Rigging Elo Ratings: Achieving the Highest Possible Rating Through Game Manipulation

Core Concepts

It is possible to achieve an extremely high Elo rating by rigging a large number of games, with the maximum rating depending on the number of players and games.

Abstract

The paper explores the problem of how high a player's Elo rating can be maximized through rigging games, given n players and a total of k games played.
The key insights are:

For n = 2 players, the highest achievable rating is bounded by a function of the pot function σ used in the Elo system. Specifically, the highest rating is Θ(f^-1(2k)), where f is defined in terms of σ.

For n > 2 players, there is a phase transition at n = k^(1/3). For n = Ω(k^(1/3)), the highest achievable rating is Θ(k^(1/3)), with the exact constant depending on the pot function σ. For n = o(k^(1/3)), the highest rating is Θ(n).

The paper provides both lower and upper bounds on the highest achievable rating for general n and k. The tightness of the bounds depends on the specific pot function σ used.

The key quantity governing the bounds is the left-tail behavior of the pot function σ, as captured by the function f defined in the paper.

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Key Insights Distilled From

Achieving the Highest Possible Elo Rating

by Rikhav Shah at arxiv.org 04-16-2024

https://arxiv.org/pdf/2203.14567.pdf

Deeper Inquiries

How would the results change if the players were allowed to collude or coordinate their game outcomes, rather than just rigging individual games

Allowing players to collude or coordinate their game outcomes would significantly impact the results discussed in the context. If players can work together to manipulate the outcomes of multiple games, they could potentially boost their ratings more effectively than by rigging individual games. This collusion could lead to a more strategic approach to maximizing Elo ratings, as players could plan a series of games to ensure certain players achieve the highest possible ratings. The optimization of ratings in a coordinated manner would likely result in a different upper bound for the highest achievable Elo rating, as the dynamics of the system would change with collusion involved.

What are the practical implications of these findings for real-world Elo rating systems

The findings presented in the paper have important practical implications for real-world Elo rating systems, especially in competitive environments where rating manipulation can occur. Organizers of such systems need to be aware of the vulnerabilities highlighted in the research and take steps to mitigate the risks of rating manipulation. Some potential adaptations that organizers could consider include:

Implementing stricter monitoring and oversight during games to detect any suspicious patterns or collusion among players.
Introducing randomization or additional factors into the rating calculation process to make it more difficult to predict or manipulate outcomes.
Setting limits on the number of games that can contribute to a player's rating within a certain time frame to prevent rapid rating inflation.
Conducting regular audits or reviews of player performance data to identify any anomalies or irregularities that may indicate manipulation.

By incorporating these measures, organizers can help safeguard the integrity of their Elo rating systems and maintain a fair and competitive environment for all participants.

How might organizers adapt their systems to mitigate the risks of rating manipulation

There is indeed a connection between the maximum achievable Elo rating and other optimization problems, such as the maximum overhang problem mentioned in the paper. Both problems involve maximizing a certain metric or value within a set of constraints. Insights from one problem can inform the other by providing new perspectives on optimization strategies and approaches.
For example, the strategies developed to maximize Elo ratings by manipulating game outcomes could potentially be adapted or applied to solve other optimization problems where maximizing a specific outcome is the goal. Similarly, the techniques used to analyze the phase transition and bounds in the Elo rating context could inspire new approaches to tackling similar optimization challenges in different domains.
By exploring the parallels between these problems and drawing on the methodologies and insights from one to inform the other, researchers and practitioners can enhance their problem-solving capabilities and potentially discover novel solutions to complex optimization problems.