Computing and Approximating Equilibria : How… …and What’s the Point?

1. Computing and Approximating Equilibria: How… …and What’s the Point? Yevgeniy Vorobeychik Sandia National Laboratories

2. Who am I? • Ph.D. CS, University of Michigan, advised by Michael Wellman • approximating/estimating equilibria in simulation-based games; computational mechanism design • Postdoc, University of Pennsylvania, advised by Michael Kearns • behavioral experiments on social networks (e.g., networked battle-of-the-sexes, network formation, etc) • Currently: Sandia National Labs • game theoretic analysis of complex systems

3. Is Computing a Nash equilibrium Hard? • PPAD complete – seems pretty hard • Leveraging graphical structure helps • AGGs (action-graph games), graphical games, etc • still hard… • Custom solvers for special cases: • e.g., Stackelberg games for security • Simple search methods • often really good: most games in GAMUT have equilibria with very small support size (most have a pure strategy Nash equilibrium)

4. What about GIANT games? • Infinite strategy spaces? Bayesian games? Dynamic games? • Yikes! • Heuristics seem to work really well at approximating Nash equilibria • Variations on iterative best response (TABU best response, keeping track of game theoretic regret, etc) • min-regret-first heuristic (explore deviations from lowest-regret profiles) • Noisy payoff function evaluations? • Take lots of samples • compute the next best sample (previous work based on KL divergence of before/after probability distributions of minimum regret profiles) • EVI (expected value of information)-based heuristic

5. Great, we can solve games. Now what? • Stackelberg games for security: • Compute optimal protection decisions against an intelligent adversary • Implemented by airports, federal air marshal • Solve other, or more complex,security related games…

6. Great, we can solve games. Now what? • Mechanism design: • can make policy decisions, solving game induced by a policy choice to “predict” strategic outcomes • example: • government can make or subsidize infrastructure investments in the electric grid • can determine the development of grid network; goal: facilitate development of renewable sources (e.g., wind) • decisions about building wind farms and generating electricity are based on grid development; done by an imperfectly competitive market

7. Great, we can solve games. Now what? • Computational “characterizations” • map out a “strategic landscape” for a complex game theoretic model • example A: what happens in a keyword auction (appropriately stylized) when market conditions change (e.g., increased/decreased number of competitors; increased/decreased number of search engines; changing ranking/pricing rules) • qualitative AND quantitative illustrations • multi-unit auctions example: know that bid under value; underbidding increases with quanitity; can we quantify this in specific settings? • somewhat related to “mechanism design”, but not entirely

8. Beyond Nash Equilibria • We want a predictive model of behavior • humans • or computers • Try to use data from multiple sources (game models, actual behavior) to predict behavior in future settings • Consider principled models of non-financial motivations; maybe alternative representations of preferences (prospect theory, goals-plans) • people care about a variety of things ($$, social capital, fairness, etc)

9. Thank you for listening!