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Some conditions on equilibria in symmetric two player discontinuous games
Economics Letters 72 (2001) 175–180 www.elsevier.com / locate / econbase
Some conditions on equilibria in symmetric two player discontinuous games Alex Coram Department of Political Science, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia Received 11 May 2000; accepted 29 August 2000
Abstract This paper examines equilibria in discontinuous symmetrical two player games of winning and losing. I show that the equilibrium for these games is given by a mixed strategy that places weight on an even number of symmetrical points. 2001 Elsevier Science B.V. All rights reserved. Keywords: Discontinuous games; Symmetrical games; Dasgupta and Maskin theorems JEL classification: C720
1. Introduction Social scientists frequently have to deal with games that have discontinuous payoffs, such as duels and other games of timing, in which the aim is to act as late as possible but before an opponent acts. Related to these games are competitions or tournaments where the payoffs are win, draw, or lose (Owen, 1982). These win–lose payoffs may have different characteristics from payoffs in games of maximizing even though they depend on the same underlying function. Think of a game where there are two types of resources and the payoff depends on their combination. An example is a battle in which there are tanks and planes. Suppose player one has picked a certain proportion of planes and tanks. As player two transfers resources from tanks to planes, the balance improves and the underlying function increases. For equal resources the payoff is the same for both players. As more planes and less tanks are used the underlying function jumps down 1 . It then increases as more planes are built and they are able to defend themselves. Alternatively, think of a political party transferring resources from single issue to general interest voters. As more resources are transferred from single issue voters support from general interest voters increases. At the point where less is given to single issue voters E-mail address: [email protected] (A. Coram). 1 Maybe there are not enough tanks to defend the planes and not enough planes to defend themselves. 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00424-4
2001 Elsevier Science B.V. All rights reserved.
A. Coram / Economics Letters 72 (2001) 175 – 180
176
total support jumps down. What is of interest is the win–lose payoff derived from the underlying function 2 . The most important theorems on the existence of equilibria in discontinuous games are those of Dasgupta and Maskin (1986). The purpose of this paper is to suggest an addition to these theorems and to clear up an ambiguity in their formulation. I am particularly concerned with symmetrical two player games of winning and losing that have a discontinuity in the payoff function at all points where the payoffs from some underlying game are equal. These seem to arise in a natural manner in many situations. I show that the equilibrium for these games may be given by a mixed strategy that places weight on an even number of symmetrical points.
2. Discontinuous symmetrical games with win–lose payoffs Assume that we have a symmetric conflict between two players with continuous strategies and model this as a game on the square. All the prize goes to the player who does best. Each player has a strategy s i [ A, i 5 1, 2, where A is the interval [0,1]. The points of discontinuity are elements of the set A* # A3 . (s i , s j ) [ A* , i, j 5 1, 2 and i ± j if s i can be expressed as a continuous function of s j . Since we have a symmetrical game all payoffs are given for player one. The underlying payoff function is a real valued function f(s 1 , s 2 ): f : [0,1] ^ [0,1] → R
1
(1)
Assume, without loss of generality, that f1 is continuous and monotonically increasing in s1 , except at the set of points s [ A** # A* given by the points on the diagonal s 1 5 s 2 . This will give something like the example in Fig. 1. Player one wins if it does better than player two, draws if it gets the same payoff and loses otherwise. The win–lose payoff for the game is thus of the form:
Fig. 1. Example of the discontinuous payoff function f.
2 3
See Fig. 2 for an illustration. I follow Dasgupta and Maskin’s notation here.
