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Learning by doing in contests

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Abstract

We introduce learning by doing in a dynamic contest. Contestants compete in an early round and can use the experience gained to reduce effort cost in a subsequent contest. A contest designer can decide how much of the prize mass to distribute in the early contest and how much to leave for the later one in order to maximize total efforts. We show how this division affects effort at each stage, and present conditions that characterize the optimal split. There is a trade off here, since a large early prize increases first period efforts leading to a substantial reduction in second round effort cost; on the other hand, there is less of the prize mass to fight over in the second round, reducing effort at that stage. The results are indicative of the fact that the designer often prefers to leave most of the prize mass for the second contest to reap the gains from the learning by doing effect.

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Notes

  1. Examples include Moldovanu and Sela (2006) and Fu and Lu (2011).

  2. See Konrad (2009: ch. 8), as well as papers by Klumpp and Polborn (2006), Konrad and Kovenock (2009), and Clark et al. (2011). These papers typically do not discuss contest design, an exception being Clark et al.’s (2011) analysis of how a principal should split prizes across rounds in a situation with repeated contests and dynamic win advantages.

  3. One can, of course, also envision there being effort disadvantages, i.e., today’s effort having a negative effect on tomorrow’s win chances, for example because of fatigue. This is discussed in single-contest, multi-stage settings by Harbaugh and Klumpp (2005) and Ryvkin (2011).

  4. This type of activity is termed learning before doing by Pisano (1994).

  5. See Baye and Hoppe (2003) and Fullerton and McAfee (1999).

  6. See Johansson et al. (2007) for more on the Lisbon agenda.

  7. Studies of how manufacturing productivity is affected by learning and experience have a long history. Wright (1936) and Alchian (1963) are early examples of attempts to quantify a learning curve in the manufacture of aircraft. Yelle (1979), and more recently Anzanello and Fogliatto (2011) provide literature surveys in this area. Conley (1970) extends the concept to that of an experience curve, applying this to managerial decision making.

  8. (A1) is a stronger assumption than necessary, but facilitates the main points of the analysis.

  9. This is a common modeling strategy in contests that extend Tullock’s framework. See, in particular the discussion around Fig. 1 in Runkel (2006: 220–221).

  10. Contests in which efforts affect the contest prize often need parameter restrictions in order to guarantee that the symmetric equilibrium exists (see Chung 1996). Additionally such contests may admit asymmetric equilibria (see Chowdhury and Sheremeta 2011).

  11. Note that this does not necessarily follow in the case of a corner solution M=1.

  12. This is the same functional form used by Fu and Lu (2009) in their analysis of pre-contest investment.

  13. This occurs for \(s>4 ( 1+\sqrt{2} \,) \approx9.66\).

  14. We disregard cases where players choose their only available corner solution, a=0, as well as cases where their optimum efforts are not well defined. This means that a player’s equilibrium action is found through the first-order condition.

  15. The concavity of the second-period profit is ensured by noting that \(c_{i}''-3 \frac{ ( c_{i}' )^{2}}{C}=\frac{s^{2}}{2(1+sa)^{3}} >0\).

  16. The second order condition is satisfied for \(\frac{s^{2}+8s+32}{12s}>M\) which always holds since the left-hand side always exceeds 1.

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Acknowledgements

We have benefited from fruitful discussions with Mads Greaker, Atle Seierstad and Jan Yngve Sand. We are grateful for helpful comments by two anonymous referees. Clark’s research has received funding from the Research Council of Norway (RCN) through the project “The knowledge based society” (grant 172603/V10). Nilssen’s research has received funding from the RCN through the Ragnar Frisch Centre for Economic Research (project “R&D, Industry Dynamics and Public Policy”), and through the ESOP Centre at the University of Oslo.

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Correspondence to Derek J. Clark.

Appendix A

Appendix A

1.1 A.1 Proof of Proposition 1

Players’ equilibrium efforts in period 1, supposing second-order conditions are satisfied, are found through the first-order condition:Footnote 14

$$\frac{a_{j}}{A^{2}}(1-M)-1-2\frac{c_{i}^{\prime }c_{j}^{2}}{C^{3}}M=0,\quad i,j\in \{ 1,2 \} ,\ i\neq j. $$
(15)

We focus on a symmetric equilibrium in which a 1=a 2=a, c 1=c 2=c, and C=2c. This means that the above equation can be written as:

which can be solved implicitly for a as in (4).

