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Graphical Models for Strategic and Economic Reasoning

Graphical Models for Strategic and Economic Reasoning. Michael Kearns Computer and Information Science University of Pennsylvania BNAIC 2003. Joint work with: Sham Kakade, John Langford, Michael Littman, Luis Ortiz, Satinder Singh. Probabilistic Reasoning.

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Graphical Models for Strategic and Economic Reasoning

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  1. Graphical Models forStrategic and Economic Reasoning Michael Kearns Computer and Information Science University of Pennsylvania BNAIC 2003 Joint work with: Sham Kakade, John Langford, Michael Littman, Luis Ortiz, Satinder Singh

  2. Probabilistic Reasoning • Need to model a complex, multivariate distribution • Dimensionality is high --- cannot write in “tabular” form • Examples: joint distributions of alarms and earthquakes, diseases and symptoms, words and documents • The world is not arbitrary: • Not all variables (directly) influence each other • True for both causal and stochastic influences • Many probabilistic independences hold • Interaction has (network) structure • Should ease modeling and inference • The answer: graphical models for probabilistic reasoning

  3. Structure in Probabilistic Interaction Engineered “Natural” [Horvitz 93] [Frey&MacKay 98]

  4. International Trade [Krempel&Pleumper] Embargoes, free trade, technology, geography…

  5. Corporate Partnerships [Krebs]

  6. Internet Connectivity [CAIDA]

  7. Structure in Social and Economic Analysis • Trade agreements and restrictions • Social relationships between business people • Reporting and organizational structure in a firm • Regulatory restrictions on Wall Street • Shared influences within an industry or sector • Geographical dispersion of consumers • Structural universals (Social Network Theory) • Strategic Reasoning: • Variables are players in a game, organizations, firms, countries… • Interactions characterized by self-interest, not probability • Foundations: game theory and mathematical economics Goal: Replicate the power of graphical models for problems of strategic reasoning.

  8. Outline • Graphical Games and the NashProp Algorithm • [K., Littman & Singh UAI01]; [Ortiz & K. NIPS02] • Correlated Equilibria, Graphical Games, and Markov Networks • [Kakade, K. & Langford EC03] • Arrow-Debreu and Graphical Economics • [Kakade, K. & Ortiz 03]

  9. Graphical Games and NashProp

  10. Basics of Game Theory • Have players 1,…n (think of n as large) • Each has actions 1,…,k (think of k as small) • Action chosen by player i is a_i • Vector ais population joint action • Player i receives payoff M_i(a) • (Note: M_i(a) has size exponential in n!) • (Nash) equilibrium: • Choice of mixed strategies for each player • No player has a unilateral incentive to deviate • Mixed strategy: product distribution over a • Exists for any game; may be many

  11. Graphical Models for Game Theory • Undirected graph G capturing local (strategic) interactions • Each player represented by a vertex • N_i(G) : neighbors of i in G (includes i) • Assume: Payoffs expressible as M_i(a’), where a’ over onlyN_i(G) • Graphical game: (G,{M’_i}) • Compact representation of game; analogous to graph + CPTs • Exponential in max degree (<< # of players) • As with Bayes nets, look for special structurefor efficient inference • Related models: [Koller & Milch 01] [La Mura 00] 8 7 3 2 1 5 4 6

  12. Message-passing, tables of “conditional” Nash equilibria Approximate (all NE) and exact (one NE) versions, efficient for trees NashProp: generalization to arbitrary topology (belief prop) Junction tree and cutset generalizations [Vickrey & Koller 02] The NashProp Algorithm U1 U2 U3 T(w,v) = 1 <--> $ an “upstream” Nash where V = vgivenW = w <--> $u: T(v,u_i) = 1 for all i, and v is a best response to u,w V W

  13. Table dimensions are probability of playing 0 • Black shows T(v,u) = 1 • Ms want to match, Os to unmatch • Relative value modulated by parent values • t=0.01, e= 0.05

  14. Experimental Performance computation time number of players

  15. Correlated Equilibria, Graphical Games and Markov Networks

  16. The Problems with Nash • Technical: • Difficult to compute (even in 2-player, multi-action case) • Conceptual: • Strictly competitive • No ability to cooperate, form coalitions, or bargain • Can lead to suboptimal collective behavior • Fully cooperative game theory: • Somewhat of a mathematical mess • Alternative: correlated equilibria

