KRANNERT GRADUATE SCHOOL OF MANAGEMENT

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1 KRANNERT GRADUATE SCHOOL OF MANAGEMENT Purdue University West Lafayette, Indiana A Comment on David and Goliath: An Analysis on Asymmetri Mixed-Strategy Games and Experimental Evidene by Emmanuel Dehenaux Dan Koveno Volodymyr Lugovsyy Paper No. 116 August 003 Institute for Researh in the Behavioral, Eonomi, and Management Sienes

2 A Comment on David vs. Goliath: An Analysis of Asymmetri Mixed-Strategy Games and Experimental Evidene Emmanuel Dehenaux Dan Koveno Volodymyr Lugovsyy August 19, 003 Abstrat In this note, we haraterize the full set of equilibria of the -firm patent rae analyzed by Amaldoss and Jain (Management Siene, 48(8), August 00, pp ). Contrary to Amaldoss and Jain s (00) laim, we show that the equilibrium is not always unique and that the set of equilibria is non-robust to hanges in the (disrete) set of available strategies. In some equilibria, the qualitative results are the reverse of those in the only equilibrium Amaldoss and Jain identify. Our findings have important impliations for the analysis of the data from Amaldoss and Jain s experiments, as well as other experiments appearing in the literature. Keywords: All-Pay Aution, Contests, Experimental Eonomis, Competitive Strategy, R&D. Department of Eonomis, Purdue University. Corresponding author. Department of Eonomis, Purdue University, 403 W. State St, West Lafayette, IN oveno@mgmt.purdue.edu. 1

3 1 Introdution In a reent paper appearing in this journal, Amaldoss and Jain (00) develop a simple model of a ontest. Two firms ompete to win a patent, the prize, by simultaneously investing in R&D. The firm that invests the higher amount wins the prize and reeives a payoff equal to its value of the prize minus its investment. The firm that invests the lower amount loses an amount equal to its investment. Both firmsfaeasymmetrifinanial onstraint whih prevents them from investing above a ertain amount. This ap on investment is lower than thevalueoftheprizeforeitherfirm. Although the authors fous on a partiular appliation, the game analyzed is a type of all-pay aution with (idential) bid aps. The all-pay aution has been applied in the literature on rent seeing and it is often noted that it an be used as a redued form to model R&D raes. 1 In fat, a very similar all-pay aution has been analyzed by Che and Gale (1998) to model lobbying with expenditure aps. There are two major differenes between the Amaldoss-Jain (heneforth A-J) and the Che-Gale (heneforth C-G) treatments. One is that C-G assume that in the event of a tie in expenditure the prize is either split or alloated with a fair randomizing devie (heneforth partial dissipation ), whereas A-J s main theoretial treatment assumes that in the event of a tie in expenditure the value of the prize is ompletely dissipated and no one reeives the prize (heneforth full dissipation ). 3 A seond major differene lies in the fat that A-J assume a disrete (pure) strategy spae, whereas the strategy spae in C-G is ontinuous. 4 In this note we demonstrate that several of the results obtained by A-J are erroneous, inluding Proposition 1, the one and only proposition in the artile. Contrary to the laim of A-J s Proposition 1, we show that under full dissipation the equilibrium is not always unique. We provide an exhaustive haraterization of the set of equilibria and show that 1 See for instane Dasgupta (1986), Rosen (1988), and Baye, Koveno and de Vries (1996). Baye and Hoppe (003) demonstrate that many other models of patent raes appearing in the literature an be formulated as Tullo rent seeing games, of whih the all-pay aution is a speial ase. Che and Gale (1996a, 1996b) extend the analysis of the all-pay aution with finanially onstrained bidders and a ontinuous strategy spae to the ase of inomplete information and N bidders. 3 A-J also note that other tie breaing rules an be used and analyze an example that uses the prize splitting rule employed by C-G. We omment further on this ase below. 4 For a treatment of the all-pay aution with a disrete strategy spae in the absene of bid aps see Baye, Koveno and de Vries (1994).

