Revealing Positive and Negative Role Models to Help People Make Good Decisions

Abstract

We consider a setting where agents take action by following their role models in a social network, and study strategies for a social planner to help agents by revealing whether the role models are positive or negative. Specifically, agents observe a local neighborhood of possible role models they can emulate, but do not know their true labels. Revealing a positive label encourages emulation, while revealing a negative one redirects agents toward alternative options. The social planner observes all labels, but operates under a limited disclosure budget that it selectively allocates to maximize social welfare (the expected number of agents who emulate adjacent positive role models). We consider both algorithms and hardness results for welfare maximization, and provide a sample-complexity guarantee when the planner observes a sampled subset of agents. We also consider fairness guarantees when agents belong to different groups. It is a technical challenge that the ability to reveal negative role models breaks submodularity. We thus introduce a proxy welfare function that remains submodular even when revealed targets include negative ones. When each agent has at most a constant number of negative target neighbors, we use this proxy to achieve a constant-factor approximation to the true optimal welfare gain. When agents belong to different groups, we also show that each group's welfare gain is within a constant factor of the optimum achievable if the full budget were allocated to that group. Beyond this basic model, we also propose an intervention model that directly connects high-risk agents to positive role models, and a coverage radius model that expands the visibility of selected positive role models. Lastly, we conduct extensive experiments on four real-world datasets to support our theoretical results and assess the effectiveness of the proposed algorithms.

Figures (17)

• Figure 1: An unweighted bipartite graph in which the LHS nodes $\{x_1, x_2\}$ represent agents, the RHS nodes $\{t_1, t_2, t_3\}$ represent targets, and edges connect agents to targets they can emulate. Initially, agents do not know whether a target is positive or negative.
• Figure 2: Comparative social welfare returned by random $F(S_{\mathrm{r}})$ and classical greedy $F(S_{\mathrm{g}})$ algorithms on Adult dataset for $K =\{1,5\}$. Black lines mark the maximum $F(S_{\mathrm{full}})$ and minimum $F(S_{\mathrm{o}})$ welfare. Classic greedy consistently outperforms random when executed at the same target reveal budget $K$.
• Figure 3: Comparison of pre-reveal ($\Delta_{F}(ig,g)$) and post-reveal ($\Delta_{F}(gi,g))$) intervention gains, for $K\!=\!5$ with $B\!=\!\{1,3\}$ across $k$NN graphs on the Productivity dataset. Gains increase with decrease in $K$, and pre-reveal gains may be negative.
• Figure 4: Analysis of the training ($\mathrm{tr}$) and testing ($\mathrm{ts}$) performance $(\mathrm{Perf}_2, \mathrm{Perf}_3)$ scores when Algorithm \ref{['alg:greedy_lbreveal']} is run under budget $K=\{1,5\}$ on $k$NN graphs (see Appendix Table \ref{['tab:math_kmax_r_stats']}) from the Math dataset. When each agent in train/test sets has at most one neighbor, $\mathrm{Perf}_3$ is zero (\ref{['fig:math_learn_knn_perf3']}). Both $\mathrm{Perf}_2$ and $\mathrm{Perf}_3$ increase with $K$.
• Figure 5: A bipartite graph where $n= 4, \mathop{\mathrm{\mathit{m}^{-}}}\nolimits = 2$ and $\mathop{\mathrm{\mathit{m}^{+}}}\nolimits = 4$.

Theorems & Definitions (71)

• Proposition 1
• Definition 2.1: Submodularity
• Proposition 2
• Proposition 3
• proof : Proof sketch
• Theorem 1
• Definition 3.1: Proxy social welfare function
• Definition 3.2: Proxy welfare gain
• Theorem 2
• proof