# Detection of Decision-Making Manipulation in the Pairwise Comparison Method
## 1 Introduction  
When it comes to decision-making, especially in high-stakes situations such as medical diagnoses, financial investments, political elections, or major sports tournaments, it is essential to ensure the fairness and transparency of the process [1]. Manipulation of the decision-making process can have serious consequences that can negatively affect individuals [2], society, organizations [3], or tournament results [4]. Prejudice, external pressures, bribery, or multiple factors can influence decision-makers, leading to suboptimal outcomes or harm. In political elections, propaganda, disinformation, or bribery can manipulate voters and influence elections [5], leading to long-term societal consequences. In the context of sports, various forms of cheating, including doping, collusion, and matching fixing [6], as well as coordinated fights [7], are considered unethical and have a negative financial impact on individuals who place bets. We can take various measures to prevent manipulation and ensure transparency [1] and objectivity in decision-making. For example, the number of decision-makers can be increased to make manipulation more difficult [8], or the decision-making processes may be subject to external oversight (or review) to ensure compliance with ethical and legal standards [9]. Another critical aspect is data-driven decision-making, where objective and reliable data support decision-making. By analyzing these data, decision-makers can identify patterns and anticipate and evaluate alternatives more objectively, reducing the influence of biases and external pressures. Researchers have proposed many methods to reduce subjectivity and increase transparency, ensuring the decision-making process is open, visible, and not manipulable [10; 11]. Many of them are based on well-defined mathematical models [12; 13; 14]. [PERSON] et al. [14] defined a form of optimal manipulation for a pair of alternatives. Knowing it allows one to estimate the difficulty of the manipulation for a particular model. In this study, we would like to consider the problem of manipulative behavior for one of the most popular decision-making methods, the quantitative pairwise comparison method, and its best-known multiple criteria implementation, the AHP (Analytic Hierarchy Process). It is a commonly used approach that breaks down complex decisions into smaller, more manageable components, allowing decision-makers to evaluate alternatives based on multiple criteria [15]. The quantitative pairwise comparison method emerged at the beginning of the XX century and is attributed to [PERSON] [16]. It has inspired many researchers such as [PERSON] [17] and [PERSON] [18]. [PERSON]'s seminal paper [19], published in 1977, in which he proposed the AHP method, contributed to its growing popularity. It continues to inspire and be the research subject for those involved in the study of decision-making methods. The research concerns synthesis and aggregation of weights [20; 21], inconsistencies and inconsistency indexes [22; 23], incompleteness [24; 25; 26], preference representation using different representations such as interval, fuzzy, or grey numbers [27; 28; 29], and exploring new areas and methods including BWM [30], MACBETH [31], and HRE [32]. In this context, outranking decision-making methods such as PROMETHEE and ELECTRE [33] or hybrid solutions [34; 35] are worth mentioning. Using multiple-criteria decision-making methods (MCDM) can be particularly challenging when bribery and corruption are prevalent. Bribery, corruption, and manipulation can undermine the integrity of the decision-making process [36], leading to decisions based on personal or financial interests rather than on the merits of the evaluated alternatives. This can have severe consequences for stakeholders' welfare and the sustainability of the decision. As a response to this intricate issue, researchers and practitioners have embraced innovative methodologies to scrutinize, comprehend, and alleviate the influence of manipulation.