Econ.

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Econ.

  1. 1. Economic Foundations and Game Theory Peter Wurman
  2. 2. Presentation Overview <ul><li>Economics </li></ul><ul><li>Economics of Trading Agents </li></ul><ul><li>Economic modeling </li></ul><ul><li>General Equilibrium and its Limitations </li></ul><ul><li>Mechanism design </li></ul><ul><li>Introduction to Game Theory </li></ul><ul><li>Pareto Efficiency and Dominant strategy </li></ul><ul><li>Nash Equilibrium </li></ul><ul><li>Mixed Strategies </li></ul><ul><li>Extensive Form and Sub-game Analysis </li></ul><ul><li>Advanced Topics in Game Theory </li></ul>
  3. 3. Economics <ul><li>Study of the allocation of limited resources in a society of self-interested agents. </li></ul><ul><li>Essential features: </li></ul><ul><ul><li>Agents are rational; </li></ul></ul><ul><ul><li>Decisions concern the use of resources; </li></ul></ul><ul><ul><li>Prices significantly simplify the allocation process. </li></ul></ul><ul><li>Note: agents are not assumed to be software entities here. </li></ul>
  4. 4. Trading Agents <ul><li>Agent : software to which we ascribe </li></ul><ul><ul><li>Beliefs and knowledge; </li></ul></ul><ul><ul><li>Rationality; </li></ul></ul><ul><ul><li>Competence; </li></ul></ul><ul><ul><li>Autonomy. </li></ul></ul><ul><li>Trading agent : software that participates in an electronic market and </li></ul><ul><ul><li>Is governed in its decision-making by a set of constraints (budget) and preferences; </li></ul></ul><ul><ul><li>Obtains the above from a user; </li></ul></ul><ul><ul><li>Acts in the world by making offers (bids) on the user’s behalf. </li></ul></ul>
  5. 5. Economics of Trading Agents <ul><li>We will consider economics of trading agents as software entities. </li></ul><ul><li>Elements of an Economic Model </li></ul><ul><ul><li>Resources; </li></ul></ul><ul><ul><li>Agents; </li></ul></ul><ul><ul><li>Market Infrastructure. </li></ul></ul>
  6. 6. Resources <ul><li>Resources </li></ul><ul><ul><li>Limited; </li></ul></ul><ul><ul><li>Consumed (private) or shared (public). </li></ul></ul><ul><li>Formalization </li></ul><ul><ul><li>N is the number of resources types; </li></ul></ul><ul><ul><li>x i is an amount of resource i; </li></ul></ul><ul><ul><li>x is a N -vector of quantities. </li></ul></ul>
  7. 7. Two Types of Agents <ul><li>Consumers </li></ul><ul><ul><li>Derive value from owning/consuming resources. </li></ul></ul><ul><li>Producers </li></ul><ul><ul><li>Have technologies to transform resources; </li></ul></ul><ul><ul><li>Goal is to make money (distributed to shareholders). </li></ul></ul><ul><li>Both have private information. </li></ul>
  8. 8. Consumer Preferences <ul><li>Preferences ( >, ≥ ) </li></ul><ul><ul><li>Total preorder over all bundles x in X </li></ul></ul><ul><ul><ul><li>x ≥ x’ or x’ ≥ x (completeness) </li></ul></ul></ul><ul><ul><ul><li>x ≥ x’ and x’ ≥ x” implies x ≥ x” (transitivity) </li></ul></ul></ul>
  9. 9. Consumer Preferences (2) <ul><li>Often, we assume convexity </li></ul><ul><ul><li>For all  in [0,1], x ≥ x” and x’ ≥ x” and x ≠ x’ implies [  x + (1-  ) x’ ] ≥ x” </li></ul></ul>x 1 x 2 x x ’ x ”
  10. 10. Preferences Expressed as Utility <ul><li>Generally, we express preferences as a utility function: </li></ul><ul><ul><li>u j ( x ) assigns a numeric value to all bundles </li></ul></ul><ul><li>Often, we assume that utility is quasi-linear in one resource: </li></ul><ul><ul><li>u j ( x ) = v j ( x ) + m, where m is money </li></ul></ul>
  11. 11. Consumer Endowments <ul><li>Consumers generally begin with some resources, denoted e j. </li></ul><ul><li>Often, these endowments do not maximize the agent’s utility. </li></ul><ul><li>Agents engage in economic activities. </li></ul>
