Games, Algorithms, and Social Good

My slides for my presentation "Games, Algorithms, and Social Good" presented at PyData Global on November 15, 2020.

Transcript

1. Games, Algorithms, and Social Good Manojit Nandi @mnandi92
2. Game Theory: Review
3. Prisoner’s Dilemma • In NASH EQUILIBRIUM, no player wants to change their decision (fixing everyone else’s actions). • If Player 2 chooses to Stay Silent Player 1 will want to Confess (0 vs -1). • If Player 2 chooses to Confess Player 1 will want to Confess(-3 vs -5). • The Nash Equilibrium of this game is (Confess, Confess). -1 -5 0 -3 Silent Confess Silent Confess -1 0 -3 -5
4. Stag Hunt Game • This game has two Nash Equilibriums: (Stag, Stag) and (Hare, Hare). • This is an important problem of evolutionary game theory ◦ Stag Hunt shows the evolution of trust and the value of communication. ◦ (Stag, Stag) is a better equilibrium than (Hare, Hare), but this requires me to trust the other player. 5 0 2 2 Stag Hare Stag Hare 5 2 2 0
5. Cooperative Game Theory
6. Cooperative Games • In cooperative games, the unit of analysis is a coalition, a group of players. • The goal of cooperative game theory is to build a team of players to achieve a common result. ◦ Ensemble Learning has a lot of theoretical basis in cooperative game theory. • With the recent interest in model interpretability, researchers have applied cooperative game theory to problem of evaluating feature importance in a model. ◦ The features of the model are seen as the “players” of the game, and the model’s output is “result” of the game.
7. Shapley Values • Key Idea: Can we measure each player’s contribution to the team. • Heuristic: If we remove a player from the team and the outcome doesn’t change, then the player wasn’t useful. • To compute the Shapley Value for a player, we compute each outcome where the player was present, and compare it to the counterfactual outcome where the player wasn’t present. Player i present Player i not present
8. SHAP for Model Interpretability • TreeShap is an implementation of SHAP integrated with XGBoost, a popular gradient boosting tree algorithm. • Brute-forcing all 2N feature combinations is inefficient, so TreeSHAP uses the structure of decision trees to quickly approximate the Shapley Values. • Python Library: https://github.com/slundberg/shap (Age, Education, Marital Status) (Age, Education, Marital Status, Occupation) (Age, Education, Marital Status, Workclass) (Age, Education, Marital Status, Occupation, Workclass)
9. Equitable Valuation of Data • One new problem of interest in algorithmic fairness is how to compensate individuals for their data*. • Game Theoretic Set-up of the Problem: ◦ We view the training data points D as the “players” of the game, and given some learning algorithm A and performance metric V, we want to optimize the result V(S, A) where S is a subset of the data. ◦ This is not a universal framework: The result is dependent on the choice of the algorithm A and performance metric V. • Goal: We want to compute the marginal contribution of each data point Φ i (V) to the performance metric. • Gradient Shapley Algorithm ◦ Generate a random permutation of the training data. ◦ Update model using gradient descent, and compute the marginal contribution as the change in performance. Source: Ghorbani and Zou (2019), http://proceedings.mlr.press/v97/ghorba ni19c/ghorbani19c.pdf
10. Mechanism Design
11. Mechanism Design: Intuition Game Theory: Starting with an economic incentive structure, Find the optimal* behavior for each agent. Mechanism Design: Starting with the desired behavior for each agent, Find an incentive structure that induces this behavior. Economic Incentives Optimal Behavior Game Theory Mech. Design
12. Cake-Cutting: Problem • Alice and Bob are given a cake, and they must decide how to split the cake between themselves. • Assume each of them prefer larger portions of cake (monotonically increasing utility), how can we cut the cake to appease both of them? • Formally, we want the cake allocation to be envy-free (Alice doesn’t want Bob’s cake more than her own).