A. Coram / Economics Letters 72 (2001) 175 – 180
if f1 . f2 if f1 5 f2 if f1 , f2
1 v1 (s 1 , s 2 ) 5 1 / 2 0
*
177
(2)
The existence of an equilibrium for this game is guaranteed by the following theorem by Dasgupta and Maskin (1986). To avoid confusion with the example in this paper let the payoffs in Dasgupta and Maskin’s argument be u. Theorem. (Dasgupta and Maskin, 14) Let o u i be upper semi-continuous and u i bounded and weakly lower semi-continuous in s i . Then there exists a mixed strategy equilibrium. Dasgupta and Maskin also give a theorem that specifies the equilibrium in an N person symmetrical game with the property that, for all s [ A* , each point on the diagonal is a point of discontinuity. That is, if we have s i : s i 5 s 2i then s i [ A** . This theorem requires that u i satisfies a strong version of weak lower semi-continuity called property a. This is that for each s 9i [ A** there exists an l [ [0,1] such that for s 2i [ A** (s 9i )4 we have:
l lim inf s 2i →s i 9 u i (s i , s 2i ) 1 (1 2 l) lim inf s i 1
→s i 9
u i (s i , s 2i ) $ u i (s 9i , s 2i )
with strict inequality for s 2i 5 s i9 . Theorem. (Dasgupta and Maskin, 19) Suppose that u i : S → R 1 is continuous except on A** . Let o u i be upper semi-continuous and for all i, u i be bounded and satisfy property a. Then there exists a symmetric mixed strategy equilibrium (s, . . . ,s ) with the property that for all i and s i [ A** , s (hs 9i j) 5 0. Proof. (Sketch) The essential part of Dasgupta and Maskin’s proof is to suppose a non-zero weight is assigned to s i9 in equilibrium. Without loss of generality set: lim inf s 2i →s i 9
E u (s , s i
i
2i
) ds2i .
E u (s9, s i
i
2i
) ds2i
This contradicts condition a. The problem is that this theorem is only correct where s i : s i 5 s 2i is the only element of A** . Consider the case where there are other elements of discontinuity and player i puts positive weight on points s 9i and s 99i where s i99 5 s i99 (s i9 ). Then it is not necessarily the case that: i
E u (s9, s
2i
.
4
E u (s , s ) ds 1 lim inf ) ds 1 E u (s 99 , s ) ds
l[lim inf s 2i →s i 9 i
i
2i
i
2i
2i
i
i
2i
s2 i →s i 9
E u (s99, s i
i
2i
) ds2i ]
2i
Note that Dasgupta and Maskin do not specify that this is the only point of discontinuity (19).
A. Coram / Economics Letters 72 (2001) 175 – 180
178
Fig. 2. Payoffs in the win–lose game with underlying function f.
This is because it is possible that u i jumps up at s i99 and: lim inf s 2i →s i 9
E u (s99, s i
i
2i
) ds2i ] ,
E u (s99, s i
i
2i
) ds2i
It follows that the theorem does not apply to the win–lose game with payoff v given in the example because there are points in the set A** other than s 1 5 s 2 . This is illustrated in Fig. 2. So where does this leave us for games of this type with more than one point of discontinuity? I set out below a theorem for two player symmetrical games.
3. Equilibria in discontinuous win–lose games on the square The win–lose game is specified by the functions f and v above and it is assumed that there is no solution in pure strategies. This eliminates trivial cases. The regions where player one has a winning strategy will be reflected around the diagonal by symmetry. Set the horizontal axis as s 1 and express the boundaries between a region in which player one wins and loses by the continuous function s 2 5 w i (s 1 ). Since w : s 1 → R, we can also express s 1 8 5 w (s i1 ). An example is given in Fig. 3. To save cluttering the notation, label the first boundary above the diagonal w (s 1 ). The following definitions will be useful. Definitions. (a) w symmetrical. A game is w symmetrical if w (0) 5 w 21 (1). (b) Condition g. s 1i meets condition g if there exists an s i1 5 hs 1 : v1 [s i1 , 0] 1 v1 [w i (s i1 ), 0] ± 0, and v1 [s i1 , 1] 1 v1 [w i (s i1 ), 1] ± 0j.
w symmetrical imposes a strong condition on the game and would only be met in special cases. Condition g means that there is a pair of strategies s i1 , and s 1 8 5 w i (s i1 ) such that there is no s 2 5 0 or s 2 5 1 that can defeat both s i1 and w i (s 1i ) simultaneously. Theorem. There is a mixed strategy equilibrium for player one given by any number n . 0 of the pairs s *1 5 (1 / 2n, 1 / 2n)[s i1 , w (s i1 )], (1 / 2n, 1 / 2n)[s j1 , w (s j1 )], . . . when s i1 and w i (s i1 ) meet condition g. This equilibrium is unique when the game is not w symmetrical.
