledge'' and costs c1 > 0; and high-e€ort, which provides two units of knowledge and costs c2 > c1 . Following Fudenberg et al. (1983) we assume that c2 > 2c1 : 2:1† A strategy S is a sequence of e€orts (``costs'') at every stage t; 1 6 t 6 T , where T is the time when the ®rm terminates its participation in the race. On the foregoing our assumptions, an alternative way to describe a strategy S is as a sequence of knowledge units at obtained at each stage t; 1 6 t 6 T , i.e., S ˆ a1 ; a2 ; . . . ; aT †, where at ˆ f0; 1; 2g depending on whether the ®rm exerts zero-, low- or high-e€ort at each stage t. Let us consider the possible strategies in more details. Among them is ``doing nothing'', i.e., not participating in the race. This corresponds to the ®rmÕs decision to make zero-e€ort at every stage. Denote this strategy as S ˆ 0†: 2:2† In this case the ®rmÕs payo€ does not depend on the choices of the rival; it is nil, since the ®rm spends nothing but also earns nothing. 4 Any strategy S ˆ a1 ; a2 ; . . . ; aT † that leads to the ®rmÕs completing the race at time T p S† ˆ T X ai 2:3† iˆ1 denotes the accumulated knowledge of the ®rm playing the strategy, and T X cai d S† ˆ 2:4† iˆ1 denotes the cost. 2 The discrete nature of the relationship between money and knowledge embodies the assumption that technological progress takes place through jumps in the knowledge, more than by a continuous relationship between the two variables. 3 There are many ways in which ®rms may determine their initial relative position. One is the quality of the assets and of the products of the ®rms, another may be the past performance of the ®rms in the ®eld. Note, however, that the qualitative results of the paper continue to hold even if ®rms do not know their initial positions and make decisions on the basis of conjectures about the initial position of the rival. 4 This corresponds to the assumption that both ®rms are new entrants in the product market. This assumption does not hold if one ®rm is already established in the market. V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 417 Any strategy S such that p S† < N (non-®nishing strategy) leads to the ®rmÕs dropping out of the race. This strategy provides to the ®rm a payo€ ÿd(S) independent of the actions of the rival. If T > 1, i.e., S 6ˆ S , then ÿd S † < 0 and it follows that S strictly dominates S. In other words, once started it is worthwhile to complete the race. As a consequence, we can eliminate from the analysis all the non-®nishing strategies except S . Thus, we consider ``®nishing'' strategies that reach the given knowledge level N. Note that for any ®nishing strategy S we have N 6 p S† 6 N ‡ 1: 2:5† Note further that the payo€ of each ®rm depends on its rivalÕs strategy. So if ®rm i, i ˆ a; b; pursues strategy S and its rival pursues S  (both ®nishing) the payo€ of player i is 8 < V ÿ d S† . . . if . . . T S† < T S  † . . . or . . . T S† ˆ T S  † . . . and . . . p S† > p S  ††;  Ui S; S † ˆ V =2 ÿ d S† . . . . . . . . . if . . . T S† ˆ T S  † . . . and . . . p S† ˆ p S  †; : ÿd S† . . . if . . . T S† > T S  † . . . or . . . T S† ˆ T S  † . . . and . . . p S† < p S  ††: We can rewrite the expressions for the accumulated knowledge p(S) and the expenditure d(S) as p S† ˆ q1 ‡ 2q2 ; 2:6† d S† ˆ q1 c1 ‡ q2 c2 ; 2:7† where q1 indicates the number of times that ®rm i exerts low-e€ort and q2 the number that it exerts highe€ort under strategy S. It is evident that the number of steps for S T S† ˆ q1 ‡ q2 : 2:8† Several strategies that achieve p(S) in the same period T(S) can exist. They di€er in the order of exerting the same levels of e€ort, but, because there is imperfect information between players, 5 q1 and q2 coincide for all the strategies. As an example for the case N ˆ 4, we can indicate the strategies S 0 ˆ 1; 1; 2†, S 00 ˆ 1; 2; 1†; S 000 ˆ 2; 1; 1†. Formally they are di€erent but all lead to the same ®nal result. Thus, each ®nishing strategy is determined by two parameters q1 and q2 only, and we can restrict our analysis to strategies that di€er with respect to these two parameters. So without loss of generality we consider the strategy applying ®rst all q1 (low-e€ort) and subsequently the remaining q2 steps with maximum e€ort as a representative of the entire group (S 0 in the above example). We denote this strategy as S ˆ q1 ; q2 †: 2:9† Let us de®ne the payo€ Ui S; S  † of the ®rm i adopting S ˆ q1 ; q2 † while the rival pursues S  ˆ q1 ; q2 † as 8 V ÿ q1 c1 ÿ q2 c2 if q1 ‡ q2 < q1 ‡ q2 or > > > > q1 ‡ q2 ˆ q1 ‡ q2 and q1 ‡ 2q2 > q1 ‡ 2q2 †; > > > <   2:10† Ui S; S  † ˆ V =2 ÿ q1 c1 ÿ q2 c2 if q1 ˆ q1 and q2 ˆ q2 ; > > > > >   > > : ÿq1 c1 ÿ q2 c2 if q1 ‡ q2 > q1 ‡ q2 or  q1 ‡ q2 ˆ q1 ‡ q2 and q1 ‡ 2q2 < q1 ‡ 2q2 †: 5 Of course, this assumption is not any more valid if ®rms know their positions in the race. In this case, the order of play is crucial to their subsequent positions and decisions in the race. For the analysis of this case, see Fudenberg et al. (1983) and Harris and Vickers (1985). 418 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 In this setting, we next consider the conditions for the existence of equilibrium. A Nash equilibrium for this game consists of a strategy Si for each ®rm such that U Si ; Sj † P U Si ; Sj † for each ®rm and for every Si 6ˆ Si ; i; j ˆ a; b. To ®nd the possible Nash equilibrium, we ®rst eliminate dominated strategies and then determine the conditions for the existence of Nash equilibrium with respect to the parametric space of the model (i.e., V, N, c1 and c2 ) and the relative position of the ®rms at the start of the race. To this end, we construct all ®nishing strategies and distribute them in decreasing order with respect to q1 , indicating the corresponding number of steps T to arrive at the ®nish line. That is ˆ S N ; 0†; T ˆ N; ˆ S N ÿ 1; 1†; T ˆ N; ˆ S N ÿ 2; 1†; T ˆ N ÿ 1; 1|‚‚{z‚‚}


