The Contest Game for Crowdsourcing Reviews

Abstract

We consider a contest game modelling a contest where reviews for a proposal are crowdsourced from n players. Player i has a skill $s_{i}$, strategically chooses a quality $q \in \{ 1, 2, \ldots , Q \}$ for her review and pays an effort $\textsf{f}_{q} \ge 0$, strictly increasing with q. Under voluntary participation, a player may opt to not write a review, paying zero effort; mandatory participation does not provide this option. For her effort, she is awarded a payment per her payment function, which is either player-invariant, like, e.g., the popular proportional allocation, or player-specific; it is oblivious when it does not depend on the numbers of players choosing a different quality. The utility to player i is the difference between her payment and her cost, calculated by a skill-effort function $\mathsf{\Lambda } (s_{i}, \textsf{f}_{q})$. Skills may vary for arbitrary players; anonymous players means $s_{i} = 1$ for all players i. In a pure Nash equilibrium, no player could unilaterally increase her utility by switching to a different quality. We show the following results about the existence and the computation of a pure Nash equilibrium:

• We present an exact potential to show the existence of a pure Nash equilibrium for the contest game with arbitrary players and player-invariant and oblivious payments. A particular case of this result provides an answer to an open question from [6]. In contrast, a pure Nash equilibrium might not exist (i) for player-invariant payments, even if players are anonymous, (ii) for proportional allocation payments and arbitrary players, and (iii) for player-specific payments, even if players are anonymous; in the last case, it is $\mathcal{N}\mathcal{P}$-hard to tell. These counterexamples prove the tightness of our existence result.

• We show that the contest game with proportional allocation, voluntary participation and anonymous players has the Finite Improvement Property, or FIP; this yields two pure Nash equilibria. The FIP carries over to mandatory participation, except that there is now a single pure Nash equilibrium. For arbitrary players, we determine a simple sufficient condition for the FIP in the special case where the skill-effort function has the product form $\mathsf{\Lambda } (s_{i}, \textsf{f}_{q}) = s_{i}\, \textsf{f}_{q}$.

• We introduce a novel, discrete concavity property of player-specific payments, namely three-discrete-concavity, which we exploit to devise, for constant Q, a polynomial-time $\varTheta (n^{Q})$ algorithm to compute a pure Nash equilibrium in the contest game with arbitrary players; it is a special case of a $\varTheta \left( n\, Q^{2}\, \left( {\begin{array}{c}\textstyle n+Q-1\\ \textstyle Q-1\end{array}}\right) \right)$ algorithm for arbitrary Q that we present. Thus, the problem is ${\mathcal{X}\mathcal{P}}$-tractable with respect to the parameter Q. The computed equilibrium is contiguous: players with higher skills are contiguously assigned to lower qualities. Both three-discrete-concavity and the algorithm extend naturally to player-invariant payments.

The first author is supported by research funds at University of Cyprus. The second author is supported by ESPRC grant EP/P02002X/1.