Discussion Paper No.313

A Dynamic Model of Asymmetric Contest
Akio Matsumoto
Chuo University

Ferenc Szidarovszky
Corvinus UniversityApril 2019

INSTITUTE OF ECONOMIC RESEARCH
Chuo UniversityTokyo, JapanA Dynamic Model of Asymmetric Contest∗
Akio Matsumoto†

Ferenc Szidarovszky‡

Abstract
Asymmetric contest games are examined under conditions which guarantee the existence of a unique pure Nash equilibrium. Conditions arederived for the local asymptotical stability of the equilibrium under continuous and discrete dynamics with gradient adjustments. In both cases,a crucial assumption is the nonexistence of a dominant player at the equilibrium level. In the case of continuous time scales, this is suﬃcient forstability, and in the discrete case, the speeds of adjustments have to be
suﬃciently small. As special cases, symmetric and semisymmetric games
are analysed in details.
Keywords: Asymmetric contests, Dynamic models, Discrete dynamics, Continuous dynamics, Stability analysis∗ The ﬁrst author highly acknowledges the ﬁnancial supports from the Japan Society for
the Promotion of Science (Grant-in-Aid for Scientiﬁc Research (C) 16K03556) and ChuoUniversity (Grant for Special Research). The usual disclaimers apply.
† Professor, Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji,Tokyo, 192-0393, Japan; akiom@tamacc.chuo-u.ac.jp‡ Professor, Department of Mathematics, Corvinus University, Budapest, Fövám tér 8,1093, Hungary; szidarka@gmail.com

1

1

Introduction

Contest games model situations when the players invest in order to increase
the probability of winning a given prize. Contest games are closely related tooligopolies with hyperbolic price functions (Bischi et al., 2010), market shareattraction games (Hanssens et al., 1990), rent seeking games (Tullock, 1980), tomention only a few. Many studies have examined contest and related games withexogenous prize, Pérez-Castrillo and Verdier (1992), Szidarovszky and Okuguchi(1997), Cornes and Hartley (2005) and Yamazaki (2008).
However in many cases (such as R&D contest, war, armament) the size ofthe prize depends on the eﬀorts of the players. For example, higher eﬀorts makemore valuable prize available to the players. Chung (1996) was the ﬁrst who
examined rent-seeking games with an endogenous prize (rent in this case) which
was assumed to be an increasing function of the aggregate eﬀort of all players.
For this case, Okuguchi (2005) and Corchón (2007) proved the existence of aunique symmetric pure Nash equilibrium. In the model developed by Shaﬀer(2006), an increased eﬀort has a decreasing eﬀect on the value of the prize. In
these studies identical players were assumed in the valuation of the prize as well
as abilities.
The heterogeneity of players can be divided into three types: in diﬀerent
valuations of the prize (e.g., Hillman and Riley, 1989), in diﬀerent abilities toconvert higher expenditures to higher productivity (e.g., Baik, 1994) and also
in diﬀerent ﬁnancial constraints (e.g., Che and Gale, 1997).
Szidarovszky and Okuguchi (1997) proved the existence of a unique pure
Nash equilibrium in special asymmetric rent-seeking games. This result waslater generalized for contest games by Hirai (2012) and Hirai and Szidarovszky(2013).
Dynamic rent-seeking games were ﬁrst examined in Okuguchi and Szidarovszky(1999) where the local asymptotic stability of the pure Nash equilibrium wasproved by using linearization around the equilibrium. This result was reconsidered to discrete time scales and the global stability of the equilibrium wasanalyzed in both discrete and continuous time sales in Bischi et al. (2010). In
these studies instantaneous information was assumed about the actions of thecompetitors as well as on available data and expectations of the players. Timedelays were introduced in many variants of the oligopoly model including thecases of hyperbolic price function in Matsumoto and Szidarovszky (2018).
In this paper the model of Hirai and Szidarovszky (2013) is reconsidered byintroducing its dynamic extensions in both discrete and continuous time scales.
The stability analysis is more general than in earlier studies. The paper isdeveloped as follows. In Section 2, the basic model is described. Sections 3 and
4, discuss the stability properties of the pure Nash equilibrium with continuousand discrete time scales. Conclusions and future research directions are given
in Section 5.

