- Email: [email protected]
A pure strategy nash equilibrium for a 3-firm location game on a sphere
Pergamon
PII:s~9(97)oooo4-l
Location Science,Vol. 4, No. 4, pp. 247-250, 1996 0 1997 Elsevia ScienceLtd All rightsreserved.Printed in Great Britain 0966-8349/97s17.00+0.00
A PURE STRATEGY NASH EQUILIBRIUM FOR A 3-FIRM LOCATION GAME ON A SPHERE VICIU KNOBLAUCH Department of Economics, Vassar College, Poughkeepsie, NY 12604, U.S.A.
(Received 7 Febnmy 1997)
Abstract-There are only two known pure strategy Nash equilibria for multi-firm location games in dimension two. This paper presents an infinite family of pure strategy Nash equilibria for a 3-firm location game on a sphere. 8 1997 Elsevier Science Ltd. Key words: Nash equilibrium, location game.
1. INTRODUCTION Location games on sets of dimension two can be used to model different types of spatial competition. In one type, firms locate geographically, in a city (a subset of ,%‘) or around the world (on a sphere). In another type, firms compete via product characteristic selection for a product with two important characteristics. In a third type, candidates choose positions in a two-issue election. Shaked (1975) proved there are no pure strategy Nash equilibria for 34~1 location games in the plane with a wide variety of consumer distributions. Eaton and Lipsey (1976) used numerical methods to produce strong (but not conclusive) evidence that if consumers are distributed uniformly in the plane, then a square array of firms, a hexagonal array, and many rectangular arrays are Nash equilibria. Many theoretical location game studies begin with one of these presumed equilibria and proceed to discuss a subsequent pricing game or entry round; for examples, see Braid (1991) Eaton and Lipsey (1976) and Hartwick (1973). After nearly twenty years of such studies, Okabe and Aoyagi (1991) produced a proof that, for the location game in the plane with infinitely many players and consumers distributed uniformly, a square array of players is a pure strategy Nash equilibrium as is a regular hexagonal array. Proposition 1 below presents an infinite family of pure strategy Nash equilibria for a 3-firm location game on a sphere with consumers distributed uniformly. 1.1. A location game on S2
Suppose consumers are distributed uniformly with density one on S2, a sphere of radius one, and Firms 1, 2, and 3 then locate simultaneously on S2 at points a,b, and c respectively. The market set of a location is defined to be the set of consumers nearer it than any other location; the distance between two points of S2 is the length of the shortest arc in S2 joining the two points. The payoff to Firm 1, written a,(a,b,c), is defined to be the area of the market set of a divided by the number of firms located at a. The payoffs n2(a,b,c) and n,(a,b,c) are defined similarly. Here is an example of an application for location games on a sphere. 247
V. KNOBLAUCH
248
Example1. Each of n firms chooses a location on the surface of the earth for a satellite communication station. Subsequently, every satellite with communications capabilities stays in constant contact with its owner by communicating through the nearest communication station. Each month, each satellite owner pays each of the 12firms a fee based on the time the owner’s satellite spent communicating through the firm’s station. Since the expected number of fly-overs per month above one acre on the earths surface is about equal to the expected number of fly-overs above another, for the purposes of calculating a firm’s monthly total revenue the satellites can be thought of as uniformly distributed over the earth’s surface. Therefore, each firm’s monthly revenue is equal to its payoff in an n-person location game on a sphere with consumers distributed uniformly. Proposition 1. The location triple (u,b,c) is a Nash equilibrium of the 3-firm location game on S* if and only if ??a,
b and c lie on a great circle S’ of S*, and
??(a$,~)
is a Nash equilibrium of the 3-firm location game on S’ (consumers distributed uniformly).