4 there is more than one equilibrium under fairly general onditions. We suspet that this failure to reognize the existene of other equilibria stems from an erroneous laim used in the proof of Proposition 1, namely that both firms always earn an expeted profit of zero in a non-degenerate mixed-strategy equilibrium. We also examine some of the impliations of our analysis for the onlusions of the experimental investigation undertaen by A-J. Speifially, we show that the error of imposing a zero profit onstraint arries over to the partial dissipation experiment analyzed by A-J. The vetors of probabilities presented as equilibrium strategies by A-J on page 984 of their artile do not onstitute a Nash equilibrium of the game. In equilibrium, under partial dissipation, the firmwiththelowervalue does not neessarily invest more aggressively than the firm with the higher value as the authors laim on page 977 (even with the additional assumptions mentioned on that page). Below, we begin by introduing some notation. Then, we demonstrate that Amaldoss and Jain s major theoretial laims are erroneous and orret them. We end by disussing the empirial impliations of our results. The model Suppose there are two firms indexed by j, j {H, L}. Firmj s pure strategy spae is given by X j {0,/,/, 3/,..., } with generi element x j. Thus firms have idential strategy spaes. Let r j denote firm j s value for the prize. Following A-J, let r H >r L >.Firmj s payoff from playing x j when the other firm plays x j is denoted by Π j (x j,x j ). Let t denote the fration of the prize eah firm obtains when both firmsinvestthesameamount. We have: Π j (x j,x j ) r j x j if x j >x j, tr j x if x j x j x, x j if x j <x j. In the mixed extension of the game, let p j (x) denote a probability distribution over the elements of X j. Π j (p j,p j ) is then firm j 0 sexpetedpayoff. 3

5 3 Equilibria under the full dissipation assumption We show that equilibria that are qualitatively distint from the equilibrium suggested by A-J exist under fairly general onditions in the game they examine. This diretly ontradits their Proposition 1 and has important impliations for empirial analyses of this type of all-pay aution. We begin by restating A-J s Proposition 1 and its assumptions as they appear in the paper. Reall that the result below assumes full dissipation (t 0) sothatfirm j s payoff is given by: Π j (x j,x j ) r j x j if x j >x j, x j otherwise. Proposition 1 (Amaldoss and Jain) If > 1, the unique equilibrium of this disrete game is for firm H to invest i/ disrete units of apital (where i 1,..., ) with probability: µ p H i Similarly, for firm L we have: µ p L i 1 r L if i 0, 1,..., 1, 1 r L if i. 1 r H if i 0, 1,..., 1, 1 r H if i. The orresponding.d.f. for firms H and L onverges to (3) and (4), respetively, as where: x r F H (x) L 0 x< (3) 1 x. F L (x) 0 x< 1 x. x r H Claim 1 below provides a omplete haraterization of the set of equilibria of the disrete full dissipation game with disrete strategy spae. It shows that equilibria that do not satisfy (1) and () exist fairly generally. These equilibria have an alternating struture, where one firm s support ontains the even points of the strategy spae and the other firm s 4 (1) () (4)

