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Markov perfect equilibria in an N-player war of attrition
economics letters ELSEVIER
Economics Letters 47 (1995) 149-154
Markov perfect equilibria in an N-player war of attrition Sandeep Kapur Department of Economics, Birkbeck College, Gresse Street, London WIP 1PA, UK Received 11 April 1994; accepted 6 June 1994
Abstract
A symmetric finite-player war of attrition is studied in discrete time using the notion of Markov perfect equilibria. Considering the limit of the discrete time model as the decision interval becomes arbitrarily small allows a simple characterization of the mixed strategy equilibrium.
Keywords: War of attrition; Waiting games JEL classification: C70
I. Introduction
The war of attrition is a classic game of timing introduced by Maynard Smith (1974). In a stylized version of the two-player game, each player must choose when to stop fighting (i.e. yield to the other player), and the player who outlasts the other receives a higher payoff. Additionally, since fighting is costly for each player, the value of the payoff to each is decreasing in her own stopping time. This creates a structure in which each player prefers that the other stops first but, given the order in which they stop, it makes sense for each to stop sooner rather than later. In a natural generalization of this to a finite number of players, we consider the case where the rewards are monotone in the order in which the players stop. We assume that a player obtains the payoff only when she stops, and that the value of the payoff is strictly increasing in the number of other players who have stopped earlier in the game. The players discount the future so that a delay in stopping is costly. This structure may be useful in understanding behaviour in situations in which there is some advantage in being the follower rather than the leader, and the advantage is cumulatively increasing in the number of predecessors. For simplicity we consider a symmetric N-player war of attrition with complete information, and for tractability we consider only Markov Perfect Equilibria. The model is set up in discrete time but our concern lies principally with the limit of the discrete time game as the decision interval becomes arbitrarily small. 0165-1765/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1765(94)00532-X
S. Kapur / Economics Letters 47 (1995) 149-154
150
2. The game
Each of N identical players must choose a stopping time t E {0, A, 2 A , . . . } . At any time t the action set for a player is A = {stop, don't stop} if the player has not stopped before t, and comprises the trivial action 'do nothing' otherwise. Clearly, once a player stops she ceases to have further influence on the game. The payoffs are symmetric. A player receives the payoff only when she stops, and since all players discount the future (with a c o m m o n discount factor), the value of the payoffs is decreasing in the actual stopping time. However, the payoff to a player is strictly increasing in the number of other players that have stopped before she does. Without loss of generality, we order the players according to their stopping times. If t i denotes the stopping time for player i, we have t i -< ti+ ~ for i = 1, 2 , . . . , N - 1. Note that the inequality is weak: we do not rule out the possibility that two or more players stop simultaneously. For any t, define y ( t ) to be the number of players who have stopped before time t. That is, let 0, y(t) =
max{jltj
for t --
T h e n n(t) - N - y ( t ) denotes the number of players who have n o t stopped by time t. Given an arbitrary sequence of stopping times t = { t l , t 2 , . . , tN} , and the rate of interest p > 0, the payoff to player i equals Pi(ti, t - i ) = e x p ( - p t i ) T r , ( t i ) .
This specification implies that if two or more players stop simultaneously they receive identical payoffs. Also, given that the interest rate is positive, lim,i~=Pi(t i, t _ i ) = 0. The game is stationary in the sense that the current value of the payoff, 7r, depends only on the n u m b e r of players that have not stopped earlier, and does not vary with the stopping time. For the game to be a war of attrition, we need Assumption
1. exp(-pA)rr,_ 1 > ~-n -- 0, for n = 2 , . . . , N.
