February 11, JEL Classification: C72, D43, D44 Keywords: Discontinuous games, Bertrand game, Toehold, First- and Second- Price Auctions

Transcription

1 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets. Gian Luigi Albano and Aleander Matros Department of Economics and ELSE Universit College London Februar 11, 2001 Abstract. We prove the eistence of mied strateg equilibria for a class of discontinuous two-plaer games with non-compact strateg sets. We appl this result to obtain continuum of new mied strateg equilibria in Bertrand game with two firms. Moreover, we construct continuum of mied strateg equilibria in the first- and second-price auctions with toeholds. JEL Classification: C72, D43, D44 Kewords: Discontinuous games, Bertrand game, Toehold, First- and Second- Price Auctions

2 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.1 1. Introduction The aim of this paper is to show the eistence of Nash equilibria in mied strategies for a class of two-plaer discontinuous games with complete information in which the strateg sets are non-compact. The problem of the eistence of equilibria in discontinuous games has been alread addressed b Dasgupta and Maskin (1986a, 1986b), Maskin (1986), Simon (1987), Simon and Zame (1990), and, more recentl b Ren (1999). Unlike the eisting papers, we construct Nash equilibria in mied strategies whenever the plaers strateg set coincides with the set of real numbers. We provide a class of games which fit into our theoretical framework: Auctions with toeholds. Two bidders compete for an object. Each of them owns a (strictl positive) share of the object. Their valuations and their shares are common knowledge. Both bidders submit simultaneousl sealed bids, the higher bidder gets the object and bus her competitor s share at the selling price. The relevant feature of this game is that each bidder is a buer and a seller at the same time. Discontinuit comes from a tie breaking rule. If ties are broken through an random device such that a bidder gets the object with probabilit strictl less than one, plaers best responses are not well defined. We suggest a wa out of the resulting discontinuit b opening the plaers strateg space. We let plaers submit an real number. However, the eistence of mied strateg Nash equilibria is not guaranteed b an fied point theorem since the strateg set is not compact. We show that a continuum of mied strateg equilibria do eist in our model. An interesting co-product of our analsis is that we construct a new class of mied strateg equilibria in the classical Bertrand game. This class of equilibria differ from the one proposed b Klemperer (2000). Moreover, the epected profit ofbothfirms is strictl positive in all our mied strateg equilibria. The rest of the paper is organized as follows. The net section describes our assumptions and states the main eistence results. We provide some eamples in Section3. Section 4 concludes. 2. The Model We consider two classes of games and prove eistence of mied equilibria in each of them. Let Γ A ({i, j}, R R, (u i,u j )) be a two-plaer game. Assume that if plaer i chooses strateg R and plaer j plas strateg R, thenpaoff functions u k : R R R, k i, j, are (v i (),w j ()), if > (u i (, ),u j (, )) A (w i (),v j ()), if <, (αv i ()+[1 α] w i (), [1 α] w j ()+αv j ()) if

3 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.2 where α [0, 1]. Here α is a tie breaking rule. The model allows an tie breaking rule. We make the following assumptions about functions v k (t) andw k (t), k i, j. A1. w k (t) is differentiable, k i, j. A2. w 0 k (t) 0, k i, j. This assumption shows that if a plaer chooses a number (strateg) less than the opponent s number (strateg) then she would prefer to pick up a number as closed as possible to the opponent s strateg. This folds, for eample, in the auctions with toeholds, when each plaer is a buer and a seller at the same time. A3. There eists t R, suchthatw k (t) v k (t) > 0, for all t t and k i, j. The eplanation for this assumption is as follows. If one plaer chooses a ver high number (a price in the Bertrand model or a bid in the auction with toeholds), then the other plaer will prefer to choose a smaller number. A4. Z + t wk 0 (t) dt w k (t) v k (t) +, fork i, j. Assumptions A1-A4 guarantee our main result. Theorem 1. Suppose that assumptions A1-A4 hold. Then there eists continuum of equilibria in mied strategies in game Γ A.Moreover,foran t, thefollowing probabilit distribution constitutes a mied equilibrium: ( 0, if t<t F j (t) h 1 ep R i t w 0 i (s)ds w i (s) v i if t, (1) (s) where i 6 j. Proof. Notice first that distribution function F j (t) is a positive, strictl increasing function which satisfies F j ( )0andF j (+ ) 1, because of assumption A4. We show now that the distribution functions from (1) constitute a mied strateg equilibrium. Suppose that plaer j, i 6 j, uses the c.d.f. F j (t) above, then there are two cases. Case 1. If plaer i chooses a strateg [, + ), then i s epected paoff is: E [u i (, F j ())] Z v i (s) f j (s) ds + Z + w i () f j (s) ds, (2)

