Graphical Models for Strategic and Economic Reasoning Michael Kearns Computer and Information Science University of Pennsy...
Probabilistic Reasoning <ul><li>Need to model a complex, multivariate distribution </li></ul><ul><li>Dimensionality is hig...
Structure in Probabilistic Interaction [Frey&MacKay 98] [Horvitz 93] Engineered “ Natural”
International Trade [Krempel&Pleumper ] Embargoes, free trade, technology, geography…
Corporate Partnerships [Krebs ]
Internet Connectivity [CAIDA]
Structure in Social and Economic Analysis <ul><li>Trade agreements and restrictions </li></ul><ul><li>Social relationships...
Outline <ul><li>Graphical Games and the NashProp Algorithm </li></ul><ul><ul><li>[K., Littman & Singh UAI01]; [Ortiz & K. ...
Graphical Games and NashProp
Basics of Game Theory <ul><li>Have players 1,…n (think of n as large) </li></ul><ul><li>Each has actions 1,…,k (think of k...
 
Graphical Models for Game Theory <ul><li>Undirected  graph  G  capturing  local (strategic) interactions </li></ul><ul><li...
The NashProp Algorithm <ul><li>Message-passing, tables of “conditional” Nash equilibria </li></ul><ul><li>Approximate (all...
<ul><li>Table dimensions are probability of playing 0 </li></ul><ul><li>Black shows T(v,u) = 1 </li></ul><ul><li>Ms want t...
 
Experimental Performance number of players computation time
Correlated Equilibria, Graphical Games and Markov Networks
The Problems with Nash <ul><li>Technical: </li></ul><ul><ul><li>Difficult to compute (even in 2-player, multi-action case)...
Correlated Equilibria in Games [Aumann 74] <ul><li>Recall Nash equilibrium is a product distribution  P( a ) </li></ul><ul...
Advantages of CE <ul><li>Technical: </li></ul><ul><ul><li>Easier to compute: linear feasibility formulation </li></ul></ul...
Graphical Games and Markov Networks <ul><li>Let  G  be the graph of a graphical game ( strategic  structure) </li></ul><ul...
From Micro to Macro: Arrow-Debreu and Graphical Economics
Arrow-Debreu Economics <ul><li>Both a generalization (continuous vector actions) and specialization (form of payoffs) of g...
Graphical Economics <ul><li>Again wish to capture  structure,  now in  multi-economy  interaction </li></ul><ul><li>Repres...
Graphical Economics: Results <ul><li>Graphical  equilibria always exist (under ADE condition analogues) </li></ul><ul><ul>...
Conclusion <ul><li>Use of game-theoretic and economic models rising </li></ul><ul><ul><li>Evolutionary biology </li></ul><...
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tutorial_kearns

  1. 1. 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
  2. 2. Probabilistic Reasoning <ul><li>Need to model a complex, multivariate distribution </li></ul><ul><li>Dimensionality is high --- cannot write in “tabular” form </li></ul><ul><li>Examples: joint distributions of alarms and earthquakes, diseases and symptoms, words and documents </li></ul><ul><li>The world is not arbitrary: </li></ul><ul><ul><li>Not all variables (directly) influence each other </li></ul></ul><ul><ul><li>True for both causal and stochastic influences </li></ul></ul><ul><ul><li>Many probabilistic independences hold </li></ul></ul><ul><ul><li>Interaction has (network) structure </li></ul></ul><ul><ul><li>Should ease modeling and inference </li></ul></ul><ul><li>The answer: graphical models for probabilistic reasoning </li></ul>
  3. 3. Structure in Probabilistic Interaction [Frey&MacKay 98] [Horvitz 93] Engineered “ Natural”
  4. 4. International Trade [Krempel&Pleumper ] Embargoes, free trade, technology, geography…
  5. 5. Corporate Partnerships [Krebs ]
  6. 6. Internet Connectivity [CAIDA]
  7. 7. Structure in Social and Economic Analysis <ul><li>Trade agreements and restrictions </li></ul><ul><li>Social relationships between business people </li></ul><ul><li>Reporting and organizational structure in a firm </li></ul><ul><li>Regulatory restrictions on Wall Street </li></ul><ul><li>Shared influences within an industry or sector </li></ul><ul><li>Geographical dispersion of consumers </li></ul><ul><li>Structural universals (Social Network Theory) </li></ul>Goal: Replicate the power of graphical models for problems of strategic reasoning. <ul><li>Strategic Reasoning: </li></ul><ul><ul><li>Variables are players in a game, organizations, firms, countries… </li></ul></ul><ul><ul><li>Interactions characterized by self-interest, not probability </li></ul></ul><ul><ul><li>Foundations: game theory and mathematical economics </li></ul></ul>