  12. 12. Simple Exchange Economy <ul><li>Suppose all participants are consumers </li></ul><ul><li>How do we determine resources to exchange? </li></ul><ul><li>What is a “good” allocation? </li></ul>Agent 1 Agent 2 Agent 3 Agent 1 Agent 2 Agent 3
  13. 13. Price Systems <ul><li>Associate a price p i with each resource i </li></ul><ul><li>Prices specify resource exchange rates: </li></ul><ul><ul><li>One unit of i can be exchanged for p i /p h units of h. </li></ul></ul><ul><li>Present a common scale on which to measure resource value. </li></ul><ul><li>Very compact representation of value </li></ul>
  14. 14. Solutions <ul><li>An allocation assigns quantities of each resource to each consumer </li></ul><ul><li>Feasible allocations satisfy </li></ul><ul><ul><li>Material balance which requires that, for all i ,  x i,j =  e i,j ; </li></ul></ul><ul><ul><li>Other feasibility constraints. </li></ul></ul>
  15. 15. Solution Quality <ul><li>Pareto efficiency </li></ul><ul><ul><li>There is no other solution in which </li></ul></ul><ul><ul><ul><li>one agent is strictly better off, and </li></ul></ul></ul><ul><ul><ul><li>no agent is worse off. </li></ul></ul></ul><ul><li>Global efficiency (when utility is quasilinear) </li></ul><ul><ul><li>Corresponds to maximizing  j u j ( x j ); </li></ul></ul><ul><ul><li>Unique. </li></ul></ul>
  16. 16. Equilibrium <ul><li>General Definition </li></ul><ul><ul><li>A state from which no agent wishes to deviate. </li></ul></ul><ul><li>Equilibrium concepts make assumptions about </li></ul><ul><ul><li>Agent knowledge; </li></ul></ul><ul><ul><li>Agent behaviors. </li></ul></ul><ul><li>Equilibrium questions </li></ul><ul><ul><li>Do equilibria exist? </li></ul></ul><ul><ul><li>How many? </li></ul></ul><ul><ul><li>Do they support efficient solutions? </li></ul></ul>
  17. 17. Classic Agent Behavior <ul><li>Competitive assumption </li></ul><ul><ul><li>Agents solve optimization problem: </li></ul></ul><ul><ul><ul><li>Find a bundle that maximizes agent’s utility, x i * = argmax x u j ( x ); </li></ul></ul></ul><ul><ul><ul><li>Subject to agent’s budget,  p i e i,j ; </li></ul></ul></ul><ul><ul><ul><li>Assuming prices are given. </li></ul></ul></ul><ul><ul><li>Agents truthfully state their demand (supply) </li></ul></ul><ul><ul><ul><li>z i = x i * - e i . </li></ul></ul></ul>
  18. 18. General Equilibrium <ul><li>Definition: A price vector and allocation such that </li></ul><ul><ul><li>All agents are maximizing their utility with respect to the prices; </li></ul></ul><ul><ul><li>No resource is over demanded. </li></ul></ul><ul><li>Also called Competitive or Walrasian equilibrium. </li></ul>
  19. 19. General Equilibrium Existence <ul><li>A competitive equilibrium exists in an exchange economy if </li></ul><ul><ul><li>There is a positive endowment of every good; </li></ul></ul><ul><ul><li>Preferences are continuous, strongly convex, and strongly monotone. </li></ul></ul><ul><li>One sufficient condition for existence is gross substitutability </li></ul><ul><ul><li>Raising the price of one good will not decrease the demand of another. </li></ul></ul>
  20. 20. Production Economies <ul><li>We allow agents to transform resources from one type to another. </li></ul><ul><li>Competitive Equilibrium exist if </li></ul><ul><ul><li>Production technologies have convex or constant returns to scale. </li></ul></ul>
  21. 21. Fundamental Theorems <ul><li>First Welfare Theorem </li></ul><ul><ul><li>Any competitive equilibrium is Pareto efficient. </li></ul></ul><ul><li>Second Welfare Theorem </li></ul><ul><ul><li>If preferences and technologies are convex, any feasible Pareto solution is a Competitive equilibrium for some price vector and set of endowments. </li></ul></ul>