In order to capture the essential characteristic of a war of attrition, namely that those who stop after others do better, it would be more appropriate to require only that rrn > 7rn+~. However, in our discrete time version of the game, the earliest that one player can stop after another entails a lag A. Our assumption guarantees that the time discount associated with this lag, e x p ( - p A ) , is sufficiently small so as not to disturb the essential feature of the war of attrition. Note that in the limiting case that is of principal interest, namely where A becomes arbitrarily small, this discount factor approaches unity. Furthermore, we assume that 7rn -> 0 to ensure that the payoff is indeed decreasing over time. If, for instance, 7rn < 0, a player will choose never to stop since this minimizes the loss. We consider the Markov Perfect Equilibria (MPE) of this game. As shown by Maskin and Tirole (1988), for instance, games in which the previous history at any stage influences current play only through its effect on some 'state' variables can be analysed in terms of Markov
S. Kapur / Economics Letters 47 (1995) 149-154
151
strategies. These are mappings from the set of all possible states to (the space of probability distributions over) the set of feasible actions. The state is a summary of the payoff-relevant history, where what is relevant and what is not depends, of course, on the specification of the game. In this game the relevant aspects of the history are the elapsed time t, and the number n(t) of players who are currently active. Let the pair In(t), t] define the state. It is analytically convenient to decompose the game into a sequence of stage games, where a stage is defined to coincide with the time interval over which n(t) remains unchanged. Within each stage game the number of players is fixed so that the elapsed time t provides a complete description of the relevant history in that stage game. Loosely speaking, the stage games correspond to subgames in the original game, and if we can specify Nash equilibrium strategies for each stage game, these taken over all possible stage-games would constitute a MPE of the original game. Let Ga(n ) denote the nth stage game, parameterized by the size of the decision interval a. Assume that the stage game begins at t,, and that it ends as soon as at least one player stops. Let the players active in this stage game be indexed by j, j = 1, 2 , . . . , n. Each active player must choose a stopping time t E {tn, t, + A, t, + 2 A , . . . } . Behaviour strategies can be defined for the stage game as follows. For player j, let/3a.j(n, t) denote the probability of stopping at time t, conditional on reaching period t. We drop the subscript A where it does not affect the argument. The time profile of the (conditional) stopping probabilities amounts to a behaviour strategy in the stage game. Notice, however, that due to the assumed stationarity, whenever time t + a is reached, the game looks the same as it did at time t. Consequently, any Nash equilibrium involves only stationary strategies so that, in equilibrium,/3j(n, t) =/3 7 (n) for all t. A Nash equilibrium for the stage game is written as [/3~(n),/3 2 ( n ) , . . . , / 3 * ( n ) ] . This game has a multiplicity of equilibria. We do not attempt a complete characterization of these but notice that a strategy combination that involves exactly one player choosing 'stop immediately', and the others choosing 'never stop' in response, is a Nash equilibrium in the stage game. This is so because the players that never stop at this stage obtain the value of the continuation stage game, which is at least exp(-pa)cr,_ 1- Given our assumption, this is higher than ~-,, the payoff from stopping in the stage game. And, clearly, if all other players choose 'never stop', it makes sense for the first one to stop immediately. Next, we consider the symmetric mixed-strategy Nash equilibria for the stage game. To review the familiar intuition of the mixed-strategy equilibrium in this context, in a war of attrition a player delays the stopping decision in the hope that some other player would stop first, thereby leading to a higher payoff. However, per se, the delay is costly because players discount the future. At the mixed-strategy equilibrium, the randomization is such that these two effects balance each other and each player is indifferent between alternative stopping times. We extend this intuition to the n-player stage game. Not surprisingly, the value of the equilibrium randomization depends on A, but the limiting case as A---~0 affords explicit results. P r o p o s i t i o n 1. S u p p o s e A s s u m p t i o n 1 holds f o r s o m e n > 1. L e t the s y m m e t r i c m i x e d strategy equilibrium be given as [/3a*(n), fl] (n), . . . , /3 a (n)]. Then, we have lim /3 a*(n) _ (n - 1) _
.o)
=
S. Kapur / Economics Letters 47 (1995) 149-154
152
Proof. The argument is in distinct stages. We first fix A and show that a unique non-trivial
mixed-strategy equilibrium exists. Then we examine the limiting case as a tends to 0. Fix a for the moment so that the symmetric probability that a typical rival player will stop at time t is written as /3. For randomization to be an equilibrium strategy, a player must be indifferent between the stopping times in support of her randomization. In particular, in this infinite-horizon game, a player must obtain the same expected payoff if she stops at any time t, or at t + a. We require that
[
e-OtTr. = e p(t+A) ( 1 __ /3)n--13Tn ..l_ E
(k-1)/3k(1 __ [ ~ ) n - l - k c k , _ k
k=l
]
,
(1)
where, to define ~b,_k, a little explanation is needed. The left-hand side is merely the value of stopping at time t; the right-hand side represents the consequences of waiting for exactly one additional period prior to stopping. It is the probability-weighted sum of being either in the situation where no player stops at time t (so that our reference player gets the same return as if she stopped at t, but discounted additionally for the lag A), or of being in the situation where k players have stopped at t. Here, ~b,_k is the current value of the continuation stage game that begins at t + A if k players stop simultaneously at t. The rest of the proof is in three steps. First, we show that, given Assumption 1, there exists a value of/3* in the open interval (0, 1) for which Eq. (1) holds. Then it is argued that the value Cn-k of the continuation game n - k equals 7r,_ k, at least if we confine attention to the symmetric mixed strategy equilibria in each stage game. This allows us, inter alia, to argue that the solution/3* is unique. Lastly, we show that, as A---~0, Eq. (1), which is of degree n - 1 in /3, is approximated by a linear equation in the neighbourhood of/3". This is then solved for a ( n ) . Lemma
1. c~1 = 7r~.