1 22 ˆ S N ÿ 3; 2†; T ˆ N ÿ 1; 2


2 1


1 |‚‚{z‚‚} |‚‚{z‚‚} ˆ S N ÿ 2k; k†; T ˆ N ÿ k; ˆ S N ÿ 2k ÿ 1; k ‡ 1†; T ˆ N ÿ k; 1|‚‚{z‚‚}


1 N 1|‚‚{z‚‚}


12 N ÿ1 1|‚‚{z‚‚}


12 N ÿ2 N ÿ3 N ÿ2k .. . k


2 1


1 2|‚‚{z‚‚} |‚‚{z‚‚} N ÿ2kÿ1 k‡1 2


2 |‚‚{z‚‚} .. . ˆ S 0; kN †; T ˆ kN ; kN where kN ˆ  N =2 N ‡ 1†=2 for N even; for N odd: So, we have N ‡ 1 ®nishing strategies, which can be summarized by the kN pairs corresponding to the same number of steps T. (If N is even, the strategy S(0, kN ) will be unique). In the pair for T ˆ N ÿ k the ®rst strategy S N ÿ 2k; k† reaches the discovery strictly and the strategy S N ÿ 2k ÿ 1; k ‡ 1† exceeds the required amount of knowledge N. Proposition 1. Any ®nishing strategy S ˆ q1 ; q2 † exceeding the ®nal level of knowledge N except the strategy S 0; kN † ˆ 2|‚‚{z‚‚}


2 for N odd is dominated. kN Proof. Consider a ®nishing strategy S ˆ q1 ; q2 † such that p S† ˆ N ‡ 1 and q1 P 1. For this strategy it is always possible to match the strategy S 0 ˆ q01 ; q02 † with q01 ˆ q1 ÿ 1, q02 ˆ q2 . Note that T S 0 † ˆ T S† ÿ 1; p S 0 † ˆ N (this strategy reaches the discovery strictly, without exceeding). Compare the payo€s of the strategies S and S 0 to ®rm i, i ˆ a; b; for all possible strategies S  of the rival. If T  ˆ T S  † > T S† or T  ˆ T S† and p S  † ˆ N then Ui S; S  † ˆ V ÿ q1 c1 ÿ q2 c2 ; 2:11† 419 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 Ui S 0 ; S  † ˆ V ÿ q1 ÿ 1†c1 ÿ q2 c2 ; 2:12† whence Ui S; S  † < Ui S 0 ; S  †; i ˆ a; b: 2:13† In the case S  ˆ S we have Ui S; S  † ˆ V =2 ÿ q1 c1 ÿ q2 c2 and comparing the last equality with (2.12), the inequality (2.13) follows. If T  < T S† the payo€ Ui S; S  † ˆ ÿq1 c1 ÿ q2 c2 but the payo€ Ui S 0 ; S  † ˆ V =2 ÿ q1 ÿ 1†c1 ÿ q2 c2 in the situation S  ˆ S 0 , or Ui S 0 ; S  † ˆ ÿ q1 ÿ 1†c1 ÿ q2 c2 in the remaining cases, and the relation (2.13) is again ful®lled.  Thus, the proposition shows that pursuing a strategy that brings about a level of knowledge exceeding required knowledge N is not optimal. As a consequence we can build up the matrix of the game from the set of strategies T comprising the strategy S , all the strategies S N ÿ 2k; k† for 0 6 k 6 kN , if N is even or for 0 6 k 6 kN ÿ 1, if N is odd, and the strategy S 0; kN † at N odd. Let us consider all the strategies of the set T in an ``increasing'' order in accordance with the following rule. A ®nishing strategy S q1 ; q2 † 2 T is supposed to be ``worse'' than a ®nishing strategy S 0 ˆ q01 ; q02 † if q2 < q02 . Eliminating all dominated strategies from the initial full set, we get the reduced matrix of the game by the strategies ordered as follows: S < S N ; 0† < S N ÿ 2; 1† <


< S N ÿ 2k; k† <


< S 2; kN ÿ 1† < S 0; kN † 2:14† if N is even, and S < S N ; 0† < S N ÿ 2; 1† <


< S N ÿ 2k; k† <


< S 1; kN ÿ 1† < S 0; kN †: 2:15† If N is odd. For any strategy S q1 ; q2 † 2 T such that S 6ˆ S and S 6ˆ S 0; kN † for N odd we have q1 ˆ N ÿ 2q2 ; p S† ˆ N ; T S† ˆ N ÿ q2 and, therefore, for any such strategies S q1 ; q2 † and S 0 ˆ q01 ; q02 †, S < S 0 if and only if q2 < q02 : In the case of N odd and S ˆ S 0; kN †, p S† ˆ N ‡ 1; T S† ˆ kN ˆ N ‡ 1†=2: Note that under N odd there is a strategy S 1; kN ÿ 1† for which T S† ˆ kN as well, but p S† ˆ N . 420 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 Taking into account the above relations we can write (2.14) and (2.15) in the following uni®ed and shorter notation: S < S0 < S1 <


