Fixed-budget and Multiple-issue Quadratic Voting

Abstract

Quadratic Voting (QV) is a social choice mechanism that addresses the "tyranny of the majority" of one-person-one-vote mechanisms. Agents express not only their preference ordering but also their preference intensity by purchasing $x$ votes at a cost of $x^2$. Although this pricing rule maximizes utilitarian social welfare and is robust against strategic manipulation, it has not yet found many real-life applications. One key reason is that the original QV mechanism does not limit voter budgets. Two variations have since been proposed: a (no-budget) multiple-issue generalization and a fixed-budget version that allocates a constant number of credits to agents for use in multiple binary elections. While some analysis has been undertaken with respect to the multiple-issue variation, the fixed-budget version has not yet been rigorously studied. In this work, we formally propose a novel fixed-budget multiple-issue QV mechanism. This integrates the advantages of both the aforementioned variations, laying the theoretical foundations for practical use cases of QV, such as multi-agent resource allocation. We analyse our fixed-budget multiple-issue QV by comparing it with traditional voting systems, exploring potential collusion strategies, and showing that checking whether strategy profiles form a Nash equilibrium is tractable.

Figures (2)

• Figure 1: Strategy based on $V$ and $W = W^- \cup W^+$. The green nodes represent outcomes that become the possible winners after agent $i$ casts their votes. The red nodes are the outcomes that receive negative support from agent $i$, in order to be excluded from winning the election. The agent does not cast any votes for the gray outcomes.
• Figure 2: The functions $g_\omega$ and the intersections described by $V \in G_I$. If $V \in [-\infty, V_{\phi\pi}]$, then $g_\pi\geq g_\phi \geq g_\psi$, however, if $V \in [V_{\phi\pi}, V_{\psi\pi}]$ then $g_\phi \geq g_\pi \geq g_\psi$, and so on. In this case, $G_I = \{-\infty, V_{\phi\pi}, V_{\psi\pi}, V_{\phi\psi}, \infty\}.$

Theorems & Definitions (39)

• Definition 2.1: No-budget multiple-issue QV
• Example 2.1
• Definition 3.1: Fixed-budget multiple-issue QV
• Theorem 3.1
• Definition 3.2
• Example 3.1
• Theorem 3.2
• Theorem 3.3
• proof : Proof sketch
• Definition 4.1