Proof. Once all players but one have stopped, the last player faces a single-agent optimization
problem. The best it can do is to stop immediately, which yields ~'1. To obtain a solution to Eq. (1) in the unit interval, we rewrite it as
[(1 - / 3 1 "-I
-
e
pA
n-I
1"1"i" n --~ E
k=l
( kn -)1/ 3
k (1 _ ~1,2l, ]n - l - k ~). - k = 0 "
(2)
This is of the form g(/3) = O, where g(/3) is a (n - 1)th degree polynomial in/3. L e m m a 2. For A > 0, there exists a / 3 * E (0, 1) such that g ( / 3 * ) = O. Proof. For A > 0 ,
we have g ( 0 ) = ( 1 - eP'a)Tr, < 0 , and g ( 1 ) = z r 1 -ePa~r, > 0 ; the second relation uses Lemma 1 and Assumption 1. By continuity, there exists a/3* E (0, 1) such that g(/3*) = 0 . L e m m a 3. c~n_k = Ir -k f o r all k.
S. Kapur / Economics Letters 47 (1995) 149-154
153
Proof. The existence argument of L e m m a 2 holds for an arbitrary n. In the subsequent stage games, the equilibrium strategy is either 'stop immediately' or it involves a randomization over all possible stopping times. In either case, the expected payoff equals ~n-~" L e m m a 4. /3* is unique. Proof. Using L e m m a 3, and differentiating g(/3) with respect to/3, we get g'(/3 ) = (n __
[~.~n--2R_0
__ ]J )
[ 71"n_ k _ 1 -- ,Wn _ k
1]
.
By assumption, [Tr,_k_ 1 -- 7rn_k] > 0 for all k. Hence, g(/3) is strictly monotonic in the unit interval. The claim follows. We now consider what happens to this solution as the decision interval shrinks. Notice that g(/3a) = 0 is of the form K n _ l / 3 n-1 + Kn_2/3 n-2 + . . " + g l / 3 + Ko(ZI ) = O,
(3)
where, in particular, K 1 = ( n - 1)(Tr,_ 1 - 7 r , ) ; and K0(za) = ( 1 - e°a)zr, is the only coefficient that varies with a. As a---> 0, K0(a ) goes to zero and therefore so does/3,]. However, it is easy to check by L'Hopital's rule that their ratio is well-defined. Therefore we can divide Eq. (3) by K0(Zi), and obtain Kn_l(~o
) n - 1 (K0) "-2 + . . . + K 2( ~ o ) 2
+ K, (--'~o) K 0 + 1 = O.
(4)
As K0(A)---~ 0, all but the last two terms on the LHS vanish, and the equation reduces to K1
( 0t
+ 1 = 0,
or that/3,] -
K1
Taking limits we have,
/3]
1
p~'.
a(n) =laim° A - (n - 1) ~r,_, - 7r, as stated.
[]
3. Comments The limiting case described here corresponds to the situation in which, at the equilibrium, each firm chooses an exponential distribution over stopping times. The parameter a(n) has the usual interpretation of a hazard rate, and here this rate is found to be increasing in the cost, p'nn, of continuing in the game for one additional period and decreasing in the incremental gain from delaying the stopping decision by exactly one stage. Clearly, the two-player game is
154
s. Kapur / Economics Letters 47 (1995) 149-154
a special case of this general form. The greater generality of the N-player game m a y be of s o m e use in understanding multi-agent strategic interaction, especially w h e r e the additional restrictions used here are not particularly inconvenient.
Acknowledgements I thank J D u t t a for comments.
References Maskin, E. and J. Tirole, 1988, A theory of dynamic oligopoly I: Overview and quantity competition with large fixed costs, Econometrica 56, 549-570. Maynard Smith, J., 1974, The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology 47, 209-221.