22The Basic Model

There are n players in a contest game, which are risk-neutral. If xi denotes theexpenditure of player i and ϕi (xi ) is its production function for lotteries, thenthe probability that player i wins the prize ispi =ϕi (xi ).
nj=1 ϕj (xj )Let Li denote the budget of player i implying that its set of feasible strategies
is the closed interval [0, Li ]. In Hirai and Szidarovszky (2013) the following
assumptions were made:Assumption 1. For all players i, function ϕi is twice diﬀerentiable,
ϕi (0) = 0, ϕ′i (xi ) > 0 and ϕ′′i (xi ) < 0 for xi ∈ [0, Li ].
The special form of ϕi (xi ) = ai xi was studied earlier by several authors(e.g., Skaperdas, 1996, Clark and Riis, 1998). Introducing the new variables
yi = ϕi (xi ) which represent the eﬀective eﬀort of player i, the expected payoﬀof player i can be given as follows:Πi = Ri (yi + Qi )whereyi− gi (yi )yi + Qi(1)nyj , gi (yi ) = ϕ−1i (yi )Qi =j=i

and the prize as the function of the aggregate eﬀort isRi (yi + Qi ).
Notice that Assumption 1 implies thatgi (0) = 0, gi′ (yi ) > 0 and gi′′ (yi ) > 0 for all yi ∈ [0, ϕi (Li )].
Function gi (yi ) can be considered as the total cost of player i to generate level
yi of eﬀort. Letn

Q=yi
i=1

be the aggregate eﬀort of all players.
Assumption 2. For all players i, the prize Ri (Q) is twice diﬀerentiable,nRi (Q) > 0 and Ri′′ (Q) ≤ 0 for all Q ∈ [0,ϕi (Li )].
i=13

This assumption allows both positive and negative externalities of the aggregate eﬀort as the linear function
Ri (Q) = ai + bi Q
shows with bi > 0 and bi < 0. Notice in addition that Assumptions 1 and 2 imply
that Πi (yi .Qi ) is strictly concave in yi . Hirai and Szidarovszky (2013) provedthat under Assumptions 1 and 2, there is a unique pure Nash equilibrium. The
asymptotic properties of this equilibrium will be examined in the next sections.

3

Continuous Dynamics

Considering gradient adjustments, we notice ﬁrst that by diﬀerentiation,
yiQi∂Πi= Ri′ (yi + Qi )+ Ri (yi + Qi )− gi′ (yi ),∂yiyi + Qi(yi + Qi )2(2)

which will be denoted by fi (yi , Qi ) for notational simplicity. Then the gradient
dynamics is described by the following system of ordinary diﬀerential equations:ẏi = Ki fi (yi , Qi ) (i = 1, 2, . . . , n)(3)where Ki > 0 is the speed of adjustment of player i. The local asymptotic
behavior of the equilibrium can be examined by linearization. Notice that′′′′∂fi(Ri yi + Ri )(yi + Qi ) − Ri′ yi Ri Qi (yi + Qi )2 − 2Ri Qi (yi + Qi ) ′′=+−gi (yi ),∂yi(yi + Qi )2(yi + Qi )4(4)and with j = i,′′′′∂fi
R yi (yi + Qi ) − Ri yi(R Qi + Ri )(yi + Qi )2 − 2Ri Qi (yi + Qi )= i+ i. (5)2∂yj(yi + Qi )(yi + Qi )4
Let Si and Ti denote the right hand sides of (4) and (5), respectively. Noticeﬁrst that
Si = Ti + ui(6)withui =′′Ri′ Q − Ri− gi .
Q2(7)We will now prove that ui < 0. Consider function′hi (Q) = Ri (Q)Q − Ri (Q).
′′Clearly, hi (0) = −Ri (0) ≦ 0 and h′i (Q) = Ri (Q)Q ≦ 0, therefore hi (Q) ≦ 0 for′′all Q ≧ 0. Since gi (yi ) > 0, the value of ui is always negative.

4

Next we will ﬁnd conditions which guarantee that Ti ≦ 0, implying that
Si < 0. Its numerator can be rewritten as′′′Ri yi Q2 − Ri yi Q + Ri′ Qi Q + Ri Q − 2Ri Qi′′′= Ri yi Q2 − Ri Q(yi − Qi ) + Ri (Q − Qi ) − Ri Qi′′′= Ri yi Q2 + (Qi − yi )(Ri Q − Ri ).
′′′Since Ri ≦ 0 and Ri Q − Ri ≦ 0, the value of Ti is less than or equal to zero if
Qi ≧ yi for all i, that is, there is no dominant player.
The Jacobian of system (4) at the equilibrium has the special formK1 S1 K1 T1 . . . K1 T1
 K2 T2 K2 S2 . . . K2 T2 JC =  .
(8)..
..  .
 ..
.
. Kn TnKn Tn. . . Kn Sn

where Si and Ti are now at their equilibrium levels.
By introducing

K1 T1


D = diag(K1 u1 , . . . , Kn un ), a =  ...  and 1T = (1, . . . , 1).
Kn Tn

Here ui denotes its equilibrium level. We have

J C = D + a1T(9)with characteristic polynomial
ϕC (λ) = det(D + +a1T − λI)= det(D − λI) det(I + (D − λI)−1 a1T )n= Πni=1 (Ki ui − λ) 1 +Ki TiKi ui − λ
i=1where we used the well known fact that if u and v are n-element column vectors
and I is the n × n identity matrix, then det(I + uv T ) = 1 + v T u, (Okuguchiand Szidarovszky (1999)). The potential eigenvalues are the Ki ui < 0 quantities
and the roots of equationnKi Ti+ 1 = 0.
Ki ui − λi=1(10)Let kC (λ) denote the left hand side, then clearly
lim kC (λ) = 1,λ→±∞lim