6 ontains the odd points. In these alternating equilibria, one firm earns a stritly positive expeted profit, and the other firm earns zero. Claim 1 Assume > 1. If is odd, there exist three Nash equilibria. One equilibrium is haraterized by (1) and () and satisfies Π H Π L 0. The two other equilibria are given by (p j,p j), j {H, L} where: and p j p j µ i µ i if i 1,3..., ³ if i 1 ³ 1 0 if i 0,,..., 1 1 ³ ³ 1 r j if i 0 r j if i, 4,..., 1 0 if i 1, 3,..., In suh equilibria, expeted payoffs areequaltoπ j r j >0 and Π j 0, j {H, L}. If is even, there exists a unique equilibrium haraterized by (1) and (). The proof of laim 1 appears in the Appendix. The reason why A-J laim uniqueness of equilibrium in their Proposition 1 whereas our Claim 1 demonstrates nonuniqueness is that the proof of A-J s Proposition 1, provided in part A of their Tehnial Appendix, ontains an error. 5 The statement diretly preeding Equation (A1) in A-J s Lemma A4 is not orret. Showing that the system (A1) has a unique solution is not suffiient to prove uniqueness of a Nash equilibrium in mixed strategies. The system (A1) is written with an equality in every row, while it should be written with an inequality ( ) ineveryrow,sinewhena pure strategy is played with probability zero it may indeed yield an expeted payoff that is stritly below the speified equilibrium level. This is exatly the reasoning used in the proof of Claim 1 in this note, whih follows the general programming framewor for solving for the Nash equilibria of all-pay autions with disrete strategy spaes developed in Baye, Koveno, and De Vries (1994). 6 5 The Tehnial Appendies an be found on the INFORMS website at 6 The seond part of A-J s Proposition 1 onerns the ontinuous strategy spae equilibrium that is the limit of the disrete equilibria that they identify as the mesh of the disrete grid of feasible expenditures (5) (6) 5

7 4 Empirial impliations Based on Claim 1, we now reonsider Results and 3 appearing on Page 975 of Amaldoss and Jain (00). Result states that [o]n average, firm L invests more than firm H. Result 3 states that [f]irm L is more liely to win the patent. These two results are not true in the equilibrium where the high-value firm earns a stritly positive expeted profit. Therefore, the ounterintuitive result that the low-value firm invests more aggressively in the patent rae than the high-value firm is speifi to a partiular lass of equilibria in whih the high-value firm earns zero expeted profit. Claim There exists an equilibrium in whih Results and 3 of A-J are reversed. Suppose that >1 and is an odd number: (i) In the equilibrium in whih Π H > 0, firm L invests less than firm H on average and is less liely to win the patent than firm H. (ii) In all equilibria in whih Π H 0, firm L invests more than firm H on average and is more liely to win the patent than firm H. Suppose that >1 and is an even number: (iii) In the unique equilibrium, firm L invests more than firm H on average and is more liely to win the patent than firm H. The proof of Claim appears in the Appendix. In Amaldoss and Jain (00), Proposition 1 is used as a theoretial predition against whih data from experiments are evaluated. The strategy spae hosen by the authors for the experiments is suh that. In this partiular ase, we showed in Claim 1 that the equilibrium is unique and oinides with the one desribed in Proposition 1 of A-J. However, we question whether an all-pay aution with only three possible levels of investment is apt for modeling a patent rae. More speifially, for the parameters hosen for the sessions of the experiments with full dissipation, the equilibrium predits that firms mixed strategies yield a tie in investment, goes to zero. Corresponding to the equilibria that we identify are ontinuous strategy spae equilibria that have the property that one firm plaes a mass point at zero and the other firm a mass point at. A orret analysis of the ontinuous ase under partial dissipation may be found in Che and Gale (1998). 6