< Sk <


< SkN ; 2:16† where Sk ˆ S N ÿ 2k; k†; k ˆ 0; k 6 kN for N even or k 6 kN ÿ 1 for N odd, and SkN ˆ S 0; kN † for N odd. Moreover, we can represent the payo€s (2.10) in a simpler form. First, if S ˆ S , then Ui S; S  † ˆ 0 for any rival strategy S  . In the case S 6ˆ S 8 if S  < S; < V ÿ q1 c 1 ÿ q 2 c 2  2:17† Ui S; S † ˆ V =2 ÿ q1 c1 ÿ q2 c2 if S  ˆ S; : if S  > S: ÿq1 c1 ÿ q2 c2 When N is even or when N is odd and S 6ˆ SkN , we can write the payo€ as 8 if q2 < q2 ; < V ÿ N ÿ 2q2 †c1 ÿ q2 c2  Ui S; S † ˆ V =2 ÿ N ÿ 2q2 †c1 ÿ q2 c2 if q2 ˆ q2 ; : ÿ N ÿ 2q2 †c1 ÿ q2 c2 if q2 > q2 and for N odd and S ˆ SkN we have  V ÿ N ‡ 1†c2 =2 if q2 < q2 ; Ui S; S  † ˆ V =2 ÿ N ‡ 1†c2 =2 if q2 ˆ q2 : 2:18† 2:19† Thus, we have de®ned the possible strategies of the game (they are enumerated in (2.16)) and the payo€s for all possible pairs of strategies. 3. The Nash equilibria of the symmetric game The aim of the present section is to establish the conditions under which the Nash equilibrium exists, assuming that the ®rms start the race in the same position. We consider the case in which ®rms start in di€erent positions in the following section. Let us ®rst prove two useful lemmas. The ®rst establishes that provided one ®rm is the winner of the race it is pro®table for it to make the lowest e€ort necessary to achieve the discovery. In formal terms, we have Lemma 1. If kN ˆ m > k ˆ 0, then Ui Sk ; S  † > Ui Sm ; S  † for any S  < Sk and i ˆ a; b. Proof. Note that m > k implies Sm > Sk . Since c2 > 2c1 then m ÿ k†c2 > 2 m ÿ k†c1 ; 2kc1 ÿ kc2 > 2mc1 ÿ mc2 ; V ÿ Nc1 ‡ 2kc1 ÿ kc2 > V ÿ Nc1 ‡ 2mc1 ÿ mc2 ; V ÿ N ÿ 2k†c1 ÿ kc2 > V ÿ N ÿ 2m†c1 ÿ mc2 ; i.e., (3.1) is ful®lled for the strategies Sk ˆ S N ÿ 2k; k† and Sm ˆ S N ÿ 2m; m†. 3:1† V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 421 If N is odd and m ˆ kN then SkN ˆ S 0; kN † and Ui SkN ; S  † ˆ V ÿ kN c2 ; S  < Sk N : Since c2 ÿ c1 > 0, then V ‡ c2 ÿ c1 ÿ kN c2 > V ÿ kN c2 ; V ÿ c1 ÿ kN ÿ 1†c2 > V ÿ kN c2 ; Ui SkN ÿ1 ; S  † > Ui SkN ; S  †; S  < SkN ; i.e., (3.1) holds for m ˆ kN as well.  The second lemma states that if the strategy of doing nothing dominates one ®nishing strategy, then it dominates all ®nishing strategies. Lemma 2. If the strategy S dominates a strategy Sk ; k P 0, then S dominates all the strategies Sm ; m > k. Proof. First recall that Ui S ; S  † ˆ 0 for any S  . For a strategy Sq ; q P 0, max Ui Sq ; S† ˆ Ui Sq ; S † 3:2† Ui Sq ; S† ˆ Ui Sq ; S †; 3:3† S since S < Sq ; Ui Sq ; S† ˆ Ui Sq ; S † ÿ V =2; Ui Sq ; S† ˆ Ui Sq ; S † ÿ V ; S ˆ Sq ; 3:4† S > Sq : 3:5† If S dominates Sk , then Ui Sk ; S † 6 0. But, from (3.1), for m > k, Ui Sm ; S † < Ui Sk ; S † 6 0 and from (3.2) we have Ui Sm ; S† < 0 for any S.  With the foregoing lemmas, we can prove the following theorem. Theorem 1. In the game under consideration there exist the following types of Nash equilibria in pure strategies: (i) if V 6 Nc1 ; 3:6† then the pair (S ; S ) implements the unique Nash equilibrium; (ii) in the cases Nc1 < V < minf2Nc1 ; N ÿ 2†c1 ‡ c2 g; N P 2; 3:7† 422 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 or c1 < V < 2c1 ; N ˆ 1; 3:8† the matrix of the game contains two (pure-strategy) Nash equilibria, namely, S ; S0 † and S0 ; S †; and one mixed strategy Nash equilibrium on the two pure-strategy equilibria. (iii) Under conditions c2 P N ‡ 2†c1 ; or N P 2; 2Nc1 6 V 6 2 c2 ÿ 2c1 †; 2c1 6 V 6 2c2 ÿ 2c1 ; 3:9† N P 2; 3:10† N ˆ 1; 3:11† there is the unique Nash equilibrium S0 ; S0 †; (iv) the unique Nash equilibrium (SkN ; SkN ) is also guaranteed by the inequality V P 2kN c2 : 3:12† Proof. See Appendix A. The theoretical results of the theorem can be interpreted as follows. The e€ort made by ®rms starting in the same position depends on the value of the prize relative to its cost. First, ®rms engage in R&D only if the value is greater than Nc1 , i.e., the cost of achieving the discovery exerting low-e€ort in each period. Moreover, when the value of prize is low (within the double inequalities (3.7) or (3.8)), there are three Nash equilibria: two pure-strategy ones ((S ,S0 ) and (S0 ,S )), and one mixed-strategy combining the two pure strategies. However, since the payo€ of the ®rms di€ers in the pure strategies, if there is no coordination both ®rms want to do R&D and both make losses. So the only plausible Nash equilibrium for the relatively low-value prize is the mixed strategy equilibrium that makes the ®rms indi€erent between the two purestrategy equilibria. As the prize becomes more attractive relative to the cost (condition (3.10)), or the cost c2 is signi®cantly greater than c1 (condition (3.9)), the optimal behavior for both ®rms is to move to the ®nish line at lowe€ort. Finally, if the prize exceeds the