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+FRACTIONAL SAMPLING RATE CONVERSION IN THE 3RD ORDER
+CUMULANT DOMAIN AND APPLICATIONS
+
+Anastasios Delopoulos, Maria Rangoussi and Demetrios Kalogeras
+
+Computer Science Division,
+Department of Electrical Engineering,
+National Technical University of Athens,
+Athens GR-15780, GREECE
+e-mail: {adelo, maria, dkalo}@image.ece.ntua.gr
+
+ABSTRACT
+
+In a variety of problems a random process is observed at different resolutions while knowledge of the corresponding scale conversion ratio usually contains useful information related to problem-specific quantities. A method is proposed which exploits cumulant domain relations of such signals in order to yield fractional estimates of the unknown conversion ratio. The noise insensitivity and shift invariance property of the cumulants offers advantages to the proposed method over signal domain alternatives. These advantages are discussed in two classes of practical problems involving 1-D and 2-D scale converted signals.
+
+1. INTRODUCTION
+
+Fractional sampling rate (or scale) conversion of D-
+dimensional processes arises in a variety of signal pro-
+cessing contexts involving signals observed at two dif-
+ferent resolutions. The conversion of a signal from an
+original resolution to a lower one may be due either to
+the digital signal processing method employed (e.g.,
+deliberate downsampling) or to the sampling mech-
+anism employed for A/D conversion. As fractional
+numbers are dense in real numbers, fractional reso-
+lution conversion ratios can approximate arbitrarily
+close any resolution ratio. Rate conversion problems
+arise, for example, in pattern recognition or classi-
+fication applications where one has to compare in-
+coming signals, acquired at a given (test) resolution,
+to pre-stored data, acquired at a different (usually
+higher) reference resolution. The ratio of these two
+scales may be unknown. The processing required can
+be carried out in the signal domain itself; feature ex-
+traction or statistics estimation is normally employed,
+however, in order to move to a domain where com-
+parisons are possible and they can be performed at
+
+reduced computational cost.
+
+Third order cumulants are examined as a candi-
+date domain in the present work, as a special case
+of the more general k-th order cumulant domain, ad-
+dressed in [5]. Motivation for this provide the estab-
+lished *noise robustness* and *phase information reten-
+sion* properties of the third order cumulant, rendering
+it an attractive candidate either for feature selection
+or for statistics estimation. See, e.g., [1] and [8] for
+definitions, properties and applications of third order
+cumulants and bispectra. Specifically in this paper
+relations are established between the third order cu-
+mulants (and bispectra) of the signal at the high and
+the low resolution. These relations can be used either
+to estimate the third order cumulants of the low res-
+olution signal directly from the cumulants of the high
+resolution signal or to estimate the ratio of the two
+resolutions (scales), when unknown.
+
+The obtained relations are used in two representa-
+tive applications involving sampling rate conversions
+of 1-D and 2-D stochastic signals respectively.
+
+The first application estimates the velocity of a
+moving source by exploiting the Doppler effect. The
+moving object, whose velocity is sought, may either
+reflect an 1-D narrowband signal (e.g., a tone) trans-
+mitted by the detector, or independently emit a gen-
+erally wideband signal. The signal received at the
+detector site is a fractionally converted version of the
+original signal, and the fractional conversion ratio is
+related to the velocity of the target. Therefore, the
+target velocity can be obtained via estimation of the
+scale ratio between the original and the received ver-
+sions of the signal. Traditional methods rely on the
+assumption that the measured signals are strictly of
+narrowband nature (tones). Our approach focuses on
+the alternative assumption of stochastic signals, pos-
+sibly measured in low signal to noise ratio conditions.

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+In the second application, the distance from a textured surface is computed by comparing the resolution of a captured image to that of a reference image. A typical example of such problems is the estimation of the flight altitude of a plane from landscape images taken during flight. Resolution conversion in this case involves two 2 × 2 integer, non-singular decimalation and expansion matrices.
+
+## 2. THIRD ORDER STATISTICS RELATIONS
+
+Let $y(n)$ denote a rate-converted, discrete variable, $D$-dimensional process obtained from the original process $x(n)$ after fractional rate conversion:
+
+$$y(n) = \sum_{k} h(M_{D \times D}n - L_{D \times D}k) x(k), \quad (1)$$
+
+where the rate conversion is obtained through a "downsample by matrix $L_{D \times D}$ filter by $h(n)$ - expand by matrix $M_{D \times D}$" operation and $L_{D \times D}, M_{D \times D}$ are $D \times D$ integer, non-singular, commutative and coprime matrices, [2]. Of practical interest are the cases with $det(M_{D \times D}) > det(L_{D \times D})$, meaning that $y(n)$ is observed at a resolution lower than the original (reference) resolution of $x(n)$.
+
+Equation (1) covers the general case where resampling is possibly accompanied by rotation and azimuth changes (see [9], chap. 12). In [5] input-output cumulant expressions are established for the general case of equation (1) with full integer matrices $L_{D \times D}, M_{D \times D}$. For the purposes of the present work, we restrict ourselves to the special case of diagonal matrices $L_{D \times D} = L_{D \times D}, M_{D \times D} = M L_{D \times D}$, where $L, M$ are coprime integers. This corresponds to rescaling by the scalar factor $M/L$ alone, thus preserving the view point from which the signal is observed. This choice is made in order to simplify the mathematical notations, since here the focus is on the use of these relations for the estimation of the resampling ratio in the aforementioned applications. It is straightforward to rewrite the obtained relations, however, for the general case of general diagonal or full matrices $L_{D \times D}, M_{D \times D}$.
+
+Equation (1) is equivalent to a resampling of the continuous signal $x_c(t)$ from which $x(n)$ was originally obtained through sampling, provided that the filter $h(n)$ employed has an ideal lowpass transfer function $H(\omega)$ with gain $L$ and cutoff frequencies $\pi/M$ over all dimensions $d = 1, ..., D$.
+
+The third order cyclic cumulant of $y(n)$, defined as $[5]$
+
+$$\tilde{c}_{3,y}(m_1, m_2) \triangleq \frac{1}{L^D} \sum_{n \in [0..L-1]^D} c_{3,y}(m_1, m_2; n), \quad (2)$$
+
+where $c_{3,y}(m_1, m_2; n)$ is periodic in $(n)$ with period $[0..L-1]^D$, is related to $c_{3,x}(m_1, m_2)$ through the following equation:
+
+$$\tilde{c}_{3,y}(m_1, m_2) = \frac{1}{L^D} \sum_{s_1} \sum_{s_2} c_{3,x}(s_1, s_2) \\ \times h_3(Mm_1 - Ls_1, Mm_2 - Ls_2)(3)$$
+
+where $h_3(m_1, m_2) \triangleq \sum_n h(n)h(n+m_1)h(n+m_2)$ is the triple correlation of the decimation $D$-dimensional filter $h(n)$. See [3], [6] for definitions and properties of cyclic moments and cumulants.
+
+Equation (3) can be transformed to the frequency domain, to yield relations between the cyclic bispectrum of the low resolution signals and the bispectrum of the reference resolution signals, by exploiting the coprimeness of $L, M$:
+
+$$\begin{aligned} \tilde{C}_{3,y}(\omega_1, \omega_2) &\equiv \frac{1}{M^D} \sum_{l_1} \sum_{l_2} \sum_{\substack{n \\ 0 < l_1 l_2 < [0..M-1]^D}} \\ &\quad \times C_{3,x}(\frac{l_1\omega_1 + 2\pi L l_1}{M}, \frac{l_2\omega_2 + 2\pi L l_2}{M}) \\ &\quad \times H_3(\frac{\omega_1 + 2\pi l_1}{M}, \frac{\omega_2 + 2\pi l_2}{M}) \end{aligned} \quad (4)$$
+
+The RHS of equation (4) is a summation of frequency shifted replicas of $\tilde{C}_{3,x}(\omega_1, \omega_2)$, each replica shrunk by the scaling factor $M/L$. It is interesting to notice the resemblance that equation (4) bears to the corresponding input-output relation between the Fourier transforms of deterministic signals.
+
+## 3. ESTIMATION OF THE RESOLUTION CONVERSION RATIO
+
+The input-output relations given in the previous section allow for the computation of the resolution conversion ratio, $L/M$, provided that both $x(n)$ and $y(n)$ are available and the blurring mechanism $h(n)$ is known. The method proposed in the sequel for the computation of $L/M$ relies on matching the cumulants of the measured signal $y(n)$ to successive resolution-converted versions of the cumulants of the original signal $x(n)$. Although computationally demanding, this method is shown to converge to the true resolution conversion ratio.

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+Before proceeding to the description of the proposed method, it should be emphasized that addressing this problem in the third order statistics rather than the signal domain offers two advantages.
+
+1. Statistical similarity is exploited, which means that valid results are obtained even if $y(\mathbf{n})$ and $\mathbf{x}(\mathbf{n})$ correspond to different realizations of a stochastic process. This feature is of great importance in situations where the reference (high resolution) and the test (low resolution) signals are not acquired simultaneously. This is the case, e.g., with pattern classification applications that use pre-stored data.
+
+2. The well known immunity of the third order statistics to a wide class of additive noises, makes the proposed method appropriate for situations where only noisy data is available.
+
+### Proposed Method:
+
+step 1: Estimate the cumulant of the reference signal $\mathbf{x}(\mathbf{n})$, $\hat{c}_{3x}(\mathbf{m}_1, \mathbf{m}_2)$, and the cyclic cumulant of the test signal $y(\mathbf{n})$, $\hat{c}_{3y}(\mathbf{m}_1, \mathbf{m}_2)$. The asymptotically consistent estimator of the cyclic cumulant proposed in [3] can be employed. This estimator in the present set up takes the form
+
+$$ \hat{c}_{3y}(\mathbf{m}_1, \mathbf{m}_2) = \frac{1}{T_y} \sum_{t \in W_y} y(t)y(t+\mathbf{m}_1)y(t+\mathbf{m}_2), \quad (5) $$
+
+where $W_y$ is the set of all available data samples of signal $y(\mathbf{n})$, and $T_y$ is the cardinal number of $W_y$. Note that in practice the conventional estimator $\hat{c}_{3x}(\mathbf{m}_1, \mathbf{m}_2)$ for the (non-cyclic) cumulant of $\mathbf{x}(\mathbf{n})$ is implemented in the same way.
+
+step 2: Define a partition $\{q_n\}$, $n = 1, 2, \dots, N$ of the scale (resolution) interval $(0, 1]$ with $q_n$ chosen as fractional numbers $L_n/M_n$, where $L_n \le M_n$ and $(L_n, M_n)$ are coprime integers.
+
+step 3: For $n=1, 2, \dots, N$,
+
+1. Compute the triple correlation $h_3^{(n)}(\mathbf{m}_1, \mathbf{m}_2)$ of the resampling filter $h(\mathbf{n})^{(n)}$, which should have gain $L_n$ and cutoff frequency $\pi/M_n$.
+
+2. Estimate the cyclic cumulant $\hat{c}_{3y}^{(n)}(\mathbf{m}_1, \mathbf{m}_2)$ of $y^{(n)}(\mathbf{n})$, which is a resampled version of the reference signal $\mathbf{x}(\mathbf{n})$ at a sampling rate $M_n/L_n$ times lower than that of $\mathbf{x}(\mathbf{n})$.
+
+This estimator can be implemented as in equation (3), using $L_n, M_n, h_3^{(n)}$ in place of $L, M, h_3$.
+
+3. Compute the similarity index
+
+$$ f\left(\frac{L_n}{M_n}\right) \triangleq \sum_{\mathbf{m}_1, \mathbf{m}_2} |\hat{c}_{3,y}^{(n)}(\mathbf{m}_1, \mathbf{m}_2) - \hat{c}_{3,y}^{(n)}(\mathbf{m}_1, \mathbf{m}_2)|^2. \quad (6) $$
+
+step 4: Obtain an estimate of the conversion ratio $L/M$ as the point $q_n = L_n/M_n$ of the global minimum of index $f(q_n)$ over all $n = 1, 2, \dots, N$. The estimate of the conversion ratio $L/M$ can be drawn arbitrarily close to the true ratio $L/M$ by repeatedly refining the partition $q_n$ of $(0, 1]$. Local refinement can be used, in a process of zooming into the neighborhood of the initial the global minimum.
+
+### Comment:
+
+The similarity index $f(q_n)$ is in general a non-convex function of $q_n$. However, it has been observed that $f(q_n)$ exhibits a deep global minimum at $L/M$. Also in the neighborhood of the global minimum it assumes a convex form. Therefore, in the neighborhood of the global minimum the refinement process can be driven by fast minimization algorithms such as the Fibonacci and Golden Section methods, ([7]).
+
+# 4. APPLICATIONS
+
+Use of the proposed method for the estimation of the unknown sampling rate conversion ratio $L/M$ is investigated in the sequel, in an 1-D and a 2-D problem.
+
+## 4.1. Velocity estimation via Doppler effect
+
+A signal $x(t)$ reflected by an object moving with velocity $\mathbf{v}$ is observed at the source (detector) site as $y(t) = x(\alpha t)$, where $\alpha = \frac{c+v}{c}$ and $\mathbf{c}$ is the velocity of transmitted signal. This corresponds to a rescaling of the transmitted signal $x(t)$ by a factor $\alpha$. A similar relation with $\alpha = \frac{c-v}{c}$ holds in the case that the moving object itself is emitting the signal $x(n)$, rather than reflecting it.
+
+Conventional methods assume that $x(t)$ is a narrowband signal, usually a sinusoid, and measure the frequency shift between the transmitted and the received signals as a means to compute $\alpha$ and then $\mathbf{v}$. This narrowband assumption is not always met, either because of physical constraints of the emitters, or when transmission of sinusoidal signals is to be avoided for security reasons.

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+In such cases the method proposed here can be
+used, approximating a arbitrarily close by a fraction
+M/L. The proposed method offers the advantages of
+(i) being immune to a class of additive noises, as a
+result of its third order cumulant basis, and (ii) al-
+lowing the widespreading of the frequency contents of
+the transmitted signal, necessary under certain appli-
+cations.
+
+## 4.2. Relative distances from textured images
+
+Pictures of a textured surface, acquired from vary-
+ing distances, represent versions of the surface un-
+der different space scalings. The scale ratio between
+two such pictures (a reference and a test one) can be
+used to obtain the distance of the camera from the
+surface, for the test picture. This problem arises in
+various image processing tasks, as for example: (i)
+when landscape images are taken from different al-
+titudes, which are unknown at the processing time,
+(ii) in motion information extraction from video se-
+quences depicting objects that move towards/off the
+camera, (iii) in medical imaging application where
+tissue images are processed, etc.
+
+If $I(t)$ is the analog image and $x(t)$ the analog
+form of $I(t)$ observed from distance $d_x$, then $x(t) =
+I(\frac{d_x}{f} t)$, where $f$ is the focal distance of the camera.
+Assuming that the test picture $y(t)$ is taken from an
+unknown distance $d_y$, then $y(t) = I(\frac{d_y}{f} t) =
+x(\frac{d_y}{f} t)$. The proposed method can be used to approximate ar-
+bitrarily close the ratio $d_y/d_x$ by a fractional number.
+Therefore, if $d_x$ is known, $d_y$ can be computed.
+
+Comments:
+
+Scale registration methods applied directly to the
+image rather than to the cumulant domain can be
+also used along the lines of the proposed method. The
+advantages of using cumulant statistics, though, are:
+
+1. The fact that statistical similarity is exploited allows to obtain valid distance estimates even when the test and the reference images do not depict the same surface, provided that they correspond to images possessing similar statistical structure.
+
+2. The shift invariance of the cumulant domain allows for the comparison of images that are not necessarily aligned in space.
+
+Possible generalization of the algorithm to address problems involving general diagonal or non-diagonal decimation and interpolation matrices *L*, *M* will make the comparisons insensitive to azimuth / rotation changes.
+
+5. CONCLUSIONS
+
+A cumulant based method for the estimation of frac-
+tional scale conversion ratio of signals is proposed in
+the present work. Expressions relating the higher-
+order statistics of signals observed at two different
+resolutions (scales) are quoted and used in a prac-
+tical algorithm for scale conversion ratio estimation.
+Application of the proposed algorithm in problems
+involving 1-D and 2-D signals are outlined, and rela-
+tive merits of the proposed approach due to the use
+of higher order statistics are discussed.
+
+6. REFERENCES
+
+[1] D.R.Brillinger, "Time series: Data Analysis and Theory," McGraw-Hill, 1981.
+
+[2] T.Chen, P.P.Vaidyanathan, "Commutativity of D-Dimensional Decimation and Expansion Matrices, and Application to Rational Decimation Systems," Proc. ICASSP'92, vol. IV, pp. 637-640, San Francisco, USA, March 1992.
+
+[3] A.V.Dandawate, G.B.Giannakis, "Asymptotic theory of mixed time averages and kth-order cyclic moment and cumulant statistics," IEEE Trans. on Information Theory, vol. 41, pp. 216-232, January 1995.
+
+[4] A.Delopoulos, S.Kollias, "Optimal Filterbanks for Signal Reconstruction from Noisy Subband Components," IEEE Trans. on Signal Proc., vol. 44, no. 2, pp. 212-224, Feb. 1996.
+
+[5] A.Delopoulos, M.Rangoussi, "Cumulant relations of multidimensional processes observed at rationally related resolutions," Proc. Int'l Workshop on Sampling Theory and Applications, Aveiro, Portugal, June 1997 (to be presented).
+
+[6] G.B.Giannakis, G.Zhou, "Parameter estimation of cyclostationary amplitude modulated time series with application to missing observations," IEEE Trans. on Signal Processing, vol. 42, no. 9, pp. 2408-2419, Sept. 1994.
+
+[7] G.Karmanov, "Mathematical Programming." Mir Publishers, Moscow, 1987.
+
+[8] J.Mendel, "Tutorial on Higher-Order Statistics (Spectra) in Signal Processing and System Theory: Theoretical Results and some Applications," Proc. of the IEEE, vol. 79, no. 3, pp. 278-305, March 1991.
+
+[9] P.P.Vaidyanathan, "Multirate Systems and Filter banks," Prentice Hall, Englewood Cliffs, New Jersey. 1993.