8 and thus no firm wins the patent rae, in roughly 43% of the ases. Hene, the full dissipation assumption provides a player with a strong inentive to avoid investing the same amount as the other player. We argue that suh inentives do not seem to reflet the reality of a patent rae. 7 In support of the qualitative nature of their findings with a three point strategy spae, theauthorsnote(a-j,p. 978,footnote7)thatrelatedworbyRapoportandAmaldoss (000) shows that the results are robust to inreases in the set of feasible strategies. Indeed, Rapoport and Amaldoss (000) examine a game with six feasible strategies, a spae that, ontrary to the laims of the authors, yields multiple Nash equilibria. However, in the Rapoport and Amaldoss (000) experiments, players plae on average higher probability at a 0 investment than is predited by the only equilibrium they identify (whih orresponds to that in Proposition 1 of A-J with r H r L ). In a separate note (Dehenaux, Koveno and Lugovsyy 003), we examine how the experimental findings of Rapoport and Amaldoss (000) appear onsistent with at least a subset of players sometimes playing asymmetri equilibria of the type derived in this note. Above we argued that it is the failure to allow for the possibility that a single player need not play all feasible pure strategies with positive probability in equilibrium that auses A-J to arrive at the false onlusion (A-J, p. 976) that if a firm gets zero from not investing, then its expeted payoffs frominvesting apositiveamountmustalsobezero. Thisleads A-J to erroneously onlude that equilibrium profits must be zero for both firms. Indeed, the error of assuming that both firms must earn zero profit in equilibrium arries over to A-J s treatment of the partial dissipation ase, where even a zero expenditure does not generally lead to a zero expeted profit. The authors examine the behavioral impliations of the partial dissipation ase where,r L.,r H.9 and t 1. A-J laim that the equilibrium solutions for this ase are (A-J, p. 984): p H (p H (0),p H (1),p H ()) (0.0909, 0.77, ), p L (p L (0),p L (1),p L ()) (0.3103, , 0.607). We laim that the statement above is erroneous, as under the assumptions made by the authors, (p H,p L ) does not onstitute an equilibrium of the game. In a Nash equilibrium, if 7 The literature on ontest approahes to patent raes also generally assumes partial dissipation in the event of a tie and not full dissipation. See Dasgupta (1986). 7

9 x and z are in the support of firm j s mixed strategy: Π j (x, p j )Π j (z, p j ). But then, using the probabilities suggested by the authors: Π L (0,p H ) > Π L (1,p H ) On page C1 of A-J s Tehnial Appendix, the statement preeding equations (C1), (C) and (C3) is inorret. If both firms play 0 with positive probability, they obtain a stritly positive expeted profit at0. Thus, the system of equations that follows the statement does not haraterize an equilibrium. We now ompute the orret equilibrium probabilities assuming that every pure strategy is in the support of eah firm for a general and t 1. Reall that a firm s payoff under partial dissipation is given by: Π j (x j,x j ) r j x j if x j >x j, 1 j x if x j x j x, x j if x j <x j. A-J fous on a lass of equilibria in whih eah firm s support ontains all pure strategies. Knowing that in equilibrium at eah point of its support a firm earns the same expeted payoff, Π j, we an haraterize this type of equilibrium by the following system of equations for j {L, H}: Π j 1 p j(0)r j Π j [p j (0) + 1p j( )]r j... Π j P 1 i0 p j (i )r j + 1p j()r j. In addition, we now that all probabilities are nonnegative and the probabilities sum to one for eah firm j {L, H}: X p j (i )1. (8) i0 Now from (7) we an express all probabilities for firm j in terms of its probability of bidding zero, p j (0): p j (i ) (1 + ( 1)i+1 ) +( 1) i p j (0). (9) 8 (7)

10 By plugging this result into (8) we obtain: X i0 p j (i ) + 1 ( 1) + 1+( 1) p j (0) 1. (10) For odd, ³ 1+ 1 r j should hold. But this ontradits the fat that r H >r L and thus an equilibrium with the speified properties does not exist for odd. For even, p j (0) 1 so for eah firm: µ p j i 1 ³ + Using (11) to alulate the equilibrium solution yields: for i even, (11) 1 for i odd. p H (p H (0),p H (1),p H ()) (0.0909, , ), p L (p L (0),p L (1),p L ()) (0.3103, , ). On page 977 A-J laim that the qualitative impliations of their theoretial results derived for t 0hold for every t [0, 1 ] given that, in equilibrium, both firms randomize over all pure strategies. One of these impliations is that firm L invests more on average than firm H. Using (11) we an alulate expeted investment for eah firm j, j {H, L}: E[x j ] [ X i0 à + p j (i µ ) i à + 1! 1] ( )! ( ). (1) As we an see, in equilibrium both firms invest an equal amount on average, whih ontradits the laim made by A-J. 5 Conlusion In this note, we have applied methods developed in Baye, Koveno, and de Vries (1994) to illustrate some of the pitfalls of haraterizing the omplete set of Nash equilibria in 9