Economics Letters 47 (1995) 149-154
Markov perfect equilibria in an N-player war of attrition Sandeep Kapur Department of Economics, Birkbeck College, Gresse Street, London WIP 1PA, UK Received 11 April 1994; accepted 6 June 1994
Abstract
A symmetric finite-player war of attrition is studied in discrete time using the notion of Markov perfect equilibria. Considering the limit of the discrete time model as the decision interval becomes arbitrarily small allows a simple characterization of the mixed strategy equilibrium.
Keywords: War of attrition; Waiting games JEL classification: C70
I. Introduction
The war of attrition is a classic game of timing introduced by Maynard Smith (1974). In a stylized version of the two-player game, each player must choose when to stop fighting (i.e. yield to the other player), and the player who outlasts the other receives a higher payoff. Additionally, since fighting is costly for each player, the value of the payoff to each is decreasing in her own stopping time. This creates a structure in which each player prefers that the other stops first but, given the order in which they stop, it makes sense for each to stop sooner rather than later. In a natural generalization of this to a finite number of players, we consider the case where the rewards are monotone in the order in which the players stop. We assume that a player obtains the payoff only when she stops, and that the value of the payoff is strictly increasing in the number of other players who have stopped earlier in the game. The players discount the future so that a delay in stopping is costly. This structure may be useful in understanding behaviour in situations in which there is some advantage in being the follower rather than the leader, and the advantage is cumulatively increasing in the number of predecessors. For simplicity we consider a symmetric N-player war of attrition with complete information, and for tractability we consider only Markov Perfect Equilibria. The model is set up in discrete time but our concern lies principally with the limit of the discrete time game as the decision interval becomes arbitrarily small. 0165-1765/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1765(94)00532-X
S. Kapur / Economics Letters 47 (1995) 149-154
150
2. The game
Each of N identical players must choose a stopping time t E {0, A, 2 A , . . . } . At any time t the action set for a player is A = {stop, don't stop} if the player has not stopped before t, and comprises the trivial action 'do nothing' otherwise. Clearly, once a player stops she ceases to have further influence on the game. The payoffs are symmetric. A player receives the payoff only when she stops, and since all players discount the future (with a c o m m o n discount factor), the value of the payoffs is decreasing in the actual stopping time. However, the payoff to a player is strictly increasing in the number of other players that have stopped before she does. Without loss of generality, we order the players according to their stopping times. If t i denotes the stopping time for player i, we have t i -< ti+ ~ for i = 1, 2 , . . . , N - 1. Note that the inequality is weak: we do not rule out the possibility that two or more players stop simultaneously. For any t, define y ( t ) to be the number of players who have stopped before time t. That is, let 0, y(t) =
max{jltj
for t --
T h e n n(t) - N - y ( t ) denotes the number of players who have n o t stopped by time t. Given an arbitrary sequence of stopping times t = { t l , t 2 , . . , tN} , and the rate of interest p > 0, the payoff to player i equals Pi(ti, t - i ) = e x p ( - p t i ) T r , ( t i ) .
This specification implies that if two or more players stop simultaneously they receive identical payoffs. Also, given that the interest rate is positive, lim,i~=Pi(t i, t _ i ) = 0. The game is stationary in the sense that the current value of the payoff, 7r, depends only on the n u m b e r of players that have not stopped earlier, and does not vary with the stopping time. For the game to be a war of attrition, we need Assumption
1. exp(-pA)rr,_ 1 > ~-n -- 0, for n = 2 , . . . , N.