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+Onset of chaotic phase synchronization in complex networks of coupled heterogeneous oscillators
+
+Francesco Ricci,¹ Roberto Tonelli,² Liang Huang,³,⁴ and Ying-Cheng Lai⁴,⁵
+
+¹Department of Physics, University of Cagliari, I-09042 Monserrato, Italy
+
+²Dept. of Electrical and Electronic Engineering, University of Cagliari, piazza d'Armi, I-09123 Cagliari, Italy
+
+³Institute of Computational Physics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
+
+⁴School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
+
+⁵Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
+
+(Received 25 February 2012; published 14 August 2012)
+
+Existing studies on network synchronization focused on complex networks possessing either identical or nonidentical but simple nodal dynamics. We consider networks of both complex topologies and heterogeneous but chaotic oscillators, and investigate the onset of global phase synchronization. Based on a heuristic analysis and by developing an efficient numerical procedure to detect the onset of phase synchronization, we uncover a general scaling law, revealing that chaotic phase synchronization can be facilitated by making the network more densely connected. Our methodology can find applications in probing the fundamental network dynamics in realistic situations, where both complex topology and complicated, heterogeneous nodal dynamics are expected.
+
+DOI: 10.1103/PhysRevE.86.027201
+
+PACS number(s): 05.45.Xt, 89.75.Hc
+
+Synchronization in complex networks has been a problem of continuous interest [1–10]. For networks having identical nodal dynamics, e.g., a network of coupled identical oscillators, the question of whether the network is synchronizable can be answered through the approach of master-stability function (MSF) [11,12]. In particular, due to the fact that the nodal dynamics are all identical, synchronization among all nodes in the network is a mathematical solution of the system, defining a synchronization manifold in the phase space of the whole system. If the solution is stable with respect to perturbations in all subeigenspaces that are transverse to the manifold, synchronization is physically observable or realizable. The MSF, an invariant property determined solely by the nodal dynamics, provides a computationally feasible way to determine the transverse stabilities of the synchronization manifold in terms of the network structure. This approach has been applied to analyzing the synchronizability of various complex networks, such as small-world networks [2–4], scale-free networks [5,6], weighted complex networks [7], adaptive complex networks [8], complex clustered networks [9], and complex gradient networks [10]. Particularly, in Ref. [6] the authors found that on the example of coupled Rossler systems, that small network size *N* and dense network connection favor synchronization. The approach has also been extended to situations where the nodal dynamics are slightly nonidentical [13].
+
+In real-world networks heterogeneity in the nodal dynamics is expected. In such a case, an exact synchronization manifold cannot be defined. To probe into the fundamental synchronization dynamics for heterogeneous networks, a viable approach is to reduce the complexity of the nodal dynamics. In this regard, the Kuramoto model [14,15] has been studied extensively in which the nodal dynamics is given by that of a uniform rotation: $\dot{\theta} = \omega$, where $\theta$ is a phase variable and $\omega$ is the frequency. Heterogeneity can be modeled by assuming that the frequency for each node is distinct and can be drawn from a random distribution. For complex networks hosting the Kuramoto phase dynamics, transition to synchronization can be understood fairly comprehensively [16,17]. For example, say *K* is a general coupling parameter. As *K* is increased
+
+through a critical value $K_c$, partial synchronization in the network in the form of synchronous clusters can arise, where $K_c$ is determined by the network topology and the frequency distribution of the oscillators. Transition to complete or global synchronization, where all nodes in the network are synchronized, has also been investigated [18]. In other systems of heterogeneous dynamics, like two-dimensional nonidentical neuron-like maps, phase synchronization has been investigated [19].
+
+Despite vast literature on complete synchronization of identical chaotic oscillators in complex dynamical systems, the problem of phase synchronization in complex networks with heterogeneous nodal dynamics that are more complicated than uniform rotation has remained to be outstanding. The specific question that we ask is, if the nonidentical individual nodal dynamics are chaotic, what determines the transition to synchronization? Since the nodal dynamics are not identical, chaotic phase synchronization is expected to arise [20], especially in the weakly coupling regime. To be concrete and realistic, we shall then focus on the onset of chaotic phase synchronization in complex networks, with the goal to uncover scaling relation of the critical coupling strength $K_c$ required for this type of synchronization to occur in the entire network. A heuristic analysis reveals an algebraic scaling relation between $K_c$ and a parameter characterizing the link density of the network, indicating that as the network becomes more densely connected, the threshold coupling value required for chaotic phase synchronization decreases. To verify the scaling relation, we develop a computational procedure to accurately determine $K_c$ for large networks in an extremely efficient manner. Our result has the following significance. In complex dynamical systems complete synchronization is difficult to be realized, as enormous coupling may be required. However, phase synchronization, a weaker type of synchronization, can be typical and finds applications in many areas of significant interest such as epileptic seizures [21]. The scaling relation uncovered in this paper can be used to assess, for complex networks of given topology, size, and linkage, when phase synchronization can be expected.

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+We consider networks of coupled, nonidentical oscillators, mathematically described by
+
+$$
+\frac{d\mathbf{x}_i}{dt} = \mathbf{F}_i(\mathbf{x}_i) - \epsilon \sum_{j=1}^{N} G_{ij} \mathbf{H}(\mathbf{x}_j), \quad (1)
+$$
+
+where N is the number of oscillators (nodes) in the network, **x**<sub>i</sub> is a d-dimensional vector of the dynamical variables of node i, **F**<sub>*i*</sub>(**x**<sub>*i*</sub>) is the vector field of node *i*, *ε* is a coupling parameter, and **G** is a coupling matrix determined by the connection topology. The elements of **G** are G<sub>ij</sub> = -1, *i* ≠ *j* if oscillators *i*, *j* are coupled and G<sub>ij</sub> = 0 if they are not. The diagonal elements are given by G<sub>ii</sub> = -∑<sub>j≠i</sub> G<sub>ij</sub> in order to satisfy the condition ∑<sub><i>j</i>=1</sub><sup><i>N</i></sup> G<sub>ij</sub> = 0 for any *i*, where *N* is the network size. When all the oscillators are identical, a complete synchronized state defined by x<sub>1</sub> = x<sub>2</sub> = ... = x<sub>N</sub> = s is an exact solution of Eq. (1). The coupling matrix G, as determined by the network topology, can be diagonalized with a set of real eigenvalues {λ<sub>i</sub>, *i* = 1, ..., *N*} and the corresponding set of eigenvectors e<sub>1</sub>, e<sub>2</sub>, ..., e<sub>N</sub>. Full connectivity of the network ensures that there is one zero eigenvalue and the eigenvalues can be sorted as 0 = λ<sub>1</sub> < λ<sub>2</sub> < ... < λ<sub>N</sub>. The variational equations governing the time evolution of the set of infinitesimal vectors transverse to the synchronization manifold, δ**x**<sub>*i*</sub>(*t*) ≡ **x**<sub>*i*</sub>(*t*) − **s**(*t*), are dδ**x**<sub>*i*</sub>/dt = **DF**(**s**) · δ**x**<sub>*i*</sub> − ε ∑<sub><i>j</i>=1</sub><sup><i>N</i></sup> G<sub>ij</sub> **DH**(**s**) · δ**x**<sub>*j*</sub>, where **DF**(**s**) and **DH**(**s**) are the d × d Jacobian matrices of the corresponding vector functions evaluated at **s**(*t*). The transform δ**y** = **Q**<sup>-1</sup> · δ**x**, where **Q** is a matrix whose columns are the set of eigenvectors of **G**, leads to the block-diagonally decoupled form of the variational equation: dδ**y**/dt = [**DF**(**s**) − ελ<sub>*i*</sub> **DH**(**s**)] · δ**y**<sub>*i*</sub>. Let K<sub>*i*</sub> = ελ<sub>*i*</sub>(*i* = 2, ..., *N*) be a specific set of values of a normalized coupling parameter K. All blocks of the decoupled equation are structurally the same with only the factor of K<sub>*i*</sub> being different, leading to the following generic form for all the decoupled blocks: dδ**y**/dt = [**DF**(**s**) − **KDH**(**s**)] · δ**y**.
+
+The largest Lyapunov exponent of the above block-diagonal variational equation is the MSF Ψ(K). If Ψ(K) is negative, a small disturbance from the synchronization state will diminish exponentially so that the synchronous solution is stable, at least when the oscillators are initialized in its vicinity. The synchronous solution is unstable and cannot be realized physically if Ψ(K) is positive because small perturbations from the synchronous state will lead to trajectories that diverge from the state. For the coupled oscillator network Eq. (1), a necessary condition for synchronization is then that all normalized coupling parameters Kᵢ (*i* = 2, . . . , N) fall in an interval on the K axis where Ψ(K) is negative. The network is more synchronizable if the spread in the set of Kᵢ values, or equivalently, the spread in the eigenvalue spectrum λᵢ, is smaller. The MSF allows the synchronization interval to be determined, which depends only on the coupling function (H) but is independent of the network topology. In particular, the condition for complete synchronization is given by ελ₂ ≃ K₁, and the critical parameter for the onset of synchronization is K_c ≃ K₁/λ₂, where K₁ is the value of K at which Ψ(K) becomes negative from the positive side.
+
+In Ref. [13], a stability analysis for synchronization of nearly identical oscillators was carried out, which was based
+
+on the following extended master-stability function (eMSF):
+
+$$
+\dot{\xi} = [D_{\omega}f - KDH] \cdot \xi + D_{\mu}f \cdot \psi. \quad (2)
+$$
+
+The term in the square parentheses is the same as that in
+the block-diagonal form of the variational equation in the
+case of identical nodal dynamics. In the second term, D_μ is
+the Jacobian matrix with respect to the parameter vector μ
+and ψ = Σ_{j=1}^N u_j μ_j, where δμ_i = μ_i - μ̄ is the parameter
+mismatch with respect to mean of μ_i over all oscillators, and
+u_i is the ith eigenvector of G. The stability of Eq. (2) can
+then be determined as a function of the two parameters K
+and ψ, and we can decompose the problem into two separate
+parts: one that depends only on the nodal dynamics and the
+coupling function H, and another determined by the parameter
+mismatch among the oscillators.
+
+Let Ψ(K, ψ) be the largest Lyapunov exponent of Eq. (2).
+In general, the value of K₁ depends on ψ. To be specific, we
+fix the number of oscillators and variation in the parameter
+mismatch, and focus on the scaling relation between K_c and
+the link density. In this case, δμᵢ obeys the same statistic
+as that for ψ in Eq. (2). For different link density, since the
+mismatch is bounded in the same range, the value of K₁ can be
+regarded as a constant, which is slightly larger than K₁|ψ=0.
+As the amount of mismatch is increased, i.e., with a larger
+standard deviation σ as in our case, K₁ also increases, which
+has been verified numerically. Since the oscillators are not
+identical, K₁ is not the critical coupling strength for complete
+synchronization, but the value for which the variation for the
+dynamical variables are bounded [13], so it is essentially the
+onset point of phase synchronization. Approximately, we can
+use K₁|ψ=0 to represent K₁ with parameter mismatches. In
+fact, we find numerically that the product K_cλ₂ for different
+networks is nearly constant and the values are comparable with
+those of identical oscillators. We thus have K_cλ₂ ≃ K₁.
+
+For complex topologies we consider random and scale-free
+networks. For the former, the link density is determined
+by p, the probability that an arbitrary pair of oscillators in
+the network are coupled. For the latter, we consider those
+generated by the preferential-attachment mechanism [22]:
+starting from a small number m₀ of completely connected
+oscillators, a new oscillator is introduced into the network
+with m links according to the preferential-attachment rule.
+The parameter m thus determines the link density. For
+both types of networks, estimates of the value of λ₂ are
+available [23]. In particular, for random networks, we have
+λ₂ ~ Np - 2√Np(1-p), while for scale-free networks, we
+have λ₂ ~ Cm. The relationship K_cλ₂ ≃ K₁ gives, for random
+and scale-free networks, respectively, the following scaling
+law governing the onset of phase synchronization:
+
+$$
+K_c \sim \begin{cases}
+\frac{K_1}{Np - 2\sqrt{Np(1-p)}}, & \text{random networks,} \\
+\frac{K_1}{C_m}, & \text{scale-free networks.}
+\end{cases} \tag{3}
+$$
+
+The scaling laws (3) are the main result of this paper. We note that, for large random networks that satisfy $Np \gg 1$, the scaling law becomes $K_c \sim p^{-1}$.
+
+To provide numerical verification of Eq. (3), we consider networks of coupled Rössler chaotic oscillators. The vector field of the *i*-th oscillator (node *i*) is given by $\mathbf{F}_i(\mathbf{x}) = [-(\omega_i y + z), \omega_i x + 0.165y, 0.2 + z(x - 10)]$, where the parameter $\omega_i$ is

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+FIG. 1. (Color online) For random network of $N = 10^2$, $10^3$, and $10^4$ chaotic Rössler oscillators, nonlinear fit $K_c \sim A/[Np - 2\sqrt{Np(1-p)}]$, as predicted theoretically, where A is a fitting parameter.
+
+different for each oscillator and is taken from some random distribution. For oscillator *i*, a phase variable $\phi_i(t)$ can be calculated [20]. For a pair of oscillators *i* and *j*, phase synchronization is defined by $\Delta\phi_{ij} = |\phi_i(t) - \phi_j(t)| < 2\pi$. The average phase-synchronization time $\tau_{ij}$ is the average time interval during which the phase difference is bounded within $2\pi$. As the coupling parameter *K* is increased toward $K_c$, $\tau_{ij}$ increases and obeys the scaling law [24]: $\tau_{ij} \sim \exp[C(K_c - K)^{-\gamma}]$, where *C* and $\gamma$ are positive constants and $\tau_{ij}$ diverges for $K \ge K_c$. For a network of *N* oscillators, as $K_c$ is reached, all $N(N-1)/2$ values of $\tau_{ij}$ (one for each distinct pair) diverge. Computationally the behavior of $\tau_{ij}$ can thus be used to determine the onset of phase synchronization on the network. In particular, $\tau_{ij}$ can be regarded as an element of the $N \times N$ symmetric matrix, $\Gamma$. For finite time series measured from oscillators in the network, when the diagonal elements of $\Gamma$ are chosen properly, the determinant of the matrix provides an effective way to determine the onset of chaotic phase synchronization globally [21].
+
+To calculate the determinant is computationally costly. To remedy this difficulty, we propose the following quantity:
+
+$$P = \prod_{i=1}^{N-1} \sqrt[N]{\tau_{i,i+1}}, \quad (4)$$
+
+FIG. 2. (Color online) For random networks of $N = 10^2$, $10^3$, and $10^4$ nodes, numerically obtained scaling of $K_c$ with the linking probability $p$. The scaling is algebraic in the regime of large $Np$ values, as predicted by our theory.
+
+FIG. 3. (Color online) For scale-free networks of $N = 10^2$ and $10^3$ oscillators, numerically obtained scaling of $K_c$ with the network-generating parameter $m$.
+
+which can be computed extremely efficiently. As phase synchronization is approached, $P$ tends to diverge. The computational load is only proportional to $N$, the network size. We have verified that the values of critical coupling parameter $K_c$ computed using $P$ and the previous method of determinant coincide with high accuracy.
+
+Figure 1 shows for random networks of $N = 10^2$, $10^3$, and $10^4$ nodes, the relationship between $K_c$ and $p$, fitted according to the theoretical prediction in Eq. (3). We observe a good agreement between numerics and theory. For the regime of large $Np$ values, the scaling becomes algebraic, as shown in Fig. 2 for networks of three different sizes ($N = 10^2$, $10^3$, and $10^4$). Figure 3 shows for scale-free networks of $N = 10^2$ and $10^3$ oscillators, numerically obtained scaling of $K_c$ with the network-generating parameter $m$. The scaling is algebraic, as
+
+FIG. 4. (Color online) For random (upper panel) and scale-free (lower panel) networks, scaling of $K_c$ with $p$ and $m$, respectively, for different distributions of the frequency parameter $\omega_i$ of the chaotic oscillators. The network size is $N = 10^3$.

