CC BY
3Andre A. Keller
Universite de Haute Alsace, France E-mail: [email protected]
Abstract Research efforts and outcomes are generally private information of innovative firms : research inputs are unobservable and the value of innovations is difficult to evaluate . This is the reason why rank-order tournaments are more adequate incentive schemes rather than a conventional contracting.
The typical model considers a risk-neutral sponsor (commonly governments or private corporations) and a number of risk-neutral or risk-averse contestants, such as research teams, startup companies. The contestants are competing to find the ”best” innovation. The winner obtains the prize and the losers get nothing in a ”winner-take-all” game. The prize is thus awarded by the sponsor on the basis of relative rank rather than on the absolute performance. An innovation tournament belongs to the class of dynamic n-player two-stage games of imperfect information : at the ”entry stage” each firm decides whether to participate, at the ’’contest stage” each contestant decides whether to invest in each period without knowing the rivals’ choices.
The game is solved by backward induction. Provided the objective function is quasiconcave, the tournament subgame has a unique symmetric equilibrium in pure strategies. This contribution reviews the innovation tournament models for different probability distributions of shocks.
Keywords: research tournament, innovation race, two-stage game, incomplete information.
Research efforts and outcomes are generally private information of innovative firms : research inputs are unobservable and the value of innovations is difficult to evaluate. This is the reason why rank-order tournaments are more adequate incentive schemes rather than a conventional contracting (Taylor, 1995). The theory of rank-order tournaments was pioneered by Lazear and Rosen, 1981 and promoted as optimum labor contracts. The tournaments refer to incentive compensation schemes which pay according to an individual’s ordinal rank rather than on output levels. The research competition may be analyzed in a labor tournament framework (Carmichael, 1983, Ma, 1988, Roy Chowdhury, 2005, Zhou, 2006). The typical model considers a risk-neutral sponsor (commonly governments or private firms) and a number of risk-neutral or risk-averse contestants (such as research teams, startup companies). The contestants are competing to find the ”best” innovation. The winner obtains the prize and the losers get nothing in a ”winner-take-all” game. The prize is thus awarded by the sponsor on the basis of relative rank rather than on the absolute performance (Green and Stokey, 1986). Following Taylor, 1995, research tournaments and innovation races differ fundamentally : in tournaments the terminal date is fixed and the quality of innovations will vary, whereas in innovation races the date of discovery is unknown and the quality standard is fixed. An innovation tournament belongs to the class of dynamic n-player two-stage games of imperfect information : at the ”entry stage” each firm decides whether to participate, at the ’’contest stage” each contestant decides whether to
invest in each period without knowing the rivals’ choices. The game is solved by backward induction : the tournament is first solved for given prizes, then the sponsor’s expected profit is computed and the optimal prize is deduced. Provided the objective function is quasiconcave, the tournament subgame has a unique symmetric equilibrium in pure strategies (Taylor, 1995,Wolfstetter, 1999). In most innovation tournaments (Fullerton and McAfee, 1999, Taylor, 1995), the value of the winner prize is exogenous. In new tournament models (Baye and Hoppe, 2003), research inputs not only the determine the probability of winning but also the value of the winner prize. This contribution reviews the innovation tournament models for different probability distributions of shocks.
1. Theory of Rank-Order Tournaments
Rank-order tournaments belong to a variety of alternative compensation schemes. A rank-order tournament is a compensation scheme in which a contest earning will depend on the rank order of contestants rather than on their absolute outputs. The winner is paid more than his marginal product. This compensation scheme motivates a greater effort among the contestants. We will present the elementary model and solve it for two and more contestants 1.
1.1. Contracting framework
A tournament is played between several agents (workers of a firms, firms of an industry, etc). These agents are identical and perform similar but independent tasks. The agents and the firm (the principal) are supposed to be risk-neutral. The agents compete for fixed prizes by the firm with their efforts or actions.