λ→Ki ui −0kC (λ) = −∞,5lim

λ→Ki ui +0kC (λ) = +∞andn′(λ) =kCKi Ti<02(Ki ui − λ)i=1unless all Ti = 0. In this case the eigenvalues are Ki Si < 0 for all i. The poles ofkC (λ) are the Ki ui < 0 values. Figure 1 shows the graph of kC (λ) by assumingthat n = 4 and the players are numbered so that K1 u1 < K2 u2 < . . . < Kn un .
If some of the Ki ui values are equal, then the same argument can be used withminor modiﬁcations.

Figure 1. n = 4 and K1u1 < K2u2 < K3u3 < K4u4
Notice that (10) is equivalent to a polynomial equation of degree n. There is
one root before K1 u1 , and one root inside each interval (Ki ui , Ki+1 ui+1 ) fori = 1, 2, . . . , n − 1. So we found n real negative roots. So all eigenvalues arenegative implying the following result.
Proposition 1 The equilibrium is always locally asymptotically stable with continuous time dynamics (3), if no dominant player is present.

4Discrete Dynamics

In the case of discrete time scales system (3) is modiﬁed asyi (t + 1) = yi (t) + Ki fi (yi (t), Qi (t))(11)

J D = I + J C = I + D + a1T .

(12)with Jacobian matrix

6

The characteristic polynomial has the formϕD (λ) = det(I + D + a1T − λI)= det(I + D − λI) det(I + (I + D − λI)−1 a1T )n= Πni=1 (1 + Ki ui − λ) 1 +Ki Ti.
1 + Ki ui − λi=1The possible eigenvalues are 1 + Ki ui for i = 1, 2, . . . , n and the solutions ofequation
nKi Ti+ 1 = 0.
(13)kD (λ) =1+Ki ui − λi=1Similarly to function kC (λ) it is easy to see thatlim kD (λ) = 1,λ→±∞lim

λ→1+Ki ui −0kD (λ) = −∞,lim

λ→1+Ki ui +0kD (λ) = +∞,n′(λ) =kD

Ki Ti≦ 0,2(1+Ki ui − λ)i=1

and the poles are the 1 + Ki ui values. The graph of function kD (λ) is shown inFigure 2. There is one root before 1 + K1 u1 and a root between each interval(1 + Ki ui , 1 + Ki+1 ui+1 ). All roots are therefore less than 1, and they are larger
than −1 if for all i, 1 + Ki ui > −1 and kD (−1) > 0.

Figure 2. Graph of function kD (λ)
These conditions can be written as
Ki <−2ui

for all i,7(14)andnKi Ti> −1.
2+Ki uii=1(15)when the denominator is positive by (14).
The left hand side of (15) is negative unless all Ti = 0, in which case theeigenvalues are the 1 + Ki ui values, which are between −1 and +1 if (14) holds.
Relation (15) can be interpreted as the speeds of adjustments have to be suﬃciently small.
Proposition 2 Assume that no dominant player is present. The equilibriumwith the discrete time dynamics is locally asymptotically stable if (14) and (15)hold.
In the symmetric case Ti ≡ T , ui ≡ u, Ki ≡ K, yi (0) ≡ y(0). Clearly there is
no dominant player, so the continuous dynamics is locally asymptotically stable.
Relations (14) and (15) are simpliﬁed as
K<and−2u(16)nKT> −1,2 + Ku

that is,−2,(17)nT + uwhich is the stability condition for the discrete time system.
Consider next the semi-symmetric case where for players i (1 ≦ i ≦ m),
K<Ti ≡ T, ui ≡ u, Ki ≡ K
and for players j (m + 1 ≦ j ≦ n),Ti = T̄ , uj ≡ ū and Kj ≡ K̄.
Then relation (14) has the forms
K<−2uand K̄ <−2ū(18)

and (15) can be rewritten as
mKT(n − m)K̄ T̄
+> −1
2 + Ku2 + K̄ ū
orK(2mT + 2u) + K̄(2(n − m)T̄ + 2ū) + K K̄(mT ū + (n − m)T̄ u + uū) > −4 (19)8The coeﬃcients of K and K̄ are negative with a positive coeﬃcient of K K̄.
This relation can be rewritten asK̄ K mT ū + (n − m)T̄ u + uū + 2(n − m)T̄ + 2ū> −4 − K (2mT + 2u)orK̄ (AK − B) > −4 + CK(20)with
A = mT ū + (n − m)T̄ u + uū,B = − 2(n − m)T̄ + 2ū
andC = − (2mT + 2u)
all being positive.
Relation (20) can be rewritten as
−4 + CK > AK − BK̄−4 + CK <AK − B

It is easy to show that

if AK − B < 0(21)if AK − B > 04B
4C<and< .
CA
BAMoreover−B
2
C2>
and- − > ,uAūAfurthermore point22− ,−u ū
is on the upper part of the hyperbola. So the stability region is the lower shaded

9area in Figure 3.