11 ontests with a disrete strategy spae. We have shown that Amaldoss and Jain (00) have erroneously haraterized the set of Nash equilibria for disrete strategy spaes under full dissipation (A-J, Proposition 1) and have misspeified the equilibrium strategies in their experimental test of the partial dissipation ase. The experiments arried out by A-J for the full dissipation ase employ a game with a strategy spae with three feasible strategies and a unique Nash equilibrium. A-J justify the hoie of suh a restrited strategy set by appealing to its simpliity and to the robustness of the set of equilibria to inreases in the number of possible strategies. However, our theoretial results show that the set of equilibria is not robust to inreases in the number of possible strategies, invalidating A-J s justifiation. In a ompanion piee (Dehenaux, Koveno, and Lugovsyy 003), we examine how the experimental findings of Rapoport and Amaldoss (000) appear onsistent with at least a subset of players sometimes playing asymmetri equilibria of the type derived in this note. Sine experimental testing requires disrete strategy spaes, areful attention should be paid to ompletely haraterizing the equilibria of the game to be tested. It is somewhat troubling that the set of equilibria in the all-pay aution with ommon aps on expenditures is non-robust to the ardinality of the strategy spae. Indeed, it is hard to understand how experimental subjets ould unover (unstable) mixed strategy equilibria when it is diffiult for trained researhers in the management sienes to do so. 6 Referenes Amaldoss, W. and S. Jain (00) David vs. Goliath: An Analysis of Asymmetri Mixed- Strategy Games and Experimental Evidene, Management Siene, Vol. 48, No. 8, Baye, M.R. and H.C. Hoppe (003) The Strategi Equivalene of Rent-Seeing, Innovation, and Patent-Rae Games, Games and Eonomi Behavior, Vol. 44, No., Baye, M.R., D. Koveno and C.J. de Vries (1994) The Solution to the Tullo Rent-Seeing Game when R>: Mixed Strategy Equilibria and Mean Dissipation Rates, Publi Choie, Vol. 81, No. 3-4,

12 Baye, M.R., D. Koveno and C.J. de Vries (1996) The All-Pay Aution with Complete Information, Eonomi Theory, Vol. 8, Che, Y-K and I.L. Gale (1996a) Expeted Revenue of All-Pay Autions and First-Prie Sealed-Bid Autions with Budget Constraints, Eonomis Letters, Vol. 50, Che, Y-K and I.L. Gale (1996b) Finanial Constraints in Autions: Effets and Antidotes, inmihaelr. Baye(ed.) Advanes in Applied Miroeonomis: Autions, Vol. 6, 97-10, JAI Press In. Che, Y-K and I.L. Gale (1998) Caps on Politial Lobbying, Amerian Eonomi Review, Vol. 88, No. 3, Dasgupta, P. (1986) The Theory of Tehnologial Competition, in J.E. Stiglitz and G.F. Matthewson (eds.) New Developments in the Analysis of Maret Struture, Cambridge, MIT Press, Dehenaux, E., D. Koveno and V. Lugovsyy (003) Caps on Bidding in All-pay Autions: Comments on the Experiments of A.Rapoport and W. Amaldoss, mimeo, Purdue University. Nash, J. (1950) Equilibrium Points in n-person Games, Proeedings of the National Aademy of Sienes, Vol. 36, Rapoport, A. and W. Amaldoss (000) Mixed Strategies and Iterative Elimination of Strongly Dominated Strategies: An Experimental Investigation of States of Knowledge, Journal of Eonomi Behavior and Organization, Vol. 4, No. 4, Rosen, S. (1988) Promotions, Eletions and Other Contests, Journal of Institutional and Theoretial Eonomis, Vol. 144,

