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A cake for Cournot Morton I. Kamien* Department of Managerial Economics and Decision Sciences, J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208, USA Received 15 March 2000; accepted 15 June 2000

Abstract A cake division mechanism is presented that is equivalent in terms of the size of the pieces of the cake n players’ receive in equilibrium to the quantities that n firms in a Cournot oligopoly supply in equilibrium. This mechanism extends to equivalence between cake division and Nash’s ‘divide the dollar’ game. 2001 Elsevier Science B.V. All rights reserved. Keywords: Cournot oligopoly; Stackelberg leader; Hotelling model; Nash bargaining solution; Consistent mechanism JEL classification: C72; D43

Consider the following cake division game. The size of the homogeneous cake is T. Each of the two players prefers more cake to less and each is asked to submit a sealed envelope indicating how much of the cake she wants. The envelopes are then opened and each player receives the smaller of what she asked for and what is leftover if both of their requests are deducted from the entire cake. If the total of what they both ask for exceeds the total size of the cake then they both get nothing. In other words, if x is what player 1 asks for and y is what player 2 asks for then each gets max h0, min hT 2 x 2 y, xjj and max h0, min hT 2 x 2 y, yjj.

(1)

Now each player realizes that if the piece of the cake she asks for exceeds T 2 x 2 y she will get a smaller piece. On the other hand, if she asks for a piece of the cake smaller than T 2 x 2 y, then she will never get a larger one. Thus, it is optimal for each player to ask for a piece of the cake of size x 5 T 2 x 2 y and y 5 T 2 x 2 y.

(2)

That way each player assures herself that the piece of the cake she receives will be no smaller than *Tel.: 11-847-491-5167; fax: 11-847-467-1220. E-mail address: [email protected] (M.I. Kamien). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 00 )00384-0

2001 Elsevier Science B.V. All rights reserved.

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x 5 (T 2 y) / 2, y 5 (T 2 x) / 2;

(3)

one-half of what is left over after what the other player requested has been deducted from the entire cake. Now (2) implies directly that the players will ask for identical size pieces equal to x* 5 y* 5 T / 3.

(4)

However, it is instructive to view the expressions in (3) as the players’ reaction functions. An intuitive explanation for the form of these reaction functions is that each player knows that if she were alone in this game her dominant strategy would be to ask for a piece of size T / 2, because that is the most that she could receive. But this is true regardless of the size of the cake and must therefore be true for a cake of size T 2 y for player 1 and T 2 x for player 2, respectively. In other words, (T 2 y) / 2 and (T 2 x) / 2 just represent each player’s optimal strategy if each alone were choosing a piece from a cake of size T 2 y and T 2 x, respectively. And at the equilibrium each player perceives two-thirds of the cake to be leftover after the other player’s piece has been deducted and then takes one-half of this remaining two-thirds. Division of the cake by means of the mechanism described by (1) is fair because it provides each player the opportunity to get an equal share of the cake. It is envy free because it derives from an equilibrium that by definition means that neither player would choose another size piece of the cake. However, it is not Pareto efficient because the players could both be made better off by having the third of the cake that is leftover divided among them. But it is asymptotically efficient as the number n of identical players among whom the cake is to be divided increases to infinity since with n players the size of each players’ piece of the cake is determined by the max h0, min h[Tx(n 2 1)y], xj,

(5)

where n$1. Thus, in equilibrium the size of each player’s piece of the cake is x* 5 T /(n 1 1).

(6)

Together they receive nx* 5 nT /(n 1 1)

(7)

of the cake, and its entirety T as n approaches infinity. The entire cake can also be divided for the case of a finite number of players by applying the division mechanism (5) repeatedly to the leftover fraction of the cake. The remarkable feature of this cake cutting game is that it is equivalent in terms of the forms of the reaction functions and equilibrium quantities to a Cournot oligopoly composed of n firms with identical cost functions competing in the sale of a homogeneous product and with the perfectly competitive output, Q c being the counterpart to the size of the cake, T. For example, in the case of two firms selling the identical product at the same constant marginal cost m$0, and facing a linear inverse demand function P 5 a 2 Q 5 a 2 (q1 1 q2 ),