value 2kN c2 , the optimal strategy is to proceed to the goal at the highest e€ort level only. However, Theorem 1 does not describe all the possible outcomes of the game. There are some circumstances in which there are no Nash equilibria in pure strategies but only mixed-strategy ones. These circumstances are established in Theorem 2 below. In the proof of the theorem we will make use of the following sequence. Let us de®ne for N P 1 the sequences aN1 ; aN2 ; . . . ; aNk ; . . . ; aNkN ‡1 3:13† such that a11 ˆ 2c2 ÿ 2c1 ; a21 a12 ˆ 2c2 ˆ maxf2c2 ÿ 4c1 ; c2 g; if N ˆ 1; a22 ˆ 2c2 aN1 ˆ maxf2c2 ÿ 4c1 ; N ÿ 2†c1 ‡ c2 g; 3:14† if N ˆ 2; aN2 ˆ kN c2 ; 3:15† aN3 ˆ 2kN c2 if 3 6 N 6 4; 3:16† V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 423 and, in the general case N P 5, aN1 ˆ maxf2c2 ÿ 4c1 ; N ÿ 2†c1 ‡ c2 g; 3:17† aNk ˆ N ÿ 2k†c1 ‡ kc2 ; 3:18† 2 6 k 6 kN ÿ 1; aNkN ˆ kN c2 ; 3:19† aNkN ‡1 ˆ 2kN c2 : 3:20† Lemma 3. The sequence (3.13) is strictly increasing. Proof. For N ˆ 1 and N ˆ 2 the assertion of the lemma is obviously ful®lled. Let N P 3. Then, for k > 1, as well as for the situation k ˆ 1 and aN1 ˆ N ÿ 2†c1 ‡ c2 , the proof of N ak < aNk‡1 can be based on the technique used in Lemma 1; because of its simplicity we do not reproduce it here. There remains the case aN1 ˆ 2c2 ÿ 4c1 . If 3 6 N 6 4, then aN2 ˆ 2c2 > aN1 . Finally, for N ˆ 5 aN2 ˆ N ÿ 4†c1 ‡ 2c2 ˆ Nc1 ‡ 2c2 ÿ 4c1 > aN1 :  Theorem 2. Assume that aNk < V < aNk‡1 ; 1 6 k 6 kN : 3:21† Then, the game has a mixed-strategy Nash equilibrium combining the pure strategies S ; S0 ; . . . ; Sk . Proof. See Appendix B. Thus, for intermediate values of V given by condition (3.21), there exist only mixed-strategy Nash equilibra, in which both ®rms randomize between the ®rst k pure strategies. Now we are able to provide the full classi®cation of Nash equilibria in relation to the values of the parameters of the game: i.e., V, N, c1 , c2 when the ®rms start in the same position in the race. This classi®cation is given with respect to an increasing order of the value of the prize, V. There are two cases as regards the R&D technology: 1† 2c1 < c2 < N ‡ 2†c1 3:22† 2† c2 P N ‡ 2†c1 : 3:23† and Consider ®rst (3.22). It follows from the foregoing that if (1.1) V 6 Nc1 , we have unique Nash equilibrium S ; S † in pure strategies. For (1.2) Nc1 < V 6 N ÿ 2†c1 ‡ c2 there exist two Nash equilibria in pure strategies S ; S0 † and S0 ; S † and one mixed-strategy equilibrium. Up to (1.3) V ˆ 2kN c2 only the mixed strategies are implemented. If 424 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 (1.4) N ÿ 2k†c1 ‡ kc2 < V 6 N ÿ 2 k ‡ 1††c1 ‡ k ‡ 1†c2 ; 3:24† 1 6 k 6 kN , then we have the equilibrium state in mixed strategies over pure strategies S ; S0 ; S1 ; . . . ; Sk †. Finally, for (1.5) V > 2kN c2 ; 3:25† we again have a unique Nash equilibrium in pure strategies: ( SkN ; SkN ). In case (3.23) and (2.1) V 6 Nc1 , there exists a unique Nash equilibrium (S , S ). But under the condition (2.2) Nc1 < V 6 2Nc1 two pure-strategy equilibria S ; S0 † and S0 ; S † and one mixed-strategy one are implemented; However, if (2.3) 2Nc1 < V 6 2c2 ÿ 4c1 we again have the single Nash equilibrium S0 ; S0 †. When (2.4) 2c2 ÿ 4c1 < V 6 N ÿ 4†c1 ‡ 2c2 , there arises the mixed equilibrium state over the pure strategies S ; S0 ; S1 †. Finally, for the conditions (2.5) (3.24) and (3.25) we have the same results as in the case (3.22). This classi®cation has at least two signi®cant implications: (1) competition in a patent race is ®ercer as the value of the prize increases relative to the cost of discovery, in that more aggressive strategies are required to win; (2) the higher is c2 relative to c1 , the greater the incentive for the ®rms to play pure strategy instead of mixed-strategy Nash equilibria. It is intuitive that whatever strategies are adopted by symmetric ®rms, their expected pro®t is nil if there are no constraints on their speed. 6 To prove this, assume instead that at a Nash equilibrium one of the ®rms does get a positive pro®t. It follows that the rival, by increasing the level of R&D expenditure, can get to the discovery earlier without making negative pro®t; this in turn would lead the ®rst ®rm to increase its own R&D e€ort. Proceeding thus, we conclude that the ®rms increase expenditure up to the point where their expected pro®t is nil. The only conclusion is that in the e€ort to win the race ®rms starting in symmetric positions dissipate the rent arising from the patent. In Section 4 we show that the rent dissipation result holds even when starting positions are di€erent. 