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

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

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)

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

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…

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

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

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)

Thank you for listening!