The model consists of two main equations : the utility function and the agents’ production function. We will consider one particular agent by the subscript i and his rivals (all other contestants) by the subscript -i. The preferences of each agent over his earnings yk and his effort ak, k £ {i, -i} are represented by a von Neumann-Morgenstern utility function. This function is assumed to be additively separable in earnings y and action (or effort, or investment level) a. We then have 2
The marginal utility of earnings is positive(u' > 0) and weakly decreasing (u" < 0) with risk-averse agents. The cost c(a) or disutility from efforts is positive and increasing (c' > 0,c'' > 0). The observable agents’ production function is given by
the labor market notably by Drago and Heywood, 1989. Experimental comparison have been presented by Bull et al., 1987
2 This is the form given by Nalebuff and Stiglitz, 1983. The alternative utility function by
Lazear and Rosen, 1981 is of the form: Uk = U(yk, ak) = U^yk — c(ak)^ , for k G {i, -i},
where the random payment is defined by the two-parameter class function
U(yk,ak) = u(yu) - c(au), for k £ {i, -i}.
1 The efficiency of this compensation scheme has been compared to the piece rates on
Wl , if Xi > x-i, w2 , otherwise.
the random function of the effort level 3
Xk = x(ak, £ k) for k £ {i, -i},
where xj, x_j are the observable output of the agents and £, £_j the individual noise. The probability distribution of the random variable (RV) £ is known of the firm and agents, is zero mean and uncorrelated with effort. A simplified additive specification is
Xk = ak + £k for k £ {i, -i}.
The game consists of two stages with imperfect information. Nature draws outputs noises from the same distribution. This drawing is not revealed but the distribution of the outputs is common knowledge. At stage 1, the firm commits to pay two prizes, one (w1) is for the winner with high-output level and the other (w2) is for the low-output agent. The prizes are fixed arbitrarily. At stage 2, each agent chooses his effort when ignoring the rival’s choice. Output are observed and prizes are paid. Given the strategies of agents, the pair of prizes (w1,w2) is found such as to maximize an agent’s expected utility function, subject to the competitive zero-profit constraint by firms.
1.2. Two-contestant rank order tournaments under risk neutrality
This detailed presentation will be restricted to the case of risk-neutral agents as in (Wolfstetter, 1999). The expected profit of the firm is zero for a competitive market. The information is imperfect.
Agent problem. The two players are denoted by the subscripts i and -i. The problem of the agent i is to maximize his expected utility subject to the zero expected profit condition. We have
maxa, Wj = P(aj,a-j) wi + ^1 - P(aj,a_j)^ w2 - c(aj),
s.t.
wi + w2 - p.(aj + a_j) = 0,
where P(aj,a_j) = Prob{xj > x_j} is the probability of winning the tournament and p the price of good. If the agent i wins, the realization of the £ must satisfy the inequality
aj + £j > a_j + £_j.
For a given £_j, the probability of occurrence is given by 4
1 - F(a_j - aj + £_j).
3 This is a simplification of the form given by Nalebuff and Stiglitz, 1983. The initial equation introduces a multiplicative common random environment rj of the form xi = fj.ai + j for the ith agent. The alternative utility function by Lazear and Rosen, 1981 is of the additive form: xi = ai + ji + rj with zero mean.
4
Proof. From the probability Prob{ai + ei > a-i + e-i}, we deduce easily Prob{ei >
a-i — ai + e-i} = 1 — Prob{ei > a-i — ai + e-i} = 1 — F(a-i — ai + e-i) ■
The total probability of winning is obtained by integration over all values of the £i, weighted by the density f (£_i). We have
P(ai,a-i) = y|l - F^(
1 — F ^(a_i — ai) + £-ij jf (£-i)d£-i. The pair of equilibrium efforts levels (a*, a*i) will satisfy
a*k = argmax-j^W2 + F(ak — a%)(wi — w2) — c(ak) j, for k, l £ {i, —i}, k = l
Tournament subgame. The first- and second order conditions (FOC and SOC) are
(wi - w2) a -c'(afc) = 0, for k£{i,-i},
oak
{wi - w2^ _ c"(ak) = 0, for k £ {i, -i}.
The Nash equilibrium consists of a pair of optimal efforts, where each agent chooses his investment (or effort) as the best response to the rival’s effort. The reaction function are determined by
8P{aha-i) ,
{wi - w2)---------------c (ai) = 0,
dai
d2P(ai, a-i) „
(wi - w2)------------------c (a-i) = 0.