(8)

where q1 and q2 refer to the first firm’s and the second firm’s respective outputs and a . m, it is easy to show that the firm’s respective reaction functions

M.I. Kamien / Economics Letters 70 (2001) 349 – 355

q1 5 (Q c 2 q2 ) / 2, q2 5 (Q c 2 q1 ) / 2,

351

(9)

follow from each firm choosing its output level qi , i 5 1,2, so as to maximize its profit (P 2 m)qi , assuming that the other firms output is held fixed. The perfectly competitive output occurs when P 5 m. From (9) it follows that 2q1 1 q2 5 Q c 5 2q2 1 q1 and that the profit maximizing Cournot equilibrium outputs are q* 1 5 q* 2 5 Q c / 3.

(10)

Now it is immediately apparent that the reaction functions in (9) are the counterparts of those in the cake division game, described in (3). And the logic of these reaction functions is the same as in the cake division game in that here a firm will produce the monopoly output, Q c / 2 if its rival produces nothing. But since this strategy is optimal for all values of Q c , it is optimal for whatever quantity remains after the rival’s output has been deducted from Q c . In other words, the intuition for the reaction functions in (9) is ‘do the monopoly thing’ with whatever the size of the market there is left to serve, just as in the cake division game it is to always ask for one-half of what is left of the cake. Moreover, as the number of competing firms grows, each produces Q c /(n 1 1) in equilibrium and their total output nQ c /(n 1 1) approaches Q c . In the Cournot oligopoly game with linear demand and constant marginal costs each firms’ profit is proportional to the square of its output. Thus, to complete the analogy between the cake division game and the Cournot oligopoly game it can be supposed that a piece of cake can be sold at a profit equal to the square of its size. The cake division participants would then have an incentive to collude to limit their total take of the cake to T / 2, just as the firms in a Cournot oligopoly have an incentive to limit their output to the monopoly level Q c / 2, so as to maximize their joint profits. And of course, they would have the same incentives to deviate from an agreement to collude as the firms in a Cournot oligopoly do. The equivalence between the cake division game and the Cournot model applies if the monopoly output is Q c / 2 even if the inverse demand function is not linear and the firms’ marginal costs are not constant providing a Cournot equilibrium exits, see Novshek (1985). Moreover, the cake division game can be modified to maintain equivalence with a Cournot oligopoly for a monopoly output that is any fraction of Q c . Thus, if the monopoly output equals kQ c , where 1 . k . 0, then letting the size of each player’s piece of the cake be determined by max h0, min hk[T 2 x 2 (n 2 1)y] /(1 2 k), xjj,

(11)

maintains the equivalence between the Cournot oligopoly equilibrium and the cake division game because each player’s reaction function is of the form x 5 k[T 2 (n 2 1)y].

(12)

In equilibrium, each player receives a piece of the cake of size x* 5 kT / [1 1 k(n 2 1)],

(13)

that with T replaced by Q c yields each firm’s equilibrium output in the Cournot model. Moreover, if

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the managers of identical firms in a Cournot oligopoly are strategically delegated to maximize a convex combination of profits and sales as in Fershtman and Judd (1987), then according to Saracho (2000, private communication) each firm’s equilibrium output is identical to the equilibrium size of a players’ piece of cake determined by max h0, min h[n(n 1 1)T /(n 2 1 1)] 2 x 2 (n 2 1)y, xjj.

(14)

The cake division game can also be modified to provide equivalence with the Cournot oligopoly without identical firms. For example, if the firms in a Cournot duopoly with a linear inverse demand function have different marginal costs then their respective reaction functions are q1 5 (Q c1 2 q2 ) / 2, and q2 5 (Q c2 2 q1 ) / 2,

(15)

where Q c1 , Q c2 refer to the perfectly competitive outputs corresponding to the first firm’s constant marginal cost and the second firm’s constant marginal cost, respectively. If the second firm’s constant marginal cost exceeds the first firm’s constant marginal cost then Q c1 .Q c2 . The firms’ respective equilibrium outputs are q1 5 (2Q c1 2 Q c2 ) / 3 and q2 5 (2Q c2 2 Q c1 ) / 3,