4. The asymmetric game We now consider a situation in which one ®rm has an advantage over the other ®rm at the beginning of the race, due to such factors as di€erences in size, market position or assets, and this advantage is common knowledge to both. Let us assume at the outset that ®rm i is k P 1 steps ahead of ®rm j; i; j ˆ a; b. That is, ®rm i needs to accumulate only N ÿ k > 0 units of knowledge when j must accumulate N units to get to the discovery. The following theorem then holds. Theorem 3. If ®rm i is k P 1 steps ahead of ®rm j, there always exists a strategy allowing i to win the race, and therefore forcing j not to play, i; j ˆ a; b. 6 For example, for the case V P 2kN c2 , the ®rms make positive pro®ts at a Nash equilibrium. However, this result depends on the constraint on available options to the ®rms they can play only {0, 1, 2}. Without this constraint the rent dissipation would be total. V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 425 Proof. Assume the ®rms are at t ˆ 1. Further, without loss of generality, assume that at the beginning of the race ®rm a is k-steps closer to the discovery than ®rm b. Thus, ®rm a wins the game if it accumulates an amount of knowledge equal to N ÿ k, that is q1 ‡ 2q2 ˆ N ÿ k and it arrives before ®rm b, i.e., q1 ‡ q2 < q1 ‡ q2 ; where, of course, q1 ‡ 2q2 ˆ N : It follows from the above inequalities that ®rm a wins if q2 > q2 ÿ k: 4:1† Firm b wants to catch up with ®rm a and will run at its top possible speed insofar as it does not make losses; that is, it pursues strategy q1 ; q2 †, where q2 ˆ maxfq2 : V ÿ c1 q1 ÿ c2 q2 P 0; q1 ‡ 2q2 ˆ N g: 4:2† Let us ®rst consider the case when V is high enough to allow to ®rm b to run at top speed, i.e., q2 ˆ I N =2† and V ÿ c1 q1 ÿ c2 q2 P 0; where I(r) is the integer part of r. Thus, it follows from (4.1) and k P 1 that there exists a strategy (q1 , q2 ) for ®rm a 0 6 q2 6 I N =2†† such that q2 > I N =2† ÿ k; and a wins the game because b cannot catch up and therefore will not participate. When the value V is not high enough to allow to ®rm b to run at top speed, its speed is bounded by the inequality V ÿ c1 q1 ÿ c2 q2 P 0: Thus, ®rm a wins again, because given (4.1) and k P 1, it is in a position to make the highest e€ort enough times that it is uneconomic for b to participate. So at t ˆ 1 ®rm b drops out and ®rm a makes the highest e€ort.  In order to demonstrate the stability of the solution reached, we show that ®rms do not deviate from the above equilibrium conditions. Assume at t ˆ 1 that ®rm a is closer to the discovery than ®rm b by one step. 7 At t ˆ 1 ®rm b does not participate and ®rm a makes the highest e€ort (see Theorem 3). Therefore, at time t ˆ 2 ®rm a is three steps ahead of ®rm b, and consider the behavior of the ®rms at this stage. To show that also in this stage ®rm a makes the highest e€ort, let assume that at t ˆ 2 ®rm a changes its strategy to low-e€ort. In this case, ®rm b would enter the race at the outset and catch up at time t ˆ 2. So, in order to keep ®rm b from participating, ®rm a must make the highest e€ort also in the second period. By induction, assume this is true up to stage T ÿ 2, and consider the behavior of the ®rms at stage 7 This result holds a fortiori if k > 1. 426 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 T ÿ 1. Assume ®rm a plays one at this stage. In this case, ®rm b would choose to enter at the beginning of the race and make the highest e€ort in each period, because this strategy guarantees a neck-and-neck (®nish). It follows that a ®rm that has a one-step headstart, cannot deviate from the highest e€ort level even in the last period. 8 So when one ®rm is ahead of its rival, it runs to the ®nish line at the highest possible speed. The main implication of the theorem is thus that, with incomplete information, the leader must dissipate all or a signi®cant part of the rent from the discovery in order to win the race. This result contrasts sharply with the case of complete information in patent races (see, for example Harris and Vickers, 1985, 1987; Fudenberg et al., 1983). On the other hand, it is in line with the recent result of Nti (1997). In a symmetric patent race with complete and perfect information, the latter proves that the pro®ts of the ®rms go to 0 as the number of rivals increase. In this paper we have shown that if there is incomplete information this result is also valid when there are only two ®rms. 9 Finally, let us consider the e€ects of an increase in the value of the prize and in the R&D costs on the speed of R&D when ®rms are in an asymmetric position. By conditions (4.1) and (4.2), it is straightforward to conclude that an increase in V determines an increase in the number of times that ®rm b can make the highest e€ort. It follows that an increase in the value of the prize intensi®es competition, in that it induces the leader