Provided the objective function is quasiconcave, the best replies are unique 5and the tournament has a unique equilibrium in pure strategies. Due to the symmetry of the reaction functions, the Nash equilibrium is such that ai = a_i = a* with the same probability of success P(aj,a_j) = P(a_j,aj) = The winner of the competition is designed by Nature. At the symmetric choice the player’s increasing chance of giving more effort is
-----~da■---- = ~da- J ^ _ F{a-i ~ ai + f{£—i)d£—i = J F'aif(£-i)d£-i.
Hence, for a symmetric effort, we have
dP(a, a)
= / f (£_i)f (£_i )d£_
da
= E[f (£)] = f (0).
According to the FOC, the optimal effort is implicitly defined by 6
(wi — W2 )f (0) = c' (a*).
The optimal effort is then strict monotone increasing in the prize spread w1 — w2.The chosen effort does not depend on the total amount of prizes.
5 The concavity of the distribution function F does not assure the quasiconcavity of the objective function. But if the cumulative distribution function (cdf) F is concave, the objective function is also concave and then quasiconcave.
6 This condition assumes a local optimum. To reach a global optimum, one supplementary incentive condition is necessary. According to that condition, the agent will be insured
Proposition 1. The principal (or firm) can induce any feasible effort as an equilibrium of the tournament subgame by choosing an appropriate prize spread.
Equilibrium prizes. Using the solution of the tournament subgame, we can determine the equilibrium prizes. The firm chooses those prizes that maximize the expected value of profits subject to the agents participation condition. We have
maxtoj.tijj l^2(p.a*(w1, w2) - -(wi + w2) j,
w2 + - w2) - c( a*(w1,w2) I > 0
1
2(
s.t.
1
21
Substituting the constraint into the objective function, we have the simplified decision problem
maxa 2 (p.a — c(a) + w2)
The equilibrium effort a then solves c'(a) = p. The equilibrium outcome (a* ,w* ,w*) of the tournament game is obtained from the conditions
c'(a) = P,
(w 1 — w2 )f (0) = c' (a),
^(wi +w2) = p.a*.
The optimal values of the pair of prizes (w * ,w2*) is given by
c' (a*)
w* = p.a +
2f(0)
* * c'(«*) w2 = P-a
2f(0)
These optimal prizes consist of a common part p.a* equal to the expected output Wl+W2 and a prime of zb 2/(0) = ± wi+w? wjnner anc[ to the looser respectively.
The agents are then not necessary paid to their marginal productivity. The prime is the incitation to participate to the tournament.
Numerical examples. Let us consider (see Wolfstetter, 1999) two different distributions of the RVs £ : a normal distribution N(0, a2) and a uniform distribution £ ^ N(0, a2), then d = £_i — £i ^ N(0, 2a2). The probability density
to receive an utility level, superior to the utility reservation. For a minimal effort a with probability P(a,a*), this condition implies Wi(a*) > Wi(a-i,aj). Then, we have |(wi — W2) — c(a*) > P(a-i, a*j)(w 1 — W2) — c(a-i) with (wi — W2)f(0) = c'(a*). The constraint is
Q - P(a-i, a*_i)j > c(a*) - c(a_i).
Then, this constraint is easier to satisfy with higher minimal effort. Moreover, since f (0) ^ to when the variance a2 ^ 0, a global optimum is achieved for higher variance.
function (pdf) is 7 f(0) = 1/V4cr27rexp(—02/Aa2) and /(0) = 1/V4cr27r. We then have the optimum
P> 2o-\/7r), ^p(p - 2o-v/7r)
If e-w then [-1,1]^.The pdf is 8
1 + 6 for 6 £ ( —1, 0],
f (6) = ^1 — 6 for 6 £ [0,1),
otherwise.