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+predicted. The scaling laws (3) are independent of the intrinsic
+properties of the oscillators, as shown in Fig. 4.
+
+In summary, we have demonstrated, by using a heuristic
+analysis and numerical computations, that the onset of phase
+synchronization in complex networks of coupled, heteroge-
+neous chaotic oscillators can be facilitated by increasing the
+link density. This is substantiated by a scaling law relating the
+critical coupling parameter required for phase synchronization
+among all oscillators in the network to a parameter charac-
+terizing the network linkage. Computational detection of the
+onset of global chaotic phase synchronization is made possible
+by an efficient numerical method to calculate the pairwise
+average phase-synchronization time. Our work treats both
+complex network topology and complicated heterogeneous
+nodal dynamics at the same time, versus existing works, e.g.,
+on complex networks of simple Kuramoto phase oscillators.
+
+The complexity of our problem renders infeasible any analytic treatment at a comprehensive level, but nonetheless we are able to obtain quantitative results on chaotic phase synchronization in the network. Networked systems in real applications are often heterogeneous, complicated in both topology and nodal dynamical processes. The methodology developed in our work can be useful and further developed to probe into the fundamentals of the network dynamics with significant applications.
+
+This work was partly supported by a grant from R.A.S.
+(Regione Autonoma della Sardegna) awarded to F. Ricci,
+PO Sardegna FSE 2007-2013, L.R.7/2007 "Promotion of the
+scientific research and technological innovation in Sardinia".
+Y.C.L. and L.H. were supported by AFOSR under Grant No.
+FA9550-10-1-0083. L.H. was also supported by NSF of China
+under Grants No. 11005053 and No. 11135001.
+
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+[18] X.-G. Wang, L. Huang, S.-G. Guan, Y.-C. Lai, and C. H. Lai, Chaos **18**, 037117 (2008).
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+[19] C. A. S. Batista, A. M. Batista, J. A. C. de Pontes, R. L. Viana, and S. R. Lopes, Phys. Rev. E **76**, 016218 (2007).
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+It can immediately be seen that expressions (1) and (2) are equal when $e_i = e_k = e \in \text{int } E$ since, in this case, $g_{e_k}^{-1}(g_{e_i}(x)) = g_{e_i}^{-1}(g_{e_k}(x)) = x$ and $\frac{d}{de_k}g_{e_i}^{-1}(g_{e_k}(x)) = \frac{d}{de_i}g_{e_k}^{-1}(g_{e_i}(x))$. Thus, the sufficient condition for the existence of a symmetric equilibrium in Lemma 1 is satisfied. Hence, we have the following theorem.
+
+**Theorem 1.** There exists a symmetric equilibrium in which both players choose the same level of effort.
+
+*Proof.* See Appendix A.2. □
+
+The theorem states that, even if the players are asymmetric (i.e., $f_1 \neq f_2$), there always exists a symmetric equilibrium of the contest game. This key result allows a tractable analysis of contests between asymmetric players in a variety of different settings. We define $a_e : \mathbb{R} \to \mathbb{R}$ by
+
+$$a_e(x) = \left. \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(x)) \right|_{e_i=e_k=e} = \frac{\partial g(x,e)}{\partial e} / \frac{\partial g(x,e)}{\partial x} = \text{MRTS}(x,e), \quad (3)$$
+
+where the equality follows from an application of the inverse function theorem and MRTS($x,e$) denotes the marginal rate of technical substitution between skill and effort in a symmetric equilibrium.¹⁵ Recognizing that the two players have the same cost function $c(e)$, we can write the (identical) first-order condition for effort for the two players in a symmetric equilibrium as
+
+$$V \int_{\mathbb{R}} a_{e^*}(x) f_k(x) f_i(x) dx = c'(e^*). \quad (4)$$
+
+The key observation necessary to understand the intuition behind (4) is that a player has a positive marginal incentive to supply effort if and only if $g(\theta_k, e_k) = g(\theta_i, e_i)$. In a symmetric equilibrium where $e_k = e_i$ this implies that $\theta_k = \theta_i$. The reason a player only has a marginal incentive to exert effort when $\theta_k = \theta_i$ is that this is the only situation in which a marginal increase in output would be pivotal to winning the contest. Accordingly, equation (4) contains the “collision density” $f_k(x)f_i(x)$ that describes how likely it is that the skill realizations of the two competing players are the same. The fact that this term is the same for both players is due to our assumption of symmetric uncertainty. Furthermore, the fact that $a_e(x)$ is the same for both players follows directly from the assumption that the production function $g(\theta, e)$ is the same for both players,
+
+¹⁵To see this, notice that $\left. \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(x)) \right|_{e_i=e_k=e} = \frac{1}{g'_{e_k}(g'_{e_k}(g_{e_i}(x)))} \left. \frac{d}{de_i} g_{e_i}(x) \right|_{e_i=e_k=e} = \frac{\frac{d}{de} g_{e_i}(x)}{g'_{e_i}(x)}$.

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+and depends only on the level of effort *e* and the skill *θ*, both of which are the same for both players in situations where players have a marginal incentive to supply effort in symmetric equilibrium.
+
+The function $a_e(x)$ describes how a marginal increase in effort by a player increases output relative to his or her rival and is equal to the marginal rate of technical substitution between skill and effort. The purpose of raising effort is to beat players with higher skill. The MRTS determines the range of additional types that the player can win against through a small effort increase. The lower is the sensitivity of output to skill in the production function, the smaller is the advantage of marginally more skilled rivals, and the higher is the marginal incentive to exert effort. A direct implication is that the marginal incentive to exert effort is higher in environments in which players' outputs depend to a large degree on effort than in those in which the output is mainly determined by players' skills. The reason is that the MRTS tends to be larger in the former than in the latter environments, implying a greater impact of effort on the winning probability, as just explained.
+
+Additional intuition can be provided by considering specific functional forms. For example, if $g(\theta, e) = e + \theta$ we have that $a_e(x) = 1$ since in this case both the numerator and denominator are equal to unity. If, instead, $g(\theta, e) = \theta e$, we have that $a_e(x) = x/e$ because of the complementarity between skill and own effort in the production function. The fact that $a_e(x)$ is an increasing function of $x$ reflects that it is in this case more valuable to increase effort the higher is the skill of the player. The fact that $a_e(x)$ is decreasing in $e$ reflects that the marginally more able individual is harder to beat the higher is the baseline (symmetric) level of effort because of the complementarity between skill and effort.
+
+We end this section with an illustrative example. Consider the multiplicative production technology $g(\theta_i, e_i) = \theta_i e_i$ and the cost function $c(e_i) = e_i^2/2$. Assume further that the skill distribution of player 1 follows a Uniform distribution on [1, 2], and the skill distribution of player 2 is given by the Student's t-distribution on support $(-\infty, \infty)$, with one degree of freedom, such that:
+
+$$ f_1(x) = \begin{cases} 1 & 1 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}, \quad f_2(x) = \frac{1}{\pi(1+x^2)}, x \in \mathbb{R}. $$
+
+The event of player 1 winning is described by $g(\theta_1, e_1) > g(\theta_2, e_2) \iff \theta_2 < g_{e_2}^{-1}(g_{e_1}(\theta_1)) =$

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+players and can be written as $g_i(\theta_i, e_i) = \tilde{g}(h_i(\theta_i), e_i)$, $i \in \{1,2\}$. Then a symmetric equilibrium of the contest game exists with effort determined by
+
+$$V \int_{\mathbb{R}} a_{e^*}(x) \tilde{f}_i(x) \tilde{f}_k(x) dx = c'(e^*),$$
+
+where $h_i$ is a real-valued function, $\tilde{f}_i$ denotes the pdf of the random variable $\tilde{\Theta}_i := h_i(\Theta_i)$ and $a_{e^*}$ is calculated based on the production function $\tilde{g}(\tilde{\Theta}_i, e_i)$.
+
+**Proof.** See Appendix A.3. □
+
+As a specific example, consider the additive production function $g_i(\theta_i, e_i) = \eta_i(\theta_i) + \kappa(e_i)$, with $\eta_i$ and $\kappa$ being two strictly increasing functions. In this case, Corollary 1 can be applied since we can write $\tilde{g}(h_i(\theta_i), e_i) = h_i(\theta_i) + \kappa(e_i)$, with $h_i(\theta_i) = \eta_i(\theta_i)$. Likewise, in the case of a multiplicative production function of the form $g_i(\theta_i, e_i) = \eta_i(\theta_i)\kappa(e_i)$ (imposing the additional assumption that $\eta_i$ and $\kappa$ are non-negative), we can apply the corollary noting that $\tilde{g}(h_i(\theta_i), e_i) = h_i(\theta_i)\kappa(e_i)$, with $h_i(\theta_i) = \eta_i(\theta_i)$. Notice that in both examples, the component $\kappa(e_i)$ is the same for both players.
+
+Next, we consider a situation with heterogeneous prizes given by $V_1 = sV$ and $V_2 = V$ with $s > 0$. In this case, provided that the cost functions are homogeneous, the contest can be transformed into an equivalent contest, where the players have the same prizes, but different production functions. This is shown in the following proposition.
+
+**Proposition 1.** Consider a situation with heterogeneous prizes given by $V_1 = sV$ and $V_2 = V$ with $s > 0$, and let $c$ be homogeneous of degree $\delta > 0$. Then, the contest can be transformed into a contest where the players have the same prizes, but different production functions. This is achieved by considering the transformed effort variables $\xi_1 = e_1/s^{1/\delta}$ and $\xi_2 = e_2$. Denote the equilibrium of the transformed contest by $\xi_1^*$ and $\xi_2^*$. Then, the equilibrium of the original contest is given by $e_1^* = s^{1/\delta}\xi_1^*$ and $e_2^*$, where $\xi_1^*$ and $e_2^*$ maximize
+
+$$\int_{\mathbb{R}} F_2(g_{e_2}^{-1}(g_{s^{1/\delta}\xi_1}(x))) f_1(x) dxV - c(\xi_1)$$
+
+and
+
+$$\int_{\mathbb{R}} F_1(g_{s^{1/\delta}\xi_1}^{-1}(g_{e_2}(x))) f_2(x) dxV - c(e_2),$$
+
+respectively.
+
+**Proof.** See Appendix A.4. □
+
+Proposition 1 shows that, given homogeneity of the cost function, the equilibrium of a contest with heterogeneous prizes is characterized by conditions abiding a structure

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+which is very similar to the structure of the conditions used to characterize the equilib-
+rium in Theorem 1, although the equilibrium is in general no longer symmetric. How-
+ever, as we shall see below, if the conditions of Corollary 1 are satisfied, we can derive a
+very simple expression for the relationship between the equilibrium effort levels. Before
+turning to this result, we first demonstrate that, in some situations, a contest with differ-
+ent cost functions is equivalent to a contest with different prizes (in terms of equilibrium
+effort choices) and hence can, according to Proposition 1, be transformed into a contest
+with different production functions. This is formalized in the following remark.¹⁶
+
+**Remark 1.** Suppose that players have heterogeneous cost functions that take the form $c_i(e_i) = \omega_i c(e_i)$, with $\omega_i > 0$. Then, the objective of player i can be written as:
+
+$$
+\int_{\mathbb{R}} F_k(g_{e_k}^{-1}(g_{e_i}(x))) f_i(x) dx V_i - \omega_i c(e_i) = \omega_i \left( \int_{\mathbb{R}} F_k(g_{e_k}^{-1}(g_{e_i}(x))) f_i(x) dx \frac{V_i}{\omega_i} - c(e_i) \right).
+$$
+
+This objective is equivalent to one in which prizes are given by $\frac{V_i}{\omega_i}$, but the cost functions
+are the same for both players.
+
+Proposition 1 and Remark 1 highlight how, in certain cases, contests with different prizes or different cost functions can be transformed into equivalent contests with different production technologies. We now turn to showing that, if the assumptions underlying Corollary 1 are satisfied, these contests can be reinterpreted as contests with different skill distributions, allowing us to apply the equilibrium characterization from our baseline case.
+
+Proposition 2 below provides results for two well-known production technologies that satisfy the assumptions of Corollary 1. We first consider the Cobb-Douglas production technology. In this case, we can show that the ratio of equilibrium efforts $e_1^*/e_2^*$ depends only on the ratio of prizes and the degree of homogeneity of the cost function. We then consider the additive production technology. In this case, we obtain a similar characterization which applies to the ratio of efforts subject to an exponential transformation, $\exp(e_1^*)/\exp(e_2^*)$.
+
+**Proposition 2.** Let the two prizes be given by $V_1 = sV$ and $V_2 = V$ with $s > 0$.
+
+(i) If the production technology is given by $g(\theta,e) = \theta^\alpha e^\beta$, with $\alpha, \beta > 0$, and $c$ is homogeneous of degree $\delta > 0$, an equilibrium exists with efforts given by $e_1^* = s^{1/\delta} e_2^*$ and
+
+¹⁶Similar transformations between prizes and cost functions are standard in the literature.

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+**Assumption 2.** The primitives of the model are such that $q:E \to \mathbb{R}$, defined by
+
+$$q(e) = V \int_{\mathbb{R}} r_{e,i}(x)f_k(x)dx - c'(e),$$
+
+is strictly decreasing.
+
+As $c$ is strictly convex, Assumption 2 is not very strong and is always satisfied if $\int_{\mathbb{R}} r_{e,i}(x)f_k(x)dx$ is non-increasing in $e$. To give a specific example, consider the CES production function $g(\theta_i, e_i) = (\alpha\theta_i^\rho + \beta e_i^\rho)^{\frac{1}{\rho}}$, with $\alpha, \beta > 0$ and $\rho \le 1$. Here $a_e(x) = \frac{\beta}{\alpha}(\frac{x}{e})^{1-\rho}$, implying that $\int_{\mathbb{R}} a_e(x)f_1(x)f_2(x)dx = e^{\rho-1}\int_{\mathbb{R}} \frac{\beta}{\alpha}x^{1-\rho}f_1(x)f_2(x)dx$. For this specification, Assumption 2 is satisfied in all cases where players have an incentive to exert positive effort (i.e., $\int_{\mathbb{R}} \frac{\beta}{\alpha}x^{1-\rho}f_1(x)f_2(x)dx > 0$). Furthermore, the assumption ensures that effort is always increasing in the prize and that the considered equilibrium is unique in the class of symmetric equilibria (the latter result follows from the assumption ensuring that there is a unique $e$ solving equation (5)).
+
+## 6.1 First-Order Stochastic Dominance
+
+A standard result in contest theory is that heterogeneity among players with respect to their skills reduces the incentive to exert effort (see, e.g., Schotter and Weigelt 1992, or Observation 1 in the survey by Chowdhury, Esteve-Gonzalez, and Mukherjee 2019). In our model, this standard result is potentially reversed, as we will now show.
+
+Consider a contest with two players with skills drawn from two distributions with expected values $\mu_k$ and $\mu_i$, respectively. If, from the outset, $\mu_k \ge \mu_i$ and the difference $\mu_k - \mu_i$ is increased, then the two players become more heterogeneous in terms of their expected skill. Based on this idea, we proceed by investigating the consequences of making players more heterogeneous in the sense of first-order stochastic dominance, as captured by the following definition.
+
+**Definition 1.** Let $\mu_k$ and $\mu_i$ refer to the expected values of the skill distributions ($F_k, F_i$) in an initial contest. Players in a contest with skill distributions ($\tilde{F}_k, F_i$) are said to be more heterogeneous (with respect to their skills) relative to players in the initial contest with skill distributions ($F_k, F_i$), in a first-order sense, if either of the following conditions hold:
+
+(i) $\mu_k \ge \mu_i$ and $\tilde{F}_k$ dominates $F_k$ in the sense of first-order stochastic dominance.
+
+(ii) $\mu_k \le \mu_i$ and $\tilde{F}_k$ is dominated by $F_k$ in the sense of first-order stochastic dominance.

