Economics
Letters
0165-1765/94/$07.00
45 (1994) 181-183
0 1994 Elsevier
181
Science
B.V. All rights
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A simple extension of the DasguptaMaskin existence theorem for
discontinuous games with an
application to the theory of
rent-seeking
Chun-Lei Yang*
Department of Economics,
Received
Accepted
28 July 1993
30 September
University of Dortmund,
44221 Dortmund,
Germany
1993
Abstract
I state that upper-semi-continuity
of the sum of payoffs is not necessary for the classical existence result in symmetric
games by Dasgupta
and Maskin. This is then applied to the theory of rent-seeking
to show that there always exists some
Nash equilibrium
in mixed strategies in the simultaneous-move
game by Tullock in which no ‘overdissipation
can occur.
JEL
classification: C72
Let r = (Ai, Ui)L=, denote an N-person, symmetric, normal form game where Ai C R’ compact
is the action space for player i and Vi: A : = $!=, Aj-+ R’ his payoff function, i = 1, . . . , N.
Symmetry requires Ai = Aj and Ui(u) = Uj(p,j(u)), where pii
E A is a permutation of a with
changes in the ith and jth components only, i, j = 1, . . . , N. Dasgupta and Maskin (1986,
Theorem 6) tell us that, under some conditions to be stated in the following lemma, if CL1 Ui(a)
is upper-semi-continuous
, then r possesses a symmetric Nash equilibrium which is atomless on
any discontinuity point. In this paper I will show that it is sufficient for their result to require that
the payoffs can be properly extended to meet the upper-semi-continuity
condition for the sum
wherever it was not satisfied in the original game. Formally, we can state the following simple
extension of Theorem 6 by Dasgupta and Maskin (hereafter D-M):
Lemma. Let (A,, Ui)E 1 be a game where A i c R’ compact, Vi bounded, and the set A**(i) c A =
ni”,, Ai of discontinuity points of U, is of the type required by D-M. Suppose there is a modified
which satisfies the requirements of
gume (Ai> ui);“=ly where ci(u) # U,(u) implies a E A**(i),
Theorem 6 of D-M, i.e. Ui(ui, a_,) be bounded and strictly weakly lower semi-continuous in ui
* I am grateful
SSDI
to Wolfgang
0165-1765(93)00392-2
Leininger
for detailed
comments
and suggestions.
182
C.-L.
Yang I Economics
Letters 45 (1994) 181-183
(Property (a), p. 19, D-M), and c y=, U, be upper-semi-continuous. Then, (Ai, U,);“=, possesses a
symmetric mixed-strategy equilibrium (t.~, . . . , k) such that p({a,}) = 0 whenever (ai, a_,) E A**(i)
for some a_, E A _i.
Remark. The set A**(i) of discontinuity points of Vi is, roughly speaking, a subset of some
continuous manifold in IWNthat is at most of dimension N - 1 (D-M, p. 7). This implies also that
A**(i) has a Lebesgue measure zero.
Proof. Theorem 6 by D-M ensures the existence of some symmetric equilibrium strategy
(I-&. . . >F) such that &{a,}) = 0 whenever (a,, a-,) E A**(i) for some api E A_i in the modified
game (Aj, U,)i. It remains to show that j.~ is i’s best response to a”_,= (p, . . . , /A) in the original
game (Ai, I!I~)~,too. Since UE, A**(i), the set of discontinuity points, has a Lebesgue measure
zero, i’s expected payoff has the property Ui(ai; a”_,) = U,(a,; Li)Vaj E Aj, which completes the
proof.
0
This quite simple extension enlarges the applicability of Theorem 6 by D-M significantly. We
apply it here to solve an extension problem that has upset rent-seeking theorists for a long time
[see Rowley et al. (1988)].
To discuss the resource-wasting consequences of rent-seeking activities for artificially contrived
rents (mostly induced by government activities), Tullock (1980) analysed the model (Ai, U,),“=,
with N potential beneficiaries where Aj = [w, denotes i’s action space and his payoff is
Ui(a,,.