to make the greatest e€ort for a longer period, thus shortening the time to discovery. On the other hand, an increase of c2 relative to c1 has the opposite e€ect, because it decreases the number of times that it is pro®table for both ®rms to make the top e€ort. 5. Conclusions We have considered a deterministic model of a patent race, in which ®rms know their initial position but are not able to monitor the progress made by their rival. In this framework, we study the nature of the R&D race in relation to the position of the ®rms and the values of the parameters. With respect to position, we have proved that ®rms that start in the same position get the discovery simultaneously, while an initial lead is enough to ensure the patent to the ®rm with the head start. However, the winner dissipates much or all of the rent from the innovation in the competition to be the ®rst. Relative to the second aspect, we have shown that competition in the patent race is the more vigorous, the greater the value of the prize. Other things equal, this leads to the conclusion that the higher the value of the prize, the shorter the time to discovery. Since the model is similar to that of Fudenberg et al. (1983), apart from the latterÕs assumption that there exists a one-period information lag (in our model the information lag is in®nite), it may be interesting to compare results. Fudenberg et al. (1983) show that when ®rms begin with equal experience the race is characterized by vigorous competition in the early stages, followed by the eventual emergence of a monopolist, and if one ®rm lags two or more steps behind, the race becomes a monopoly. In a similar framework, Harris and Vickers (1985,1987) also reach the same conclusions. One implication is that an increase in the value of the prize intensi®es competition if ®rms are in similar positions, but does not a€ect the level of R&D expenditure when a ®rm lags two or more steps behind. 8 This conclusion is strictly related to the fact that ®rms are never able to monitor the behavior of the rival during the race and revise their decisions accordingly. See also Footnote 6. 9 There are, however, some important di€erences between NtiÕs paper and ours. The former is settled in a static framework, while ours adopts a dynamic approach. A extensive discussion of many contexts in which rent dissipation takes place is provided by Fudenberg and Tirole (1987). V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 427 By contrast, our own results show that with an in®nite information lag the leader never proceeds at the monopoly pace, and the ®rms intensify their e€orts as the value of the prize increases, independently of relative position. However, the most striking result of Fudenberg et al. (1983) is that the impossibility of monitoring current decisions of the rivals in some circumstances may enable the tailing ®rm to leapfrog its rival. However, this result is no longer valid if we assume that the ®rms can never monitor the position of the rival. In the latter case, the leader preempts the rival when he has even a small advantage. Even though it is very dicult to provide evidence of the role of incomplete information in patent races, many authors (e.g., Rothwell, 1992,1994; Dodgson, 1991; Maidique and Zirger, 1985) support the view that, when a ®rm achieves success with a radical innovation, this is frequently followed by an accumulative series of further successful innovations in the same ®eld. These ®ndings may be explained by drawing on the hypothesis that incomplete information in patent races leads the leader of the race to adopt an ``o€ensive'' strategy rather than resting on its laurels and consolidating its established position (see, Freeman and Soete, 1997, chapter 11). On the other hand, were the relative positions of the ®rms in the race to be known, it would be possible for the leader which is suciently ahead to proceed at the monopoly pace (see, Fudenberg et al., 1983). Although our results seem to support the adage that ``nothing succeeds like success'', the extension of the analysis to a sequence of innovation is not a matter of replicating equilibria of the single-race game. It may well be the case that leapfrogging can occur along the sequence of possible innovations even in deterministic models of patent races, if the trailing ®rm can outweigh the losses of the earlier innovations with the gains of the later ones. A di€erent type of behavior may also arise in a framework of uncertainty over the results of R&D activity. It may well be the case that the ®rm that is behind has an incentive to enter the race because it has a chance to win. However, these two extensions are beyond the scope of the present paper. Finally, although