  8. 8. Outline <ul><li>Graphical Games and the NashProp Algorithm </li></ul><ul><ul><li>[K., Littman & Singh UAI01]; [Ortiz & K. NIPS02] </li></ul></ul><ul><li>Correlated Equilibria, Graphical Games, and Markov Networks </li></ul><ul><ul><li>[Kakade, K. & Langford EC03] </li></ul></ul><ul><li>Arrow-Debreu and Graphical Economics </li></ul><ul><ul><li>[Kakade, K. & Ortiz 03] </li></ul></ul>
  9. 9. Graphical Games and NashProp
  10. 10. Basics of Game Theory <ul><li>Have players 1,…n (think of n as large) </li></ul><ul><li>Each has actions 1,…,k (think of k as small) </li></ul><ul><li>Action chosen by player i is a_i </li></ul><ul><li>Vector a is population joint action </li></ul><ul><li>Player i receives payoff M_i( a ) </li></ul><ul><li>(Note: M_i( a ) has size exponential in n!) </li></ul><ul><li>(Nash) equilibrium: </li></ul><ul><ul><li>Choice of mixed strategies for each player </li></ul></ul><ul><ul><li>No player has a unilateral incentive to deviate </li></ul></ul><ul><ul><li>Mixed strategy: product distribution over a </li></ul></ul><ul><li>Exists for any game; may be many </li></ul>
  11. 12. Graphical Models for Game Theory <ul><li>Undirected graph G capturing local (strategic) interactions </li></ul><ul><li>Each player represented by a vertex </li></ul><ul><li>N_i(G) : neighbors of i in G (includes i) </li></ul><ul><li>Assume: Payoffs expressible as M_i( a’ ), where a’ over only N_i(G) </li></ul><ul><li>Graphical game: (G,{M’_i}) </li></ul><ul><li>Compact representation of game; analogous to graph + CPTs </li></ul><ul><li>Exponential in max degree (<< # of players) </li></ul><ul><li>As with Bayes nets, look for special structure for efficient inference </li></ul><ul><li>Related models: [Koller & Milch 01] [La Mura 00] </li></ul>2 4 3 5 8 7 6 1
  12. 13. The NashProp Algorithm <ul><li>Message-passing, tables of “conditional” Nash equilibria </li></ul><ul><li>Approximate (all NE) and exact (one NE) versions, efficient for trees </li></ul><ul><li>NashProp: generalization to arbitrary topology (belief prop) </li></ul><ul><li>Junction tree and cutset generalizations [Vickrey & Koller 02] </li></ul>U1 U2 U3 W V T(w,v) = 1 <-->   an “upstream” Nash where V = v given W = w <-->  u : T(v,u_i) = 1 for all i, and v is a best response to u ,w
  13. 14. <ul><li>Table dimensions are probability of playing 0 </li></ul><ul><li>Black shows T(v,u) = 1 </li></ul><ul><li>Ms want to match, Os to unmatch </li></ul><ul><li>Relative value modulated by parent values </li></ul><ul><li>  =  0.01,   = 0.05 </li></ul>
  14. 16. Experimental Performance number of players computation time
  15. 17. Correlated Equilibria, Graphical Games and Markov Networks
  16. 18. The Problems with Nash <ul><li>Technical: </li></ul><ul><ul><li>Difficult to compute (even in 2-player, multi-action case) </li></ul></ul><ul><li>Conceptual: </li></ul><ul><ul><li>Strictly competitive </li></ul></ul><ul><ul><li>No ability to cooperate, form coalitions, or bargain </li></ul></ul><ul><ul><li>Can lead to suboptimal collective behavior </li></ul></ul><ul><li>Fully cooperative game theory: </li></ul><ul><ul><li>Somewhat of a mathematical mess </li></ul></ul><ul><li>Alternative: correlated equilibria </li></ul>
  17. 19. Correlated Equilibria in Games [Aumann 74] <ul><li>Recall Nash equilibrium is a product distribution P( a ) </li></ul><ul><li>Suffices to guarantee existence of equilibrium </li></ul><ul><li>Now let P( a ) be an arbitrary distribution over joint actions </li></ul><ul><li>Third party draws a from P and gives a_i to player i </li></ul><ul><li>P( a ) is a correlated equilibrium: </li></ul><ul><ul><li>Conditioned on everyone else playing P( a |a_i), playing a_i is optimal </li></ul></ul><ul><ul><li>No unilateral incentive to deviate, but now actions are correlated </li></ul></ul><ul><ul><li>Reduces to Nash for product distributions </li></ul></ul><ul><li>Alternative interpretation: shared randomness </li></ul><ul><li>Everyday example: traffic signal </li></ul>
  18. 20. Advantages of CE <ul><li>Technical: </li></ul><ul><ul><li>Easier to compute: linear feasibility formulation </li></ul></ul><ul><ul><li>Efficient for 2-player, multi-action case </li></ul></ul><ul><li>Conceptual: </li></ul><ul><ul><li>Correlated actions a fact of the real world </li></ul></ul><ul><ul><li>Allows “cooperation via correlation” </li></ul></ul><ul><ul><li>Modeling of shared exogenous influences </li></ul></ul><ul><ul><li>Enlarged solution space: all mixtures of NE, and more </li></ul></ul><ul><ul><li>New (non-Nash) outcomes emerge, often natural ones </li></ul></ul><ul><ul><li>Avoid quagmire of full cooperation and coalitions </li></ul></ul><ul><ul><li>Natural convergence notion for “greedy” learning </li></ul></ul><ul><li>But how do we represent an arbitrary CE? </li></ul><ul><ul><li>First, only seek to find CE up to (expected) payoff equivalence </li></ul></ul><ul><ul><li>Second, look to graphical models for probabilistic reasoning! </li></ul></ul>
