The Contest Game for Crowdsourcing Reviews

Abstract

We consider a contest game modelling a contest where reviews for m proposals are crowdsourced from n players. Player i has a skill s iℓ for reviewing proposal ℓ; for her review, she strategically chooses a quality q ∈ {1, 2, . . . , Q} and pays an effort fq ≥ 0, strictly increasing with q. For her effort, she is given a payment determined by a payment function, which is either player-invariant, like, e.g., the popular proportional allocation, or player-specific. The cost incurred to player i for each of her reviews is the difference of a skill-effort function Λ(si, fq) minus her payment. Skills may vary for arbitrary players and arbitrary proposals. A proposal-indifferent player i has identical skills: s iℓ = si for all ℓ; anonymous players means si = 1 for all players i. In a pure Nash equilibrium, no player could unilaterally reduce her cost by switching to a different quality. We present three main results:We present a novel potential function to show that the contest game has always a pure Nash equilibrium for the model of arbitrary players and arbitrary proposals with a player-invariant payment function. A particular case of this result answers an intriguing open question from [4]. In contrast, inexistence is possible for a player-specific payment function; the corresponding decision problem is N P-complete. We exploit an increasing-differences property of the skill-effort function to devise, for constant Q, a polynomial-time Θ(n Q ) algorithm for arbitrary players and arbitrary proposals, under a player-invariant payment function, to compute a pure Nash equilibrium; it is a special case of a Θ max{Q 2 , n} n Q − 1 algorithm for arbitrary Q that we present. This settles the parameterized complexity of the problem with respect to the parameter Q. The computed equilibrium is contiguous: players with better skills are contiguously assigned to lower qualities; contiguity is the crux to bypass the exponential barrier incurred when enumerating all profiles. A Θ(max{Q, n}) algorithm for proposal-indifferent and anonymous players, under proportional allocation, for the special case where Λ(si, fq) = si fq and for a concrete scenario of mandatory participation of players in the contest. Starting with the two highest qualities, we greedily proceed to the lowest, focusing each time on a pair of qualities: maintaining players previously assigned to higher qualities, we split the players assigned to the higher of the two between the two qualities currently considered so as to enforce equilibrium.These results are complemented with extentions in various directions; for example, we devise simple Θ(1) algorithms under proportional allocation, taking Λ(si, fq) = si fq and making stronger assumptions on skills and efforts for both arbitrary and proposal-indifferent and anonymous players.