In order to capture the essential characteristic of a war of attrition, namely that those who stop after others do better, it would be more appropriate to require only that rrn > 7rn+~. However, in our discrete time version of the game, the earliest that one player can stop after another entails a lag A. Our assumption guarantees that the time discount associated with this lag, e x p ( - p A ) , is sufficiently small so as not to disturb the essential feature of the war of attrition. Note that in the limiting case that is of principal interest, namely where A becomes arbitrarily small, this discount factor approaches unity. Furthermore, we assume that 7rn -> 0 to ensure that the payoff is indeed decreasing over time. If, for instance, 7rn < 0, a player will choose never to stop since this minimizes the loss. We consider the Markov Perfect Equilibria (MPE) of this game. As shown by Maskin and Tirole (1988), for instance, games in which the previous history at any stage influences current play only through its effect on some 'state' variables can be analysed in terms of Markov
S. Kapur / Economics Letters 47 (1995) 149-154
151
strategies. These are mappings from the set of all possible states to (the space of probability distributions over) the set of feasible actions. The state is a summary of the payoff-relevant history, where what is relevant and what is not depends, of course, on the specification of the game. In this game the relevant aspects of the history are the elapsed time t, and the number n(t) of players who are currently active. Let the pair In(t), t] define the state. It is analytically convenient to decompose the game into a sequence of stage games, where a stage is defined to coincide with the time interval over which n(t) remains unchanged. Within each stage game the number of players is fixed so that the elapsed time t provides a complete description of the relevant history in that stage game. Loosely speaking, the stage games correspond to subgames in the original game, and if we can specify Nash equilibrium strategies for each stage game, these taken over all possible stage-games would constitute a MPE of the original game. Let Ga(n ) denote the nth stage game, parameterized by the size of the decision interval a. Assume that the stage game begins at t,, and that it ends as soon as at least one player stops. Let the players active in this stage game be indexed by j, j = 1, 2 , . . . , n. Each active player must choose a stopping time t E {tn, t, + A, t, + 2 A , . . . } . Behaviour strategies can be defined for the stage game as follows. For player j, let/3a.j(n, t) denote the probability of stopping at time t, conditional on reaching period t. We drop the subscript A where it does not affect the argument. The time profile of the (conditional) stopping probabilities amounts to a behaviour strategy in the stage game. Notice, however, that due to the assumed stationarity, whenever time t + a is reached, the game looks the same as it did at time t. Consequently, any Nash equilibrium involves only stationary strategies so that, in equilibrium,/3j(n, t) =/3 7 (n) for all t. A Nash equilibrium for the stage game is written as [/3~(n),/3 2 ( n ) , . . . , / 3 * ( n ) ] . This game has a multiplicity of equilibria. We do not attempt a complete characterization of these but notice that a strategy combination that involves exactly one player choosing 'stop immediately', and the others choosing 'never stop' in response, is a Nash equilibrium in the stage game. This is so because the players that never stop at this stage obtain the value of the continuation stage game, which is at least exp(-pa)cr,_ 1- Given our assumption, this is higher than ~-,, the payoff from stopping in the stage game. And, clearly, if all other players choose 'never stop', it makes sense for the first one to stop immediately. Next, we consider the symmetric mixed-strategy Nash equilibria for the stage game. To review the familiar intuition of the mixed-strategy equilibrium in this context, in a war of attrition a player delays the stopping decision in the hope that some other player would stop first, thereby leading to a higher payoff. However, per se, the delay is costly because players discount the future. At the mixed-strategy equilibrium, the randomization is such that these two effects balance each other and each player is indifferent between alternative stopping times. We extend this intuition to the n-player stage game. Not surprisingly, the value of the equilibrium randomization depends on A, but the limiting case as A---~0 affords explicit results. P r o p o s i t i o n 1. S u p p o s e A s s u m p t i o n 1 holds f o r s o m e n > 1. L e t the s y m m e t r i c m i x e d strategy equilibrium be given as [/3a*(n), fl] (n), . . . , /3 a (n)]. Then, we have lim /3 a*(n) _ (n - 1) _
.o)
=
S. Kapur / Economics Letters 47 (1995) 149-154
152
Proof. The argument is in distinct stages. We first fix A and show that a unique non-trivial
mixed-strategy equilibrium exists. Then we examine the limiting case as a tends to 0. Fix a for the moment so that the symmetric probability that a typical rival player will stop at time t is written as /3. For randomization to be an equilibrium strategy, a player must be indifferent between the stopping times in support of her randomization. In particular, in this infinite-horizon game, a player must obtain the same expected payoff if she stops at any time t, or at t + a. We require that
[
e-OtTr. = e p(t+A) ( 1 __ /3)n--13Tn ..l_ E
(k-1)/3k(1 __ [ ~ ) n - l - k c k , _ k
k=l
]
,
(1)
where, to define ~b,_k, a little explanation is needed. The left-hand side is merely the value of stopping at time t; the right-hand side represents the consequences of waiting for exactly one additional period prior to stopping. It is the probability-weighted sum of being either in the situation where no player stops at time t (so that our reference player gets the same return as if she stopped at t, but discounted additionally for the lag A), or of being in the situation where k players have stopped at t. Here, ~b,_k is the current value of the continuation stage game that begins at t + A if k players stop simultaneously at t. The rest of the proof is in three steps. First, we show that, given Assumption 1, there exists a value of/3* in the open interval (0, 1) for which Eq. (1) holds. Then it is argued that the value Cn-k of the continuation game n - k equals 7r,_ k, at least if we confine attention to the symmetric mixed strategy equilibria in each stage game. This allows us, inter alia, to argue that the solution/3* is unique. Lastly, we show that, as A---~0, Eq. (1), which is of degree n - 1 in /3, is approximated by a linear equation in the neighbourhood of/3". This is then solved for a ( n ) . Lemma
1. c~1 = 7r~.
