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Economics of Governance

Erschienen in:

18.03.2019 | Original Paper

Step-by-step group contests with group-specific public-good prizes

verfasst von: Katsuya Kobayashi

Erschienen in: Economics of Governance | Ausgabe 2/2019

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Abstract

The achievements reached by group members’ effort, which accompanies the possibility of members’ free-riding, affect the outcomes of competition among groups. In some cases, each achievement has the binary characteristic of “one or nothing.” For example, research groups face the challenge of making a scientific finding or not. The groups compete for a scientific breakthrough by making the related findings in a “step-by-step” manner. One finding could fail because of a mistake caused by a slight lack of effort by one member. Such a characteristic of “one or nothing” motivates group members without any incentive scheme. This study analyzes group contests with group-specific public-good prizes, in which we introduce a step function with the characteristics of “one or nothing” and “step-by-step” as a group impact function. We show the existence of the Nash equilibrium at which no group member free-rides on the others and at which more achievements than those reachable by a single member are reached.

Vorheriger Artikel Selection rates and bureaucratic performance

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The process of making findings would bring the group members skills and expertise to attain a breakthrough earlier than rival groups. In addition, their publications on these findings would provide credibility to their breakthrough.

See the detailed explanation at www.nobelprize.org.

Chowdhury and Topolyan (2016) summarize these studies.

See also the original works of Hirshleifer (1983; 1985).

With regard to the coalition-proof Nash equilibrium, Bernheim et al. (1987) originally defined the equilibrium concept. However, the coalition-proof Nash equilibrium of Lee (2012) and Chowdhury and Topolyan (2016) is different from the original concept of Bernheim et al. (1987). Quartieri and Shinohara (2016) provide the details of the difference and redefine the equilibrium concept of Lee (2012) as the group-proof Nash equilibrium.

In other words, we employ the Tullock-form contest success function (Tullock 1980).

We allow that, in each group, some members’ valuations are equal to or higher than the costs of achievement, namely \(v_j^i \ge m_k\), and the others have lower valuations, namely \(v_h^i< m_k\), \(h \ne j\).

This is the same definition as the contest success function for group i in Baik (2008).

This assumption does not affect the following results intrinsically. We obtain the same results even if any group obtains nothing when \(X_i = 0\) and \(f(X_i)=0\) for all i. See also the Introduction of Quartieri and Shinohara (2016).

If j expends any effort less than \(x_j^i\), j has to pay this effort cost on the same achievement level \(\alpha - 1\). Thus, in this case, j expends nothing.

The results of Baik (2008) depend on the assumption of an individual constant marginal cost of effort. If we assume an increasing marginal cost of effort in Baik’s model, members with small valuations also expend effort at the Nash equilibrium. Then, the reachable achievement level is the one at the equilibrium. While our model has the same individual constant marginal cost of effort as Baik (2008), the Nash equilibria contain the characteristics that anyone can expend effort because of the step-by-step structure. In addition, multiple reachable achievement levels can exist at the Nash equilibria, as we show in the following.

Regarding the discrete public good, Palfrey and Rosenthal (1984; 1988), Gradstein (1994) and Menezes et al. (2001) are instructive works.

Even if we assume that no group obtains anything when \(X_i=0\) for all i, j does not deviate from \(x_j^i=0\) because \(m_1> v_j^i> x_j^i > 0\).

In competition in research, for example, the principal is a professor and the agents are her group members such as research fellows, research assistants, and graduate students.

The reward must cover the cost of stepping up from the default achievement level \(\alpha ^*-t\) to the target level \(\alpha ^*\): \(\displaystyle b_j^i \ge \frac{v_j^i}{\sum _{j=1}^{n_i}v_j^i} \sum _{k=\alpha ^*-t+1}^{\alpha ^*} m_k - \frac{t \beta }{(\alpha ^* + \beta )(\alpha ^*-t+\beta )} v^i_j\). The first term on the right-hand side is the cost shared by Expense Rule A. The second term is the rise in expected valuation brought about by this step up. Here, the default achievement level \(\alpha ^* - t\) viewed as a reservation is any reachable achievement level at Nash equilibria under the individual decisions.

If (2) is non-negative only in \((\alpha , \beta ) = (1,1)\), both cases are at the same level at a Nash equilibrium.

See Bernheim et al. (1987) and Quartieri and Shinohara (2016) for the strict definitions of the coalition-proof Nash equilibrium and group-proof Nash equilibrium, respectively.

Chowdhury and Topolyan (2016) use a concept with a similarity of the group-proof Nash equilibrium as their coalition-proof equilibrium. However, they do not share the concepts of Bernheim et al. (1987) or Quartieri and Shinohara (2016).

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Titel: Step-by-step group contests with group-specific public-good prizes
verfasst von: Katsuya Kobayashi
Publikationsdatum: 18.03.2019
Verlag: Springer Berlin Heidelberg
Erschienen in: Economics of Governance / Ausgabe 2/2019
Print ISSN: 1435-6104
Elektronische ISSN: 1435-8131
DOI: https://doi.org/10.1007/s10101-019-00224-4

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