13 7 Appendix ProofofClaim1: We prove Claim 1 through a series of Lemmata. It follows from Nash (1950) that an equilibrium exists. Throughout the proof, let S {0,,,...,} and let (p H,p L) be an equilibrium of the game. For a given equilibrium (p H,p L), let p j(x) be the assoiated probability that firm j plays x, S j the assoiated support of firm j s distribution, and Π j its expeted profit in that equilibrium, j {H, L}. Lemma 1 In any equilibrium (p H,p L), S H S L S. Consequently, Π j 0for at least one j, j {H, L}. Proof. First, we show that (i) if a point v,v>0, is in the support of at least one firm, then all points n where 0 n<vmustbeinthesupportofatleastonefirm. This implies that (ii) 0 is in the support of at least one firm. Then, we use (ii) to show that (iii), is in the support of at least one firm. Combining (i) and (iii) ompletes the proof of the lemma. We first show (i). Let v be a stritly positive integer with v S j for some j {H, L}. Suppose ontrary to our laim that there exists an n<vsuh that n / S H S L. This implies that there exists some u v suh that u S l for some l {H, L} and (u 1) / S H S L. Then, firm l s expeted payoff of playing u is given by: Π l (u )X i<u p l(i )r l u, and firm l s expeted profit ofplaying(u 1) is given by: Π l ((u 1) ) X i<u 1 p l(i )r l (u 1). But sine P i<(u 1) p l(n )P i<u p l(n ),wehaveπ l(u ) < Π l((u 1) ). Therefore, u annotbeinthesupportoffirm l s equilibrium distribution, a ontradition. Thus, we have established (i). We now turn to (ii). First note that firm j s expeted profit ofplaying0 is, regardless of its opponent s strategy: Π j (0) 0. 1

14 Suppose to the ontrary that 0 / S L S H.Leti min i {i i S H S L }.Supposewithout loss of generality that i S l 0 for some l0 {H, L}. It follows from i > 0, that: Π l 0(i )X n<i p l 0(n )r l 0 i 0 i < 0, sine p l 0(n )0, n <i, a ontradition to the fat that i S l 0.Thus0 S L S H. To omplete the proof of the lemma, we show (iii). Suppose to the ontrary that / S H S L. Then, any firm j s expeted profit fromplaying is equal to: Π j () X p j(n n< )r j r j >0, for j {H, L}, whereas its expeted profit fromplaying0 is: Π j (0) 0. But, by (ii) above, 0 must be in the support of at least one firm j, a ontradition. Hene, we have established (iii). Combining (i) and (iii), we have shown that S H S L S. Consequently,Π j Π j (0) 0 for at least one j, whih proves the lemma. Q.E.D. Lemma In any equilibrium (p H,p L), if p H(i ) > 0 and p L(i ) > 0, thenp H(n ) > 0 and p L(n ) > 0, for every n suh that 0 n<i. Proof. Suppose to the ontrary that there exists an investment v suh that p j(v ) > 0, j {H, L} and n<vsuh that p l ((v 1) )0for some l {H, L}. This implies that there exists u, u v suh that u S H S L andsuhthat(u 1) / S l for some l {H, L}. From Lemma 1 it follows that sine p l ((u 1) )0,thenp l((u 1) ) > 0. Furthermore, firm l s expeted profit atu is equal to: Π l (u )X n<u p l (n )r l u, while its expeted profit at(u 1) is: Π l ((u 1) ) X n<u 1 p l (n )r l (u 1). Sine p l ((u 1) )0, P n<(u 1) p l (n )P n<u p l (n ). It then follows that Π l((u 1) ) > Π l (u ), ontraditing u S l. Q.E.D. 13