  22. 22. Limitations of G.E. Model <ul><li>When are the assumptions violated? </li></ul><ul><ul><li>When agents have market power </li></ul></ul><ul><ul><li>When prices are nonlinear </li></ul></ul><ul><ul><li>When agent preferences have </li></ul></ul><ul><ul><ul><li>Externalities; </li></ul></ul></ul><ul><ul><ul><li>nonconvexities (discreteness); </li></ul></ul></ul><ul><ul><ul><li>Complementarities. </li></ul></ul></ul>
  23. 23. G.E. Summary <ul><li>General Equilibrium Theory provides </li></ul><ul><ul><li>Some conditions under which competitive equilibria exist and are unique. </li></ul></ul><ul><ul><li>Justification for price systems. </li></ul></ul><ul><li>But... </li></ul><ul><ul><li>We have said nothing about how to reach equilibrium </li></ul></ul>
  24. 24. Tatonnement <ul><li>Tatonnement is the iterative price adjustment scheme proposed by Leon Walras (1874) </li></ul><ul><ul><li>Auctioneer announces prices; </li></ul></ul><ul><ul><li>Agents respond with demands; </li></ul></ul><ul><ul><li>Auctioneer adjusts price of most overdemanded resource. </li></ul></ul><ul><li>Convergence of tatonnement iterative price adjustment guaranteed if gross substitutability holds. </li></ul>
  25. 25. Mechanism Design <ul><li>General Definition </li></ul><ul><ul><li>An allocation mechanism is a set of rules that define </li></ul></ul><ul><ul><ul><li>Allowable agent actions; </li></ul></ul></ul><ul><ul><ul><li>Information that is revealed. </li></ul></ul></ul><ul><li>Examples </li></ul><ul><ul><li>Tattonement; </li></ul></ul><ul><ul><li>Auctions; </li></ul></ul><ul><ul><li>Fixed pricing. </li></ul></ul>
  26. 26. Protocols <ul><li>A protocol is a combination of a mechanism and assumptions on the agents’ behavior; </li></ul><ul><ul><li>Tatonnement & competitive assumption = Walrasian protocol. </li></ul></ul><ul><li>Protocols allows us to analyze systems when </li></ul><ul><ul><li>General Equilibrium conditions do not hold; </li></ul></ul><ul><ul><li>Competitive assumptions are violated; </li></ul></ul><ul><ul><li>Perfect rationality is intractable. </li></ul></ul>
  27. 27. Two Sides of the Same Coin <ul><li>Given assumptions about the agents, how do we design an allocation mechanism? </li></ul><ul><li>Given an allocation mechanism, how do we design an agent to participate in it? </li></ul>
  28. 28. Game Theory <ul><li>Game theory is a general tool for </li></ul><ul><ul><li>analyzing mechanisms </li></ul></ul><ul><ul><li>synthesizing strategies </li></ul></ul>
  29. 29. Summary <ul><li>The design of trading agents should be informed by economics. </li></ul><ul><li>General Equilibrium is the foundation of modern economic theory. </li></ul><ul><li>Competitive behavior is a simple form of competence. </li></ul><ul><li>But there is much more to the story… </li></ul>
  30. 30. A Game <ul><li>Players </li></ul><ul><li>Actions </li></ul><ul><li>Payoffs </li></ul><ul><li>Information </li></ul><ul><li>Finite game : has finite number of players and finite number of decision alternatives for each player. </li></ul><ul><ul><li>We will consider examples of two-person games. </li></ul></ul><ul><li>Zero-sum game : the sum of players’ payoffs equals zero. </li></ul><ul><li>Two-person-zero-sum games: one player’s loss is the other player’s gain. </li></ul>
  31. 31. Example <ul><li>Players: Red & Blue </li></ul><ul><li>Actions </li></ul><ul><ul><li>Red: join or pass </li></ul></ul><ul><ul><li>Blue: join or pass </li></ul></ul><ul><li>Payoffs </li></ul>Red’s payoffs Blue’s payoffs
  32. 32. Play the Game Red’s payoffs Blue’s payoffs
  33. 33. Normal (Strategic) Form Red’s payoffs Blue’s payoffs “ Prisoners’ Dilemma”
  34. 34. Pareto Efficiency <ul><li>Pareto Efficiency: </li></ul><ul><ul><li>There is no other solution in which </li></ul></ul><ul><ul><ul><li>An agent is strictly better off; </li></ul></ul></ul><ul><ul><ul><li>No agent is worse off. </li></ul></ul></ul>
