Adaptive Manipulation for Coalitions in Knockout Tournaments

TL;DR

This work introduces Adaptive Constructive Coalition Manipulation for Knockout Tournaments (ACCM-KT), where coalitions adaptively decide which matches to throw based on round-by-round progress under probabilistic outcomes. It provides a unified framework spanning ACCM-KT and its generalized form ACCM-GKT, along with best-response variants BR-ACCM-KT and BR-ACCM-GKT, and a non-adaptive CCM-KT; the paper situates these problems within classical and parameterized complexity. The authors prove strong hardness results (PH-hardness for ACCM-KT, PSPACE-hardness for ACCM-GKT; NP-hardness for BR-ACCM-KT and CCM-KT) via quantified-Boolean-formula reductions using various gadgets, and they develop a dynamic-programming approach that yields tractable results when parameterized by coalition size and a minimum random game cover, placing ACCM-GKT in XP and ACCM-KT in FPT under relevant parameters, with PSPACE containment. The generalized, possibly imbalanced tournament setting is analyzed, showing how adaptiveness interacts with structural flexibility and enabling efficient best-response computations in many cases. Overall, the paper advances the theoretical understanding of adaptive manipulation in probabilistic knockout tournaments and lays groundwork for further exploration of budgeted variants and alternative parameter regimes with practical implications in sports, elections, and decision processes.

Abstract

Knockout tournaments, also known as single-elimination or cup tournaments, are a popular form of sports competitions. In the standard probabilistic setting, for each pairing of players, one of the players wins the game with a certain (a priori known) probability. Due to their competitive nature, tournaments are prone to manipulation. We investigate the computational problem of determining whether, for a given tournament, a coalition has a manipulation strategy that increases the winning probability of a designated player above a given threshold. More precisely, in every round of the tournament, coalition players can strategically decide which games to throw based on the advancement of other players to the current round. We call this setting adaptive constructive coalition manipulation. To the best of our knowledge, while coalition manipulation has been studied in the literature, this is the first work to introduce adaptiveness to this context. We show that the above problem is hard for every complexity class in the polynomial hierarchy. On the algorithmic side, we show that the problem is solvable in polynomial time when the coalition size is a constant. Furthermore, we show that the problem is fixed-parameter tractable when parameterized by the coalition size and the size of a minimum player set that must include at least one player from each non-deterministic game. Lastly, we investigate a generalized setting where the tournament tree can be imbalanced.

Figures (8)

• Figure 1: A tournament tree $T$ with 8 players: coalition players $\{c_1,c_2,c_3\}$ and non-coalition players $\{e_1,\ldots,e_5\}$. The tournament seeding is $(c_1,e_1,e_2,c_2,c_3,e_3,e_4,e_5)$. Here, for example, $c_{1}$ and $e_{1}$ are siblings of each other.
• Figure 2: An Existential Variable Gadget $T^{\exists}_{i,j}$.
• Figure 3: A Clause Gadget $T_{c}$ corresponding to a clause $c$.
• Figure 4: A schematic depiction of different types of gadgets, including enlarged versions of existential gadgets, universal gadgets, clause gadgets, dummy players, and $e^{\star}$, arranged together in tournament $T^{\star}$. Here, only the initial set of existential gadgets and the final set of universal gadgets, which alternate in the arrangement, are shown.
• Figure 5: Illustration of a Generic Selection Gadget $T_S$ for some set $S=\{s_1,s_2,\ldots,s_\ell\}$.

Theorems & Definitions (26)

• Definition 1
• Proposition 2
• Theorem 3
• proof
• Claim 4
• Claim 5
• Claim 6
• Claim 7
• Remark 1
• Theorem 8