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+distributions collide at larger values of $x$ (see Figure 2 for an illustration). This would
+have no effect on the incentive to exert effort if $a_e(x)$ would be constant, as in Example
+2. However, in the current example, we have that $a_e(x) = x/e$. Thus, taking into account
+the three terms in (4), the fact that the two distributions collide at larger values of $x$
+increases the incentive to exert effort for both players. Intuitively, given that the only
+relevant situations (where players have a positive marginal incentive to supply effort)
+now occur at larger values of skill, the fact that there is a complementarity between skill
+and effort in the production function implies that the incentive to supply effort is higher
+for both players.
+
+Figure 2: Illustration of Example 3
+
+Concluding this section, we note that the conditions in Proposition 3 are sufficient,
+but not necessary for the result that effort can be higher when contestants are more
+heterogeneous. To illustrate this, we present an additional result based on normal dis-
+tributions where we first determine the marginal winning probability in a situation with
+symmetric effort.
+
+**Proposition 4.** Suppose that $\Theta_i \sim N(\mu_i, \sigma_i^2)$, $\Theta_k \sim N(\mu_k, \sigma_k^2)$, and $g(\theta, e) = \theta \cdot e$. Then the marginal winning probability when $e_1 = e_2 = e$ is
+
+$$ \frac{\partial P_i(e_i, e_k)}{\partial e_i} \Big|_{e_i=e_k=e} = \frac{(\mu_i \sigma_k^2 + \mu_k \sigma_i^2) \exp\left(-\frac{(\mu_i - \mu_k)^2}{2(\sigma_i^2 + \sigma_k^2)}\right)}{e^{(2\pi)^{\frac{1}{2}}} (\sigma_i^2 + \sigma_k^2)^{\frac{3}{2}}}. $$
+
+*Proof.* See Appendix A.7. □

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+In the upcoming example, it can be verified that $r_{e,i}(x) = a_e(x)f_i(x)$ is neither always increasing nor always decreasing, by virtue of the multiplicative production technology combined with the bell-shaped normal distribution. Nonetheless, equilibrium effort increases as players become more heterogeneous in the sense of increasing the distance $|\mu_i - \mu_k|$.
+
+**Example 4.** Consider Proposition 4 and assume that $(\sigma_i, \sigma_k) = (1,1)$, $(\mu_i, \mu_k) = (\frac{1}{2}, \frac{1}{2})$, $V=1$, and $c(e) = \frac{e^2}{2}$. Then equilibrium effort is $e^* = (2\pi^{\frac{1}{4}})^{-1} \approx 0.38$. If we increase $\mu_i$ from $\frac{1}{2}$ to $\frac{3}{2}$, keeping $\mu_k$ constant, equilibrium effort increases to $\tilde{e}^* = (\sqrt{2}\exp(\frac{1}{8})\pi^{\frac{1}{4}})^{-1} \approx 0.47$.
+
+## 6.2 Second-Order Stochastic Dominance
+
+The studies by Hvide (2002), Kräkel and Sliwka (2004), Kräkel (2008), Gilpatric (2009), and DeVaro and Kauhanen (2016) investigate how “risk” or “uncertainty” affects players’ incentive to exert effort in contests. One result that is common to all of these analyses is that in contests between equally able players, higher risk (as measured by a higher variance of the random variables capturing the uncertainty of the contest outcome) leads to lower efforts. We revisit this result in the context of our model, and show that effort may increase as the skill distribution of one of the players becomes more uncertain.
+
+The economic literature has identified different ways to conceptualize risk or uncertainty. We follow Rothschild and Stiglitz 1970 by using second-order stochastic dominance to measure the uncertainty regarding players’ skill distributions.¹⁸
+
+**Definition 2.** The skill distribution $\tilde{F}_i$ is said to be more uncertain than the distribution $F_i$ if $\tilde{F}_i$ is a mean-preserving spread of $F_i$. This is equivalent to $\tilde{F}_i$ being dominated by $F_i$ in the sense of second-order stochastic dominance.
+
+Equipped with this definition, we can use well-known results from decision theory to obtain our next proposition:
+
+**Proposition 5.** Consider two contests with skill distributions $(\tilde{F}_k, F_i)$ and $(F_k, F_i)$ where $\text{supp}(\tilde{f}_k)$ and $\text{supp}(f_k)$ both are subsets of $\text{supp}(f_i)$. Let $\tilde{e}^*$ and $e^*$ denote, respectively, the (symmetric) equilibrium efforts associated with these contests. Suppose that $\tilde{F}_k$ is more uncertain than $F_k$. Then, the following results hold:
+
+¹⁸Gerchak and He (2003) analyze how effort in two-player contests is determined by the Rényi entropy, and Drugov and Ryvkin (2020) generalize their insights to the case of more than two players. Their results, however, rely on the assumptions of an additive production function and homogeneous players (i.e., players with the same skill distributions). It is not obvious how these results transfer to a model with general production technologies and asymmetric skill distributions, which is the primary focus here.

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+(i) If $r_{e,i}(x)$ is strictly convex on $\text{supp}(f_i)$ for all $e \ge 0$, then $\tilde{e}^* > e^*$.
+
+(ii) If $r_{e,i}(x)$ is linear on $\text{supp}(f_i)$ for all $e \ge 0$, then $\tilde{e}^* = e^*$.
+
+(iii) If $r_{e,i}(x)$ is strictly concave on $\text{supp}(f_i)$ for all $e \ge 0$, then $\tilde{e}^* < e^*$.
+
+*Proof*. See Appendix A.8. □
+
+The key insight needed to understand Proposition 5 is that applying a mean-preserving
+spread to the distribution $F_k$ shifts probability mass from the center to the tails of the
+distribution, and the impact of this change on the incentive to exert effort depends on
+the curvature of $r_{e,i}(x)$. Notice that Proposition 5 also holds if players have the same
+expected skill, namely $\mu_i = \mu_k$. This means that, in a contest with two players who are
+expected to be equally able, higher uncertainty regarding players’ skills may increase
+the incentive to exert effort.
+
+Next, we illustrate and provide intuition for Proposition 5 by presenting an example
+set in the context of the Lazear-Rosen model with an additive production technology. The
+example demonstrates that increasing the uncertainty of the contest while keeping the
+expected skill of both players unchanged, can increase equilibrium effort.
+
+**Example 5.** Consider a contest with the additive production function $g(\theta, e) = \theta + e$, the parameter $V = 1$, and the cost function $c(e) = \frac{e^2}{2}$. Suppose $\Theta_i \sim \text{Exp}(1)$ and $\Theta_k \sim U[\frac{1}{2}, \frac{3}{2}]$ (implying $\mu_i = \mu_k = 1$). Equilibrium effort is then $e^* = \frac{\exp(1)-1}{\exp(\frac{3}{2})} \approx 0.38$. Now, consider a mean-preserving spread of the skill distribution of player k, enlarging the support of the uniform distribution, such that $\tilde{\Theta}_k \sim U[0,2]$. Then effort increases to $\tilde{e}^* = \frac{\exp(2)-1}{2\exp(2)} \approx 0.43$.
+
+In Example 5, we have imposed the additive production technology which implies
+$a_e(x) = 1$. Thus, the convexity of $r_{e,i}(x)$ referred to in part (i) of Proposition 5 is de-
+termined by the convexity of $f_i(x)$. To understand how the shift from $f_k$ to $\tilde{f}_k$ affects
+the incentive to exert effort, we need to study how the integral in (4) is affected. Sim-
+ilar to Example 2, given that $a_e(x) = 1$, it is sufficient to compare $\int f_i(x)f_k(x)dx$ with
+$\int f_i(x)\tilde{f}_k(x)dx$. The shift from $f_k$ to $\tilde{f}_k$ entails an enlargement of the support of the uni-
+form distribution. This implies that the density decreases for intermediate values of x,
+but increases for low and high values of x (see Figure 3 for an illustration). Given that
+$f_i(x)$ is strictly decreasing, the part of the skill distribution of player k that is stretched
+out to the left will collide with relatively large values of $f_i$, whereas the part of the
+skill distribution of player k that is stretched out to the right will collide with relatively
+small values of $f_i$, creating a trade-off. The fact that $f_i$ is not only strictly decreasing,

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+but also convex, resolves this trade-off, implying that the overall effect of the shift is to
+increase the value of the integral expression. Thus, both players have a higher incentive
+to exert effort as a result of the move from $f_k$ to $\tilde{f}_k$. Intuitively, due to the change in
+the distribution of player k, situations where the competing players have the same skill
+become "more likely", implying an increase in equilibrium effort.
+
+Figure 3: Illustration of Example 5
+
+We conclude this section by defining contestant heterogeneity in a *second-order* sense and we follow the structure of the corresponding definition of heterogeneity in a first-order sense (Definition 1). In Definition 1, we used the ranking of players' mean skills to characterize the initial situation. In the new definition, we do so through the variances of the skill distributions of the competing players (restricting attention to statistical distributions with finite variance). Notice, however, that variance is not always a good measure of uncertainty or risk (see, e.g., Rothschild and Stiglitz 1970). Therefore one should keep in mind, when applying the definition below, that higher variance entails higher uncertainty only for certain skill distributions (e.g., the normal distribution).
+
+**Definition 3.** Let $Var_k$ and $Var_i$ refer to the variances of the skill distributions ($F_k, F_i$) in an initial contest. Players in a contest with skill distributions ($\tilde{F}_k, \tilde{F}_i$), are said to be more heterogeneous (with respect to their skills) relative to players in the initial contest with skill distributions ($F_k, F_i$), in a second-order sense, if either of the following conditions hold:
+
+(i) $Var_k \ge Var_i$ and $F_k$ dominates $\tilde{F}_k$ in the sense of second-order stochastic dominance.

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+(ii) $\mathrm{Var}_k \le \mathrm{Var}_i$ and $F_k$ is dominated by $\tilde{F}_k$ in the sense of second-order stochastic dominance.
+
+Combining Proposition 5 with Definition 3, we have the following corollary.
+
+**Corollary 3.** *Effort can be higher when contestants are more heterogeneous in a second-order sense.*
+
+## 6.3 Implications for Optimal Team Composition
+
+The results in the preceding two subsections have implications for optimal team composition and organizational design.¹⁹ In particular, our results suggest that employers could find it desirable to employ a more heterogeneous workforce as an instrument to induce higher effort. In Section 6.1, we analyzed the effects of increasing the heterogeneity in players’ expected skills, and showed how this can increase equilibrium effort. This means that a firm could benefit (from the perspective of inducing higher effort) by hiring some workers with a high expected skill and some with a low expected skill, based on, for example, signals such as the quality of the institution where a college graduate received his or her degree. In Section 6.2, we showed how increased uncertainty regarding skills of some players can increase equilibrium effort. Thus, a firm could benefit from hiring a mix of experienced workers (for whom the uncertainty regarding skills is relatively small) and inexperienced workers (for whom the uncertainty regarding skills is relatively large).
+
+To see this more formally, suppose a firm already employs a worker with skill distribution $F_1$ and considers to hire another worker with skill distribution $F_2$. Moreover, assume that $r_{e,1}(x)$ is strictly decreasing and strictly convex (for example, by assuming that the production function is given by $g(\theta, e) = \theta + e$ and skills are Exponentially distributed with parameter $\lambda$).²⁰ Then the firm may gain from hiring another worker with a lower expected skill ($\mu_2 < \mu_1$), but where $F_2$ is more uncertain (meaning that worker 2’s skill is drawn from a more uncertain distribution). This finding can be understood from the perspective of Proposition 3, that tells us that effort will be higher due to the lower expected skill of worker 2, combined with Proposition 5, which tells us that effort will
+
+¹⁹See, e.g., Gershkov, Li, and Schweinzer (2009, 2016) and Fu, Lu, and Pan (2015).
+
+²⁰An alternative skill distribution that would also be decreasing and convex would be a normal distribution that is truncated to the left at a point to the right of the second inflection point. Such a distribution could be motivated by the observation that skills are often normally distributed and that, when employing worker 1, the firm tried to hire the most able applicant, meaning that skills in the higher end of the distribution are most relevant (see, e.g., Aguinis and O’Boyle Jr. 2014).

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+be higher due to the larger uncertainty regarding the skill of worker 2. In other words,
+hiring a worker with a lower expected skill, drawn from a more uncertain distribution,
+can induce higher effort. Proposition 3 and Proposition 5 also have other managerial im-
+plications as they indicate that employers may want to hire workers who have worked on
+different tasks in the past (or on similar tasks in a different firm or industry), to create
+uncertainty about workers’ skills. In a similar vein, it might be desirable to implement
+some kind of job rotation.
+
+# 7 Extensions
+
+## 7.1 The Case of More Than Two Players ($n > 2$)
+
+We now turn to the case of $n > 2$ contestants which allows us to address the interesting question of how effort depends on the number of players competing in a contest.²¹ In Section 7.1.1, we show that the existence of a symmetric equilibrium generally cannot be extended to the case of $n > 2$ heterogeneous players. In Section 7.1.2, we consider the case of $n$ homogeneous players. Section 7.1.3 examines a special case of our model where $n - 1$ homogeneous players compete against a player who is more highly skilled (e.g., as in Brown 2011 and Krumer, Megidish, and Sela 2017), which serves to demonstrate that a symmetric equilibrium can exist when players are heterogeneous and the number of players is greater than two. In all these sections, we maintain the generality of the production technology.
+
+### 7.1.1 The $n = 2$ Result Does Not Extend to $n > 2$
+
+In the case of $n > 2$ players with different skill distributions, the equilibrium in our model
+is generally no longer symmetric. A player *i* will only win the contest if he or she beats all
+of his or her opponents. Essentially, each player is thus competing against the best of the
+other players, that is, the highest order statistic, and therefore faces a different “relevant
+rival” in the contest. This introduces an asymmetry into the model that was absent in
+the two-player case, and which generally leads to an asymmetric equilibrium. To see
+this formally, suppose, for simplicity, that $g(\theta_i, e_i) = \theta_i + e_i$, implying that $a_e(x) = 1$ (the
+following intuition also holds for general production technologies). Then, using a similar
+reasoning as in the two-player case (see Section 4), the marginal probability of winning
+
+²¹Contests with more than two players have been studied by, e.g., Tullock (1980), Nalebuff and Stiglitz (1983), Hillman and Riley (1989), Zábojník and Bernhardt (2001), Chen (2003), Zábojník (2012), and Ryvkin and Drugov (2020).

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+for player *i* and player *k* in a symmetric equilibrium can be written, respectively, as:
+
+$$ \frac{\partial P_i(e_1, e_2, \dots, e_n)}{\partial e_i} |_{e_1 = \dots = e_n = e} = \int_{\mathbb{R}} f_i(x) \frac{d}{dx} \left( F_k(x) \prod_{j \neq i, k} F_j(x) \right) dx $$
+
+and
+
+$$ \frac{\partial P_k(e_1, e_2, \dots, e_2)}{\partial e_k} |_{e_1 = \dots = e_n = e} = \int_{\mathbb{R}} f_k(x) \frac{d}{dx} \left( F_i(x) \prod_{j \neq i, k} F_j(x) \right) dx. $$
+
+Applying the product differentiation rule on the RHS of the above expressions, we obtain:
+
+$$ \int_{\mathbb{R}} \left( f_i(x)f_k(x) \prod_{j \neq i,k} F_j(x) + f_i(x)F_k(x) \frac{d}{dx} \left( \prod_{j \neq i,k} F_j(x) \right) \right) dx \quad (6) $$
+
+and
+
+$$ \int_{\mathbb{R}} \left( f_k(x)f_i(x) \prod_{j \neq i,k} F_j(x) + f_k(x)F_i(x) \frac{d}{dx} \left( \prod_{j \neq i,k} F_j(x) \right) \right) dx. \quad (7) $$
+
+The first term in (6) and (7) corresponds to the situation in which all players $j \in \{1, \dots, n\}, j \neq i, k$ perform worse than players $i$ and $k$ so that the $n$-player contest collapses to a contest between players $i$ and $k$. For this subcontest, the marginal winning probabilities are the same as shown in the analysis of the two-player contest. The second term in (6) corresponds to the situation in which player $i$ outperforms his or her rival $k$, such that the contest boils down to a contest between player $i$ and the strongest of the players $j \in \{1, \dots, n\}, j \neq i, k$. The interpretation of the second term in (7) is analogous, with the role of $i$ and $k$ interchanged.
+
+Setting expression (6) equal to expression (7), we obtain:
+
+$$ 
+\begin{align*}
+& \int_{\mathbb{R}} f_i(x) F_k(x) \frac{d}{dx} \left( \prod_{j \neq i, k} F_j(x) \right) dx = \int_{\mathbb{R}} f_k(x) F_i(x) \frac{d}{dx} \left( \prod_{j \neq i, k} F_j(x) \right) dx \\
+\vspace{0.5em}
+& \Leftrightarrow \int_{\mathbb{R}} \left( \frac{f_i(x)}{F_i(x)} - \frac{f_k(x)}{F_k(x)} \right) F_i(x) F_k(x) \frac{d}{dx} \left( \prod_{j \neq i, k} F_j(x) \right) dx = 0.
+\end{align*}
+$$
+
+Notice that *i* and *k* were arbitrarily selected. Hence, in order for a symmetric equilibrium to exist, it must be the case that the above condition holds for all $i, k \in \{1, \dots, n\}, i \neq k$. We conclude that the condition above is generally violated when the skill distributions of the competing players are distinct, which implies that a symmetric equilibrium generally does not exist in the case of *n* > 2 players.²²
+
+²²A sufficient condition for a symmetric equilibrium to exist is that $\frac{f_i(x)}{F_i(x)} = \frac{f_k(x)}{F_k(x)}$ for all $x \in \mathbb{R}$ and all $i, k \in \{1, \dots, n\}, i \neq k$. However, since $\frac{f_i(x)}{F_i(x)} = \frac{d}{dx}\log F_i(x)$ is the reversed hazard rate, which completely characterizes the properties of symmetric equilibria.