Note that Eaton and Lipsey (1975) showed that (a&) is a Nash equilibrium on a circle S’ (consumers distributed uniformly) if, and only if, no semicircle contains u,b and c in its interior. Proposition 1 produces a continuum of equilibria, in fact a two-dimensional set of equilibria. To see this, suppose S’C S* is a longitudinal great circle, x&’ is at the north pole, the small subarc Y of S’ is near the equator and the small subarc 2 of S’ is four ninths of the way from x to Y along the longer arc from x to Y. If y,y’eY, z,z’eZ and y#y’ or t#z’, then the two Nash equillibria (x,y,z) and (x,y’,.?) are distinct in the strong sense that no combinations of rotations of S* can take the set {x,y,z) onto the set {x,y’,z’}. The set of distinct equilibria {xl x Y x 2 is two-dimensional. 1.2. Proofof proposition 1 Proof. It is useful to assert that if a,b,c.sC G S* where C is a circle with radius r,, then nf(a,b,c) = Ir;‘(a,b,c) i = 1,2,3
(I)
where the superscripts indicate the set on which a location game is played, and for the game on C consumers are distributed uniformly with density 2/rc. To verify this claim, think of C as a latitudinal circle of S*. The location triple (a&) divides the sphere into market sets bounded by lines of longitude that also divide C into market sets for (a,b,c). (=) Suppose a,b,c& C S*, S’ is a great circle of S*, and (a&,~) is a Nash equilibrium for the location game on S’ (consumers distributed uniformly). Suppose firm 1 moves from a to a’. We will show &a,b,c)
2 ~$(d,b,c)
First a’, b and c are contained in a circle CC S* (C is not unique unless a’, b and c are distinct). If a, 6, c and a’ are not distinct, C can be taken to be a great circle that also contains a and the claim (t) is easily established using hypothesis (2) and equation (1). Otherwise, let G be the subarc of S’ with end points 6 and c such that ft2a&. By Eaton and Lipsey’s characterization given above of Nash equilibria on a circle, the length d(bc) is the distance in S2 from b to c. Let b& be the subarc of C with end points b and c such that
A pure strategy Nash equilibrium
249
a’#&. Then @c)a&bc) since k&c) cannot be less than the distance in S2 from b to c. For location games on S’ and C, distribute consumers uniformly with density 2 and 2/rc respectively. Then &a,b,c)
= 7&&c) = (4n - 2@))/2 3 (41~- 2/(&)/r,)/2 = rc~,‘(a’,b,c) = r&‘,b,c)
(a) Suppose a,b,c~S’, S’ is a great circle of S*, but (a&z) is not an equilibrium location game on S’. Without loss of generality, assume a is not an optimal response in the location game on S’. Then there is an u’ES’ with rr”‘(a’,b,c) > n”;‘(a,b,c). Therefore,
of the to (b,c)
n;‘(a’,b,c) = $‘(a’,b,c) > r;‘(a,b,c) = nf(a,b,c)
and (a$,~) is not a Nash equilibrium of the location game on S2. Next suppose a,b,cK and C is not a great circle of S*. If (a,b,c) is not a Nash equilibrium of the location game on C, proceed as above. If (a,b,c) is a Nash equilibrium of the location game on C, without loss of generality, assume b#c. We will produce an a’ such that xf(d,b,c)>&(a,b,c). Let G be a subarc of C with end points b,c such that a is not in the interior of GE. Let S’ be a great circle of S2 with b&3’, let & be a subarc of S’ with end points b and c such that @x) is the distance in S* from b to c, and let a’ be a point in S’ -[I&. Then P(G) > d(k) s&e C is not a great circle, and [(S’ -&)e > (C -&)/rc. Therefore, nf(d,b,c) = x.;‘(a’,b,c) = P(S’-bc) > /(C - bc)lr, = n:(a,b,c) = rf(a,b,c)
and again (u,b,c) is not a Nash equilibrium
for the location game on S2.
2. CONCLUDING
REMARK
Pure strategy Nash equilibria for location games are of interest to those researchers who have pondered the meaning of the dearth of examples of pure sfrategy equilibria for economics-related continuous strategy games. Acknowledgements-This work was partially supported by a College of Commerce and Business Administration Summer Research Award at the University of Alabama. I would like to thank Frank Plastria for bringing the Okabe and Aoyagi (1991) paper to my attention.
250
V. KNOBLAUCH
REFERENCES Braid, R. (1991) Two-dimensional Bertrand competition: block metric, Euclidean metric, and waves of entry. Joumol of Regional Science, 31, 35-48. Eaton, B. & Lipsey, R. (1975) The principle of minimum differentiation reconsidered: some new developments in the theory of spatial competition. Review of Economic Studies, 42, 27-50. Eaton, B. & Lipsey, R. (1976) The non-uniqueness of equilibrium in the Loschian location model. The American Economic Review, 66,77-93.
Hartwick, J. (1973) Losch’s theorem of hexagonal market areas. Journal of Regional Science, 13,213-222. Okabe, A. & Aoyagi, M. (1991) Existence of equilibrium configurations of competitive firms on an infinite twodimensional space. Journal of Urban Economics, 29,349-370. Shaked, A. (1975) Non-existence of equilibrium for the 2-dimensional 3-firms location problem. Review of Economic Studies, 42, 5 l-56.