A. Coram / Economics Letters 72 (2001) 175 – 180
179
Fig. 3. Example of a game on the square that is not w symmetrical.
Proof. The payoffs to one for any strategy s 2 are on a horizontal line divided into 0, 1 segments. Without loss of generality the proof is for n 5 1. (a) Existence. Show that there is always an s i1 that meets condition g. Consider the case where v1 [s 1 , 0] 5 1. Then we are done. Hence s 2 5 0 must be divided into 0, 1 segments. By the construction of f there can be only two such segments. Since v2 [0, s 2 ] is the transpose of v1 [s 1 , 0] we must have the 0, 1 sequence on v1 [s 1 , 1] in the same order as v1 [s 1 , 0]. Hence there must be an s1 that satisfies v1 [s 1 , 0] 5 1 and v1 [s 1 , 1] 5 1. Without loss of generality consider the game with the sequence on the horizontal s 2 5 0 beginning at 1. Choose s i1 so that s i1 , w (s i1 ) meets condition g. By symmetry for s 2 . w (s 1 ), v1 (s 1 ) 5 1. For s 1 , s 2 , w (s 1 ), v1 (w (s 1 )) 5 1. For s 2 , s 1 v1 (s 1 ) 5 1. Let E be expected value. Then E[s *1 ] $ 1 / 2. (b) Uniqueness. Suppose that the game is not w symmetrical and there is a continuous strategy for player one given by the probability density function p(x) on the interval [0,1]. Then E[ p, s 2 ] 5 E[ p, s 2 1 e ] and E[ p, s 2 ] 5 E[ p, s 2 2 e ] for all s 2 . Choose s 2 5 w (0) and let w (0) , w 21 (1). Then: w (s 2 )
s2
E E
E[ p, s 2 ] 5 0 1
1
p 1
s2
0
E
E
E[ p, s 2 1 e ] 5
p1
From (3)–(5)
E
01
s2
s 22e
s 21e
p1
E s2
w (s 2 )
E 01 E s2
w 21 (s 2 1 e )
E 01 E 0
s 21e
s2
0
s 22e
(3)
w (s 2 )
w 21 (s 2 1 e )
E[ p, s 2 1 e ] 5
0
E
s 21e
E
p1
s 21e
w (s 2 2 e )
p1
w (s 2 1 e )
w (s 2 )
1
p1
E
0
(4)
w (s 2 1 e )
1
p1
E
w (s 2 2 e )
0
(5)
A. Coram / Economics Letters 72 (2001) 175 – 180
180
w 21 (s 2 1 e )
E
E[ p, s 2 1 e ] 2 E[ p, s 2 ] 5
w (s 2 1 e )
E
p1
w (s 2 )
E
As e → 0
E
E
p2
p→
E
p50
w (s 2 )
p
p50
(6)
s2
(7)
s 22e
s2
E
and
s 22e
s2
E
s2
w (s 2 2 e ) s 21e
p2
w (s 2 )
0
E[ p, s 2 2 e ] 2 E[ p, s 2 ] 5
s 21e
w (s 2 2 e )
w (s 2 1 e )
p→
E
p
w (s 2 )
Hence from (6) and (7) we get for e → 0 w (s 2 1 e )
E 0
w (s 2 1 e )
E
p1
w (s 2 )
w (s 2 )
p→
E
w (s 2 2 e )
w (s 2 1 e )
p→
E
p
w (s 2 )
4. Contradiction Consider any other mixed strategy with a discrete probability distribution o s ji s ji where the distribution is not given by matched pairs [s j1 , w (s j1 )]. Pick any s 1 8 [ ⁄ [s j1 , w (s j1 )]. Then it is possible to pick an s 2 that intersects two of (s 1 8, s 1j , w (s 1j ) ) at v1 (s) 5 0. Thus s 8 . 0 is not optimal. The rest follows by induction. h
References Dasgupta, P., Maskin, E., 1986. The existence of equilibrium, in discontinuous games I: theory. The Review of Economic Studies LII, 1–26. Owen, G., 1982. Game Theory. Academic Press, New York.