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+### 7.1.2 The Case of Homogeneous Players
+
+Suppose that all players share the same skill distribution, i.e., $f_1 = f_2 = \dots = f_n = f$, and define $r_e(x) := a_e(x)f(x)$.
+
+**Proposition 6.** In an *n*-player contest with homogeneous skill distributions, a symmetric Nash equilibrium with $e_1 = e_2 = \dots = e_n = e^*$ exists and is characterized by
+
+$$V \int_{\mathbb{R}} r_{e^*}(x)(n-1)(F(x))^{n-2}f(x)dx = V \int_{\mathbb{R}} r_{e^*}(x) \frac{d}{dx}((F(x))^{n-1})dx = c'(e^*). \quad (8)$$
+
+*Proof.* See Appendix A.9. $\square$
+
+Notice that $(F(x))^{n-1}$ describes the cdf of the highest order statistic out of a group of $n-1$ players. The condition from the proposition therefore illustrates what was mentioned before: the *n*-player contest boils down to a two-player contest, in which every player competes against the strongest of the other players.
+
+A particular focus in the literature has been on the relation between effort and the number of competitors. Early studies of the *n*-player Tullock contest with $\gamma_1 = \dots = \gamma_n$, $m=1$, and linear effort costs found that equilibrium effort is given by $e^* = \frac{n-1}{n^2}V$, so that effort is decreasing in *n* (e.g., Tullock 1980, Hillman and Riley 1989). With a convex cost function (as in our setting), the condition would change to $e^*c'(e^*) = \frac{n-1}{n^2}V$, but effort would still be decreasing in *n*. The result can be explained by a discouragement effect. If a player competes against many rivals, his or her chance of winning is relatively low and the player reduces effort in turn.
+
+In what follows, we study the relationship between effort and the number of competitors in our model. To do so, we need to extend Assumption 2 to the *n*-player case.
+
+**Assumption 3.** The primitives of the model are such that $q_n : E \to \mathbb{R}$, defined by
+
+$$q_n(e) = V \int_{\mathbb{R}} r_e(x) \frac{d}{dx}((F(x))^{n-1}) dx - c'(e),$$
+
+is strictly decreasing.
+
+We observe that, in addition to the discouragement effect mentioned before, there is also an encouragement effect, inducing players to increase their effort as they compete against more players. This is reflected by the factor $(n-1)$ in $\int_{\mathbb{R}} r_e(x)(n-1)(F(x))^{n-2}f(x)dx$
+
+terizes a statistical distribution, this condition would only hold for all $x \in \mathbb{R}$ if $F_i$ and $F_k$ refer to identical distributions.

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+in Proposition 6 above. As we will show, the encouragement effect might dominate, opening up for the possibility that effort increases in the number of competitors (for a related result, see also Ryvkin and Drugov 2020). In our proof, we make use of the fact that increasing $n$ leads to a distribution of the highest order statistic that first-order stochastically dominates the original distribution. We can thus invoke Proposition 3 to study the effects of an increase in $n$ on equilibrium effort.²³
+
+**Proposition 7.** Consider the *n*-player contest with homogeneous skill distributions and let $e^*$ denote the symmetric Nash equilibrium effort. Then the following statements hold:
+
+i) If $r_e(x)$ is strictly increasing for all $x \in \text{supp}(f)$ and all $e \ge 0$, then $e^*$ increases in $n$.
+
+ii) If $r_e(x)$ is strictly decreasing for all $x \in \text{supp}(f)$ and all $e \ge 0$, then $e^*$ decreases in $n$.
+
+iii) If $r_e(x)$ is constant for all $x \in \text{supp}(f)$ and all $e \ge 0$, then $e^*$ does not depend on $n$.
+
+*Proof.* See Appendix A.10. $\square$
+
+We conclude this subsection with an example to illustrate the potentially positive relationship between effort and the number of players in the context of the well-known Lazear-Rosen model.
+
+**Example 6.** Consider a contest with an additive production function $g(\theta, e) = \theta + e$, $V = 1$, and cost function $c(e) = \frac{e^2}{2}$. Suppose each $\Theta_i$ is distributed according to the modified reflected exponential distribution with mean $\mu = 1$ and pdf $f(x) = \frac{1}{2}\exp(\frac{1}{2}(x-3))$ for $x \le 3$ and zero otherwise (see, e.g., Rinne 2014). With two players, equilibrium effort is $e^* = \frac{1}{4}$. With three players, equilibrium effort increases to $\tilde{e}^* = \frac{1}{3}$.
+
+### 7.1.3 A Contest With One Player Who Is More Highly Skilled
+
+We now turn to a special case of our contest model with $n > 2$ players where we obtain a symmetric equilibrium even when players are asymmetric in the sense of having different expected skills. For this purpose, suppose that $\Theta_i = t_i + \mathcal{E}_i$ for $i = 1, \dots, n$ where $t_1 > t_2 > \dots = t_n = t$ and the $\mathcal{E}_i$, $i = 1, \dots, n$, are i.i.d. according to the reflected exponential distribution with cdf $H(x) = \exp(\lambda x)$ defined on $(-\infty, 0]$ with $\lambda > 0$ (Rinne 2014). In this case, we have:
+
+$$ f_i(x) = \begin{cases} \lambda \exp(\lambda(x - t_i)), & \text{for } x \le t_i \\ 0, & \text{for } x > t_i \end{cases} $$
+
+²³Notice that similar to what was mentioned in connection to Corollary 2, there is a small caveat to part (i) of Proposition 7. If $e^*$ is increasing in $n$, a symmetric equilibrium in which all players exert positive effort will fail to exist if $n$ becomes so large that $V/n < c(e^*)$, as (some) players would prefer to choose an effort of zero.

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+cludes the well-known models by Tullock (1980) and Lazear and Rosen (1981) as special cases.
+
+The skill distributions of the competing players (including the expected values) are assumed to be common knowledge, whereas the exact skill realizations are generally (symmetrically) unknown (as, e.g., in Holmström 1982). These assumptions realistically reflect that in a promotion contest, for example, the expected skill of a player may be commonly known (e.g., the education, prior work experience, or CV of a player), whereas the exact skill level for the particular job is unknown (e.g., there might be uncertainty regarding how education translates into workplace performance and job match).
+
+The main contribution of the paper is to show the existence of a symmetric (pure-strategy equal-effort) equilibrium in a general two-player contest setting where players have heterogeneous skill distributions, and how simple equilibria can emerge also when considering heterogeneity in additional dimensions, such as production technologies, prizes, and cost functions. We also make two additional contributions.
+
+First, we provide intuition regarding how the different components of the contest interact to determine the incentive to exert effort. More specifically, we highlight the interaction between three factors. The first factor relates to the production technology and is the ratio of the marginal product of effort and the marginal product of skill. The intuition behind this factor is that the purpose of a marginal effort increase for an individual player is to beat marginally more able rivals. The ratio describes how effective a marginal effort increase is to overcome the output advantage of marginally more skilled players. The second and third factors are represented by the product of the densities of the skill distributions of the two competing players, evaluated at the same point. The reason for the presence of this product is that a player only has a marginal incentive to exert effort in cases where the skill realizations of the two players are exactly the same, and the product describes the “likelihood” of this event to happen.
+
+Second, we use the simple structure of the equilibrium to construct a link between our contest model and standard models of decision-making under risk (expected utility theory), allowing us to revisit important comparative statics results of contest theory. In particular, we analyze how equilibrium effort is affected by making the skill distributions of the competing players more heterogeneous, investigating both the role of differences in expected skill (conceptualized by first-order stochastic dominance) and the role of differences in the uncertainty of the skill distributions of the competing players (conceptualized by second-order stochastic dominance), and how these relationships are

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+and
+
+$$F_i(x) = \begin{cases} \exp(\lambda(x - t_i)), & \text{for } x \le t_i \\ 1, & \text{for } x > t_i, \end{cases}$$
+
+implying that $\frac{f_i(x)}{F_i(x)} = \lambda$ on the support of $f_i$ which is $(-\infty, t_i]$. Consider the condition
+
+$$\int_{\mathbb{R}} \left( \frac{f_i(x)}{F_i(x)} - \frac{f_k(x)}{F_k(x)} \right) F_i(x) F_k(x) \frac{d}{dx} \left( \prod_{j \neq i, k} F_j(x) \right) dx = 0,$$
+
+that we derived in Subsection 7.1.1.²⁴ It is satisfied for all $i,k \in \{2,...,n\}$ since in this case, $\frac{f_i(x)}{F_i(x)} = \frac{f_k(x)}{F_k(x)} = \lambda$ on the common support $(\infty,t]$ of $f_i$ and $f_k$ (since we assumed from the outset that $t_2 = t_3 = \cdots = t_n = t$). Consider now the case where $i=1$ and $k \in \{2,...,n\}$. In this case, we have that $\frac{f_1(x)}{F_1(x)} = \frac{f_k(x)}{F_k(x)} = \lambda$ for $x \le t$. For $x > t$, we have $\prod_{j \neq 1, k} F_j(x) = 1 \Rightarrow \frac{d}{dx} \left( \prod_{j \neq 1, k} F_j(x) \right) = 0$. Hence, we conclude that the condition is satisfied for all $i,k \in \{1,...,n\}$, $i \neq k$ and all $x \in \mathbb{R}$, and that the marginal winning probabilities are the same. This implies that a symmetric equilibrium exists in which all players choose the same equilibrium effort $e^*$.
+
+Next, we compute an example with a multiplicative production technology and show
+that the marginal winning probability is increasing in the number of players.
+
+**Proposition 8.** Consider the contest described above. Suppose the production technology takes the form $g(\theta, e) = \theta e$ and assume that $t$ is chosen sufficiently large so that $n\lambda t - 1 > 0$. Then, the marginal winning probability given equal effort $e$ is equal to:
+
+$$\Psi(n) = \frac{(n-1)}{e} \exp(-\lambda(t_1-t)) \frac{(n\lambda t - 1)}{n^2}, \quad \text{with} \quad \Psi'(n) > 0.$$
+
+*Proof*. See Appendix A.11. $\square$
+
+## 7.2 Privately Known Skills
+
+We now turn to examine how our analysis is affected by assuming that players have private information regarding their own skill. For this purpose, we assume that each player $i \in \{1, ..., n\}$ observes his or her own skill realization $\theta_i$ before choosing effort $e_i$. This means that each player chooses a strategy consisting of a function $e_i(\theta_i)$ that specifies the effort level for each value of $\theta_i$. Everything else in our model remains unchanged.
+
+²⁴This condition was derived under the assumption of an additive production technology. In Appendix B.1, we provide a proof for the existence of a symmetric equilibrium in the general case.

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+In particular, all the opponents' skills $\Theta_k$, $k \in \{1,...,n\}$, $k \neq i$ remain uncertain, as in the main model, and their distributions are common knowledge.
+
+This private-information assumption effectively implies that player *i* can, in a deterministic manner, choose output $g(\theta_i, e_i)$ by making the appropriate effort choice $e_i$. The decision problem of player *i* can therefore, equivalently, be expressed as the specification of optimal effort $e_i(\theta_i)$ or the choice of optimal output $z_i(\theta_i) := g(\theta_i, e_i(\theta_i))$, as a best response to the opponents' choice of effort or output. Assuming that optimal output is strictly increasing in skill (this will be confirmed in our examples), $z_i$ is invertible with inverse $z_i^{-1}$.
+
+In the two-player case, where player *i* competes against another player *k*, player *i* wins the contest for given realizations of $\Theta_i$ and $\Theta_k$ if and only if the following condition holds
+
+$$g(\theta_k, e_k) < g(\theta_i, e_i) \iff z_k(\theta_k) < z_i(\theta_i) \iff \theta_k < z_k^{-1}(z_i(\theta_i)).$$
+
+Taking into account that, from the perspective of player *i*, the uncertainty of the contest only concerns the skill realization of player *k*, we have that equilibrium efforts $e_i(\theta_i)$ and $e_k(\theta_k)$ satisfy:
+
+$$e_i(\theta_i) \in \arg\max_{e_i} \left\{ F_k(z_k^{-1}(z_i(\theta_i)))V - c(e_i) \right\},$$
+
+$$e_k(\theta_k) \in \arg\max_{e_k} \left\{ F_k(z_i^{-1}(z_k(\theta_k)))V - c(e_k) \right\}.$$
+
+It can thus immediately be seen that the first-order condition for player *i* only involves the skill distribution of the opposing player *k*, whereas the first-order condition for player *k* only involves the skill distribution of the opposing player *i*. Hence, the symmetry that was present in the main model, where the first-order condition for each player involved the product of $f_i$ and $f_k$ (see equation (4)), vanishes when skills are privately known. We thus conclude that the equilibrium effort functions $e_i(\theta_i)$ and $e_k(\theta_k)$ generally are not symmetric.
+
+The *n*-player case with privately known skills is handled in an almost identical fashion. Instead of competing against player *k*, player *i* can be viewed as competing against the strongest of the opponents $j \in \{1,...,n\}$, $j \neq i$, in the sense of the highest order statistic. We analyze the *n*-player case with symmetric skill distributions below. Our analysis generalizes existing contest models with privately informed players with respect to the production technology. For instance, when imposing a multiplicative production func-

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+tion, our model is equivalent to the single-prize version of Moldovanu and Sela (2001). Likewise, in the case of an additive production function, our model matches the nondiscriminatory contest in Pérez-Castrillo and Wettstein (2016). We note, however, that these models are more general in other respects. Whereas Moldovanu and Sela (2001) allow for multiple prizes, Pérez-Castrillo and Wettstein (2016) allow for the single prize to differ between contestants.
+
+**The case of *n* homogeneous players.** We revisit the setting with *n* homogeneous players considered in Section 7.1.2 and introduce the private-information assumption. The exposition also serves to illustrate the two-player case with symmetric players and private information.
+
+In a symmetric setting, we naturally expect symmetric equilibria in which players employ the same effort function, $e(\theta_i)$. Thus, equal types imply equal effort and output even when individuals are privately informed about their own type. As we did in Section 7.1.2, we analyze the *n*-player case by analyzing how player *i* competes against the highest order statistic of his or her opponents. We denote the distribution function of this order statistic by $F^{(n-1)}$ with the associated probability density function $f^{(n-1)}$. Given that players are assumed to have independent skill distributions, $F^{(n-1)}(x) = F(x)^{n-1}$ and $f^{(n-1)}(x) = (n-1)F(x)^{n-2}f(x)$.
+
+To solve for a symmetric equilibrium, we consider the problem of player *i* maximizing his or her expected payoff when all his or her rivals adopt the common effort function $e(\theta_k)$, or equivalently, the common output function $z(\theta_k) := g(\theta_k, e(\theta_k))$, $i \neq k$ with corresponding inverse $z^{-1}$. The equilibrium effort of player *i* is thus given by:
+
+$$e_i(\theta_i) \in \arg\max_{e_i} \left\{ F^{(n-1)}(z^{-1}(g(\theta_i, e_i)))V - c(e_i) \right\}.$$
+
+For each value of $\theta_i$, there is an associated first-order condition:
+
+$$f^{(n-1)}(z^{-1}(z_i(\theta_i))) \frac{1}{z'(z^{-1}(z_i(\theta_i))))} \frac{\partial g(\theta_i, e_i(\theta_i))}{\partial e_i} V = c'(e_i(\theta_i)),$$
+
+where $\frac{\partial g(\theta_i, e_i(\theta_i))}{\partial e_i}$ is the partial derivative of $g(\theta_i, e_i(\theta_i))$ with respect to the second argument. In a symmetric equilibrium, we can drop the index *i*, thus the first-order condition in equilibrium can be written as
+
+$$f^{(n-1)}(\theta) \frac{\partial g/\partial e}{z'(\theta)} V = c'(e(\theta)).$$

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+The above condition implicitly defines the symmetric equilibrium effort function $e(\theta)$.
+
+Note that since
+
+$$z'(\theta) = \frac{dg(\theta, e(\theta))}{d\theta} = \frac{\partial g}{\partial \theta} + \frac{\partial g}{\partial e} \frac{de(\theta)}{d\theta},$$
+
+we have that the first-order condition can be written as:
+
+$$f^{(n-1)}(\theta) \frac{\frac{\partial g}{\partial e}}{\frac{\partial g}{\partial \theta} + \frac{\partial g}{\partial e} e'(\theta)} V = c'(e(\theta)). \quad (9)$$
+
+Condition (9) has an intuitive interpretation. The LHS is the marginal probability of winning times the prize V in a symmetric equilibrium from the perspective of a player who knows that his or her skill is $\theta$. Given that a player only has a marginal incentive to exert effort when the strongest opponent (the highest order statistic) has the same skill, $f^{(n-1)}(\theta)$ is the “likelihood” of this situation. There are two main differences with respect to the corresponding condition for the case of symmetric uncertainty (equation (4)). First, because players know their own skill level, there is no need to integrate over all possible realizations of a considered player’s own skill. Second, instead of $a_e(x) =$
+
+$$\frac{\partial g(x,e)}{\partial e} / \frac{\partial g(x,e)}{\partial x} \text{ (which appeared inside the integral of (4)), we now have the factor } \frac{\frac{\partial g}{\partial e}}{\frac{\partial g}{\partial \theta} + \frac{\partial g}{\partial e} e'(\theta)}$$
+
+which includes the new term $\frac{\partial g}{\partial e}e'(\theta)$ in the denominator. This new term arises because effort is a function of skill in the private information case.
+
+Recall that when we discussed the intuition behind $a_e(x)$ in equation (4), we explained that the purpose of a marginal effort increase is to beat rivals who have marginally higher skill. In the current setting, the output advantage of marginally more able rivals is not only determined by $\frac{\partial g}{\partial \theta}$ (which is positive) but also by the additional term $\frac{\partial g}{\partial e}e'(\theta)$ which generally has an ambiguous sign. If $\frac{\partial g}{\partial e}$ and $e'(\theta)$ are both strictly positive, more highly skilled rivals are harder to beat not only because of their skill advantage, but also because they exert higher effort, reducing the marginal incentive to exert effort by any player.
+
+In the following example, we compute the equilibrium effort for a specific skill distribution and production function.²⁵
+
+**Example 7.** Consider a contest with *n* symmetric players with privately known skills independently drawn from the uniform distribution on [0, 1]. The production function is given by $g(\theta, e) = \theta e$, and the cost function is $c(e) = \frac{e^2}{2}$. Then, the symmetric equilibrium
+
+²⁵The detailed derivations for this example are provided in Appendix B.2.