4 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.3 where the first integral in the right-hand side is plaer i s epected paoff if her strateg is greater than the opponent s strateg, and the second integral is plaer i s epected paoff if her strateg is smaller than the opponent s strateg. From the probabilit distribution function F j (t), from (1), it is immediate to get the densit function f j (t): ( 0, if t<t f j (t) h wi 0(t) ep w i (t) v i R i t w 0 (t) i (s)ds w i (s) v i, if t. (s) The epected i s paoff (2) can be rewritten as Z wi 0 () E [u i (, F j ())] v i () t w i () v i () ep Z + wi 0 () +w i () w i () v i () ep Using assumption A4, we get Z + E [u i (, F j ())] w i () ep wi 0 () [v i () w i ()+w i ()] w i () v i () ep Plaer i s epected paoff becomes: or Z E [u i (, F j ())] w i () ep Z wi 0 ()ep d+ E [u i (, F j ())] w i () ep Z Z Z Z Z d+ d. tz + Z d. wi 0 () w i () w i () v i () ep wi 0 Z (z) dz w i ()ep Z wi 0 (z) dz +

5 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.4 +w i ( ) Z Z + w i () and we finall obtain w 0 Z i () dz w i () w i () v i () ep w 0 i () w i () v i () ep Z d, w 0 i (z) dz E [u i (, F j ())] w i ( )foran [, + ). d+ Case 2. If plaer i chooses a strateg (, ), then i s epected paoff is: E [u i (, F j ())] w i (), (3) because <in this case. From assumption A2, it follows that w i () w i ( ) for an (, ). Hence, an [, + ) is a best repl to the probabilit distribution function F j (t), from (1). The same argument is valid for plaer j. Hence the distribution functions from (1) characterizes a mied equilibrium. End of proof. We turn to the second possibilit now. Consider the game Γ B ({i, j}, R R, (u i,u j )). Assume that if plaer i chooses strateg R and plaer j plas strateg R, then paoff functions u k : R R R, k i, j, are (u i (, ),u j (, )) B (v i (),w j ()), if > (w i (),v j ()), if < (αv i ()+[1 α] w i (), [1 α] w j ()+αv j ()) if where α (0, 1). We make the following assumptions B1. v k (t) is differentiable, k i, j. B2. vk 0 (t) 0, k i, j. This assumption describes propert of the first price auction: given that ou win, ou would like to announce our bid as small as possible. B3. There eists t R, such that v k (t) w k (t) > 0, for all t t and k i, j. This assumption describes the fact that if our opponent chooses a number (strateg) lower than some level, then it is alwas better to pla slightl higher than her strateg, than to pla lower than her number. This situation takes place in auctions., B4. Z t vk 0 (t) dt +, fork i, j. w k (t) v k (t)