  35. 35. Pareto Efficiency <ul><li>Pareto Efficiency: </li></ul><ul><ul><li>There is no other solution in which </li></ul></ul><ul><ul><ul><li>An agent is strictly better off; </li></ul></ul></ul><ul><ul><ul><li>No agent is worse off. </li></ul></ul></ul>
  36. 36. Dominant Strategy <ul><li>Dominant Strategy: </li></ul><ul><ul><li>A strategy for which the payoffs are better regardless of the other player’s choice. </li></ul></ul>
  37. 37. Dominant Strategy Equilibrium <ul><li>Dominant Strategy: </li></ul><ul><ul><li>A strategy for which the payoffs are better regardless of the other player’s choice; </li></ul></ul><ul><ul><ul><li>Red plays join; </li></ul></ul></ul><ul><ul><ul><li>Blue plays join. </li></ul></ul></ul>
  38. 38. Iterated Strict Dominance <ul><li>Repeatedly rule out strategies until only one remains </li></ul>
  39. 39. Iterated Strict Dominance <ul><li>Repeatedly rule out strategies until only one remains </li></ul>Dominates
  40. 40. Iterated Strict Dominance <ul><li>Repeatedly rule out strategies until only one remains </li></ul>Dominates
  41. 41. Iterated Strict Dominance <ul><li>Repeatedly rule out strategies until only one remains </li></ul>
  42. 42. Dominant Strategy Evaluation <ul><li>When they exist, they are conclusive (unique). </li></ul><ul><li>Often they don’t exist. </li></ul>
  43. 43. No Dominant Strategy equilibrium. <ul><li>A solution exists if the game is played repeatedly. </li></ul>“ Matching pennies” <ul><li>Dominant strategy equilibrium does not exist for pure strategies. </li></ul><ul><li>Zero-sum game. </li></ul>
  44. 44. Nash Equilibrium <ul><li>An outcome is a Nash equilibrium if each player’s strategy is an optimal response given the other players’ strategies. </li></ul>“ Battle of the Sexes”
  45. 45. Nash Equilibrium <ul><li>An outcome is a Nash equilibrium if each player’s strategy is an optimal response given the other players’ strategies. </li></ul><ul><ul><li>If red plays B , blue should play B . </li></ul></ul><ul><ul><li>If blue plays B , red should play B . </li></ul></ul>
  46. 46. Nash Equilibrium <ul><li>An outcome is a Nash equilibrium if each player’s strategy is an optimal response given the other players’ strategies. </li></ul><ul><ul><li>If red plays F , blue should play F . </li></ul></ul><ul><ul><li>If blue plays F , red should play F . </li></ul></ul>
  47. 47. Strategies <ul><li>Strategy space </li></ul><ul><ul><li>S i = { s i 1 , s i 2 ,… s i n } </li></ul></ul><ul><li>Pure strategy </li></ul><ul><ul><li>A single action, s i j </li></ul></ul><ul><li>Mixed strategy </li></ul><ul><ul><li>A probability distribution over pure strategies </li></ul></ul><ul><ul><li> i = {( p i 1 , s i 1 ), ( p i 2 , s i 2 ),…( p i n , s i n )} where  j p i j = 1 </li></ul></ul><ul><li>Von Neumann’s Discovery: every two-person zero-sum game has a maximin solution, in pure or mixed strategies. </li></ul>
  48. 48. Mixed-Strategy Equilibrium <ul><li>A mixed-strategy equilibrium </li></ul><ul><ul><li>Red plays {(1/3, F )(2/3, B )} </li></ul></ul><ul><ul><li>Blue plays {(2/3, F )(1/3, B )} </li></ul></ul><ul><ul><li>E(u red ) = 2/3, E(u blue ) = 2/3 </li></ul></ul><ul><li>No other combination of probabilities is a Nash equilibrium </li></ul>
  49. 49. Mixed Strategy equilibrium <ul><li>Every finite strategic-form game has a mixed-strategy equilibrium (Nash, 1950). </li></ul><ul><ul><li>No pure-strategy equilibrium. </li></ul></ul><ul><ul><li>Mixed-strategy equilibrium: </li></ul></ul><ul><ul><ul><li>Red plays {(1/2, H )(1/2, T )}; </li></ul></ul></ul><ul><ul><ul><li>Blue plays {(1/2, H )(1/2, T )}. </li></ul></ul></ul>“ Matching Pennies”