  19. 21. Graphical Games and Markov Networks <ul><li>Let G be the graph of a graphical game ( strategic structure) </li></ul><ul><li>Consider the Markov network MN(G): </li></ul><ul><ul><li>Form cliques of the local neighborhoods of G </li></ul></ul><ul><ul><li>Introduce potential function  c on each clique c </li></ul></ul><ul><ul><li>Joint distribution P( a ) = (1/Z)  c  c( a ) </li></ul></ul><ul><li>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) </li></ul><ul><li>Preservation of locality </li></ul><ul><li>Direct link between strategic and probabilistic reasoning in CE </li></ul><ul><li>Computation: In trees (e.g.), can compute a CE efficiently </li></ul><ul><ul><li>Parsimonious LP formulation </li></ul></ul>
  20. 22. From Micro to Macro: Arrow-Debreu and Graphical Economics
  21. 23. Arrow-Debreu Economics <ul><li>Both a generalization (continuous vector actions) and specialization (form of payoffs) of game theory </li></ul><ul><li>Have k goods available for consumption </li></ul><ul><li>Players are: </li></ul><ul><ul><li>Insatiable consumers with utilities for amounts of goods </li></ul></ul><ul><ul><li>“ Price player” (invisible hand) setting market prices for goods </li></ul></ul><ul><li>Liquidity emerges from sale of initial endowments </li></ul><ul><ul><li>Alternative model: labor and firms </li></ul></ul><ul><li>At equilibrium (consumption plans and prices): </li></ul><ul><ul><li>Each consumer maximizing utility given budget constraint </li></ul></ul><ul><ul><li>Market clearing: supply equals demand for all goods </li></ul></ul><ul><ul><li>May also allow supply to exceed demand at 0 price (free disposal) </li></ul></ul><ul><li>ADE always exists </li></ul><ul><li>Very little known computationally </li></ul><ul><ul><li>[Devanur, Papadimitriou, Saberi, Vazirani 02]: linear utility case </li></ul></ul>
  22. 24. Graphical Economics <ul><li>Again wish to capture structure, now in multi-economy interaction </li></ul><ul><li>Represent each economy by a vertex in a network </li></ul><ul><ul><li>“ Economies” could be represent individuals or sovereign nations </li></ul></ul><ul><ul><li>From international relations to social connections </li></ul></ul><ul><li>Same goods available in each economy, but permit local prices </li></ul><ul><li>Interpretation: </li></ul><ul><ul><li>Allowed to shop for best prices in neighborhood </li></ul></ul><ul><ul><li>Utility determined only by good amounts, not their sources </li></ul></ul><ul><li>Stronger than ADE: graphical equilibrium </li></ul><ul><ul><li>Consumers still maximize utility under budget constraints </li></ul></ul><ul><ul><li>Local clearance in all goods (domestic supply = incoming demand) </li></ul></ul>2 4 3 5 8 7 6 1
  23. 25. Graphical Economics: Results <ul><li>Graphical equilibria always exist (under ADE condition analogues) </li></ul><ul><ul><li>does not follow from AD due to zero endowments of foreign goods </li></ul></ul><ul><ul><li>appeal to Debreu’s quasi-rationality: zero wealth may ignore zero prices </li></ul></ul><ul><ul><li>Wealth Propagation Lemma: spread of capital on connected graph </li></ul></ul><ul><ul><li>relative gridding of prices and consumption plans </li></ul></ul><ul><li>ADProp algorithm: </li></ul><ul><ul><li>computes controlled approximation to graphical equilibrium </li></ul></ul><ul><ul><li>message-passing on conditional prices and inbound/outbound demands </li></ul></ul><ul><ul><li>efficient for tree topologies and smooth utilities </li></ul></ul>2 4 3 5 8 7 6 1
  24. 26. Conclusion <ul><li>Use of game-theoretic and economic models rising </li></ul><ul><ul><li>Evolutionary biology </li></ul></ul><ul><ul><li>Behavioral game theory and economics NYT 6/17 </li></ul></ul><ul><ul><li>Neuroeconomics </li></ul></ul><ul><ul><li>Computer Science </li></ul></ul><ul><ul><li>Electronic Commerce </li></ul></ul><ul><li>Many of these uses are raising </li></ul><ul><ul><li>Computational issues </li></ul></ul><ul><ul><li>Representational issues </li></ul></ul><ul><li>Well-developed theory of graphical models for GT/econ </li></ul><ul><ul><li>Structure of interaction between individuals and organizations </li></ul></ul><ul><li>What about structure in </li></ul><ul><ul><li>Utilities, actions, repeated interaction, learning, states,… </li></ul></ul>[email_address] www.cis.upenn.edu/~mkearns

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