FIgure 3. Stability region in the (K, K̄) plane

5Conclusions

This paper examined the local asymptotical stability of the unique Nash equilibrium of asymmetric contest games with continuous and discrete time scales
under gradient adjustments. In both cases the critical condition was that thegame had no dominant player at the equilibrium level, which guaranteed thelocal asymptotical stability at the equilibrium in continuous dynamics. In thediscrete case, in addition, the speeds of adjustments had to be suﬃciently small
to guarantee local asymptotical stability. Symmetric and semisymmetric cases
were analyzed in detail.
Further research can continue in two diﬀerent directions. First, the globalasymptotical stability of the equilibrium can be examined, and second, timedelays can be introduced into the models and an interesting question is to see
how the delays can inﬂuence the asymptotical behavior of the equilibrium.
Conﬂict of Interest: The authors declare that they have no conﬂict of interest.

10References
Baik, K. H. (1994) Eﬀort levels in contests with two asymmetric players, Southern Economic Journal, 61, 367-378.
Bischi, G-I., C. Chiarella, M. Kopel and F. Szidarovszky (2010) Nonlinear
Oligopolies, Stability and Bifurcations, Springer-Verlag, Berlin/Heidelberg.
Che, Y. K. and I. Gale (1997) Rent dissipation when rent seekers are budget
constrained, Public Choice, 92, 109-126.
Chung, T. Y. (1996) Rent-seeking contest when the prize increases with aggregate eﬀorts. Public Choice, 87, 55-66.
Okuguchi, K. (2005) Existence of Nash equilibrium in endogeneous rent-seeking
games, in A.S. Nowak and S. Krzysztof (eds), Advances in Dynamic Games;
Applications in Economics, Finance, Optimization and Stochastic Control,
Birkhäuser, Boston.
Okuguchi, K. and F. Szidarovszky (1999) The Theory of Oligopoly with Multiproduct Firms (2nd ed.), Springer-Verlag, Berlin/Heidelberg.
Clark, D. J. and C. Riis (1998) Contest success functions: A survey, Economic
Theory, 11. 201-204.
Corchón, L. C. (2007) The theory of contests: A survey, Review of Economic
Design, 11, 69-100.
Cornes, R. and R. Hartley (2005) Asymmtric contests with general technologies, Economic Theory, 26, 923-946.
Hanssens, D. M., L. J. Parsons and R. L. Schultz (1990) Market Response
Models: Economometric and Time Series Analysis, Kluwer, Dordrecht.
Hillman, A. L. and J. C. Riley (1989) Political contestable rents and transfers,
Economic Politics, 1, 17-39.
Hirai, S. and F. Szidarovszky (2013) Existence and uniqueness of equilibrium inasymmetric contests with endogenous prices, International Game Theory
Review, 15(1), doi:10.1142/S0219198913500059.
Hirai, S. (2012) Existence and uniqueness of pure Nach equilibrium in asymmetric contests with endogeneous prizes, Economic Bulletin, 32, 27442751.
Matsumoto, A. and F. Szidarovszky (2018) Dynamic Oligopolies with Time
Delays, Springer-Verlag, Singapore.
Pérez-Castrillo, J. D. and T. Verdier (1992) A general analysis of rent-seeking
games, Public Choice, 73, 335-350.

11

Shaﬀer, S. (2006) War, labor tournamerts and contest payoﬀs, Economic Letters, 92, 250-255.
Skaperdas, S. (1996) Contest success funcions, Economic Theory, 7, 283-290.
Szidarovszky, F. and K. Okuguchi (1997) On the existence and uniquenessof pure Nash equilibrium in rent-seeking games, Games and Economic
Behavior, 18, 135-140.
Tullock, G. (1980) Eﬃcient rent-seeking games, in M. Buchanan, R. D. Tollison
and G. Tullock (eds), Toward a Theory of the Rent-Seeking Society, Texas
A&M, College Station.
Yamazaki, T. (2008) On the existence and uniqueness of pure-strategy Nashequilibrium in asymmetric rent-seeking contests, Journal of Public Economics Theory, 10, 317-327.

12