Some conditions on equilibria in symmetric two player discontinuous games Alex Coram Department of Political Science, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia Received 11 May 2000; accepted 29 August 2000
Abstract This paper examines equilibria in discontinuous symmetrical two player games of winning and losing. I show that the equilibrium for these games is given by a mixed strategy that places weight on an even number of symmetrical points. 2001 Elsevier Science B.V. All rights reserved. Keywords: Discontinuous games; Symmetrical games; Dasgupta and Maskin theorems JEL classification: C720
1. Introduction Social scientists frequently have to deal with games that have discontinuous payoffs, such as duels and other games of timing, in which the aim is to act as late as possible but before an opponent acts. Related to these games are competitions or tournaments where the payoffs are win, draw, or lose (Owen, 1982). These win–lose payoffs may have different characteristics from payoffs in games of maximizing even though they depend on the same underlying function. Think of a game where there are two types of resources and the payoff depends on their combination. An example is a battle in which there are tanks and planes. Suppose player one has picked a certain proportion of planes and tanks. As player two transfers resources from tanks to planes, the balance improves and the underlying function increases. For equal resources the payoff is the same for both players. As more planes and less tanks are used the underlying function jumps down 1 . It then increases as more planes are built and they are able to defend themselves. Alternatively, think of a political party transferring resources from single issue to general interest voters. As more resources are transferred from single issue voters support from general interest voters increases. At the point where less is given to single issue voters E-mail address: [email protected] (A. Coram). 1 Maybe there are not enough tanks to defend the planes and not enough planes to defend themselves. 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00424-4
2001 Elsevier Science B.V. All rights reserved.
A. Coram / Economics Letters 72 (2001) 175 – 180
176
total support jumps down. What is of interest is the win–lose payoff derived from the underlying function 2 . The most important theorems on the existence of equilibria in discontinuous games are those of Dasgupta and Maskin (1986). The purpose of this paper is to suggest an addition to these theorems and to clear up an ambiguity in their formulation. I am particularly concerned with symmetrical two player games of winning and losing that have a discontinuity in the payoff function at all points where the payoffs from some underlying game are equal. These seem to arise in a natural manner in many situations. I show that the equilibrium for these games may be given by a mixed strategy that places weight on an even number of symmetrical points.
2. Discontinuous symmetrical games with win–lose payoffs Assume that we have a symmetric conflict between two players with continuous strategies and model this as a game on the square. All the prize goes to the player who does best. Each player has a strategy s i [ A, i 5 1, 2, where A is the interval [0,1]. The points of discontinuity are elements of the set A* # A3 . (s i , s j ) [ A* , i, j 5 1, 2 and i ± j if s i can be expressed as a continuous function of s j . Since we have a symmetrical game all payoffs are given for player one. The underlying payoff function is a real valued function f(s 1 , s 2 ): f : [0,1] ^ [0,1] → R
1
(1)
Assume, without loss of generality, that f1 is continuous and monotonically increasing in s1 , except at the set of points s [ A** # A* given by the points on the diagonal s 1 5 s 2 . This will give something like the example in Fig. 1. Player one wins if it does better than player two, draws if it gets the same payoff and loses otherwise. The win–lose payoff for the game is thus of the form:
Fig. 1. Example of the discontinuous payoff function f.
2 3
See Fig. 2 for an illustration. I follow Dasgupta and Maskin’s notation here.