13. Cake-Cutting: Solution • Alice cuts the cake into two portions, and Bob chooses which portion he wants. • Assuming Alice is rational*, she will realize if she cuts the cake unevenly (for example, 70/30 split), Bob can always take the larger portion. • In order to maximize her portion, Alice should cut the cake 50/50. • This frameworks in an optimal allocation that is both proportional (fair) and envy-free (neither strictly prefers the others’ allocation). *rational as defined by rational choice theory
14. Kidney Exchange Market • One key contribution of Mechanism Design is designing markets for exchange economies. ◦ Alvin E. Roth won the 2012 Nobel Prize in Economics for this and other related work. • Given a set of Donors D and recipients R, we can construct a bipartite graph from a donor d to a recipient r if they have blood-type compatibility. • With blood types, there are universal donors (blood-type O negative) and universal receivers (blood-type AB positive). ◦ Heuristic: We want to efficiently match the most number of donor/recipient pairs, so we shouldn’t waste universal donors on universal receivers. • Roth’s work allows for maximal number matchings in this Kidney exchange graph.
15. Mechanism Design for Social Good (MD4SG) Rediet Abebe
16. • MD4SG is an initiative to apply algorithms, optimization, and mathematical economics to improve access to opportunity for historically underserved and marginalized communities. • Many existing Data Science for Social Good initiatives often apply existing algorithms to problems of societal impact. • The goal of this research is to CREATE NEW mathematical and computational frameworks for optimizing resource allocation and improving information access.
17. Income Subsidies • Households with low income have different susceptibility to income shocks (unexpected medical bill, car accident). • Goal: We want to allocate income subsidies such that we reduce the risk that any household experiences financial ruin (bankruptcy) • Using ideas from ruin probabilities, the authors create a framework to minimize the likelihood that any household becomes financially ruined. Ruin Probability for Individual i: We model the probability an individual experiences financial ruin as a function of their initial income, net pay, and financial shock size and frequency.
18. Problem Formulation • Each household starts with some initial income reserve u i . • At each timestep t, the household receives some net income c i . • HOWEVER, household also randomly experience income shocks S ij during the time step • We want to keep the household’s total income R i (t) to never drop below 0. ◦ Connection to Random Walks Initial Income Net Income received Sum of Income Shocks Income at time t
19. Allocating Subsidies • Goal: We want some total budget B for income subsidies. We want to allocate some subsidy x i for each household such that • Min-Sum Objective: ◦ Each household has some weight w i representing social cost of ruin. In this formulation we want to minimize the weighted expected number of agents who experience ruin. ◦ A solution to this objective function can be approximated using the Knapsack algorithm (Value of household is societal weight and weight of household is income subsidy). • Min-Max Objective: ◦ In this social objective, we want to keep the worst ruin probability experienced by any of the households as low as possible (Save everyone) ◦ A solution to this objective function can be approximated using modified Binary Search. Guess some min-max solution p, if it is infeasible, adjust down, else adjust up. Worst ruin probability
20. Works References • A Unified Approach to Interpreting Model Predictions; Scott Lundberg, Su-In Lee. https://arxiv.org/abs/1705.07874 • Data Shapley: Equitable Valuation of Data for Machine Learning; Amirata Ghorbani, James Zou. https://arxiv.org/abs/1904.02868 • Kidney Exchange; Alvin Roth, Tayfun Sonmez, M. Utku Unver. https://web.stanford.edu/~alroth/papers/kidney.qje.pdf • Subsidy Allocation in the Presence of Income Shocks; Rediet Abebe, Jon Kleinberg, S. Matthew Weinberg. https://www.cs.cornell.edu/home/kleinber/aaai20-shocks.pdf
21. Further Resources • Courses ◦ Truth, Justice, and Algorithms (CMU) ◦ AI Methods for Social Good (CMU) ◦ Applied Mechanism Design for Social Good (UMD) • Books ◦ The Model Thinker by Scott Page ◦ Handbook of Computational Social Choice by Ariel Procaccia ◦ Algorithmic Game Theory by Nisan, Roughgarden, Tardos, and Vazirani.