6 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.5 Theorem 2. Suppose that assumptions B1-B4 hold. Then there eists continuum of equilibria in mied strategies in game Γ B.Moreover,foran t, thefollowing probabilit distribution constitutes a mied equilibrium: where i 6 j. F j (t) ep Z v 0 i (s) ds, if t t w i (s) v i (s), (4) 1, if t Proof. Notice first that distribution function F j (t) is a positive, strictl increasing function which satisfies F j ( ) 0andF j ( ) 1, because of assumption B4. We show now that the distribution functions from (4) constitute a mied equilibrium. Suppose that plaer j, i 6 j, uses the c.d.f. F j (t) above, then there are two cases. Case 1. If plaer i chooses a strateg (, ], then i s epected paoff is: E [u i (, F j ())] Z v i () f j () d + Z w i () f j () d. (5) From the probabilit distribution function F j (t), from (4), it is obvious to get the densit function f j (t): vi 0 (t) Z f j (t) w i (t) v i (t) ep, if t t w i (s) v i (s). 0, if t The epected i s paoff (5) can be rewritten as E [u i (, F j ())] v i () F j ()+ Z v 0 Z i () + [v i ()+w i () v i ()] w i () v i () ep It is equivalent to E [u i (, F j ())] v i () F j ()+ Z + v i () Z Z vi 0 ()ep vi 0 Z () w i () v i () ep d. w i (s) v i (s) d+ w i (s) v i (s) d, w i (s) v i (s)

7 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.6 or Z E [u i (, F j ())] v i ()ep and we obtain +v i ( ) Z Z + v i () vi 0 Z (s) ds v i ()ep w i (s) v i (s) v 0 Z i () v i () w i () v i () ep vi 0 () Z w i () v i () ep d+ v i (s) w i (s) d, v i (s) w i (s) E [u i (, F j ())] v i ( )foran (, ]. + w i (s) v i (s) Case 2. If plaer i chooses a strateg (, + ), then i s epected paoff is: E [u i (, F j ())] v i (), because >in this case. From assumption B2, it follows that v i () v i ( ) for an (, + ). Hence an (, ] is a best repl to the probabilit distribution function F j (t), from (4). Thesamereasoningisvalidforplaerj. Hence the distribution functions from (4) do reall constitute a mied equilibrium. End of proof. 3. Applications There are man eamples of games Γ A and Γ B,whichfit assumptions A1-A4 and B1-B4 correspondingl. Let us look at some of them The Bertrand Model. The standard two-firm Bertrand model is game Γ A, where (0,), if > (u i (, ),u j (, )) A (, 0), if <, (0.5, 0.5) if or v i (t) v j (t) 0, w i (t) w j (t) t, t 0,andα 0.5. It is eas to check that all assumptions A1-A4 are fulfilled. As a corollar of Theorem 1, we have

8 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.7 Proposition 1. There eists continuum of equilibria in mied strategies in two-firm Bertrand model. Moreover, for an p > 0, the following probabilit distribution constitutes a mied equilibrium: ½ 0, if p<p F (p) 1 p if p p. p Note that the epected profit ofeachfirm in the mied equilibrium is p > Auctions with toeholds. Another class of games that fits into the theoretical model analzed in Section 2 is a two-bidder auctions with toeholds. Two risk neutral individuals are interested in acquiring an object. Bidder i (j) has a valuation v i (v j ) and owns a share θ i (θ j 1 θ i ) > 0 of the object. Bidders values and shares are common knowledge. Bidders submit bids simultaneousl. The higher bidder gets the object and pas either her bid in the first price auction, or the opponent s bid in the second price auction. If bids are equal, then the object is allocated to bidder i with probabilit α [0, 1] and to bidder j with probabilit (1 α) [0, 1]. Thus, bidder i s paoff is v i (1 θ i )p, ifhewins,andθ i p,ifheloses,wherep is the selling price. We consider below two possible mechanisms: first- and second-price auctions. The Second-Price Auction. Suppose that the two bidders compete in a second-price auction. It is eas to see that the auction game with toeholds is eactl game Γ A,where (u i (, ),u j (, )) A (v i (1 θ i ), (1 θ i )), if > (θ i, v j θ i ), if <, (α [v i (1 θ i )]+(1 α) θ i, α(1 θ i ) +(1 α)(v j θ i )) if or v i (t) v i (1 θ i )t, v j (t) v j θ i t, w i (t) θ i t, w j (t) (1 θ i )t, and t > ma {v i,v j }. It is eas to check that all assumptions A1-A4 are fulfilled. As a corollar of Theorem 1, we have Proposition 2. There eists a continuum of mied strateg equilibria in the sealed bid second-price auction in which plaer i randomizes over bids in the interval [b, + ) according to the distribution function µ 1 θi b vj F i (b) 1,i6 j, b v j where b is an number greater than ma {v i,v j }. The interesting feature of the mied equilibria is that the epected paoff of each bidder is b > ma {v i,v j } and this paoff is independent from bidder s valuation!