15 Lemma 3 In any equilibrium (p H,p L), if Π l > 0 for some l {H, L}, then: (i) S H S L, (ii) Firm l randomizes over all pure strategies i for whih i is odd and firm l randomizes over all pure strategies i for whih i is even, (iii) Π l 0. Proof. First Π l > 0 implies 0 / S l.thus,fromlemma1,0 S l. It follows that / S l sine firm l an inrease its expeted profit bymovingmassfrom to 0. Then, from Lemma 1, S l. Suppose S l. Then whether is in S l or not, firm l an inrease its expeted payoff by moving all mass from to. Thus / S l, from whih it follows that S l. Suppose now that 3 S l. Then, whether 3 is in S l or not, firm l an inrease its expeted payoff by moving all mass from 3 to (reall that firm l puts no mass at ). Sine is finite, it is straightforward to see that the same argument applies reursively to every i. That is, suppose i S m and i / S m, then(i +1) / S m and (i +1) S m, i {0, 1,...,} and m {H, L}. (iii) follows immediately from the fat that 0 S l. Q.E.D. Lemma 4 In any equilibrium (p H,p L), if Π H Π L 0,thenS H S L S. Proof. Let i j max i {i i S j} and i max{i H, i L }. We laim that i H i L i. Suppose i l > i l. Then it is lear that Π l (i l ) > 0, a ontradition to Π l 0.Thusi H i L i. Applying Lemma 1, i. The laim then follows from Lemma. Q.E.D. Lemma 5 There exists an equilibrium (p H,p L) with payoffs Π H Π L 0. In this equilibrium S L S H S and: µ p H i 1 r L if i 0, 1,..., 1, 1 r L if i. µ p L i 1 r H if i 0, 1,..., 1, 1 r H if i. 14

16 Proof. If (p H,p L) is an equilibrium with payoffs Π H Π L 0, then from Lemma 4, it follows immediately that S L S H S. We establish existene by onstrution. Sine S L S H S and both firms obtain an expeted profit of0, if the following pair of matrix equationsissatisfied (with equality) for j H, L, we have onstruted suh an equilibrium: r j r j r j... r j r j 0 p j (0) p j( )). p j() For a given j {H, L}, (13) is a system of +1equations and +1unnowns. If the leftmost matrix is non-singular, then (13) has a unique solution. The determinant of the leftmost matrix is equal to ( 1) + (r j ) 60. Therefore (13) has a unique solution. It is now straightforward to ompute the solution to (13). We have > 0. The remaining probabilities an be solved by repeated sub- whih yields p j(0) stitution of r j p j(0)r j Π j 0, 1. (13) in plae of r j p j(0) in (13), whih yields p j(i ) 1 r j for i {0, 1,..., 1} and p j() 1 r j.siner j >>i by assumption, the solutions to (13) are probabilities. Uniqueness of an equilibrium satisfying the onditions in the lemma follows immediately from Lemma 4 and the fat that suh an equilibrium must satisfy the system (13), for j {H, L}. Q.E.D. Lemma 6 An equilibrium with Π l > 0 for some l, l {H, L}, existsifandonlyif is odd. If is odd, then there exist exatly two suh equilibria (p l,p l), l H, L, given by: p l µ i r l if i 1,3..., ³ r l if i 1 ³ 1 0 if i 0,,..., 1 and p l µ i 1 ³ ³ 1 rl if i 0 r l if i, 4,..., 1 0 if i 1, 3,..., 15

17 Proof. We prove existene by onstrution. Let l be the firm obtaining Π l > 0 and let the other firm be l. From Lemma 3, the equilibrium must be of the alternating form, and firm l must play pure strategies i for whih i is an odd number. Suppose is even. It follows that is in S l but not in S l. Therefore Π l r l >0. But this and Π l > 0 ontradit Lemma 3. Therefore no equilibria with Π l > 0 exist when is even, whih proves the only if part of the statement. Suppose is odd. Then by Lemma 3, S l {1, 3,...,} and S l {0,,..., ( 1) }. It follows that Π l r l. FromLemma3,forfirm l, the stritly positive p l(x) s must be the solution to the following system of +1 equations: Π l p l(0)r l Π l [p l(0) + p l( )]r l 3... Π l [p l(0) + p l( ) p l(( 1) )]r l This system of equations an be written in matrix form: r l r l r l r l r l r l... r l p l(0) p l( )... p l(( 1) ) Π l + Π l Π l +. (14) Note that the number of equations oinides with the number of unnowns. The determinant of the first matrix equals (r l ) (+1)/ > 0. This is suffiient to prove uniqueness of the solution. Moreover, in equilibrium Π l r l p l(0)r l. Solving the system by repeated substitution yields p l(0) 1 ³ ³ 1 rl > 0 and p l (i ) > 0 for r l i {,...,( 1) }. We he that the probabilities sum to one: p l(0) + 1 X i p l(i ( 1) )1 + r l à 1! r l 1, so they are indeed probabilities. We now turn to firm l s strategy. Note that firm l obtains an expeted payoff of 0 in equilibrium, Π l 0. From Lemma 3, the stritly positive p l (x) s must solve the following 16