and we have f (0) = 1. In this case, the optimum is
P, \p{p+^), \p{p~^j-
1.3. Multiple contestants tournaments
The Lazear and Rosen’s tournament problem. Let us consider the Lazear-Rosen specification for the utility function and the output equation with risk-averse agents Lazear and Rosen, 1981. We have
U (yk, ak) = U (yk — c(ak)), for k £ {i, —i}, with U' ,c' ,c'' > 0, U'' < 0,
xk = ak + £k + n, for k £ {i, —i},
7 The sum of i.i.d and normally distributed RVs is again normally distributed, with variance equal to the sum of variances. More generally, suppose the RV X has a continuous pdf f (x) with support S C R. Let a RV Y, whose pdf is g(y) with support T C R, be a one-by-one differentiable function f of X. The pdf of Y is
9(y) = f(^1(y)).w^
or equivalently g(y) = f (^(y)).|^'(y)|, where ^ = f-1. The proof is easily obtained from the cumulative distribution function (cdf) of Y say G(y) = Prob{Y < y}, considering both monotonic increasing (f > 0) and decreasing (f < 0) cases.
8 Suppose that the RV Y is a linear transformation of RV X with Y = A.X where A is a real-valued invertible matrix. The joint pdf of Y is
g(y) = fiA^y), 1
detA|
In this case define Y = X1 — X2 and Z = X2 then A = ^ 0 ^ J and A 1 = ^ 1 j
The pdf of Y is g(y) = f f (y + z, z)dz. If X1 and X2 are i.i.d uniformly distributed on
the support (— |), Y is triangular distributed on the support ( — 1, 1). We have the
1 + y for y £ ( — 1, 0],
g(y) = ^1 — y for y £ [0, 1),
otherwise.
where the random environment is described a common RV fj (indicating the general difficulties of a set of tasks) and by the individualistic noise £ (giving the advantage or disadvantage of doing one task in particular). Moreover, we assume E £ k = E [ek ,£l] =0 and E [fj] =1. The functions F and f are respectively the distribution function and the density of the RVs £k. The principal problem is to choose the contract parameters and the effort levels of contestants so as to maximize the expected utility function subject to the zero-expected-profit condition and the incentive compatibility condition. The expected utility is given by 9
E[Ui] = (1 — Prob{xi < x_i})U(wi,ai) + Prob{xi < x—i }U(w2 ,ai)
= (1 — F(a_i — ai))U(wi — c(ai)) + F(a_i — ai)^ U (w2 — c(ai)^ ( )
The zero-expected-profit condition is such that the total prize w + w2 equals the expected outputs. We have
p.ak = ^1 — F(a_i — ai)j wi + F(a_i — ai)w2.
The incentive compatibility condition is obtained in maximizing the expected utility function E[Ui] w.r.t. ai. We have
T^-E [Ui] = /(a-i-dijU^W!-c(ai)j - c'(aj) ^1-F(a_*-a*)^ E/^oi-c(aj)^ -
f (a_i — ai)U^w2 — c(ai)^ — c'(ai)F^a_i — a^J U' ^w2 — c(a^ (2)
At the symmetric Nash solution, with a» = a_j = a and P(a,a) = the Lazear-Rosen problem is
maxffil]ffi2,o, E[t/j] = ^ “ c(ai)^j + U ^' w2 - c(ai)^j |,
p.ai
s.t. w + w2
c'(ai) =2/(0).-
2
U w — c(ai — U w2 — c(ai)
U' w — c(ai) + U' w2 — c(ai)
Taking second-order Taylor series expansions of the utility function and of the incentive compatibility condition around Zi = Wl+W2 — c(a.j), we have the quadratic approximation (McLaughlin, 1988)
TT fwi + w2 , A 1 wwi — w2 ) 2
m&xWl,w2,ai E[Ui\ — U[ ---------------c(a>i) I + 2 (Zi)\-2---'
s.t.
w + w2 P-ai =----------^-----’
c' (ai) = f (0).(w i — w2).
Indeed, we have Prob{xi < x-i} = Prob{ii — i-i < a-i — ai} = F(a- — ai).
9
Substituting the constraints into the objective function yields the unconstraint problem
rnax^, E[E/j] = U (p.a* ^f(0)Av?j - c(a*(/(0)2W)^ + ,
where Aw = wi — w2. Ignoring terms of order 3, we derive the marginal condition
The optimal prize spread is deduced as
f(0)p
Aw*
f(0)2 +
where Aa = — U''/U' denotes the absolute risk aversion. The optimal prize spread is increasing in product prize, decreasing in the risk aversion coefficient and curvature of the cost of effort. The optimal effort a* is deduced from the last incentive compatibility condition. We have to solve the implicit equation
/ / * \ P
c (a )
1 -i- A c 1 ^ a 4/(0)2
The optimal effort is increasing in product price and decreasing in the risk aversion coefficient and the curvature of the cost of effort.