(16)

from which it follows that q2 $0 if and only if Q c2 /Q c1 $1 / 2. In other words, the second firm’s perfectly competitive output must exceed the first firm’s monopoly output, in order for it to be viable in equilibrium. The intuition for this result is that if the high cost firm’s perfectly competitive output does not exceed the low cost firm’s monopoly output, then all of its potential customers can purchase the item at a price equal to or below the lowest price it can profitably sell it for. All this can be captured in a cake division game by supposing that one of the players is entitled to less of the cake then the other and in the limit none at all. Perhaps the second born child, player 2 is entitled to less than the firstborn child, player 1. The size, x, of the piece of the cake going to the firstborn, is determined as in (1) but the size, y, of the piece going to the second born is determined by max h0, min hvT 2 x 2 y, yjj.

(17)

Now it follows by analogy with (2) that x 5 T 2 x 2 y, and y 5 vT 2 x 2 y.

(18)

But then in equilibrium x* 5 (2 2 v)T / 3, and y* 5 (2v 2 1)T / 3

(19)

which implies that x* # T / 2 and y* $ 0 if and only if v $ 1 / 2. Thus v $ 1 / 2, assures that the firstborn does not benefit from the presence of a younger sibling and that the younger sibling receives a nonnegative share of the cake. In equilibrium the share of the cake going to the firstborn declines from x* 5 T / 2 to x* 5 T / 3 while the share going to the second born grows from y* 5 0 to y* 5 T / 3 as v grows from 1 / 2 to 1. Here, it is obvious that the younger sibling envies the older one because he does not have the opportunity to receive an equal share of the cake just as the high cost firm envies the low cost firm in a Cournot duopoly because it cannot get an equal share of the market. Also, it is

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apparent that v 5 Q c2 /Q c1 $ 1 / 2, is the link between the cake division game and the Cournot duopoly with firms with different constant marginal costs. Another extension of the cake division game to an asymmetric situation is to suppose that pieces of the cake are allotted on first come first served basis. This means that the first player to show up claims T / 2 of the cake, and the second T / 4, and so on. Thus, the first player to show up has a first mover advantage, which of course is the hallmark of the Stackelberg oligopoly model. If the mechanism introduced in (1) is modified to max h0, min hT 2 y, xjj, and max h0, min hT 2 x, yjj,

(20)

then the entire cake is divided between the two players but the share going to each is indeterminate. This cake division mechanism may be thought to describe two firms engaged in Bertrand competition in which they together supply the entire perfectly competitive output or Nash’s (1953) ‘divide the dollar’ game. However, if the mechanism for determining each players’ share of the cake is max h0, min hT 2 (x 1 y) / 2, xjj and max h0, min hT 2 (x 1 y) / 2, yjj

(21)

then each player gets T / 2 of the cake in equilibrium. This situation may be thought of as the Hotelling competition game in which each competitor serves one-half of the entire market or as the symmetric Nash bargaining solution to the divide the dollar game. The generalization of (21) to n identical players each receiving T /n at the unique equilibrium, is for each player’s piece of the cake to be determined by max h0, min hT 2 (n 2 1)[x 1 (n 2 1) y] /n, xjj.

(22)

It should be noted that in this cake division mechanism having one less player to divide the cake with and having a player who does not ask for any of the cake does not yield the same equilibrium, as it does in mechanism (1) and its n player generalization (5). Thus, from (22) it is apparent that if n51, then the sole player takes the entire cake. However, for n52 and y50, player 1 receives 2T / 3. Thus, the presence of another player, even one that does not request a piece of the cake inhibits the first player’s greed. The first player’s reaction function is x 5 2(T 2 y / 2) / 3. And so if the second player requests T / 2 of the cake, then player 1 perceives 3T / 4 of the cake to be left and takes 2(3T / 4) / 3 5 T / 2 of it in equilibrium. The mechanism (22) satisfies the equal splits guarantee that according to Moulin (1989) was the one discussed in the early literature on fair division of a cake. He also indicates the conditions under which such a division of the cake is Pareto optimal. Moreover, this mechanism is unique in the sense that for any cake division game of the form max h0, min hT 2 w[x 1 (n 2 1)y], xjj

(23)

it follows that x* 5 T /(nw 1 1). But the requirement that nx* 5 T is met if and only if w 5 (n 2 1) /n. If the cake were divided sequentially on a first-come first-served basis using (22) then the jth player in the queue receives

P [2(n 2 i) 1 1], j

x j 5 (n 2 j 1 1)(n 2 1)!T /(n 2 j)!