. . ,aN)=
a:
LV-ai,
E;“=,ai
ifa,#Oforsome
0,
ifaj=O,Vj=l,...N.
j,
(1)
V denotes the value of the rent at issue and a, is contender
i’s (sunk) investment to influence its
assignment to his favour, which is represented by the probability pi = ajlC~=, ui. The fact that no
Nash equilibrium in pure strategies exists whenever r > NI(N - 1) caused Tullock to speak of an
‘intellectual swamp’ - he even pointed to the conjecture that ‘overdissipation’ may occur in those
cases.
It can be checked immediately that U, has a unique discontinuity point at the origin 0 E RT. AS
c;“= 1 Ui is not upper-semi-continuous
at 0, results by D-M (1986) cannot be applied here directly
to ensure the existence of a mixed-strategy Nash equilibrium. For the r = CCcase, which is well
known in the literature as the dollar-auction, Hillman and Samet (1987) found some Nash
equilibria where only two contenders are active using a random bid uniformly distributed over the
interval [0, V] with expected payoff of zero. Armed with the above lemma, we are able to show in
the present paper that for all r:
Proposition.
equilibrium.
The symmetric game (Ai, lJi)l”,l by Tullock
There is never ‘overdissipation’ in equilibrium.
always possesses
a symmetric
Nash
To prove this, we first observe that, since ai = 0 yields player i a payoff of zero, any bid a, > V
is strictly dominated, so that the set of Nash equilibria coincides for the games (A,, Ui)i and
(A,, Ui)i, where Ai, := [0,V]. Then let us modify the payoff function to
C.-L.
Yang I Economics
Letters 45 (1994) 181-183
183
ifu,#Oforsomej,
ifaj=O,Vj=l
,...,
N,
(2)
i.e. in the modified game the rent is split evenly among contenders,
if no bids occur, whereas in
the original one the rent can only be awarded after at least one positive bid. U,(resp.
ui) is
bounded
on A = flfi, Ai. From the lemma above, it remains to check that all conditions
of
Theorem
6 are satisfied for the modified symmetric game (Ai, I!?~),“=,.
Obviously,
c;“,, fl,(u) = V- ct I 1ai is upper-semi-continuous
in a = (a,, . . . , aN). Furthermore, Property (LX)in D-M (p.19) is satisfied as the origin is the only discontinuity
point for I!?,
= V/N. Hence, the above lemma tells us that the game by
and lim, _,, ~!?,(a,; 0, . . . , 0) = V > ai
Tullock ‘has a symmetric
Nash equilibrium
(p, . . . , p) in mixed strategies with no atom at the
zero bid.
Finally,
because
X:1 Ui(p, . . . , p) = V- c:, E,a, > 0, there can never be any expected
overdissipation,
which completes the proof.
As a final remark, we mention that the lemma here is still applicable as long a (1) is re-defined
so that U,(O) % V, i = 1, . . . , N. Baye et al. (1993) apply Theorem 6 of D-M to ensure existence of
equilibrium
in the modified game (Ai, c,)i. While they are concerned with analysis of the modified
game, we show how that relates to the original one.
References
Baye, M.R.,
D. Kovenock
and C. de Vries, 1993, The solution
to the Tullock
rent-seeking
game when R>2:
Mixed-strategy
equilibria
and mean dissipation
rates, mimeo.
Dasgupta,
P. and E. Maskin, 1986, The existence of equilibrium
in discontinuous
economic games, I: Theory, Review of
Economic
Studies LIII, l-26.
Hillman,
A.L. and D. Samet, 1987, Dissipation
of contestable
rents by small numbers of contenders,
Public Choice 54,
63-82.
Rowley, Ch.K., R.D. Tollison and G. Tullock, eds., 1988, The political economy of rent-seeking
(Kluwer,
Dordrecht).
Tullock, G., 1980, Efficient rent-seeking,
in: J.M. Buchanan,
R.D. Tollison and G. Tullock, eds., Toward a theory of the
rent-seeking
society (Texas A&M University
Press, College Station, TX) 97-112.