9 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.8 The First-Price Auction. Suppose that the selling mechanism is the firstprice auction. Then, the auction game with toeholds coincides with game Γ B,where (u i (, ),u j (, )) B (v i (1 θ i ), (1 θ i )), if > (θ i, v j θ i ), if <, (α [v i (1 θ i )]+(1 α) θ i, α(1 θ i ) +(1 α)(v j θ i )) if or v i (t) v i (1 θ i )t, v j (t) v j θ i t, w i (t) θ i t, w j (t) (1 θ i )t, and t<min {v i,v j }. It is eas to check that all assumptions B1-B4 are fulfilled. As a corollar of Theorem 2, we have Proposition 3. There eists a continuum of mied strateg equilibria in the sealed bid first-price auction in which plaer i randomizes over bids in the interval (, b] according to the distribution function µ θi vj b F i (b),i6 j, v j b where b is an number lower than min {v i,v j }. The epected paoff of bidder i in the mied strateg equilibrium is v i (1 θ i )b, where b<min {v i,v j }. Note that this paoff dependents on bidder i s valuation, which was not the case in the second price auction. 4. Conclusion The main feature of the class of games studied in this paper is the presence of eternalities between plaers. We have pointed out that the use of a random tie-breaking rule makes this game discontinuous. We have shown that, if the plaers strateg space coincides with the set of real numbers, a continuum of Nash equilibria in mied strategies do eist. One might wonder what would happen if we modifiedthegameinsuchawato allow for a deterministic tie-breaking rule. In our toeholds eample, since valuations are common knowledge, one could think of breaking a tie in favor of the bidder with the higher valuation for the object. This formulation has been analzed b Ettinger (2001). The author shows that the first- and second-price auctions admit a unique equilibrium in undominated strategies. It is eas to prove that, if bidders can pla weakl dominated strategies, the set of equilibrium outcomes both in firstand second-price auction will coincide with the interval between buers valuations. However, whenever a random tie-breaking rule is introduced, the set of equilibrium outcomes will be found outside the interval between bidders valuations.

10 Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets.9 References [1] Ettinger, D. (2001): Bidding among friends and enemies, CERAS, mimeo. [2] Dasgupta P. and E. Maskin (1986): The Eistence of Equilibrium in Discontinuous Economic Games, I: Theor, Review of Economic Studies, [3] Dasgupta P. and E. Maskin (1986): The Eistence of Equilibrium in Discontinuous Economic Games, II: Applications, Review of Economic Studies, [4] Klemperer P (2000): Wh Ever Economist Should Learn Some Auction Theor, Invited Lecture to 8th World Congress of the Econometric Societ. [5] Maskin E. (1986): The Eistence of Equilibrium with Price-Setting Firms, American Economic Review, 76, [6] Simon L. (1987): Games with Discontinuous Paoffs, Review of Economic Studies, [7] Simon L. and W. Zame (1990): Discontinuous Games and Endogenous Sharing Rules, Econometrica, 58, [8] Ren P. (1999): On the Eistence of Pure and Mied Strateg Nash Equilibria in Discontinuous Games, Econometrica, 67,