The generalization to multiple contestants tournaments. The generalization of this model is given by McLaughlin, 1988, for a single top prize wn. The distribution function for each i.i.d RVs £ _i evaluated at £i + ai — a_i is
F(£i + ai — a_i) = Prob{Xi > x_i}.
The subscript -i denotes the n — 1 other contestants. The probability of ranking nth is i 0
Fn(n) = J F(£i + ai — a_i) f (£i)ds
At the symmetric Nash equilibrium, we have
fn (n) = (n — 1) / F(£i)n_2f (£i)2d£i.
10 The probability of ranking kth from the bottom, given the effort a* of other n — 1
contestants is
n- k
M I F (?. + a — a*\k-1 v I 1 — F (?. + a — a*
k —
Fn(k) = ^n _ ^ J F(si + ai — a*)k 1 v ^1 — F(ei + ai — a*^ f (ei)dei.
4
The n-contestant tournament problem for a single prize under the Lazear-Rosen specification i i is
maxtui :wn:ai E \Ui\ = (l — —) •U -h —IJ ^'wn — ,
s.t.
a 1) 1
p.cti = (1----jwi + -wn,
nn
^l^wn — c(ai) J — U i — c(ai) J j.fn(n) c (o-i) = y r y
±U'(wn - c(cii)\ + (1 - ^)U'(w 1 - c(cii)\
Using a Taylor quadratic approximation of the problem, we deduce the optimal prize spread and effort for the n-contestant tournament. We obtain
Aw. _ /» Mp
fn(n)2 +Aa^’
/ / * \ P
c (a ) =
1 + A,
c
“4 fn(n)2
Let us consider how the solutions are affected by the tournament size. The prize spread is increasing in the number of contestants with limitn^^ Aw* = <x. Since a marginal increase in effort has a negligible effect on the probability of winning for large n, the prize spread must be high to induce enough effort. Ma, 1988 considers the problem of implementation of incentive contracts when a principal hires many agents and is not able to control their actions.When the actions are mutually observable there is a unique perfect equilibrium. When the actions are not observable, there may have multiple equilibria.
2. Innovation Tournaments
2.1. Innovation games
Incentive research competition. Governments, private corporations and even individuals are commonly sponsors of research tournaments. Taylor, 1995 relates
The multiple-prize n-contestant tournament (McLaughlin, 1988)would be
max{mfc}j0j E[£/i] = - ^ U (wk - c(<n) j k—1 V '
s.t.
1n
p.CM = - Wk, k—1
Sk—1{ U (wk — c(ai))fn(k)
c'(at) = -------^----------------
iY.l=1U'[wk - c(ai)
11
some recent famous contests i2 : Frigidaire Co and Whirlpool Corp. were selected among 14 contestants to compete for a research that would use 25 to 50 percent less electricity and no chlorofluorocarbons ... (The New York Times, 8 July 1992). Whirlpool won the contest. The Federal Communication Commission held a tournament for higher definition television, Dow and IBM sponsor annual tournaments in which the winners receive grants to develop their projects. Other examples are given by C.R. Taylor. A research tournament is an incomplete contract designed to overcome the difficulties of conventional contracts (such as bilateral contracts). Moreover, the prospect of winning a specific prize (production contract, cash,etc.) provides incentives to exert significative research efforts. Rogerson, 1982 estimates the size of the prize implicit in each production contract of 12 major aerospace major research contests held by the US Department of Defense between 1964 and 1977. Knoeber and Thurman, 1994 are testing empirically the tournaments’ theory. Their results are consistent with the predictions: unchanged prize differentials (or spread) will not affect the performance, in mixed (heterogeneous) tournaments more able agents choose less risky strategies, and tournament sponsors will attempt to discourage agents with unequal ability i3.