A. Coram / Economics Letters 72 (2001) 175 – 180
if f1 . f2 if f1 5 f2 if f1 , f2
1 v1 (s 1 , s 2 ) 5 1 / 2 0
*
177
(2)
The existence of an equilibrium for this game is guaranteed by the following theorem by Dasgupta and Maskin (1986). To avoid confusion with the example in this paper let the payoffs in Dasgupta and Maskin’s argument be u. Theorem. (Dasgupta and Maskin, 14) Let o u i be upper semi-continuous and u i bounded and weakly lower semi-continuous in s i . Then there exists a mixed strategy equilibrium. Dasgupta and Maskin also give a theorem that specifies the equilibrium in an N person symmetrical game with the property that, for all s [ A* , each point on the diagonal is a point of discontinuity. That is, if we have s i : s i 5 s 2i then s i [ A** . This theorem requires that u i satisfies a strong version of weak lower semi-continuity called property a. This is that for each s 9i [ A** there exists an l [ [0,1] such that for s 2i [ A** (s 9i )4 we have:
l lim inf s 2i →s i 9 u i (s i , s 2i ) 1 (1 2 l) lim inf s i 1
→s i 9
u i (s i , s 2i ) $ u i (s 9i , s 2i )
with strict inequality for s 2i 5 s i9 . Theorem. (Dasgupta and Maskin, 19) Suppose that u i : S → R 1 is continuous except on A** . Let o u i be upper semi-continuous and for all i, u i be bounded and satisfy property a. Then there exists a symmetric mixed strategy equilibrium (s, . . . ,s ) with the property that for all i and s i [ A** , s (hs 9i j) 5 0. Proof. (Sketch) The essential part of Dasgupta and Maskin’s proof is to suppose a non-zero weight is assigned to s i9 in equilibrium. Without loss of generality set: lim inf s 2i →s i 9
E u (s , s i
i
2i
) ds2i .
E u (s9, s i
i
2i
) ds2i
This contradicts condition a. The problem is that this theorem is only correct where s i : s i 5 s 2i is the only element of A** . Consider the case where there are other elements of discontinuity and player i puts positive weight on points s 9i and s 99i where s i99 5 s i99 (s i9 ). Then it is not necessarily the case that: i
E u (s9, s
2i
.
4
E u (s , s ) ds 1 lim inf ) ds 1 E u (s 99 , s ) ds
l[lim inf s 2i →s i 9 i
i
2i
i
2i
2i
i
i
2i
s2 i →s i 9
E u (s99, s i
i
2i
) ds2i ]
2i
Note that Dasgupta and Maskin do not specify that this is the only point of discontinuity (19).
A. Coram / Economics Letters 72 (2001) 175 – 180
178
Fig. 2. Payoffs in the win–lose game with underlying function f.
This is because it is possible that u i jumps up at s i99 and: lim inf s 2i →s i 9
E u (s99, s i
i
2i
) ds2i ] ,
E u (s99, s i
i
2i
) ds2i
It follows that the theorem does not apply to the win–lose game with payoff v given in the example because there are points in the set A** other than s 1 5 s 2 . This is illustrated in Fig. 2. So where does this leave us for games of this type with more than one point of discontinuity? I set out below a theorem for two player symmetrical games.
3. Equilibria in discontinuous win–lose games on the square The win–lose game is specified by the functions f and v above and it is assumed that there is no solution in pure strategies. This eliminates trivial cases. The regions where player one has a winning strategy will be reflected around the diagonal by symmetry. Set the horizontal axis as s 1 and express the boundaries between a region in which player one wins and loses by the continuous function s 2 5 w i (s 1 ). Since w : s 1 → R, we can also express s 1 8 5 w (s i1 ). An example is given in Fig. 3. To save cluttering the notation, label the first boundary above the diagonal w (s 1 ). The following definitions will be useful. Definitions. (a) w symmetrical. A game is w symmetrical if w (0) 5 w 21 (1). (b) Condition g. s 1i meets condition g if there exists an s i1 5 hs 1 : v1 [s i1 , 0] 1 v1 [w i (s i1 ), 0] ± 0, and v1 [s i1 , 1] 1 v1 [w i (s i1 ), 1] ± 0j.
w symmetrical imposes a strong condition on the game and would only be met in special cases. Condition g means that there is a pair of strategies s i1 , and s 1 8 5 w i (s i1 ) such that there is no s 2 5 0 or s 2 5 1 that can defeat both s i1 and w i (s 1i ) simultaneously. Theorem. There is a mixed strategy equilibrium for player one given by any number n . 0 of the pairs s *1 5 (1 / 2n, 1 / 2n)[s i1 , w (s i1 )], (1 / 2n, 1 / 2n)[s j1 , w (s j1 )], . . . when s i1 and w i (s i1 ) meet condition g. This equilibrium is unique when the game is not w symmetrical.