  50. 50. Assumptions So Far <ul><li>Complete information: </li></ul><ul><ul><li>Agents know each other’s strategy space and payoffs. </li></ul></ul><ul><li>Common knowledge: </li></ul><ul><ul><li>Moreover, each agent knows the other knows… </li></ul></ul><ul><li>No communication </li></ul><ul><li>Single round </li></ul>
  51. 51. Stage Games <ul><li>Games in which the players “take turns” </li></ul><ul><li>Actions are observable </li></ul><ul><li>Payoff received at the end of the game </li></ul>
  52. 52. Example: Matchsticks <ul><li>There are four matchsticks </li></ul><ul><li>You may take either one or two matchsticks on your turn </li></ul><ul><li>The last person to take a matchstick loses </li></ul>
  53. 53. Game Tree for 4-Matchsticks 4 3 2 2 1 1 1 1 2 2 2 2 1 1 1 1 , 0 0 , 1 0 , 1 0 , 1 1 , 0
  54. 54. Sub-game Analysis 4 3 2 2 1 1 1 1 2 2 2 2 1 1 1 1 , 0 0 , 1 0 , 1 0 , 1 1 , 0
  55. 55. Sub-game Analysis 4 3 2 2 1 1 1 1 2 2 2 2 1 1 1 1 , 0 0 , 1 0 , 1 0 , 1 1 , 0
  56. 56. Extensive Form <ul><li>Extensive form contains: </li></ul><ul><ul><li>The set of players </li></ul></ul><ul><ul><li>The order of moves </li></ul></ul><ul><ul><li>The choices at each decision point </li></ul></ul><ul><ul><li>The payoffs as a function of the moves made </li></ul></ul><ul><ul><li>The information each agent has at the decision point </li></ul></ul><ul><ul><li>The probabilities associated with exogenous events (chance) </li></ul></ul>
  57. 57. Information <ul><li>Information is imperfect when </li></ul><ul><ul><li>An agent can’t observe the other agents’ moves. </li></ul></ul><ul><ul><li>There are stochastic events that occur “in nature”. </li></ul></ul>
  58. 58. Hidden-Move Matchsticks 4 3 2 2 1 1 1 1 2 2 2 2 1 1 1 1 , 0 0 , 1 0 , 1 0 , 1 1 , 0 Information set
  59. 59. Flip-a-Coin Matchsticks 4 3 2 2 1 1 1 2 2 1 1 1 1 , 0 0 , 1 0 , 1 4 3 2 1 1 2 2 2 0 , 1 1 , 0 n
  60. 60. AI: Minimax search <ul><li>Developed in the context of zero-sum games </li></ul><ul><ul><li>Chess, matchsticks, etc. </li></ul></ul><ul><li>Equivalent to backward induction </li></ul><ul><li>Can be enhanced using an evaluation function to represent estimations of terminal node values </li></ul><ul><ul><li>Allows heuristics to guide search </li></ul></ul><ul><ul><li>Allows pruning of dominated nodes before expansion </li></ul></ul>
  61. 61. Advanced Topics in Game Theory <ul><li>Equilibrium Selection </li></ul><ul><ul><li>How do we choose among multiple Nash equilibria? </li></ul></ul><ul><ul><li>Are some inherently more likely to be chosen than others? </li></ul></ul>
  62. 62. Advanced Topics in Game Theory <ul><li>Repeated Games </li></ul><ul><ul><li>Reward is received after each round </li></ul></ul><ul><ul><li>Future rewards are discounted </li></ul></ul><ul><ul><li>Punishment is possible </li></ul></ul><ul><ul><li>Learning is possible </li></ul></ul>
  63. 63. Advanced Topics in Game Theory <ul><li>Learning in repeated games </li></ul><ul><ul><li>When an agent’s knowledge of other agent’s payoffs is incomplete </li></ul></ul><ul><ul><li>When an agent doesn’t know its own payoffs </li></ul></ul>
  64. 64. Game Theory Uses <ul><li>Models of </li></ul><ul><ul><li>Contract negotiation </li></ul></ul><ul><ul><li>Social choice </li></ul></ul><ul><ul><li>Business strategy </li></ul></ul><ul><ul><li>Auctions </li></ul></ul><ul><ul><li>Marriage </li></ul></ul><ul><ul><li>... </li></ul></ul>
  65. 65. Game Theory Conclusions <ul><li>Provides a precise description of multiagent interactions. </li></ul><ul><li>Useful solution concepts. </li></ul><ul><li>Extremely general. </li></ul><ul><li>Often inconclusive. </li></ul><ul><li>Often assumes much knowledge. </li></ul><ul><li>Extremely general. </li></ul>

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