A new game-theoretic approach for combining multiple classifiers is proposed. A short introduction in Game Theory and coalitions illustrate the way any collective decision scheme can be viewed as a competitive game of coalitions that are formed naturally when players state their preferences. The winning conditions and the voting power of each player are studied under the scope of voting power indices, as well and the collective competence of the group. Coalitions and power indices are presented in relation to the Condorcet criterion of optimality in voting systems, and weighted Borda count models are asserted as a way to implement them in practice. A special case of coalition games, the weighted majority games (WMG) are presented as a restricted realization in dichotomy choice situations. As a result, the weighted majority rules (WMR), an extended version of the simple majority rules, are asserted as the theoretically optimal and complete solution to this type of coalition gaming. Subsequently, a generalized version of WMRs is suggested as the means to design a voting system that is optimal in the sense of both the correct classification criterion and the Condorcet efficiency criterion. In the scope of Pattern Recognition, a generalized risk-based approach is proposed as the framework upon which any classifier combination scheme can be applied. A new fully adaptive version of WMRs is proposed as a statistically invariant way of adjusting the design process of the optimal WMR to the arbitrary non-symmetrical properties of the underlying feature space. SVM theory is associated with properties and conclusions that emerge from the game-theoretic approach of the classification in general, while the theoretical and practical implications of employing SVM experts in WMR combination schemes are briefly discussed. Finally, a summary of the most important issues for further research is presented.
University of Padova（帕多瓦大学）；Tulane University（杜兰大学）；IHMC（人类与机器认知研究所）；NICTA（国家信息与通信技术澳大利亚有限公司）；UNSW（新南威尔士大学）
In collective decision making, where a voting rule is used to take a collective decision among a group of agents, manipulation by one or more agents is usually considered negative behavior to be avoided, or at least to be made computationally difficult for the agents to perform. However, there are scenarios in which a restricted form of manipulation can instead be beneficial. In this paper we consider the iterative version of several voting rules, where at each step one agent is allowed to manipulate by modifying his ballot according to a set of restricted manipulation moves which are computat…
Many societal decision problems lie in high-dimensional continuous spaces not amenable to the voting techniques common for their discrete or single-dimensional counterparts. These problems are typically discretized before running an election or decided upon through negotiation by representatives. We propose a algorithm called {\sc Iterative Local Voting} for collective decision-making in this setting. In this algorithm, voters are sequentially sampled and asked to modify a candidate solution within some local neighborhood of its current value, as defined by a ball in some chosen norm, with the…
Perpetual voting was recently introduced as a framework for long-term collective decision making. In this framework, we consider a sequence of subsequent approval-based elections and try to achieve a fair overall outcome. To achieve fairness over time, perpetual voting rules take the history of previous decisions into account and identify voters that were dissatisfied with previous decisions. In this paper, we look at perpetual voting rules from an axiomatic perspective and study two main questions. First, we ask how simple such rules can be while still meeting basic desiderata. For two simple…
We study voting games on binary issues, where voters hold an objective over the outcome of the collective decision and are allowed, before the vote takes place, to negotiate their voting strategy with the other participants. We analyse the voters' rational behaviour in the resulting two-phase game, showing under what conditions undesirable equilibria can be removed and desirable ones sustained as a consequence of the pre-vote phase.