A. Coram / Economics Letters 72 (2001) 175 – 180
179
Fig. 3. Example of a game on the square that is not w symmetrical.
Proof. The payoffs to one for any strategy s 2 are on a horizontal line divided into 0, 1 segments. Without loss of generality the proof is for n 5 1. (a) Existence. Show that there is always an s i1 that meets condition g. Consider the case where v1 [s 1 , 0] 5 1. Then we are done. Hence s 2 5 0 must be divided into 0, 1 segments. By the construction of f there can be only two such segments. Since v2 [0, s 2 ] is the transpose of v1 [s 1 , 0] we must have the 0, 1 sequence on v1 [s 1 , 1] in the same order as v1 [s 1 , 0]. Hence there must be an s1 that satisfies v1 [s 1 , 0] 5 1 and v1 [s 1 , 1] 5 1. Without loss of generality consider the game with the sequence on the horizontal s 2 5 0 beginning at 1. Choose s i1 so that s i1 , w (s i1 ) meets condition g. By symmetry for s 2 . w (s 1 ), v1 (s 1 ) 5 1. For s 1 , s 2 , w (s 1 ), v1 (w (s 1 )) 5 1. For s 2 , s 1 v1 (s 1 ) 5 1. Let E be expected value. Then E[s *1 ] $ 1 / 2. (b) Uniqueness. Suppose that the game is not w symmetrical and there is a continuous strategy for player one given by the probability density function p(x) on the interval [0,1]. Then E[ p, s 2 ] 5 E[ p, s 2 1 e ] and E[ p, s 2 ] 5 E[ p, s 2 2 e ] for all s 2 . Choose s 2 5 w (0) and let w (0) , w 21 (1). Then: w (s 2 )
s2
E E
E[ p, s 2 ] 5 0 1
1
p 1
s2
0
E
E
E[ p, s 2 1 e ] 5
p1
From (3)–(5)
E
01
s2
s 22e
s 21e
p1
E s2
w (s 2 )
E 01 E s2
w 21 (s 2 1 e )
E 01 E 0
s 21e
s2
0
s 22e
(3)
w (s 2 )
w 21 (s 2 1 e )
E[ p, s 2 1 e ] 5
0
E
s 21e
E
p1
s 21e
w (s 2 2 e )
p1
w (s 2 1 e )
w (s 2 )
1
p1
E
0
(4)
w (s 2 1 e )
1
p1
E
w (s 2 2 e )
0
(5)
A. Coram / Economics Letters 72 (2001) 175 – 180
180
w 21 (s 2 1 e )
E
E[ p, s 2 1 e ] 2 E[ p, s 2 ] 5
w (s 2 1 e )
E
p1
w (s 2 )
E
As e → 0
E
E
p2
p→
E
p50
w (s 2 )
p
p50
(6)
s2
(7)
s 22e
s2
E
and
s 22e
s2
E
s2
w (s 2 2 e ) s 21e
p2
w (s 2 )
0
E[ p, s 2 2 e ] 2 E[ p, s 2 ] 5
s 21e
w (s 2 2 e )
w (s 2 1 e )
p→
E
p
w (s 2 )
Hence from (6) and (7) we get for e → 0 w (s 2 1 e )
E 0
w (s 2 1 e )
E
p1
w (s 2 )
w (s 2 )
p→
E
w (s 2 2 e )
w (s 2 1 e )
p→
E
p
w (s 2 )
4. Contradiction Consider any other mixed strategy with a discrete probability distribution o s ji s ji where the distribution is not given by matched pairs [s j1 , w (s j1 )]. Pick any s 1 8 [ ⁄ [s j1 , w (s j1 )]. Then it is possible to pick an s 2 that intersects two of (s 1 8, s 1j , w (s 1j ) ) at v1 (s) 5 0. Thus s 8 . 0 is not optimal. The rest follows by induction. h
References Dasgupta, P., Maskin, E., 1986. The existence of equilibrium, in discontinuous games I: theory. The Review of Economic Studies LII, 1–26. Owen, G., 1982. Game Theory. Academic Press, New York.