PII:s~9(97)oooo4-l
Location Science,Vol. 4, No. 4, pp. 247-250, 1996 0 1997 Elsevia ScienceLtd All rightsreserved.Printed in Great Britain 0966-8349/97s17.00+0.00
A PURE STRATEGY NASH EQUILIBRIUM FOR A 3-FIRM LOCATION GAME ON A SPHERE VICIU KNOBLAUCH Department of Economics, Vassar College, Poughkeepsie, NY 12604, U.S.A.
(Received 7 Febnmy 1997)
Abstract-There are only two known pure strategy Nash equilibria for multi-firm location games in dimension two. This paper presents an infinite family of pure strategy Nash equilibria for a 3-firm location game on a sphere. 8 1997 Elsevier Science Ltd. Key words: Nash equilibrium, location game.
1. INTRODUCTION Location games on sets of dimension two can be used to model different types of spatial competition. In one type, firms locate geographically, in a city (a subset of ,%‘) or around the world (on a sphere). In another type, firms compete via product characteristic selection for a product with two important characteristics. In a third type, candidates choose positions in a two-issue election. Shaked (1975) proved there are no pure strategy Nash equilibria for 34~1 location games in the plane with a wide variety of consumer distributions. Eaton and Lipsey (1976) used numerical methods to produce strong (but not conclusive) evidence that if consumers are distributed uniformly in the plane, then a square array of firms, a hexagonal array, and many rectangular arrays are Nash equilibria. Many theoretical location game studies begin with one of these presumed equilibria and proceed to discuss a subsequent pricing game or entry round; for examples, see Braid (1991) Eaton and Lipsey (1976) and Hartwick (1973). After nearly twenty years of such studies, Okabe and Aoyagi (1991) produced a proof that, for the location game in the plane with infinitely many players and consumers distributed uniformly, a square array of players is a pure strategy Nash equilibrium as is a regular hexagonal array. Proposition 1 below presents an infinite family of pure strategy Nash equilibria for a 3-firm location game on a sphere with consumers distributed uniformly. 1.1. A location game on S2
Suppose consumers are distributed uniformly with density one on S2, a sphere of radius one, and Firms 1, 2, and 3 then locate simultaneously on S2 at points a,b, and c respectively. The market set of a location is defined to be the set of consumers nearer it than any other location; the distance between two points of S2 is the length of the shortest arc in S2 joining the two points. The payoff to Firm 1, written a,(a,b,c), is defined to be the area of the market set of a divided by the number of firms located at a. The payoffs n2(a,b,c) and n,(a,b,c) are defined similarly. Here is an example of an application for location games on a sphere. 247
V. KNOBLAUCH
248
Example1. Each of n firms chooses a location on the surface of the earth for a satellite communication station. Subsequently, every satellite with communications capabilities stays in constant contact with its owner by communicating through the nearest communication station. Each month, each satellite owner pays each of the 12firms a fee based on the time the owner’s satellite spent communicating through the firm’s station. Since the expected number of fly-overs per month above one acre on the earths surface is about equal to the expected number of fly-overs above another, for the purposes of calculating a firm’s monthly total revenue the satellites can be thought of as uniformly distributed over the earth’s surface. Therefore, each firm’s monthly revenue is equal to its payoff in an n-person location game on a sphere with consumers distributed uniformly. Proposition 1. The location triple (u,b,c) is a Nash equilibrium of the 3-firm location game on S* if and only if ??a,
b and c lie on a great circle S’ of S*, and
??(a$,~)
is a Nash equilibrium of the 3-firm location game on S’ (consumers distributed uniformly).
