Timothy Highley | Search arXiv

arXiv2026

Carry-Over Lottery Allocation: Practical Incentive-Compatible Drafts

The NBA draft can incentivize teams to deliberately lose. We propose a draft mechanism that is practical, incentive-compatible, and favors weaker teams. The Carry-Over Lottery Allocation (COLA) framework represents a paradigm shift in evaluating team quality, replacing single season standings with multi-year playoff outcomes. In our proposed mechanism, every non-playoff team receives the same number of lottery tickets, removing incentives to lose. Lottery tickets carry over to future lotteries, but playoff success or winning a top pick diminishes a team's accumulated tickets. The lottery is familiar and preserves fan engagement. Implementation challenges are addressed to demonstrate feasibility, including transitioning to COLA, handling trades, and accommodating draft classes of varying strength. For exceptionally strong classes, teams may prefer the lottery to the playoffs. We provide a solution, employing a truth-elicitation mechanism to identify such years and expanding lottery eligibility to include as many playoff teams as necessary to preserve incentive compatibility.

cs.GT↗

arXiv2016

Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph: Barter Exchange with Multiple Items Per Agent

In a barter exchange market, agents bring items and seek to exchange their items with one another. Agents may agree to a k-way exchange involving a cycle of k agents. A barter exchange market can be represented by a digraph where the vertices represent items and the edges out of a vertex indicate the items that an agent is willing to accept in exchange for that item. It is known that the problem of finding a set of vertex-disjoint cycles with the maximum total number of vertices (MAX-SIZE-EXCHANGE) can be solved in polynomial time. We consider a barter exchange where each agent may bring multiple items, and items of the same agent are represented by vertices with the same color. A set of cycles is said to be tropical if for every color there is a cycle that contains a vertex of that color. We show that the problem of determining whether there exists a tropical set of vertex-disjoint cycles in a digraph (TROPICAL-EXCHANGE) is NP-complete and APX-hard. This is equivalent to determining whether it is possible to arrange an exchange of items among agents such that every agent trades away at least one item. TROPICAL-MAX-SIZE-EXCHANGE is a similar problem, where the goal is to find a set of vertex-disjoint cycles that contains the maximum number of vertices and also contains all of the colors in the graph. We show that this problem is likewise NP-complete and APX-hard. For the restricted case where there are at most two vertices of each color (corresponding to a restriction that each agent may bring at most two items), both problems remain NP-hard but are in APX. Finally, we consider MAX-SIZE-TROPICAL-EXCHANGE, where the set of cycles must primarily include as many colors as possible and secondarily include as many vertices as possible. We show that this problem is NP-hard.

cs.DS↗