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+effort is:
+
+$$e(\theta) = \sqrt{\frac{2(n-1)}{n+1} V^{\theta^{n-1}}}.$$
+
+Notice that for the contest in the above example, $r_e(x)$ is strictly increasing on $[0, 1]$ for all $e \ge 0$. Hence, equilibrium effort in the symmetric uncertainty case would be *increasing* in $n$ according to Proposition 7. To obtain an analogue of this result in the case of private information, we can compute the expectation of the equilibrium effort in Example 7 to obtain:
+
+$$E[e(\theta)] = \sqrt{(n-1)\left(\frac{2}{n+1}\right)^3} V. \quad (10)$$
+
+We immediately see that the expected effort in (10) is *decreasing* in $n$. Hence, Example 7 serves to demonstrate that the comparative statics results from the baseline case with symmetric uncertainty do not necessarily carry over to the private-information case.
+
+# 8 Concluding Remarks
+
+We have explored simple equilibria in contests between heterogeneous players. Under general assumptions about the production technology and the skill distributions of the competing players, we have shown that the contest has a symmetric equilibrium in which all players exert the same effort. We have also provided intuition regarding how the different components of the contest interact to determine the incentive to exert effort and revisited several important comparative statics results of contest theory, showing that standard results in the literature are not necessarily robust to generalizations of the production technology or skill distributions. In particular, we have found that making players more heterogeneous can increase the incentive to exert effort. We have also investigated the robustness of our results with respect to the assumption of symmetric uncertainty and the number of players.
+
+We would like to mention a few broader implications of our analysis. First, our main result regarding the emergence of symmetric equilibria in the presence of heterogeneous players is quite surprising, and an important message of our paper is that differences between people do not necessarily translate into different behavior in contest situations. Second, our finding that making skill distributions more heterogeneous can increase equilibrium effort, has implications for optimal team composition, as employers could

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+find it desirable to increase the diversity of the workforce by hiring a worker drawn from
+a more uncertain skill distribution, such as a minority worker, or a worker for whom less
+prior information is available.
+
+There are several possible extensions to our analysis. For instance, prior work has investigated strategic information revelation by the tournament designer (e.g., Aoyagi 2010). If the tournament designer possesses some private information about the players' skills, he or she may decide to reveal some or all of this information to trigger higher effort. Another extension would be to consider an endogenous prize structure. Finally, the implications of our model for promotion tournaments and hiring decisions could be further explored. We leave these interesting topics as avenues for future research.

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+# Appendix
+
+## A Proofs
+
+### A.1 Proof of Lemma 1
+
+Suppose that $\frac{\partial P_i(e_i, e_k)}{\partial e_i}|_{e_i=e_k=e}$ is the same for both $i \in \{1,2\}$ and all $e \in \text{int } E$. Then we have
+$\frac{\partial P_1(e_1, e_2)}{\partial e_1}|_{e_1=e_2=e} V - c'(e) = \frac{\partial P_2(e_2, e_1)}{\partial e_2}|_{e_1=e_2=e} V - c'(e)$ for all $e \in \text{int } E$. Since $\pi_i(e_i, e_k)$ is continuously differentiable, $\frac{\partial P_i(e_i, e_k)}{\partial e_i}|_{e_i=e_k=e} V - c'(e)$ is a continuous function of $e$. Furthermore, recall that there exist $\bar{e}_i, \tilde{e}_i \in \text{int } E$ such that $\frac{\partial \pi_i(e_i, e_k)}{\partial e_i}|_{e_i=e_k=\bar{e}_i} < 0$ and $\frac{\partial \pi_i(e_i, e_k)}{\partial e_i}|_{e_i=e_k=\tilde{e}_i} > 0$. Hence, by the Intermediate Value Theorem, there is some $e^* \in \text{int } E$ such that $\frac{\partial P_i(e_i, e_k)}{\partial e_i}|_{e_i=e_k=e^*} V - c'(e^*) = 0$. By Assumption 1, $e_1 = e_2 = e^*$ is a Nash equilibrium.
+
+### A.2 Proof of Theorem 1
+
+Since we wish to apply the sufficient condition from Lemma 1, we restrict attention to $e_i > 0$. Then, the function $g_e : \mathbb{R} \to \mathbb{R}$ defined by $g_e(x) = g(x,e)$ is strictly increasing and, thus, invertible. The inverse, $g_e^{-1}$, is strictly increasing as well. For the two (different) players $i,k \in \{1,2\}$, we observe
+
+$$ 
+\begin{align*}
+& g(\theta_i, e_i) < g(\theta_k, e_k) \\
+\Leftrightarrow & g_{e_i}(\theta_i) < g_{e_k}(\theta_k) \\
+\Leftrightarrow & \theta_i < g_{e_i}^{-1}(g_{e_k}(\theta_k)).
+\end{align*}
+$$
+
+Player *k* thus wins with probability
+
+$$ \int F_i(g_{e_i}^{-1}(g_{e_k}(x))) f_k(x) dx. $$
+
+Differentiating with respect to $e_k$, we obtain
+
+$$ \int f_i(g_{e_i}^{-1}(g_{e_k}(x))) \left( \frac{d}{de_k} g_{e_i}^{-1}(g_{e_k}(x)) \right) f_k(x) dx. $$
+
+According to Lemma 1, and noting that $g_{e_k}^{-1}(g_{e_i}(x)) = g_{e_i}^{-1}(g_{e_k}(x)) = x$ if $e_i = e_k$, a sufficient condition for a symmetric equilibrium to exist is that
+
+$$ 
+\begin{align*}
+& \int \left( \left. \frac{d}{de_1} g_{e_2}^{-1}(g_{e_1}(x)) \right|_{e_1=e_2=e} f_1(x) f_2(x) dx \right. \\
+& = \int \left. \left. \frac{d}{de_2} g_{e_1}^{-1}(g_{e_2}(x)) \right|_{e_1=e_2=e} f_1(x) f_2(x) dx \right.
+\end{align*}
+$$

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+for all $e \in \text{int } E$. Since $\left. \frac{d}{de_1} g_{e_2}^{-1}(g_{e_1}(x)) \right|_{e_1=e_2=e} = \left. \frac{d}{de_2} g_{e_1}^{-1}(g_{e_2}(x)) \right|_{e_1=e_2=e}$, this condition is
+always fulfilled.
+
+**A.3 Proof of Corollary 1**
+
+Define $\tilde{\Theta}_i := h_i(\Theta_i)$. The considered contest is then equivalent (in terms of equilibrium effort choices) to a contest in which skills are given by $\tilde{\Theta}_i$ and the production function $\tilde{g}(\tilde{\theta}_i, e_i)$ is the same for both players. Hence, a symmetric equilibrium exists with effort determined by the condition provided in the corollary.
+
+**A.4 Proof of Proposition 1**
+
+Player 1's objective function is given by
+
+$$\int_{\mathbb{R}} F_2(g_{e_2}^{-1}(g_{e_1}(x))) f_1(x) dx s V - c(e_1).$$
+
+Using $\xi_1 = e_1/s^{1/\delta}$, the preceding expression becomes
+
+$$ 
+\begin{align*}
+& \int_{\mathbb{R}} F_2(g_{e_2}^{-1}(g_{s^{1/\delta}\xi_1}(x))) f_1(x) dx s V - c(s^{1/\delta}\xi_1) \\
+&= \int_{\mathbb{R}} F_2(g_{e_2}^{-1}(g_{s^{1/\delta}\xi_1}(x))) f_1(x) dx s V - s c(\xi_1) \\
+&= s \left( \int_{\mathbb{R}} F_2(g_{e_2}^{-1}(g_{s^{1/\delta}\xi_1}(x))) f_1(x) dx V - c(\xi_1) \right).
+\end{align*}
+$$
+
+Maximizing this function is equivalent to maximizing
+
+$$\int_{\mathbb{R}} F_2(g_{e_2}^{-1}(g_{s^{1/\delta}\xi_1}(x))) f_1(x) dxV - c(\xi_1).$$
+
+Player 2's objective function can be stated as
+
+$$ 
+\begin{align*}
+& \int_{\mathbb{R}} F_1(g_{e_1}^{-1}(g_{e_2}(x))) f_2(x) dxV - c(e_2) \\
+&= \int_{\mathbb{R}} F_1(g_{s^{1/\delta}\xi_1}^{-1}(g_{e_2}(x))) f_2(x) dxV - c(e_2).
+\end{align*}
+$$
+
+**A.5 Proof of Proposition 2**
+
+(i) Player 1 wins if and only if
+
+$$ 
+\begin{align*}
+& \theta_1^{\alpha} e_1^{\beta} > \theta_2^{\alpha} e_2^{\beta} \\
+& \Leftrightarrow \theta_1^{\frac{\alpha}{\beta}} e_1 > \theta_2^{\frac{\alpha}{\beta}} e_2.
+\end{align*}
+$$

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+Substituting $\xi_1 = e_1/s^{1/\delta}$, the condition becomes
+
+$$ \theta_1^{\frac{\alpha}{\beta}} s^{\frac{1}{\delta}} \xi_1 > \theta_2^{\frac{\alpha}{\beta}} e_2. $$
+
+Now define $\tilde{\Theta}_1 := s^{\frac{1}{\delta}} \Theta_1^{\frac{\alpha}{\beta}}$ and $\tilde{\Theta}_2 := \Theta_2^{\frac{\alpha}{\beta}}$, and denote the corresponding pdfs and cdfs by $\tilde{f}_1$, $\tilde{f}_2$, $\tilde{F}_1$, and $\tilde{F}_2$, respectively.
+
+Player 1's objective function can be stated as
+
+$$ \int_{\mathbb{R}} \tilde{F}_2 \left( \frac{\xi_1 x}{e_2} \right) \tilde{f}_1(x) dx s V - c \left( s^{1/\delta} \xi_1 \right) \\ = s \left( \int_{\mathbb{R}} \tilde{F}_2 \left( \frac{\xi_1 x}{e_2} \right) \tilde{f}_1(x) dx V - c(\xi_1) \right). $$
+
+Maximization of the objective function is equivalent to maximization of
+
+$$ \int_{\mathbb{R}} \tilde{F}_2 \left( \frac{\xi_1 x}{e_2} \right) \tilde{f}_1(x) dxV - c(\xi_1), $$
+
+so we consider this latter problem. Since player 2's objective function can be stated as
+
+$$ \int_{\mathbb{R}} \tilde{F}_1 \left( \frac{e_2 x}{\xi_1} \right) \tilde{f}_2(x) dxV - c(e_2), $$
+
+we have transformed the contest into the form of our main model, meaning that an
+equilibrium $(e_1^*, e_2^*)$, where $e_2^* = \xi_1^* = \frac{e_1^*}{s^\delta}$ is characterized by
+
+$$ \int_{\mathbb{R}} \tilde{f}_1(x) \tilde{f}_2(x) x dx V = e_2^* c'(e_2^*), $$
+
+exists.
+
+(ii) Notice that player 1 wins if and only if
+
+$$ 
+\begin{align*}
+& \alpha\theta_1 + \beta e_1 > \alpha\theta_2 + \beta e_2 \\
+\Leftrightarrow & \frac{\alpha}{\beta}\theta_1 + e_1 > \frac{\alpha}{\beta}\theta_2 + e_2 \\
+\Leftrightarrow & \exp\left(\frac{\alpha}{\beta}\theta_1 + e_1\right) > \exp\left(\frac{\alpha}{\beta}\theta_2 + e_2\right) \\
+\Leftrightarrow & \exp\left(\frac{\alpha}{\beta}\theta_1\right)\exp(e_1) > \exp\left(\frac{\alpha}{\beta}\theta_2\right)\exp(e_2).
+\end{align*}
+$$
+
+Define $\tilde{e}_i = \exp(e_i)$ for $i \in \{1, 2\}$. From part (i) of the proposition, we know that, in equilibrium, $\tilde{e}_1 = s^{1/\delta}\tilde{e}_2$. (Note that this is equivalent to $e_i = \ln(\tilde{e}_i)$ and $\exp(e_1) = s^{1/\delta}\exp(e_2) \Leftrightarrow$

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+$e_1 = \frac{1}{\delta} \ln s + e_2$.) Hence, defining $\xi_1 = \frac{\tilde{e}_1}{s^{1/\delta}}$, the event of player 1 winning can be restated as
+
+$$ \exp\left(\frac{\alpha}{\beta}\theta_1\right) s^{1/\delta} \xi_1 > \exp\left(\frac{\alpha}{\beta}\theta_2\right) \tilde{e}_2. $$
+
+Now define $\tilde{\Theta}_1 := \exp(\frac{\alpha}{\beta}\Theta_1)s^{1/\delta}$ and $\tilde{\Theta}_2 := \exp(\frac{\alpha}{\beta}\Theta_2)$, and denote the corresponding pdfs and cdfs by $\tilde{f}_1, \tilde{f}_2, \tilde{F}_1,$ and $\tilde{F}_2$, respectively.
+
+Player 1's objective function can then be stated as
+
+$$ 
+\begin{align*}
+& \int_{\mathbb{R}} \tilde{F}_2 \left( \frac{\xi_1 x}{\tilde{e}_2} \right) \tilde{f}_1(x) dx s V - c \left( \ln \left( s^{\frac{1}{\delta}} \xi_1 \right) \right) \\
+&= \int_{\mathbb{R}} \tilde{F}_2 \left( \frac{\xi_1 x}{\tilde{e}_2} \right) \tilde{f}_1(x) dx s V - s c (\ln(\xi_1)) \\
+&= s \left( \int_{\mathbb{R}} \tilde{F}_2 \left( \frac{\xi_1 x}{\tilde{e}_2} \right) \tilde{f}_1(x) dx V - \tilde{c}(\xi_1) \right),
+\end{align*}
+$$
+
+where the first transformation used the homogeneity of the function $c \circ \ln(e)$. Maximizing
+the objective is equivalent to maximizing
+
+$$ \int_{\mathbb{R}} \tilde{F}_2 \left( \frac{\xi_1 x}{\tilde{e}_2} \right) \tilde{f}_1(x) dxV - \tilde{c}(\xi_1) $$
+
+and we consider this latter problem in what follows. Since player 2's objective function
+can be stated as
+
+$$ 
+\begin{align*}
+& \int_{\mathbb{R}} \tilde{F}_1 \left( \frac{\tilde{e}_2 x}{\xi_1} \right) \tilde{f}_2(x) dxV - c(\ln(\tilde{e}_2)) \\
+&= \int_{\mathbb{R}} \tilde{F}_1 \left( \frac{\tilde{e}_2 x}{\xi_1} \right) \tilde{f}_2(x) dxV - \tilde{c}(\tilde{e}_2),
+\end{align*}
+$$
+
+we have transformed the contest into the form of our main model, meaning that an
+equilibrium with $\xi_1^* = \tilde{e}_2^*$ characterized by
+
+$$ \int_{\mathbb{R}} \tilde{f}_1(x) \tilde{f}_2(x) x dx V = \tilde{e}_2^* c'(\tilde{e}_2^*) $$
+
+exists. The optimal efforts of the original contest are given by $e_1^* = \ln(s^{\frac{1}{\delta}}\xi_1^*)$ and $e_2^* = \ln(\tilde{e}_2^*)$.
+
+A.6 Proof of Proposition 3
+
+Suppose that Assumption 2 holds, and consider case (i), i.e., $r_{e,i}(x)$ is monotonically increasing in $x$, and $\tilde{F}_k$ first-order stochastically dominates $F_k$. Denote the equilibrium effort levels for the two contests by $\tilde{e}^*$ and $e^*$, respectively. Our goal is to show that

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+affected by the production technology. The general message is that making contest par-
+ticipants more heterogeneous can increase equilibrium effort. These findings contradict
+certain “standard” results known from the Tullock contest and the Lazear-Rosen tourna-
+ment. Thus, the comparative statics results derived from those standard models are not
+representative of the conclusions derived in the more general model.
+
+To shed further light on our results, we also provide two extensions to our main anal-
+ysis. In a first extension, we study the behavior in contests when the number of players
+$n$ is greater than two and show that the existence of a symmetric equilibrium, and the
+interpretation for the two-player case, extend to the $n$-player case when players have
+identical skill distributions. We also show that, for a specific class of skill distributions, a
+symmetric equilibrium exists when $n - 1$ identical players compete against a player who
+has a higher expected skill. Moreover, we show that increasing the number of contest-
+tants can increase equilibrium effort, exploiting the fact that a contest with $n > 2$ players
+can be interpreted as a two-player contest in which every player competes against the
+strongest (i.e., the highest order statistic) of his or her opponents.¹
+
+In a second extension, we investigate the robustness of our results with respect to the assumption of symmetric uncertainty by analyzing the consequences of letting players be privately informed about their skills. In this case, equilibria are in general not symmetric, but focusing on symmetric players, we are able to draw interesting parallels with respect to our baseline case, highlighting the role of our general production technology in influencing the marginal incentive to exert effort.
+
+We also discuss the implications of our findings for optimal team composition and certain real-world applications in the context of labor and personnel economics. For instance, our finding that efforts can increase if the skill distribution of one of the competing players becomes more uncertain (in the sense of second-order stochastic dominance) has several interesting managerial implications. It indicates that contest organizers might wish to increase the uncertainty regarding the skills of certain players in order to induce higher effort. In a worker-firm context, employers could achieve this by, for instance, hiring a worker for whom little prior information is available, or a minority worker with a skill level drawn from a distribution that generally tends to be more uncertain (as argued, e.g., by Bjerk 2008). This means that having diverse teams might be desirable from an employer's point of view.
+
+¹This result can be understood by the fact that as the number of contestants increases, the strongest opponents grow stronger in the sense of first-order stochastic dominance, allowing us to apply our results from the two-player case.