1

j 5 1, . . . ,n

(24)

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that reduces to x j 5 T / 2 j , as n approaches infinity. This sequential division of a cake might be used to allow the players to express their individual differences in the utility they derive from pieces of the cake by bidding in an auction for priority in the queue. The proceeds from the auction would be distributed to the players in reverse order from their place in the queue, i.e., those further back in the queue receive more than those ahead of them. A more explicit relationship between (22) and the Nash bargaining solution framework can be established by the observation that the first order condition to the problem max x i [(n 1 1 2 i)T /n 2 x i ] n 2i , i 5 1, . . . n,

(25)

with respect to x i , is x i 5 (n 1 1 2 i) T /n 2 (n 2 i)x i , which has the same form as the expression for maximizing (22), and also leads to x *i 5 T /n. Moreover, x i [(n 1 1 2 i)T /n 2 x i )] n 2i can be regarded as the product of the ith player’s utility from getting a piece of the cake of size x i and the (n 2 i)-fold product of the subsequent n 2 i players’ identical utilities for the availability of the remainder of the cake, (n 1 1 2 i)T /n 2 x i , i.e., after the (i 2 1)T /n of the previous i21 players’ pieces of the cake have been deducted from T and then the ith player’s piece x i is deducted. A similar interpretation of the maximand in (25) appears in Kalai (1977). Repeated application of (25) in determining each of the remaining n 2 i players’ share of the remainder of the cake yields (T /n)n as the total utility derived from the cake and is a ‘consistent’ mechanism for dividing the remainder of the cake among the remaining n 2 i players since it is unaffected by the share of the cake going to the previous i players, see Young (1994). The mechanisms (5) and (22) neither require that the players know each others utility functions nor that a planner know them in order to induce an equal division of the cake. It should also be possible to modify the mechanism (22) along the lines of (16) to accommodate players with different entitlements. Brams and Taylor (1996) discuss other mechanisms for the divide a cake and the divide the dollar game. The equivalence between the cake division game governed by the mechanism (1) and the Cournot oligopoly model may be useful in computing equilibria when the algebra in the Cournot game is complicated. It may also be easier for subjects in experiments to deal with the cake cutting game to arrive at the Nash equilibrium solutions than the Cournot oligopoly game (see Rassenti et al., 1999). The same may be true for experiments involving the divide the dollar game as well.

Acknowledgements I wish to thank Krishnan Anand, Sandeep Baliga, Steven Brams, Julian Jamison, Ulrich Kaiser, Ehud Kalai, Randall Kamien, Herve Moulin, Ana Saracho, Dan Spulber, Jeron Swinkels, Joe Swierzbinski, Yair Tauman, Mikhel Tombak, and David Ulph for their useful comments and suggestions.

References Brams, S.J., Taylor, A.D., 1996. In: Fair Division: From Cake Eating to Dispute Resolution. Cambridge University Press, Cambridge.

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Fershtman, C., Judd, K.L., 1987. Equilibrium incentives in oligopoly. American Economic Review 77, 927–940. Kalai, E., 1977. Nonsymmetric Nash solutions and replications of 2-person bargaining. International Jouranal of Game Theory 6, 129–133. Nash, J., 1953. Two person cooperative games. Econometrica 21, 128–140. Novshek, W., 1985. On the existence of Cournot equilibrium. Review of Economic Studies 52, 85–98. Moulin, H., 1989. Fair Division Under Joint Ownership: Recent Results and Open Problems, Department d’Economia I Historia Economica, Universitat Autonoma de Barcelona & Institut d’Analisi Economica CSIC, W.P. 123.89 Rassenti, S., Reynolds, S.S., Smith, V.L., Szidarovsky, F., 1999. Adaptation and convergence of behavior in repeated experimental games, Journal of Economic Behavior and Organization, in press. Young, H.P., 1994. In: Equity in Theory and Practice. Princeton University Press, Princeton, NJ.