Innovation games : patent races and innovation tournaments. There is two classes of innovation games : the innovation or patent race and the tournaments (Baye and Hoppe, 2003, Taylor, 1995). Both games differ regarding the features and modeling. Innovation races are used to model the competition to be first: the date of discovery is uncertain and the standard quality is fixed. A larger prize reduces the expected amount of time to innovate. On the contrary, innovation tournaments are used to model the competition to be best : the terminal date is fixed and the quality of innovation varies. A larger prize raises the expected quality of winning invention. patent race models were pioneered by Dasgupta and Stiglitz, 1980a, 1980b, Lee and Wilde, 1980, Loury, 1979, Reinganum, 1981,1982. A simple model is the memoryless patent race where uncertainty is formalized by an exponential distribution. In the Loury’s study, the relationship between the market structure and the R&D spending relies on this probabilistic assumption. The assumption is that the firm’s probability to discover depends on the current R&D expenses i4. On the contrary, there no need to restrict the RVs to be exponentially distributed in the tournament game. Baye and Hoppe, 2003 demonstrate the strategic equivalence of tournaments and patent-race games. It is shown that innovation tournaments produce not only negative externalities on R&D due to the incentives to win, but also positive externalities due to the competition between contestants.
12 A famous historic example of tournament is the Liverpool and Manchester Railway Company in 1829. This company offered a prize of £500 for the best transportation engine (Fullerton and McAfee, 1999).
13 Becker and Huselid, 1992 show how tournaments are incentive, using empirical tests on a panel data set from auto racing. The tournament spread exerts incentive effects on individual performances. Ehrenberg and Bognanno, 1990 focus on professional golf tournaments and also prove that level and structures of the prizes influence significantly the players’ performances.
14 Further refinements of the basic patent-race model (Tirole, 1990) introduce a choice between less or more risky technologies by the firms, learning effects such as accumulated experiences on the probability of discovering by unit of time.
The dynamic game design and model building. A dynamic game generally is played by n contestants participating to a two-stage game of imperfect information. Let us describe the two-contestant tournament Taylor, 1995, Wolfstetter, 1999. Nature draws i.i.d. output distortions. The drawing is not revealed either to the contestants before there decisions or to the sponsor. The distribution of the distortion is common knowledge. At the first step, the sponsor sets two prizes wi > w2 and commits to pay wi to the high-output R&D and w2 to the low output. At the second stage, each contestant choose his investment in R&D without observing the opponents’ choices. The prizes are then paid.The tournament game is solved by backward induction using prior beliefs. The tournament subgame is first solved for given prizes. Thereafter, the sponsor computes the expected profit and find the optimal prizes. In the tournament subgame, the two contestants choose their investment simultaneously. The contestant i’s probability of winning must be increasing in his R&D spending and decreasing in that of his rival. The equilibrium investments satisfy the reaction functions. A unique symmetric equilibrium in pure strategies is achieved, provided the quasiconcavity of the objective function. The equilibrium investment spending is strictly increasing in the prize spread. Using the solution of the tournament subgame, the sponsor chose prizes that maximize the expected profits subject to a contestant’s participation condition. The equilibrium of the whole game is then deduced. Zheng and Vukina, 2006 construct an empirical model of a rank-order tournament game. The authors estimate the structural parameters of the symmetric Nash equilibrium and simulate the model. The observed cardinal tournament with risk-averse agents fits the data better.
3. A R&D Tournament Model with Spillovers
3.1. Presentation
In the R&D tournament model by Zhou, 2006, n risk- neutral contestants are investing research inputs (or efforts) in order to produce the best research output. These contestants may be firms competing a specific award from some public or private sponsor. The contestant who produces the best R&D output obtains a reward of wi. Each other contestants who loses the contest is supposed to receive a weaker prize of w2. The prize spread is Aw = wi — w2. The i’s contestant is identified by the subscript i and the n — 1 rivals by -i with —i <G {1,2,... ,i — 1,i + 1,...,n}. The contestant i’s R&D output is given by
Xi = ai + ft ''y ' aj + £ i, (3)
j=i
where ai denotes the R&D input, ft (0 < ft < 1) the spillovers from the R&D output i5, and ei an individual noise. The i’s R&D cost c(ai) is increasing and convex in efforts such that c',c" > 0. Let denote Pi the probability that contestant i’s wins the contest. His expected payoff is
Pi wi + (1 — Pi )w2 — c(ai).