Note that Eaton and Lipsey (1975) showed that (a&) is a Nash equilibrium on a circle S’ (consumers distributed uniformly) if, and only if, no semicircle contains u,b and c in its interior. Proposition 1 produces a continuum of equilibria, in fact a two-dimensional set of equilibria. To see this, suppose S’C S* is a longitudinal great circle, x&’ is at the north pole, the small subarc Y of S’ is near the equator and the small subarc 2 of S’ is four ninths of the way from x to Y along the longer arc from x to Y. If y,y’eY, z,z’eZ and y#y’ or t#z’, then the two Nash equillibria (x,y,z) and (x,y’,.?) are distinct in the strong sense that no combinations of rotations of S* can take the set {x,y,z) onto the set {x,y’,z’}. The set of distinct equilibria {xl x Y x 2 is two-dimensional. 1.2. Proofof proposition 1 Proof. It is useful to assert that if a,b,c.sC G S* where C is a circle with radius r,, then nf(a,b,c) = Ir;‘(a,b,c) i = 1,2,3
(I)
where the superscripts indicate the set on which a location game is played, and for the game on C consumers are distributed uniformly with density 2/rc. To verify this claim, think of C as a latitudinal circle of S*. The location triple (a&) divides the sphere into market sets bounded by lines of longitude that also divide C into market sets for (a,b,c). (=) Suppose a,b,c& C S*, S’ is a great circle of S*, and (a&,~) is a Nash equilibrium for the location game on S’ (consumers distributed uniformly). Suppose firm 1 moves from a to a’. We will show &a,b,c)
2 ~$(d,b,c)
First a’, b and c are contained in a circle CC S* (C is not unique unless a’, b and c are distinct). If a, 6, c and a’ are not distinct, C can be taken to be a great circle that also contains a and the claim (t) is easily established using hypothesis (2) and equation (1). Otherwise, let G be the subarc of S’ with end points 6 and c such that ft2a&. By Eaton and Lipsey’s characterization given above of Nash equilibria on a circle, the length d(bc) is the distance in S2 from b to c. Let b& be the subarc of C with end points b and c such that
A pure strategy Nash equilibrium
249
a’#&. Then @c)a&bc) since k&c) cannot be less than the distance in S2 from b to c. For location games on S’ and C, distribute consumers uniformly with density 2 and 2/rc respectively. Then &a,b,c)
= 7&&c) = (4n - 2@))/2 3 (41~- 2/(&)/r,)/2 = rc~,‘(a’,b,c) = r&‘,b,c)
(a) Suppose a,b,c~S’, S’ is a great circle of S*, but (a&z) is not an equilibrium location game on S’. Without loss of generality, assume a is not an optimal response in the location game on S’. Then there is an u’ES’ with rr”‘(a’,b,c) > n”;‘(a,b,c). Therefore,
of the to (b,c)
n;‘(a’,b,c) = $‘(a’,b,c) > r;‘(a,b,c) = nf(a,b,c)
and (a$,~) is not a Nash equilibrium of the location game on S2. Next suppose a,b,cK and C is not a great circle of S*. If (a,b,c) is not a Nash equilibrium of the location game on C, proceed as above. If (a,b,c) is a Nash equilibrium of the location game on C, without loss of generality, assume b#c. We will produce an a’ such that xf(d,b,c)>&(a,b,c). Let G be a subarc of C with end points b,c such that a is not in the interior of GE. Let S’ be a great circle of S2 with b&3’, let & be a subarc of S’ with end points b and c such that @x) is the distance in S* from b to c, and let a’ be a point in S’ -[I&. Then P(G) > d(k) s&e C is not a great circle, and [(S’ -&)e > (C -&)/rc. Therefore, nf(d,b,c) = x.;‘(a’,b,c) = P(S’-bc) > /(C - bc)lr, = n:(a,b,c) = rf(a,b,c)
and again (u,b,c) is not a Nash equilibrium
for the location game on S2.
2. CONCLUDING
REMARK
Pure strategy Nash equilibria for location games are of interest to those researchers who have pondered the meaning of the dearth of examples of pure sfrategy equilibria for economics-related continuous strategy games. Acknowledgements-This work was partially supported by a College of Commerce and Business Administration Summer Research Award at the University of Alabama. I would like to thank Frank Plastria for bringing the Okabe and Aoyagi (1991) paper to my attention.
250
V. KNOBLAUCH
REFERENCES Braid, R. (1991) Two-dimensional Bertrand competition: block metric, Euclidean metric, and waves of entry. Joumol of Regional Science, 31, 35-48. Eaton, B. & Lipsey, R. (1975) The principle of minimum differentiation reconsidered: some new developments in the theory of spatial competition. Review of Economic Studies, 42, 27-50. Eaton, B. & Lipsey, R. (1976) The non-uniqueness of equilibrium in the Loschian location model. The American Economic Review, 66,77-93.
Hartwick, J. (1973) Losch’s theorem of hexagonal market areas. Journal of Regional Science, 13,213-222. Okabe, A. & Aoyagi, M. (1991) Existence of equilibrium configurations of competitive firms on an infinite twodimensional space. Journal of Urban Economics, 29,349-370. Shaked, A. (1975) Non-existence of equilibrium for the 2-dimensional 3-firms location problem. Review of Economic Studies, 42, 5 l-56.