some results of our paper are in line with previous results on this topic (notably, Nti, 1997; Harris and Vickers, 1985,1987; Fudenberg et al., 1983; Reinganum, 1981), our conclusions suggest that rent dissipation in patent races is a more extensive phenomenon than is generally supposed. Acknowledgements We are grateful to Vincenzo Denicol o and three anonymous referees for helpful comments on a previous version of this paper. Appendix A. Proof of Theorem 1 A.1. Assertion (i) Because of V 6 Nc1 , Ui S0 ; S † ˆ V ÿ Nc1 6 0 i ˆ a; b, and according to (3.4) and (3.5), Ui S0 ; S† < Ui S0 ; S † for all S > S , i.e., S dominates S0 and, therefore, all other strategies Sk ; kN P k P 1. Thus, there is the unique Nash equilibrium (S ; S ). A.2. Assertion (ii) Assume that N P 2. It means that there exists a strategy S1 ˆ S N ÿ 2; 1† for which Ui S1 ; S  † 6 V ÿ N ÿ 2†c1 ÿ c2 < 0 for any strategy S  of the rival. Therefore, S dominates S1 and, 428 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 according to Lemma 2, all the strategies Sk ; k > 1. After eliminating dominated strategies, we can reduce the matrix of the game up to the 2  2 matrix S S0 S S0 0, 0 V ÿ Nc1 ; 0 0; V ÿ Nc1 V =2 ÿ Nc1 ; V =2 ÿ Nc1 where the ®rst column represents the possible strategies of ®rm a and the ®rst row the possible strategies of ®rm b. So, the left-hand number in a cell contains the payo€ of player a, and the right-hand indicates the payo€ of player b. Let us denote as R1 (S) the best response of the ith player, i ˆ a; b, to a strategy S of the rival. Then, i R S † ˆ S0 , since V > Nc1 , Ri S0 † ˆ S , since V < 2Nc1 , and we have two Nash equilibria in pure strategies: S0 ; S † and S ; S0 †. If N ˆ 1, then S1 ˆ S 0; 2† and it is dominated by S , because V < 2c1 < c2 . For N ˆ 1, after elimination of S1 from the matrix we obtain the 2  2 matrix (see above). As c1 < V < 2c1 , the situation in this matrix is the same: two Nash equilibria S0 ; S † and S ; S0 †. Moreover, there always exist (see Dasgupta and Maskin, 1986) a mixed-strategy Nash equilibrium which makes each ®rm indi€erent between the two pure-strategy equilibria. A.3. Assertion (iii) Let us compare the strategy S0 ˆ S N ; 0† with the nearest strategy S1 . If N P 2, then S1 ˆ S N ÿ 2; 1†. Take an arbitrary strategy S  of the rival. If S  ˆ S , then Ui S0 ; S  † ˆ V ÿ Nc1 ; Ui Sk ; S  † ˆ V ÿ N ÿ 2†c1 ÿ c2 ; A:1† and due to c2 > 2c1 we have Ui Sk ; S  † < Ui S0 ; S  †: A:2† If S  ˆ S0 , then in accordance with (3.10) Ui S k ; S0 † ˆ V ÿ N ÿ 1†c1 ÿ c2 6 V =2 ‡ c2 ÿ 2c1 ÿ N ÿ 2†c1 ÿ c2 ˆ V =2 ÿ Nc1 ˆ Ui S0 ; S0 †: Now let S  ˆ Sk In this case Ui S0 ; S  † ˆ ÿNc1 ; Ui Sk ; S  † ˆ V =2 ÿ N ÿ 2†c1 ÿ c2 : Again from the right-hand side of (3.10) Ui Sk ; S  † 6 c2 ÿ 2c1 ÿ N ÿ 2†c1 ÿ c2 ˆ ÿNc1 ˆ u S0 ; S  †: At last, if S  > Sk , then Ui S0 ; S  † is taken as (A.3) and Ui Sk ; S  † ˆ ÿ N ÿ 2†c1 ÿ c2 ˆ ÿNc1 ÿ c2 ÿ 2c1 † < ÿNc1 ˆ Ui S0 ; S  †; i.e., (A.2) is ful®lled. A:3† V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 429 So, we have proved that strategy S0 dominates strategy S1 . Therefore, we can delete S1 from the matrix. Proceeding by induction on k, assume that for q ˆ k ÿ 1; k > 1, all the strategies Sm ˆ S N ÿ 2m; m†; m 6 q, are excluded. This means that the strategy Sk ˆ S N ÿ 2k; k†; k > 1, will be the nearest such that Sk > S0 . Let us choose the traditional way of comparing the payo€s for the strategies S0 and Sk . First, notice that V =2 ÿ k c2 ÿ 2c1 † < 0; k > 1, since V =2 6 c2 ÿ 2c1 < k c2 ÿ 2c1 †; k > 1: A:4† If S  ˆ S0 then Ui Sk ; S  † ˆ V ÿ N ÿ 2k†c1 ÿ kc2 ˆ V ÿ Nc1 ÿ k c2 ÿ 2c1 † < V =2 ÿ Nc1 ˆ Ui S0 ; S  †: In the case S  ˆ S ; Ui Sk ; S † ˆ Ui Sk ; S0 †, but Ui S0 ; S † ˆ V ÿ Nc1 > Ui S0 ; S0 † and (A.2) holds again. The next case: S  ˆ Sk . We have (A.3) for Ui S0 ; S  † and Ui Sk ; S  † ˆ V =2 ÿ N ÿ 2k†c1 ÿ kc2 : Taking into account (A.4) the inequality (A.2) holds, as well as in the situation S  > Sk when (A.3) holds for Ui S0 ; S  †, but Ui Sk ; S  † ˆ ÿ N ÿ 2k†c1 ÿ kc2 : Thus, the strategy Sk is also dominated by S0 . So, all the strategies of type Sk ˆ S N ÿ 2k; k†; k ˆ 1 are dominated, and there remain only the two strategies S ˆ S 0; 0† and S0 ˆ S N ; 0† for N even, or three strategies S0 , S1 and SkN ˆ S 0; kN † for N odd, N P 1. In the last case, we have Ui S0 ; S0 † ˆ V =2 ÿ Nc1 ; Ui Sk ; S0 † ˆ V ÿ kN c2 : But V =2 6 c2 ÿ 2c1 6 N c2 ÿ 2c1 †=2 < kN c2 ÿ 2c1 †; N P 3; A:5† i.e., Ui SkN ; S0 † ˆ V =2 ‡ V =2 ÿ kN c2 < V =2 ÿ kN c2 ‡ kN c2 ÿ 2c1 † ˆ V =2 ÿ 2kN c1 ˆ V =2 ÿ Nc1 ÿ c1 < V =2 ÿ Nc1 ˆ Ui S0 ; S0 †: The domination of Ui S0 ; S † over Ui SkN ; S † is given by Lemma 2. The case S  ˆ SkN is the duplication of the variant S  ˆ S0 because of Ui S0 ; SkN † ˆ Ui S0 ; S0 † ÿ V =2 and Ui SkN ; SkN † ˆ Ui SkN ; S0 † ÿ V =2. The situation N ˆ 1 must be considered separately, because in this case (A.5) does not hold. Compare again Ui S0 ; S0 †: ˆ V =2 ÿ c1 and Ui S1 ; S0 † ˆ V ÿ c2 . From (3.11) V =2 < c2 ÿ c1 , hence, V ÿ c2 < V =2 ÿ c1 : The domination of S0 over S1 for S  ˆ S is obvious. 