Proof. Once all players but one have stopped, the last player faces a single-agent optimization
problem. The best it can do is to stop immediately, which yields ~'1. To obtain a solution to Eq. (1) in the unit interval, we rewrite it as
[(1 - / 3 1 "-I
-
e
pA
n-I
1"1"i" n --~ E
k=l
( kn -)1/ 3
k (1 _ ~1,2l, ]n - l - k ~). - k = 0 "
(2)
This is of the form g(/3) = O, where g(/3) is a (n - 1)th degree polynomial in/3. L e m m a 2. For A > 0, there exists a / 3 * E (0, 1) such that g ( / 3 * ) = O. Proof. For A > 0 ,
we have g ( 0 ) = ( 1 - eP'a)Tr, < 0 , and g ( 1 ) = z r 1 -ePa~r, > 0 ; the second relation uses Lemma 1 and Assumption 1. By continuity, there exists a/3* E (0, 1) such that g(/3*) = 0 . L e m m a 3. c~n_k = Ir -k f o r all k.
S. Kapur / Economics Letters 47 (1995) 149-154
153
Proof. The existence argument of L e m m a 2 holds for an arbitrary n. In the subsequent stage games, the equilibrium strategy is either 'stop immediately' or it involves a randomization over all possible stopping times. In either case, the expected payoff equals ~n-~" L e m m a 4. /3* is unique. Proof. Using L e m m a 3, and differentiating g(/3) with respect to/3, we get g'(/3 ) = (n __
[~.~n--2R_0
__ ]J )
[ 71"n_ k _ 1 -- ,Wn _ k
1]
.
By assumption, [Tr,_k_ 1 -- 7rn_k] > 0 for all k. Hence, g(/3) is strictly monotonic in the unit interval. The claim follows. We now consider what happens to this solution as the decision interval shrinks. Notice that g(/3a) = 0 is of the form K n _ l / 3 n-1 + Kn_2/3 n-2 + . . " + g l / 3 + Ko(ZI ) = O,
(3)
where, in particular, K 1 = ( n - 1)(Tr,_ 1 - 7 r , ) ; and K0(za) = ( 1 - e°a)zr, is the only coefficient that varies with a. As a---> 0, K0(a ) goes to zero and therefore so does/3,]. However, it is easy to check by L'Hopital's rule that their ratio is well-defined. Therefore we can divide Eq. (3) by K0(Zi), and obtain Kn_l(~o
) n - 1 (K0) "-2 + . . . + K 2( ~ o ) 2
+ K, (--'~o) K 0 + 1 = O.
(4)
As K0(A)---~ 0, all but the last two terms on the LHS vanish, and the equation reduces to K1
( 0t
+ 1 = 0,
or that/3,] -
K1
Taking limits we have,
/3]
1
p~'.
a(n) =laim° A - (n - 1) ~r,_, - 7r, as stated.
[]
3. Comments The limiting case described here corresponds to the situation in which, at the equilibrium, each firm chooses an exponential distribution over stopping times. The parameter a(n) has the usual interpretation of a hazard rate, and here this rate is found to be increasing in the cost, p'nn, of continuing in the game for one additional period and decreasing in the incremental gain from delaying the stopping decision by exactly one stage. Clearly, the two-player game is
154
s. Kapur / Economics Letters 47 (1995) 149-154
a special case of this general form. The greater generality of the N-player game m a y be of s o m e use in understanding multi-agent strategic interaction, especially w h e r e the additional restrictions used here are not particularly inconvenient.
Acknowledgements I thank J D u t t a for comments.
References Maskin, E. and J. Tirole, 1988, A theory of dynamic oligopoly I: Overview and quantity competition with large fixed costs, Econometrica 56, 549-570. Maynard Smith, J., 1974, The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology 47, 209-221.