15 Griliches, 1992 reviews the basic model of R&D spillovers. The importance of spillovers is also shown empirically by Bernstein, 1988. The author estimates the effects of intra-and interindustry R&D investment spillovers on costs and on the production structure. The absorptive capacity and knowledge accumulation may also affect a firm’s R&D output (Zhou, 2006).
3.2. Expected payoff of the contestants
Since the winner is supposed to have the highest xi, the condition is according to equation (3)
ai + ft aj + £i > a-i + ft ak + £-i; for all — i = i.
j=i k=i
Then, we have
(1 — ft)(ai — a-i + £ > £-i, for all — i = i.
We assume that there exists a symmetric equilibrium of R&D inputsi6. Given the R&D inputs of the rivals all equal to the same amount a, the contestant i’s probability of winning is
Fn-i ^(1 — ft)(ai — a) + £^j , for a given £i.
Integrating over all realizations of £, the contestant i’s expected probability of winning is
ni
J Fn ^(1 — ft)(ai — a) + £^j f (£i)d£i.
The expected payoff is
Wi = wi J Fn ^(1 — ft)(ai — a) + £^j f (£i)d£i+
w2 ^ 1 — j Fn ^(1 — ft)(ai — a) + £-^j f (£i)d£i^ — c(ai). (4)
In condensed form, we also have
Wi = w2 + Aw J Fn ^(1 — ft)(ai — a) + £^j f (£i)d£i — c(ai). (5)
3.3. Reaction functions of the contestants
The contestant i chooses the input R&D ai to maximize the expected payoff. The FOC of this problem is
(1 — ft) Aw J ~Q^Fn 1 ^(1 — ft)(ai ~ a) + £i'j f(£i)d£i — c {ai) = 0.
We then have
(1 — ft) Aw J (n — 1)Fn 2 ^ (1 — ft)(ai — a) + £^
X f f(1 — ft)(ai — a) + £^ f (£i)d£i — d (ai) = °. (6)
16 The existence of a symmetric equilibrium in tournaments is discussed by Nalebuff and Stiglitz, 1983. A symmetric equilibrium is more likely to exist when the variance of the RVs is large.
At the symmetric equilibrium with ai = a, a contestant chooses his R&D spending so as to equalize the marginal benefit of increasing his R&D spending and the marginal cost of these research spending. We have the rule
(1 — ft)Aw j(n — 1)Fn-2(£i)f 2(£i)d£i = c'(a). (7)
A larger spillover ft will then decrease the contestant’s incentive to spend. Let us assume an exponential distribution of the type f (£ = 0e-°£. The condition f '(£) = —a2e-0s < 0 is then satisfied. Introducing this exponential pdf into 7, we have the integral
[ f-M (l-l3)(ai-a)+£i) = [ ae-aE'nd£i = Ig-aU-W»*-»)
J-™ \ ) J-™ n
We then obtain the following expected payoff according to 7
Wi = W2 + Aw X
n
The reaction functions for two players are
a(l — (3)—Aw — c'(ct) = 0. n
Fig.1. Reaction function of two contestants with exponential noise
3.4. Size of the tournament
Proposition 2. The contestant’s R&D spending varies as the pdf of the individual noise when the size of the tournament is increasing. The derivatives ^ and f'(£) have the same sign. Hence, if f '((£) < 0 the contestant’s R&D investment decreases when the number of contestants increases, if f' (£) = 0 the R&D investment remains unchanged when the size of the tournament increases and if f'(£) > 0 the R&D investment will increase with the size of the tournament.
Proof. The integration by parts of the integral in equation(7) yields
(1 — ft)Aw|f ( + to) — 2J F n i (£i )f' (£i )d£i^ = c (ai).