430 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 So, our initial matrix can be reduced to 2  2 with the strategies S and S0 . The domination of S0 over S in this matrix follows immediately from the left-hand sides of the inequalities (3.10) and (3.11), since Ui S0 ; S † ˆ V ÿ Nc1 > Ui S0 ; S0 † ˆ V =2 ÿ Nc1 P 0: Thus, the proof of assertion (iii) is complete. A.4. Assertion (iv) Consider the strategy SkN and evaluate its payo€s. Ui SkN ; SkN † ˆ V =2 ÿ kN c2 P 0; Ui SkN ; S  † ˆ V ÿ kN c2 > 0; S  < SkN ; from which it follows that SkN dominates S . Delete S from the matrix and compare the strategies SkN and S0 . As Ui S0 ; S  † ˆ ÿNc1 < 0, if S  > S0 . To prove the domination of SkN over S0 it is sucient to show that Ui S0 ; S0 † < Ui SkN ; S0 †: Recalling (3.12) we have V > 2kN c2 ÿ 2Nc1 ; V =2 ÿ kN c2 > ÿNc1 ; V ÿ kN c2 > V =2 ÿ Nc1 : The last inequality proves that SkN dominates S0 . Proceeding by induction on k < kN ÿ 1 assume that all the strategies S0 ; . . . ; Sk ; k P 0, have been excluded from the matrix (the strategy S is supposed to have been deleted from the initial matrix before the induction). For the strategy Sk‡1 Ui Sk‡1 ; S  † ˆ ÿ N ÿ 2 k ‡ 1††c1 ÿ k ‡ 1†c2 < 0; S  > Sk‡1 : Meanwhile, from (3.12) V =2 P kN c2 > kN c2 ÿ Nc1 ÿ k ‡ 1† c2 ÿ 2c1 † or, after simple transformations, V ÿ kN c2 > V =2 ÿ N ÿ 2 k ‡ 1††c1 ÿ k ‡ 1†c2 ; which implies Ui Sk‡1 ; Sk‡1 † < Ui SkN ; Sk‡1 † and the last inequality completes the proof of assertion (iv) and of the theorem.  Appendix B. Proof of Theorem 2 As a ®rst step, let us show that any strategy Sm , m > k, is dominated by the strategy S . This becomes clear immediately if we indicate that Ui Sm ; S  † ˆ V ÿ aNm 6 0; S  < Sm ; B:1† V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 431 Ui Sm ; Sm † ˆ V =2 ÿ aNm < 0; B:2† Ui Sm ; S  † ˆ ÿaNm < 0; B:3† S  > Sm ; i ˆ a; b: Eliminate all the strategies Sk‡1 ; . . . ; SkN from the matrix (if such strategies exist). Before the continuation of the proof notice that aN1 > Nc1 : B:4† For N ˆ 1 and N ˆ 2 (B.4) is true because of c2 > 2c1 , and if N P 3, then aN1 ˆ N ÿ 2†c1 ‡ c2 ˆ Nc1 ‡ c2 ÿ 2c1 > Nc1 : Now let us obtain the best responses Ri S† of player i, i ˆ 1; 2, to all possible variants S fS ; S0 ; . . . ; Sk g of rival's behavior in the new matrix. Let the rival of the player i adopt strategy S . Then, R i S † ˆ S0 since Ui S0 ; S † > 0 because of (3.21) and (B.4) and Ui S 0 ; S † < Ui S0 ; S †; S 0 > S0 , according to Lemma 1. Consider a strategy Sq of the rival, 1 6 q 6 k ÿ 1, and show that the best response Ri Sq † ˆ Sq‡1 : B:5† Indeed, Ui Sq‡1 ; Sq † ˆ V ÿ aNq‡1 ˆ V ÿ aNk > 0; i.e., Ui Sq‡1 ; Sq † > Ui S ; Sq †: Moreover, Ui Sm ; Sq † ˆ ÿaNm < 0; 0 6 m < q; and Ui Sm ; Sq † < Ui Sq‡1 ; Sq †; q ‡ 1 < m 6 kN ; due to Lemma 1. Finally, from (3.21) and (3.15)±(3.17) for N P 2, V > aNk P aN1 P 2c2 ÿ 4c1 : B:6† So, for q < k 6 kN , if N is even, and for q < k < kN , if N is odd, it follows that V =2 > c2 ÿ 2c1 ˆ N ÿ 2 q ‡ 1††c1 ‡ q ‡ 1†c2 ÿ N ÿ 2q†c1 ÿ qc2 or V ÿ aNq‡1 > V =2 ÿ aNq ; B:7† 432 V.A. Grishagin et al. / European Journal of Operational Research 129 (2001) 414±433 i.e., Ui Sq‡1 ; Sq † > Ui Sq ; Sq †: B:8† If N ˆ 1 is odd and k ˆ kN ; q ˆ kN ÿ 1, then according to (B.6) V > aNkN ˆ kN c2 ˆ 2c2 > 2c2 ÿ 2c1 ; N P3 (if N ˆ 1 then V > 2c2 ÿ 2c1 follows direct from (3.14)). But then V =2 > c2 ÿ c1 ˆ c2 ÿ N ÿ 2 kN ÿ 1††c1 or V ÿ kN c2 > V =2 ÿ N ÿ 2 kN ÿ 1††c1 ÿ kN ÿ 1†c2 ; i.e., (B.8) holds again. Thus, we have shown that the maximal payo€ of the ith player given the rival's strategy Sq ; q < k, is achieved by adopting Sq‡1 , i.e., the best response is described by (B.5). Consider the last case, when the rival plays Sk . We have for 0 6 m < k Ui Sm ; Sk † ˆ ÿ N ÿ 2m†c1 ÿ mc2 < 0: From (3.21) V 6 aNk‡1 ˆ aNk ‡ c2 ÿ 2c1 if 1 < k 6 kN ÿ 1 at N even or 1 < k < kN ÿ 1 at N odd. But c2 ÿ 2c1 < 2c2 ÿ 4c1 6 aNk ; i.e., V < 2aNk , or Ui Sk ; Sk † ˆ V =2 ÿ aNk < 0: If k ˆ 1, then V 6 aN2 ˆ N ÿ 4†c1 ‡ 2c2 < N ÿ 4†c1 ‡ 2c2 ‡ Nc1 ˆ 2‰ N ÿ 2†c1 ‡ c2 Š; whence Ui Sk ; Sk † ˆ V =2 ÿ N ÿ 2†c1 ÿ c2 < 0: Let k ˆ kN ÿ 1 and N P 3 odd. Then V 6 aNkN ˆ kN c2 < kN c2 ‡ 2c1 ‡ kN ÿ 2†c2 ˆ 2 kN ÿ 1†c2 ‡ c1 †; whence Ui Sk ; Sk † ˆ V =2 ÿ c1 ÿ kN ÿ 1†c2 < 0: Consider the last subcase k ˆ kN . Then V < 2kN c2 and Ui Sk ; Sk † ˆ V =2 ÿ kN c2 < 0: It follows that R i Sk † ˆ S : and the last condition completes the proof.  V.A. 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