  17. Correlated Equilibria in Games[Aumann 74] • Recall Nash equilibrium is a product distribution P(a) • Suffices to guarantee existence of equilibrium • Now let P(a) be an arbitrary distribution over joint actions • Third party draws a from P and gives a_i to player i • P(a) is a correlated equilibrium: • Conditioned on everyone else playing P(a|a_i), playing a_i is optimal • No unilateral incentive to deviate, but now actions are correlated • Reduces to Nash for product distributions • Alternative interpretation: shared randomness • Everyday example: traffic signal

  18. Advantages of CE • Technical: • Easier to compute: linear feasibility formulation • Efficient for 2-player, multi-action case • Conceptual: • Correlated actions a fact of the real world • Allows “cooperation via correlation” • Modeling of shared exogenous influences • Enlarged solution space: all mixtures of NE, and more • New (non-Nash) outcomes emerge, often natural ones • Avoid quagmire of full cooperation and coalitions • Natural convergence notion for “greedy” learning • But how do we represent an arbitrary CE? • First, only seek to find CE up to (expected) payoff equivalence • Second, look to graphical models for probabilistic reasoning!

  19. Graphical Games and Markov Networks • Let G be the graph of a graphical game (strategic structure) • Consider the Markov network MN(G): • Form cliques of the local neighborhoods of G • Introduce potential function f_c on each clique c • Joint distribution P(a) = (1/Z)P_c f_c(a) • Theorem: For any game with graph G, and any CE of this game, there is a CE with the same payoffs that can be represented in MN(G) • Preservation of locality • Direct link between strategic and probabilistic reasoning in CE • Computation: In trees (e.g.), can compute a CE efficiently • Parsimonious LP formulation

  20. From Micro to Macro:Arrow-Debreu and Graphical Economics

  21. Arrow-Debreu Economics • Both a generalization (continuous vector actions) and specialization (form of payoffs) of game theory • Have k goods available for consumption • Players are: • Insatiable consumers with utilities for amounts of goods • “Price player” (invisible hand) setting market prices for goods • Liquidity emerges from sale of initial endowments • Alternative model: labor and firms • At equilibrium (consumption plans and prices): • Each consumer maximizing utility given budget constraint • Market clearing: supply equals demand for all goods • May also allow supply to exceed demand at 0 price (free disposal) • ADE always exists • Very little known computationally • [Devanur, Papadimitriou, Saberi, Vazirani 02]: linear utility case

  22. Graphical Economics • Again wish to capture structure, now in multi-economy interaction • Represent each economy by a vertex in a network • “Economies” could be represent individuals or sovereign nations • From international relations to social connections • Same goods available in each economy, but permit local prices • Interpretation: • Allowed to shop for best prices in neighborhood • Utility determined only by good amounts, not their sources • Stronger than ADE: graphical equilibrium • Consumers still maximize utility under budget constraints • Local clearance in all goods (domestic supply = incoming demand) 8 7 3 2 1 5 4 6

  23. Graphical Economics: Results • Graphical equilibria always exist (under ADE condition analogues) • does not follow from AD due to zero endowments of foreign goods • appeal to Debreu’s quasi-rationality: zero wealth may ignore zero prices • Wealth Propagation Lemma: spread of capital on connected graph • relative gridding of prices and consumption plans • ADProp algorithm: • computes controlled approximation to graphical equilibrium • message-passing on conditional prices and inbound/outbound demands • efficient for tree topologies and smooth utilities 8 7 3 2 1 5 4 6

  24. Conclusion • Use of game-theoretic and economic models rising • Evolutionary biology • Behavioral game theory and economics NYT 6/17 • Neuroeconomics • Computer Science • Electronic Commerce • Many of these uses are raising • Computational issues • Representational issues • Well-developed theory of graphical models for GT/econ • Structure of interaction between individuals and organizations • What about structure in • Utilities, actions, repeated interaction, learning, states,… mkearns@cis.upenn.edu www.cis.upenn.edu/~mkearns

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