The derivation w.r.t the size of the tournament n leads to
^{/(+00) “2 [ (d) f'(£i)d£i) = C"K)^
(8)
(1 —/3)Z\w—■|/(+00) — 2 J Fn 1 (£i)f'(£i)d£i — c"(ai) ~yj~i
/do •
-\n F(ei)Fn~1 (£i)f (£i)d£i = c"(ai)
With fixed spread Aw, we have
3g”(s) - sgn/,(£-)'
Zhou, 2006 illustrates this proposition with different distributions of the random variables. The exponential distribution f (£ = 0 exp(—0£), 9 > 0 satisfies the condition f '(£) < 0 and leads to the same result as Loury, 1979. For the uniform distribution of the RVs, we have f '(£) = 0 with no change in the R&D spending when the number of opponents increases. R&D spending increase with a power
function distribution like /(e) = , 0<e<7, 0 > 0 (Zhou, 2006).
4. Further Extensions
Further extensions of the basic tournament model have been considered with endogenous payoffs, introduction of multiplicative common shocks, multiple prizes, nonlinear output in effort and heterogeneous contestants.
4.1. Endogenous payoffs
Zhou, 2006 considers a tournament where the payoffs are endogenously determined by R&D spending. It is assumed that if the contestant i’s wins the contest with R&D output is aij=i aj + £i, his reward will have the same amount. If the contestant looses, the reward will be zero. From the FOC for the contestant i’s expected payoff maximization, Zhou, 2006 shows two advantages : the payoff (conditioned of winning) is increased for a given probability of winning, and the expected probability of winning is also increased for a given reward. However, an increase of the prize to the winner may decrease each contestant’s expected payoff and discourage the R&D spending.
e
4.2. Common shocks
The stochastic environment of the game already has individual-specific components ej, for j = It may also have a common component j with cdf G(n).
The random component will then represent the level of difficulty of a set of tasks, while the random component n will be the individual’s comparative advantage or disadvantage. These random disturbances are uncorrelated. The probability distributions F(e) and G(n) are common knowledge. Both random disturbances affect the outputs. Two alternative forms may be considered for the contestant i’s output equation. The Lazear and Rosen, 1981 specification is additive
xi = ai + j + £i, ai > 0, E[e i] = E[j] = 0.
The Nalebuff and Stiglitz, 1983 specification is multiplicative with
xi = fj.ai + £i, E[e i] = 0, and E[j] = 1.
Green and Stokey, 1986 introduce common additive shocks and prove that tournaments dominate the individual-based contracts if the common shock is strong. Nalebuff and Stiglitz, 1983 use multiplicative shocks which affect the marginal product of effort and produce different results.
4.3. Multiple prizes
Nalebuff and Stiglitz, 1983 also consider tournaments with multiple prizes. The probability that the contestant i obtains the jth position up from the bottom is
i \ / \j—1
n1
Pr°b{xi = j} = y ^ f (e)F(^V(ai - «) + £^)
1 _ F (n(ai _ a) + e) | de. (9) At the symmetric equilibrium (where aia), we compute
ap TOla{Xi=j}=r)j i-n£))”5 1f{£)^
X {(j _ 1)(1 _ F{£)) _ (n _ j)F(£)}de. (10)
Thus by increasing the R&D inputs there an increased chance to have the position j.
4.4. Nonlinear outputs
When outputs are nonlinear in inputs,we may have the contestant i’s output equation
Xi = ^(fj.ai) + Si,
or
Xi = ^(fj.ai + ei).
This nonlinearities are studied by Nalebuff and Stiglitz, 1983. In the first situation, investing more at the symmetric solution increases the probability of winning by a factor proportional to <p.
4.5. Heterogeneous contestants
Relaxing the assumption of identical contestants also generalize the tournament game (Lazear and Rosen, 1981,McLaughlin, 1988). Each contestant knows his own abilities but ignores those of his opponents. Asymmetries in the knowledge of abilities produce inefficiencies. Lazear and Rosen, 1981 prove that if heterogeneous contestants do not self sort, the tournament is inefficient. Bhattacharya and Guasch, 1988 show how, by a proper design of contracts, the efficiency can be achieved with such tournaments with asymmetrically heterogeneity of agents. Jost and Krakel, 2005 consider a sequential-move tournament with heterogeneous players. The agents’ strategic behaviors differ from that one in simultaneous-move tournaments. Thus, the first playing agent may choose a preemptively high effort so that the follower will give up.
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