The second-order condition for player i’s choice of first-period effort is:

$$-(1-M) \frac{2a_{j}}{(a_{i}+a_{j})^{3}}-2c_{j}^{2} \biggl( \frac{ (c_{i}+c_{j} ) c_{i}''-3c_{i}^{\prime2}}{ (c_{i}+c_{j} )^{4}} \biggr) M<0, $$
(16)

where the first (second) term stems from the effect of a i on first-period (second-period) profit. Given that (16) is fulfilled then (15) is necessary and sufficient for maximizing player i’s payoff in period 1. Evaluated at a symmetric situation, (16) amounts to:

Inserting from (4) and simplifying, we obtain:

$$-2 ( 4c+Mc' )^{2}- \bigl[ 2cc''-3 (c' )^{2} \bigr] M(1-M)<0. $$
(17)

First-period profit is increasing and strictly concave in a i , and second-period profit is increasing and may be concave or convex in a i . If it is too convex, then the player will want a corner solution for a i . To avoid this, we need a learning-by-doing function that is sufficiently convex in a. In order to ensure that the second-order condition is satisfied, we assume (A1).

We also need to ensure that the equilibrium first-period effort, given implicitly in (4), is feasible, in particular that it is non-negative. For this to hold requires that the denominator on the right-hand side of (4) be positive, or that 4c(a)+Mc′(a)>0, for all M∈[0,1]. Since the second term here is negative, because c′<0, this condition holds for all a and M if it holds for all a at M=1, i.e., if (A2) holds.

1.2 A.2 Proof of Proposition 2

  1. (i)

    Without learning-by-doing in this model there is no connection between periods 1 and 2 and c i =c j =1, and c′(.)=0. Hence we obtain and amount of effort per player in the first period of \(\hat{a}(M)=\frac{1-M}{4}\) which is the standard Tullock contest effort with a prize of 1−M. With learning-by-doing, efforts are given by (4), and \(a(M)\geq \hat{a}(M)\) holds if 4c≥4c+Mc′(a), which holds since c′(a)<0. When M=0 the whole prize is distributed in the first period and \(a(M)=\hat{a}(M)\).

  2. (ii)

    Follows directly from (5).

1.3 A.3 Logistic learning by doing

We show here the steps involved in working out the optimal solutions for the specific functional form. With the logistic learning-by-doing function

$$c ( a ) = \frac{1}{1+sa},\quad s>0, $$
(18)

we have:

$$c' ( a ) =- \frac{s}{ ( 1+sa )^{2}}<0;\qquad c^{\prime \prime} ( a ) = \frac{2s^{2}}{ ( 1+sa )^{3}}>0. $$
(19)

This function has the interesting property that cc′′=2(c′)2, so the quadratic term in (12) vanishes. Therefore, a solution of the principal’s problem is found implicitly from that equation as

or, with insertions from (14) and (19),

The next step is to find a(M). By combining (4) and (18), we find an implicit expression for a(M):

Solving for an explicit expression, we find two solutions, out of which one is always negative and one that always is positive. We want the positive solution, implying:

$$a ( M ) = \frac{1}{8s} \Bigl( s-4+\sqrt{ ( s+4 )^{2}-16Ms}\,\Bigr) $$
(20)

Note that \(\frac{da}{dM}<0\) for all parameter values as noted in the general analysis. Moreover, it turns out that the second-period profit of a player is always concave in this case, implying that the second-order condition for a contestant’s choice of first-period effort is always satisfied.Footnote 15

We can now use (5) and (20) to find an expression for contestants’ second-period effort in this case. We have

$$b ( M ) = \frac{1}{32}M \Bigl( s+4+\sqrt{ ( s+4 )^{2}-16Ms}\,\Bigr) . $$
(21)

Note that a necessary condition for the effort functions to be defined is that the square root expression in a(M) and b(M) is defined, so that \(\frac{(s+4)^{2}}{16s}\geq M\). This is always fulfilled since the left-hand side has a minimum value of 1.

The principal aims at maximizing total expected effort, that is maximizing a(M)+b(M), where a(M) and b(M) are given by (20) and (21). This gives rise to the first-order conditionFootnote 16

Heeding the restriction that M∈[0,1], we thus have:

$$M=\left\{\begin{array} {l} \frac{1}{36s} [s^{2}+8s-32+ ( s+4 ) \sqrt {s^{2}+8s+64} \,] ,\\[3pt]\quad \mbox{if }0<s<4 (1+\sqrt{2} \,) \approx9.66,\\[3pt]1,\quad \mbox{if }s\geq 4 (1+\sqrt{2} \,) .\end{array}\right. $$
(22)

We can now put (22) into (20) and (21) to find how contestants’ effort depends on the extent of learning by doing under the principal’s optimal contest design. We have:

Figure 4 shows the profit from playing the symmetric equilibrium (whole line) and that from deviating to a=0 (dashed line). Hence one can verify that assumption (A4) is satisfied.

Fig. 4
figure 4

Profit from the symmetric equilibrium (whole line) and unilateral deviation to a=0 (dashed)

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Clark, D.J., Nilssen, T. Learning by doing in contests. Public Choice 156, 329–343 (2013). https://doi.org/10.1007/s11127-011-9905-9

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