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+$e^* > e^*$.
+
+The proof proceeds by contradiction, so suppose $\tilde{e}^* \le e^*$. Now observe that
+
+$$ 
+\begin{align*}
+V \int r_{\tilde{e}^*,i}(x) \tilde{f}_k(x) dx - c'(\tilde{e}^*) &\ge \\
+V \int r_{e^*,i}(x) \tilde{f}_k(x) dx - c'(e^*) &> \\
+V \int r_{e^*,i}(x) f_k(x) dx - c'(e^*) &= 0.
+\end{align*}
+$$
+
+The first inequality follows from $\tilde{e}^* \le e^*$ together with Assumption 2. The second inequality follows from $r_{e,i}(x)$ being monotonically increasing on $\text{supp}(f_i)$, $\tilde{F}_k$ first-order stochastically dominating $F_k$, and the fact that we have assumed that both $\text{supp}(\tilde{f}_k)$ and $\text{supp}(f_k)$ are subsets of $\text{supp}(f_i)$.²⁶ The equality follows since $e^*$ is characterized by the first-order condition $V \int r_{e^*,i}(x)f_k(x)dx - c'(e^*) = 0$. We conclude that
+
+$$ V \int r_{\tilde{e}^*,i}(x) \tilde{f}_k(x) dx - c'(\tilde{e}^*) > 0. $$
+
+This shows that the first-order condition for equilibrium effort cannot be fulfilled in the case of the distribution $\tilde{F}_k$, giving us the desired contradiction.
+
+By an analogous argument, we can show that $\tilde{e}^* > e^*$ also in case (ii) where $r_{e,i}$ is monotonically decreasing in $x$ for all $e > 0$ and $F_k$ first-order stochastically dominates $\tilde{F}_k$. In this case, $\int r_{e,i}(x)\tilde{f}_k(x)dx > \int r_{e,i}(x)f_k(x)dx$ for all $e > 0$ (see, e.g., Levy 1992, p.557).
+
+## A.7 Proof of Proposition 4
+
+Suppose that $g(\theta,e) = \theta e$. This means that
+
+$$ a_e(x) = \frac{d}{de_i} \left( g_{e_k}^{-1}(g_{e_i}(x)) \right) \bigg|_{e_i=e_k=e} = \frac{d}{de_i} \left( \frac{xe_i}{e_k} \right) \bigg|_{e_i=e_k=e} = \frac{x}{e}. $$
+
+In the considered situation, the marginal winning probability is
+
+$$ \frac{1}{2\pi\sigma_1\sigma_2} \int \frac{x}{e} \exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2} - \frac{(x-\mu_2)^2}{2\sigma_2^2}\right) dx. $$
+
+²⁶See, e.g., Levy 1992, p.557. Notice that, in decision theory, the utility function is defined for all possible payoffs and therefore no additional constraints regarding the statistical supports need to be imposed.

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+To prove the proposition, it is sufficient to show that
+
+$$
+\begin{align*}
+& \frac{1}{2\pi\sigma_1\sigma_2} \int x \exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2} - \frac{(x-\mu_2)^2}{2\sigma_2^2}\right) dx \\
+&= \frac{(\mu_1\sigma_2^2 + \mu_2\sigma_1^2) \exp\left(-\frac{(\mu_1-\mu_2)^2}{2(\sigma_1^2+\sigma_2^2)}\right)}{(2\pi)^{\frac{1}{2}} (\sigma_1^2+\sigma_2^2)^{\frac{3}{2}}}.
+\end{align*}
+$$
+
+Define
+
+$$
+Z := \frac{1}{2\pi\sigma_1\sigma_2} \int x \exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2} - \frac{(x-\mu_2)^2}{2\sigma_2^2}\right) dx
+$$
+
+and notice that:
+
+$$
+\begin{align*}
+& \frac{(x - \mu_1)^2}{2\sigma_1^2} + \frac{(x - \mu_2)^2}{2\sigma_2^2} \\
+&= \frac{\sigma_2^2 (x - \mu_1)^2 (\sigma_1^2 + \sigma_2^2) + \sigma_1^2 (x - \mu_2)^2 (\sigma_1^2 + \sigma_2^2)}{2\sigma_1^2 \sigma_2^2 (\sigma_1^2 + \sigma_2^2)} \\
+&= \frac{\sigma_2^2 (x^2 - 2x\mu_1 + \mu_1^2)(\sigma_1^2 + \sigma_2^2) + \sigma_1^2 (x^2 - 2x\mu_2 + \mu_2^2)(\sigma_1^2 + \sigma_2^2)}{2\sigma_1^2 \sigma_2^2 (\sigma_1^2 + \sigma_2^2)} \\
+&= \frac{x^2 (\sigma_1^2 + \sigma_2^2)^2 - 2x (\mu_1 \sigma_2^2 + \mu_2 \sigma_1^2)(\sigma_1^2 + \sigma_2^2) + (\mu_1^2 \sigma_2^2 + \mu_2^2 \sigma_1^2)(\sigma_1^2 + \sigma_2^2)}{2\sigma_1^2 \sigma_2^2 (\sigma_1^2 + \sigma_2^2)} \\
+&= \frac{x^2 (\sigma_1^2 + \sigma_2^2)^2 - 2x (\mu_1 \sigma_2^2 + \mu_2 \sigma_1^2)(\sigma_1^2 + \sigma_2^2) + (\mu_1 \sigma_2^2 + \mu_2 \sigma_1^2)^2}{2\sigma_1^2 \sigma_2^2 (\sigma_1^2 + \sigma_2^2)} \\
+&\qquad - \frac{(\mu_1^2 \sigma_2^4 + 2\mu_1 \sigma_2^2 \mu_2 \sigma_1^2 + \mu_1^2 \sigma_1^4 - \mu_1^2 \sigma_1^4 (\sigma_1^2 + \sigma_1^4) - \mu_1^4 \sigma_1^4 (\sigma_1^4 + \sigma_1^6))}{2\sigma_1^4 \sigma_3^4 (\sigma_1^4 + \sigma_3^4)} \\
+&= \frac{(x(\sigma_1^4 + \sigma_3^4) - (\mu_1 \sigma_3^4 + \mu_3 \sigma_1^4))^3 + (\mu_3^4 \sigma_3^4 - 6\mu_3 \sigma_3^4 (\mu_3^4 + \mu_3^6) - 9\mu_3^6 (\mu_3^4 + \mu_3^6))^3}{6\mu_3^6 (\mu_3^4 + \mu_3^6)} \\
+&= \frac{(x(\sigma_1^4 + \sigma_3^4) - (\mu_1 \sigma_3^4 + \mu_3 \sigma_1^4))^3}{6\mu_3^6 (\mu_3^4 + \mu_3^6)} + \frac{(\mu_3 - \mu_1)^3}{6(\mu_3 - \mu_1)^3} \\
+&= \frac{(x(\sigma_1^4 + \sigma_3^4) - (\mu_1 \sigma_3^4 + \mu_3 \sigma_1^4))^3}{6\mu_3^6 (\mu_3 - \mu_1)^3} + \frac{(\mu_3 - \mu_1)^3}{6(\mu_3 - \mu_1)^3}.
+\end{align*}
+$$
+
+Using this, we obtain
+
+$$
+Z = \frac{\exp\left(-\frac{(\mu_{1}-\mu_{2})^{2}}{2(\sigma_{1}^{2}+\sigma_{2}^{2})}\right)}{2\pi\sigma_{1}\sigma_{2}} \int x \exp\left(-\frac{(x(\sigma_{1}^{2}+\sigma_{3}^{2})-(\mu_{1}\sigma_{3}^{2}+\mu_{3}\sigma_{1}^{3}))^{2}}{8\sigma_{1}^{2}\sigma_{3}^{2}(\sigma_{1}^{3}+\sigma_{3}^{3})}\right) dx \\
+= \frac{\exp\left(-\frac{(\mu_{1}-\mu_{3})^{2}}{6(\sigma_{1}^{3}+\sigma_{3}^{3})}\right)}{(2\pi)^{\frac{5}{6}}(\sigma_{1}^{3}+\sigma_{3}^{3})^{\frac{5}{6}}} (\sigma_{1}^{3}+\sigma_{3}^{3}) \frac{1}{\sqrt{6\pi}\frac{\sigma_{1}\sigma_{3}}{\sqrt{\sigma_{1}^{3}+\sigma_{3}^{3}}}} \int x \exp\left(-\frac{\left(x-\frac{\mu_{1}\sigma_{3}^{3}+\mu_{3}\sigma_{1}^{3}}{\sigma_{1}^{3}+\sigma_{3}^{3}}\right)^{6}}{8\left(\frac{\sigma_{1}\sigma_{3}}{\sqrt{\sigma_{1}^{3}+\sigma_{3}^{3}}}\right)^{6}}\right) dx.
+$$

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+Now notice that
+
+$$ \frac{1}{\sqrt{2\pi} \frac{\sigma_1\sigma_2}{\sqrt{\sigma_1^2+\sigma_2^2}}} \int x \exp\left(-\frac{\left(x-\frac{\mu_1\sigma_2^2+\mu_2\sigma_1^2}{\sigma_1^2+\sigma_2^2}}{2\left(\frac{\sigma_1\sigma_2}{\sqrt{\sigma_1^2+\sigma_2^2}}\right)^2}\right)} dx $$
+
+describes the mean of a normally distributed random variable with variance $\left(\frac{\sigma_1\sigma_2}{\sqrt{\sigma_1^2+\sigma_2^2}}\right)^2$ and mean $\frac{\mu_1\sigma_2^2+\mu_2\sigma_1^2}{\sigma_1^2+\sigma_2^2}$, hence
+
+$$ \frac{1}{\sqrt{2\pi} \frac{\sigma_1\sigma_2}{\sqrt{\sigma_1^2+\sigma_2^2}}} \int x \exp\left(-\frac{\left(x-\frac{\mu_1\sigma_2^2+\mu_2\sigma_1^2}{\sigma_1^2+\sigma_2^2}\right)^2}{2\left(\frac{\sigma_1\sigma_2}{\sqrt{\sigma_1^2+\sigma_2^2}}\right)^2}\right) dx = \frac{\mu_1\sigma_2^2+\mu_2\sigma_1^2}{\sigma_1^2+\sigma_2^2}. $$
+
+We obtain
+
+$$ 
+\begin{aligned}
+Z &= \frac{\exp\left(-\frac{(\mu_1 - \mu_2)^2}{2(\sigma_1^2 + \sigma_2^2)}\right)}{(2\pi)^{0.5}(\sigma_1^2 + \sigma_2^2)^{1.5}} (\sigma_1^2 + \sigma_2^2) \frac{\mu_1\sigma_2^2 + \mu_2\sigma_1^2}{\sigma_1^2 + \sigma_2^2} \\
+&= \frac{(\mu_1\sigma_2^2 + \mu_2\sigma_1^2)\exp\left(-\frac{(\mu_1 - \mu_2)^2}{2(\sigma_1^2 + \sigma_2^2)}\right)}{(2\pi)^{1/2}(\sigma_1^2 + \sigma_2^2)^{3/2}}.
+\end{aligned}
+$$
+
+## A.8 Proof of Proposition 5
+
+Because of Assumption 2, and the condition characterizing equilibrium effort, we need to show that $\int r_{e,i}(x) \tilde{f}_k(x) dx > (=, <) \int r_{e,i}(x) f_k(x) dx$ if $r_{e,i}$ is convex (linear, concave). The proof is very similar to part a) of the proof of Theorem 2 in Rothschild and Stiglitz (1970, p.237). In the case of convex $r_{e,i}$, the inequality in their proof is reversed, while it is replaced by an equality if $r_{e,i}$ is linear.
+
+## A.9 Proof of Proposition 6
+
+Player *i* wins the contest with probability
+
+$$ \int \prod_{k \neq i} F_k(g_{e_k}^{-1}(g_{e_i}(x))) f_i(x) dx. $$

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+Differentiating with respect to $e_i$, we obtain
+
+$$ \int \left( \prod_{k \neq i} F_k(g_{e_k}^{-1}(g_{e_i}(x))) \right) \left( \sum_{k \neq i} \frac{f_k(g_{e_k}^{-1}(g_{e_i}(x))) (\frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(x)))}{F_k(g_{e_k}^{-1}(g_{e_i}(x)))} \right) f_i(x) dx. $$
+
+In a symmetric equilibrium with $e_1^* = \dots = e_n^* =: e^*$, and symmetric skill distributions, this marginal effect of effort on the probability of winning simplifies to
+
+$$ \int \left( \prod_{k \neq i} F(x) \right) \left( \sum_{k \neq i} \left( \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(x)) \right) \right|_{e_1^*=\dots=e_n^*=e^*} \frac{f(x)}{F(x)} f(x) dx, $$
+
+and must be identical for all $i$. We can restate the above expression as
+
+$$ \int r_{e^*}(x)(n-1)(F(x))^{n-2}f(x)dx = \int r_{e^*}(x)\left(\frac{d}{dx}(F(x))^{n-1}\right)dx, $$
+
+which is identical for all $i$.
+
+## A.10 Proof of Proposition 7
+
+**Part i)** As explained in the main body of the paper, the equilibrium first-order condition for an *n*-player contest is equivalent to that of a two-player contest in which the second player's skill distribution is replaced by the strongest rival's skill distribution (the highest order statistic) of the *n*-player contest. We show that $\int r_e(x) (\frac{d}{dx}(F(x))^{n-1}) dx$ is increasing in *n*. If $n_1, n_2 \in \mathbb{N}$, with $n_1 > n_2$, then $(F(x))^{n_1-1}$ first-order stochastically dominates $(F(x))^{n_2-1}$, and the result follows from Proposition 3.
+
+**Part ii)** Suppose that $r_e(x)$ is monotonically decreasing in $x$ for all $e \ge 0$, and let $n_1, n_2 \in \mathbb{N}$, with $n_1 > n_2$. It follows that $(F(x))^{n_1-1}$ first-order stochastically dominates $(F(x))^{n_2-1}$, as just mentioned, implying that
+
+$$ \int r_e(x) \left( \frac{d}{dx} (F(x))^{n_1-1} \right) dx < \int r_e(x) \left( \frac{d}{dx} (F(x))^{n_2-1} \right) dx. $$
+
+**Part iii)** If $r_e(x) = r_e$ is constant in $x$ for all $e \ge 0$, we have
+
+$$ \int r_e(x) \left( \frac{d}{dx} (F(x))^{n-1} \right) dx = r_e \int \left( \frac{d}{dx} (F(x))^{n-1} \right) dx = r_e, $$
+
+which is independent of $n$.

