Competition Between Cooperative Projects

Abstract

A paper needs to be good enough to be published; a grant proposal needs to be sufficiently convincing compared to the other proposals, in order to get funded. Papers and proposals are examples of cooperative projects that compete with each other and require effort from the involved agents, while often these agents need to divide their efforts across several such projects. We aim to provide advice how an agent can act optimally and how the designer of such a competition (e.g., the program chairs) can create the conditions under which a socially optimal outcome can be obtained. We therefore extend a model for dividing effort across projects with two types of competition: a quota or a success threshold. In the quota competition type, only a given number of the best projects survive, while in the second competition type, only the projects that are better than a predefined success threshold survive. For these two types of games we prove conditions for equilibrium existence and efficiency. Additionally we find that competitions using a success threshold can more often have an efficient equilibrium than those using a quota. We also show that often a socially optimal Nash equilibrium exists, but there exist inefficient equilibria as well, requiring regulation.

G. Polevoy—Most of this work was done at Delft University of Technology.

A Omitted Proofs

We now prove Theorem 6.

Theorem 6

Consider an equal $\theta$-sharing game with $n \ge 2$ players with budgets $B_n \ge \ldots \ge B_2 \ge B_1$, $0< \theta < 1$ (the order is w.l.o.g.), linear project functions with coefficients $\alpha _m = \alpha _{m - 1} = \ldots = \alpha _{m - k + 1} > \alpha _{m - k} \ge \alpha _{m - k - 1} \ge \ldots \ge \alpha _1$ (the order is w.l.o.g) (see Footnote 2), and success threshold $\delta$. Define $p {\mathop {=}\limits ^\mathrm{\Delta }}\left\lfloor {\frac{\alpha _m \sum _{i \in N}{B_i}}{\delta }} \right\rfloor$, as in Theorem 5.

• 1. If at least one of the following holds.

  • (a) $B_1 \ge k\theta B_n$, $k \le p$ and $\frac{1}{n}\alpha _{m - k + 1} \ge \alpha _{m - k}$,

  • (b) $B_1 \ge p\theta B_n$, $k \ge p \ge 1$ and $\alpha _m B_n < \delta$;

  Then, there exists a pure NE and there holds: ${{\mathrm{PoS}}}= 1$.

• 2. Assume $B_{n - 1} < \frac{\theta }{\left| \varOmega \right| } B_n$, all the project functions are equal, i.e. $\alpha _m = \alpha _1$. Then, there exists a pure NE and ${{\mathrm{PoS}}}= 1$. If, in addition, $\alpha _m B_n \ge \delta$, then ${{\mathrm{PoA}}}= \frac{B_n}{\sum _{i\in \left\{ 1, 2, \ldots , n \right\} }{B_i}}$.

Proof

We first prove parts 1a and 1b. According to proof of parts 1 and 2 in Theorem 5, equally dividing all the budgets among $\min \left\{ k, p \right\}$ steep projects is an NE. Therefore, ${{\mathrm{PoS}}}= 1$.

Part 2 is proven as follows. Since all the players dividing their budgets equally between any $\min \left\{ p, m \right\}$ projects constitutes an NE, we have ${{\mathrm{PoS}}}= 1$.

To treat the ${{\mathrm{PoA}}}$, we define the number of projects player n can make accepted on her own, $r {\mathop {=}\limits ^\mathrm{\Delta }}\left\lfloor {\alpha _m \frac{B_n}{\delta }} \right\rfloor$, and distinguish between the case where $m \le r$ and $m > r$. If $m \le r$, consider the profile where player n divides her budget equally between all the projects, while the other players contribute nothing at all. This is an NE, because all the projects are accepted, player n cannot increase her profit and any other player will be suppressed, if she contributes anything anywhere. On the other hand, if $m > r$, consider the profile where player n divides her budget equally between $m, m - 1, \ldots , m - r + 1$, while the other players contribute nothing at all. The only possible deviation is player $j < n$ contributing to a vacant project. However, we have $B_j \le B_{n - 1}< \frac{\theta }{\left| \varOmega \right| } B_n < \theta \delta / \alpha _m \le \delta / \alpha _m$. This means that the project would be unaccepted. Therefore, this is an NE.

Therefore, ${{\mathrm{PoA}}}\le \frac{\alpha _m B_n}{\alpha _m (\sum _{i \in N}{B_i})}$. Since $\alpha _m B_n \ge \delta$, in any NE, player n receives at least $\alpha _m B_n$, and therefore, ${{\mathrm{PoA}}}= \frac{B_n}{\sum _{i\in \left\{ 1, 2, \ldots , n \right\} }{B_i}}$.    $\blacksquare$

We finally prove Proposition 2.

Proposition 2

For any success threshold $\delta \in [0, \alpha _m B_n]$ and any number of agents $n \ge 2$ and projects $m \ge 2$, there exists a game which possesses an NE, and a game which does not.

Proof

For $\delta = 0$, which means for no threshold, the theorem follows from Theorem 3 from [13]. Therefore, we assume henceforth a positive success threshold.

A game that satisfies the conditions of Theorem 5 provides an example of the existence. Notice that the p they define is positive, since $\delta \le \alpha _m B_N$.

To find a game that does not possess an equilibrium, let $\alpha _m = \alpha _1$ and let

$$\begin{aligned} B_n > \sum _{i = 1}^{n - 1} {B_i},\end{aligned}$$

(4)

$$\begin{aligned} B_1 = \ldots = B_{n - 1} = \theta B_n \text { and } \delta = \alpha B_n. \end{aligned}$$

(5)

Because of the equality of all the project coefficients, of (4) and of the choice of the success threshold, in an equilibrium, all the agents with budgets at least $\theta B_n$ (which are $1, \ldots , B_{n - 1}$ here) will be together with n on the same single project. Then, agent n will deviate, contradictory to being in an equilibrium.    $\blacksquare$