18 system of 1 equations: Π l p l ( )r l Π l [p l ( )+p l (3 )]r l 4... Π l [p l ( ) p l (( ) )]r l ( 1) This system of equations an be written in matrix form: 4... r l p l ( ) r l r l p l (3 ). (15) r l r l r l... r l p l (( ) ) ( 1) Note that the number of equations oinides with the number of unnowns. The determinant of the first matrix equals (r l ) ( 1)/ > 0. Thisissuffiient to prove uniqueness of the solution. Solving the system by repeated substitution yields p l (i ) > 0, where r l < 1, foralli {1, 3,..., }. Using the fat that probabilities must sum to 1 to solve for p l (), weobtainp l () 1 ³ ³ 1 r l > 0. Uniqueness of an equilibrium (p j,p h) satisfying Π j > 0 and Π h 0follows from Lemma 3 and the fat that suh an equilibrium must satisfy the systems (14) and (15). Thus using Lemma 3, if is odd, there exists exatly one equilibrium in whih Π H > 0 and exatly one equilibrium in whih Π L > 0. Q.E.D. r l To omplete the proof of the laim, it suffies to note that Lemma 1 implies that there arenoequilibriainwhihbothfirms earn a stritly positive expeted profit. Therefore, all equilibria of the game are haraterized by Lemma 5 and Lemma 6. Q.E.D. ProofofClaim: The proof of (iii) an be found on Pages of A-J, below Results and 3. Uniqueness of the equilibrium in this ase follows from Claim 1. Now onsider the equilibria in whih Π j > 0 and Π j 0. First, from Claim 1, suh equilibria exist if and only if, is odd. Seond, using (5) and (6), we ompute eah firm s 17

19 expeted investment: E[x j ] " ( ) + Ã 1 1! #, (16) and E[x j ] r j µ ( ). (17) Using the fat that: we rewrite (16): E[x j ] " µ ( +1) + 4 Ã ( +1), Straightforward omputations yield: " + # ( ) E[x j ]. Now using the fat that: , 4 we rewrite (17): " ( # 1) E[x j ] r j. Using the expressions for E[x j ] and E[x j ], we obtain: E[x j ] E[x j ]! #. " + # ( ) ( 1) r j " r j + r j (1 )+ (1 ) r j " # (r j max l {r l }) > r j > 0. Therefore in the equilibrium in whih Π j > 0, firm j invests more than firm j on average, j {H, L}. Now we ompute the probability of winning for eah firm, again using (5) and (6): Pr[j wins] ( 3)/ X z0 " 1 Ã 1!Ã! 18 r j + z # +1 r j # Ã 1!Ã! r j,

20 and, Pr[ j wins] Straightforward omputations yield: ( 1)/ X z1 Pr[j wins] Pr[ j wins] 1 " z #. r j Ã!Ã r j!ã 1 Therefore in the equilibrium in whih Π j > 0, firm j is more liely to win the patent than firm j, j {H, L}. The argument for the equilibrium in whih Π H Π L 0in (ii) is similar to the one used to prove (iii). This ompletes the proof of Claim. Q.E.D.! > 0, 19