We investigate the distribution of the well-studied Shapley--Shubik values in weighted voting games where the agents are stochastically determined. The Shapley--Shubik value measures the voting power of an agent, in typical collective decision making systems. While easy to estimate empirically given the parameters of a weighted voting game, the Shapley values are notoriously hard to reason about analytically. We propose a probabilistic approach in which the agent weights are drawn i.i.d. from some known exponentially decaying distribution. We provide a general closed-form characterization of t…
Proof of Stake (PoS) protocols rely on voting mechanisms to reach consensus on the current state. If an enhanced majority of staking nodes, also called validators, agree on a proposed block, then this block is appended to the blockchain. Yet, these protocols remain vulnerable to faults caused by validators who abstain either accidentally or maliciously. To protect against such faults while retaining the PoS selection and reward allocation schemes, we study weighted voting in validator committees. We formalize the block creation process and introduce validators' voting profiles which we update …
In many real world situations, collective decisions are made using voting. Moreover, scenarios such as committee or board elections require voting rules that return multiple winners. In multi-winner approval voting (AV), an agent may vote for as many candidates as they wish. Winners are chosen by tallying up the votes and choosing the top-$k$ candidates receiving the most votes. An agent may manipulate the vote to achieve a better outcome by voting in a way that does not reflect their true preferences. In complex and uncertain situations, agents may use heuristics to strategize, instead of inc…
Many binary collective choice situations can be described as weighted simple voting games. We introduce weighted committee games to model decisions on an arbitrary number of alternatives in analogous fashion. We compare the effect of different voting weights (share-holdings, party seats, etc.) under plurality, Borda, Copeland, and antiplurality rule. The number and geometry of weight equivalence classes differ widely across the rules. Decisions can be much more sensitive to weights in Borda committees than (anti-)plurality or Copeland ones.
We consider a simple and altruistic multiagent system in which the agents are eager to perform a collective task but where their real engagement depends on the willingness to perform the task of other influential agents. We model this scenario by an influence game, a cooperative simple game in which a team (or coalition) of players succeeds if it is able to convince enough agents to participate in the task (to vote in favor of a decision). We take the linear threshold model as the influence model. We show first the expressiveness of influence games showing that they capture the class of simple…
University of Padova（帕多瓦大学）；NICTA（国家信息与通信技术研究院）；UNSW Sydney（悉尼新南威尔士大学）
We consider multi-agent systems where agents' preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents' preferences is common in many real-life settings due to privacy issues or an ongoing elicitation process. In addition, there may be uncertainty about how the preferences are aggregated. For example, the agenda (a tree whose leaves are labelled with the decisions being compared) may not yet be known or fixed. We therefore st…
Beyond Majority Voting: LLM Aggregation by Leveraging Higher-Order Information
Massachusetts Institute of Technology（麻省理工学院）；School of Engineering and Applied Sciences（工程与应用科学学院）；Harvard University（哈佛大学）；Data Science, The University of Chicago（数据科学，芝加哥大学）
Tulane University（杜兰大学）；Louisiana State University（路易斯安那州立大学）；University of West Florida（西佛罗里达大学）；Institute for Human and Machine Cognition（人机认知研究所）
In many real world situations, collective decisions are made using voting and, in scenarios such as committee or board elections, employing voting rules that return multiple winners. In multi-winner approval voting (AV), an agent submits a ballot consisting of approvals for as many candidates as they wish, and winners are chosen by tallying up the votes and choosing the top-$k$ candidates receiving the most approvals. In many scenarios, an agent may manipulate the ballot they submit in order to achieve a better outcome by voting in a way that does not reflect their true preferences. In complex…
In this paper, a gentle introduction to Game Theory is presented in the form of basic concepts and examples. Minimax and Nash's theorem are introduced as the formal definitions for optimal strategies and equilibria in zero-sum and nonzero-sum games. Several elements of cooperative gaming, coalitions, voting ensembles, voting power and collective efficiency are described in brief. Analytical (matrix) and extended (tree-graph) forms of game representation is illustrated as the basic tools for identifying optimal strategies and "solutions" in games of any kind. Next, a typology of four standard n…
Copenhagen Causality Lab and Pioneer Centre for AI, University of Copenhagen, Denmark（哥本哈根因果实验室和先锋人工智能中心，哥本哈根大学，丹麦）；Cooperative AI Foundation, United Kingdom（协作人工智能基金会，英国）
We consider an agent community wishing to decide on several binary issues by means of issue-by-issue majority voting. For each issue and each agent, one of the two options is better than the other. However, some of the agents may be confused about some of the issues, in which case they may vote for the option that is objectively worse for them. A benevolent external party wants to help the agents to make better decisions, i.e., select the majority-preferred option for as many issues as possible. This party may have one of the following tools at its disposal: (1) educating some of the agents, s…