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+A.11 Proof of Proposition 8
+
+As shown before, if $g(\theta_i, e_i) = \theta_i e_i$, we have
+
+$$ \left. \left( \frac{d}{de_i} g_{e_k}^{-1} (g_{e_i}(t_i + x)) \right) \right|_{e_1=\dots=e_n=e} = \frac{(t_i+x)}{e}. $$
+
+Thus, making use of expression (11), derived in Section B.1, and denoting $\Delta t = t_1 - t > 0$,
+
+$$ \begin{align*} & \lambda^2 \int^{-\Delta t} \left( H(x) \prod_{k \neq i} H(\Delta t + x) \right) \left( \sum_{k \neq i} \left( \frac{d}{de_i} g_{e_k}^{-1} (g_{e_i}(t_1+x)) \right) \right) \Big|_{e_1=\dots=e_n=e} dx \\ &= \lambda^2 \int^{-\Delta t} \exp(n\lambda y) \cdot \exp((n-1)\lambda(\Delta t + x)) (n-1) \frac{(t_1+x)}{e} dx \\ &= \frac{\lambda^2(n-1)}{e} \int^{-\Delta t} \exp(n\lambda y + (n-1)\lambda\Delta t)(t_1+y) dy. \end{align*} $$
+
+The map $\phi_2 : \mathbb{R}_x \to \mathbb{R}_y$ given by $x \to y = -\Delta t + x$ is a smooth diffeomorphism with $\det|\phi_2'(x)| = 1$. Applying the associated change of variables to the integral, we obtain
+
+$$ \begin{align*} & \frac{\lambda^2(n-1)}{e} \int^0 \exp(n\lambda(x-\Delta t) + (n-1)\lambda\Delta t)(t+x)dx \\ &= \frac{(n-1)}{e} \exp(-\lambda\Delta t) \lambda^2 \left( \int^0 x \exp(n\lambda x)dx + t \int^0 \exp(n\lambda x)dx \right). \end{align*} $$
+
+Notice that
+
+$$ n\lambda \int^{0} x \exp(n\lambda x) dx $$
+
+is the mean of a random variable that is distributed according to the reflected exponential distribution with parameter $n\lambda$, hence
+
+$$ n\lambda \int^{0} x \exp(n\lambda x) dx = -\frac{1}{n\lambda} $$
+
+$$ \Leftrightarrow \int^{0} x \exp(n\lambda x) dx = -\frac{1}{n^2\lambda^2}. $$
+
+Furthermore,
+
+$$ n\lambda \int^{0} \exp(n\lambda x) dx = 1 $$
+
+$$ \Leftrightarrow \int^{0} \exp(n\lambda x) dx = \frac{1}{n\lambda}. $$

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+It follows that
+
+$$
+\begin{align*}
+& \frac{(n-1)}{e} \exp(-\lambda \Delta t) \lambda^2 \left( \int^0 x \exp(n\lambda x) dx + t \int^0 \exp(n\lambda x) dx \right) \\
+= & \frac{(n-1)}{e} \exp(-\lambda \Delta t) \lambda^2 \frac{(-1+n\lambda t)}{n^2\lambda^2} \\
+= & \frac{(n-1)}{e} \exp(-\lambda \Delta t) \frac{(-1+n\lambda t)}{n^2}.
+\end{align*}
+$$
+
+Notice that the last expression is positive if and only if $n\lambda t - 1 > 0$. Taking the derivative of the expression w.r.t. $n$ results in an expression that is positive if $t > 0$, which is implied by $n\lambda t - 1 > 0$.
+
+# B Other Computations and Derivations
+
+## B.1 Additional Derivations for Section 7.1.3.
+
+Player *i* outperforms player *k* iff
+
+$$
+\begin{align*}
+g_{e_i}(t_i + \varepsilon_i) &> g_{e_k}(t_k + \varepsilon_k) \\
+\Leftrightarrow \quad & \varepsilon_k < g_{e_k}^{-1}(g_{e_i}(t_i + \varepsilon_i)) - t_k.
+\end{align*}
+$$
+
+Recall that the $E_i$ are i.i.d., following the reflected exponential distribution on $(-\infty, 0]$.
+The cdf is denoted by $H$ and the pdf by $h$. Hence, player $i$ wins the contest with probability
+
+$$
+\int \prod_{k \neq i} H(g_{e_k}^{-1}(g_{e_i}(t_i+x)) - t_k) h(x) dx.
+$$
+
+In a symmetric equilibrium with $e_1^* = \dots = e_n^* =: e^*$, the marginal effect of effort on the probability of winning,
+
+$$
+\int \left( \prod_{k \neq i} H(t_i + x - t_k) \right) \left( \sum_{k \neq i} \left( \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(t_i + x)) \right) \Big|_{e_1^* = \dots = e_n^* = e^*} \frac{h(t_i + x - t_k)}{H(t_i + x - t_k)} \right) h(x)dx,
+$$
+
+must be the same for all *i*. Denote $\Delta t = t_1 - t > 0$. For player 1, we have,
+
+$$
+\int \left( \prod_{k \neq 1} H(\Delta t + x) \right) \left( \sum_{k \neq 1} \left( \frac{d}{de_1} g_{e_k}^{-1}(g_{e_1}(t_1 + x)) \right) \Big|_{e_1^* = \dots = e_n^* = e^*} \frac{h(\Delta t + x)}{H(\Delta t + x)} \right) h(x) dx.
+$$

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+For any other player $i \in \{2, \dots, n\}$, we have
+
+$$
+\int \left( H(-\Delta t + y) \prod_{k \neq 1, i} H(y) \right) \left[ \left( \left. \frac{d}{de_i} g_{e_1}^{-1}(g_{e_i}(t+y)) \right|_{e_1^* = \dots = e_n^* = e^*} - \frac{h(-\Delta t + y)}{H(-\Delta t + y)} + \sum_{k \neq 1, i} \left( \left. \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(t+y)) \right|_{e_1^* = \dots = e_n^* = e^*} - \frac{h(y)}{H(y)} \right) h(y) \right] dy.
+$$
+
+The map $\phi_1 : \mathbb{R}_x \to \mathbb{R}_y$ given by $x \to y = \Delta t + x$ is a smooth diffeomorphism with $\det |\phi'_1(x)| =$
+
+1. Applying the associated change of variables to the preceding expression, we obtain
+
+$$
+\int \left( H(x) \prod_{k \neq 1, i} H(\Delta t + x) \right) \left[ \left. \left( \frac{d}{de_i} g_{e_1}^{-1}(g_{e_i}(t_1+x)) \right) \right|_{e_1^* = \dots = e_n^* = e^*} - \frac{h(x)}{H(x)} + \sum_{k \neq 1, i} \left( \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(t_1+x)) \right) \right] \left. \frac{h(\Delta t+x)}{H(\Delta t+x)} \right|_{e_1^* = \dots = e_n^* = e^*} h(\Delta t+x) dx.
+$$
+
+The expressions for the two types of players can be restated as
+
+$$
+\begin{align*}
+& \int \left( H(x) \prod_{k \neq 1} H(\Delta t + x) \right) \left[ \left( \sum_{k \neq 1} \left( \frac{d}{de_k} g_{e_k}^{-1}(g_{e_k}(t_1+x)) \right) \right) \right]_{e_1^* = \dots = e_n^* = e^*} \frac{h(\Delta t + x)}{H(\Delta t + x)} \frac{h(x)}{H(x)} dx, \\
+& + \sum_{k \neq 1, i} \left( \left. \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(t_1+x)) \right|_{e_1^* = \dots = e_n^* = e^*} \frac{h(\Delta t + x)}{H(\Delta t + x)} \right) \frac{h(\Delta t + x)}{H(\Delta t + x)} dx.
+\end{align*}
+$$
+
+Notice that both expressions are equal to zero for $x \ge -\Delta t$. Hence, they can be restated
+as
+
+$$
+\begin{align*}
+& \int^{-\Delta t} \left( H(x) \prod_{k \neq 1} H(\Delta t + x) \right) \left[ \left( \sum_{k \neq 1} \left( \frac{d}{de_1} g_{e_k}^{-1}(g_{e_1}(t_1+x)) \right) \right)_{e_1^* = \dots = e_n^* = e^*} - \frac{h(\Delta t + x)}{H(\Delta t + x)} \right] \frac{h(x)}{H(x)} dx, \\
+& + \sum_{k \neq 1, i} \left( \left. \frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(t_1+x)) \right|_{e_1^* = \dots = e_n^* = e^*} - \frac{h(\Delta t + x)}{H(\Delta t + x)} \right) \frac{h(\Delta t + x)}{H(\Delta t + x)} dx.
+\end{align*}
+$$
+
+For $x < -\Delta t$, we observe $\frac{h(x)}{H(x)} = -\frac{h(\Delta t + x)}{H(\Delta t + x)} = -\lambda$, and the expressions become
+
+$$
+\begin{equation}
+\begin{aligned}
+&\lambda^2 \int^{-\Delta t} \left(H(x) \prod_{k \neq 1} H(\Delta t + x)\right) \left[\left(\sum_{k \neq 1} \left(\frac{d}{de_k} g_{e_k}^{-1}(g_{e_k}(t_1+x))\right)\right]_{e_1^* = \dots = e_n^* = e^*} dx, \\
+&\lambda^2 \int^{-\Delta t} \left(H(x) \prod_{k \neq 1} H(\Delta t + x)\right) \left[\left(\sum_{k \neq i} \left(\frac{d}{de_i} g_{e_k}^{-1}(g_{e_i}(t_1+x))\right)\right]_{e_1^* = \dots = e_n^* = e^*} dx
+\end{aligned}
+\tag{11}
+\end{equation}
+$$

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+which are identical.
+
+## B.2 Computations for Example 7
+
+The first-order condition (9) is equivalent to (we use notation by writing $e$ instead of $e(\theta)$)
+
+$$c'(e) \frac{\partial g/\partial \theta}{\partial g/\partial e} - f^{(n-1)}(\theta)V + c'(e) \frac{de}{d\theta} = 0, \quad (12)$$
+
+which can be restated as
+
+$$P(\theta, e) + Q(\theta, e) \frac{de}{d\theta} = 0,$$
+
+with $P(\theta,e) := c'(e) \frac{\partial g/\partial \theta}{\partial g/\partial e} - f^{(n-1)}(\theta)V$ and $Q(\theta,e) := c'(e)$.
+
+Is there an integrating factor $\mu(\theta,e)$ such that $\frac{\partial(\mu P)}{\partial e} = \frac{\partial(\mu Q)}{\partial \theta}$? In other words, is there $\mu(\theta,e)$ such that
+
+$$\frac{\partial \mu}{\partial e} P + \mu \frac{\partial P}{\partial e} = \frac{\partial \mu}{\partial \theta} Q + \mu \frac{\partial Q}{\partial \theta}？$$
+
+The latter equation can be stated as
+
+$$\frac{\partial \mu}{\partial e} \left( c'(e) \frac{\partial g}{\partial g} + f^{(n-1)}(\theta) V \right) + \mu \left( c''(e) \frac{\partial g}{\partial g} + c'(e) \frac{\partial^2 g}{\partial g \partial \theta} \cdot \frac{\partial g}{\partial e} - \frac{\partial g}{\partial \theta} \cdot \frac{\partial^2 g}{\partial e^2} \right) = \frac{\partial \mu}{\partial \theta} c'(e).$$
+
+Now, for our example, assume $g(\theta,e) = \theta e$ and $c(e) = 0.5e^2$, and ignore the argument $\theta$ in $e(\theta)$. Then the equation simplifies to
+
+$$\frac{\partial \mu}{\partial e} \left( \frac{e^2}{\theta} - f^{(n-1)}(\theta) V \right) + \mu \left( 2 \frac{e}{\theta} \right) = \frac{\partial \mu}{\partial \theta} e.$$
+
+Suppose that $\frac{\partial \mu}{\partial e} = 0$. Then $\mu$ needs to satisfy
+
+$$\mu \frac{2}{\theta} = \frac{\partial \mu}{\partial \theta}$$
+
+and a solution is $\mu(\theta,e) = \theta^2$ (confirming $\frac{\partial \mu}{\partial e} = 0$).
+
+Using $g(\theta,e) = \theta e$ and $c(e) = 0.5e^2$, our differential equation (12) can be stated as
+
+$$\frac{e^2}{\theta} - f^{(n-1)}(\theta)V + e \frac{de}{d\theta} = 0,$$

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+and multiplication with $\mu(\theta, e) = \theta^2$ leads to
+
+$$\theta e^2 - \theta^2 f^{(n-1)}(\theta) V + e\theta^2 \frac{de}{d\theta} = 0.$$
+
+An integral is
+
+$$L(\theta, e(\theta)) = \frac{e(\theta)^2 \theta^2}{2} - V \int_0^\theta x^2 f^{(n-1)}(x) dx,$$
+
+which can easily be verified by computing $\frac{dL(\theta, e(\theta))}{d\theta}$.
+
+With a general distribution, effort is given by the solution to
+
+$$\begin{align*}
+\frac{e(\theta)^2 \theta^2}{2} - V \int_0^\theta x^2 f^{(n-1)}(x) dx &= \hat{c} \\
+\Leftrightarrow e(\theta) &= \sqrt{\frac{2V}{\theta^2} \int_0^\theta x^2 f^{(n-1)}(x) dx + \frac{2\hat{c}}{\theta^2}},
+\end{align*}$$
+
+where $\hat{c}$ is some constant.
+
+Using the assumption that skills are uniformly distributed on [0, 1] (implying $f(x) = 1$ and $F^{(n-1)}(t) = t^{n-1} \Rightarrow f^{(n-1)}(t) = (n-1)t^{n-2}$), we can compute effort and expected effort.
+
+In particular,
+
+$$\int_{0}^{\theta} x^{2} f^{(n-1)}(x) dx = (n-1) \int_{0}^{\theta} x^{n} dx = \frac{n-1}{n+1} \theta^{n+1},$$
+
+meaning that the integral becomes
+
+$$L(\theta, e) = \frac{e^2 \theta^2}{2} - V \frac{n-1}{n+1} \theta^{n+1}.$$
+
+Hence, the solution to the differential equation is given by
+
+$$\frac{1}{2} e^{2\theta} \theta^2 - V \frac{n-1}{n+1} \theta^{n+1} = \hat{c},$$
+
+where $\hat{c}$ is some constant. Solving for $e$, we obtain
+
+$$e(\theta) = \sqrt{2V \frac{n-1}{n+1} \theta^{n-1} + \frac{2\hat{c}}{\theta^2}}.$$
+
+Conjecturing $e(0) = 0$, we have $\hat{c} = 0$ and
+
+$$e(\theta) = \sqrt{2V \frac{n-1}{n+1} \theta^{n-1}}.$$

samples/texts/2875771/page_49.md

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+It follows that expected effort is
+
+$$E[e(\theta)] = \sqrt{2V \frac{n-1}{n+1}} \int_{0}^{1} x^{\frac{n-1}{2}} dx = \sqrt{8V \frac{n-1}{(n+1)^3}},$$
+
+which is strictly decreasing in $n$.
+
+It is straightforward to verify that the equilibrium effort function satisfies $e(0) = 0$ and is strictly increasing in the skill $\theta$. This implies that for any given skill $\theta$, output $g(\theta, e(\theta)) = \theta e(\theta)$ is increasing in skill as well, and the inverse $z^{-1}$ exists.

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+The paper is organized as follows. In Section 2 below, we discuss related literature. Section 3 introduces the contest model and discusses how our model nests the Tullock contest and the Lazear-Rosen tournament as special cases. Section 4 solves the two-player model when players have different skill distributions, whereas Section 5 addresses different production functions, prizes, and cost functions. In Section 6, we analyze the two-player case in greater detail and provide a set of comparative statics results. We also discuss implications for organizational design and optimal team composition. Section 7 studies the *n*-player case and takes a look at the case of privately known skills. Finally, Section 8 concludes.
+
+## 2 Related Literature
+
+There are three main approaches to the study of contests, the Tullock (or ratio-form) contest, the Lazear-Rosen tournament, and the complete-information all-pay auction.² In the Tullock contest, introduced by Tullock (1980), a player's winning probability is given by his/her contribution to the contest divided by the sum of the contributions of the competing players, and the contribution of each player is typically defined as a function of effort and sometimes also of skill.³ The Lazear-Rosen tournament assumes that the player with the highest contribution wins with certainty, and contributions depend on effort, some random factors (e.g., luck), and possibly on skills. The seminal paper is by Lazear and Rosen (1981) who apply the model in a labor-market context.⁴ The all-pay auction, finally, makes the same assumption as the Lazear-Rosen tournament, except that contest contributions are deterministic and do not depend on random factors.⁵
+
+²The theoretical contest literature has been surveyed in a number of books and papers. See, e.g., Konrad (2009) and Vojnović (2016) for recent textbooks and Chowdhury and Gürtler (2015), Chowdhury, Esteve-Gonzalez, and Mukherjee (2019), and Fu and Wu (2019) for recent surveys.
+
+³The Tullock contest has been analyzed by, e.g., Hillman and Riley (1989), Cornes and Hartley (2005), Fu and Lu (2009a,b), Corchón and Dahm (2010), Schweinzer and Segev (2012), and Chowdhury and Kim (2017). It has been axiomatized in various settings by Skaperdas (1996), Clark and Riis (1998b), and Münster (2009).
+
+⁴It has been further analyzed by, for example, Green and Stokey (1983), Malcomson (1984, 1986), O’Keeffe, Viscusi, and Zeckhauser (1984), Lazear (1989), Schotter and Weigelt (1992), Zábojník and Bernhardt (2001), Hvide (2002), Grund and Sliwka (2005), Schöttner and Thiele (2010), Gürtler and Gürtler (2015), and Imhof and Kräkel (2016).
+
+⁵A detailed equilibrium characterization of the all-pay auction was developed by Baye, Kovenock, and de Vries (1996). The complete-information all-pay auction (with mixed-strategy equilibria) is the most commonly used in contest theory, but a private-values version can be found as well. The all-pay auction has been further studied by, e.g., Clark and Riis (1998a), Barut and Kovenock (1998), Moldovanu and Sela (2001, 2006), Moldovanu, Sela, and Shi (2007), Cohen, Kaplan, and Sela (2008), Siegel (2009, 2010), Sela (2012), Morath and Münster (2013), Barbieri, Malueg, and Topolyan (2014), Olszewski and Siegel (2016), and Fang, Noe, and Strack (2020).

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+# References
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+CHOWDHURY, S. M., AND O. GÜRTLER (2015): “Sabotage in contests: A survey,” *Public Choice*, 164(1), 135–155.

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