Economic Foundations and Game Theory

1. Economic Foundations and Game Theory Peter Wurman

2. Presentation Overview • Economics • Economics of Trading Agents • Economic modeling • General Equilibrium and its Limitations • Mechanism design • Introduction to Game Theory • Pareto Efficiency and Dominant strategy • Nash Equilibrium • Mixed Strategies • Extensive Form and Sub-game Analysis • Advanced Topics in Game Theory

3. Economics • Study of the allocation of limited resources in a society of self-interested agents. • Essential features: • Agents are rational; • Decisions concern the use of resources; • Prices significantly simplify the allocation process. • Note: agents are not assumed to be software entities here.

4. Trading Agents • Agent: software to which we ascribe • Beliefs and knowledge; • Rationality; • Competence; • Autonomy. • Trading agent: software that participates in an electronic market and • Is governed in its decision-making by a set of constraints (budget) and preferences; • Obtains the above from a user; • Acts in the world by making offers (bids) on the user’s behalf.

5. Economics of Trading Agents • We will consider economics of trading agents as software entities. • Elements of an Economic Model • Resources; • Agents; • Market Infrastructure.

6. Resources • Resources • Limited; • Consumed (private) or shared (public). • Formalization • N is the number of resources types; • xi is an amount of resource i; • x is a N-vector of quantities.

7. Two Types of Agents • Consumers • Derive value from owning/consuming resources. • Producers • Have technologies to transform resources; • Goal is to make money (distributed to shareholders). • Both have private information.

8. Consumer Preferences • Preferences (>,≥) • Total preorder over all bundles x in X • x≥ x’ or x’≥x (completeness) • x≥x’ and x’≥x”implies x≥x”(transitivity)

9. Consumer Preferences (2) • Often, we assume convexity • For all a in [0,1], x≥x” and x’≥x”andx≠x’ implies [ax+ (1-a)x’] ≥x” x1 x x” x’ x2

10. Preferences Expressed as Utility • Generally, we express preferences as a utility function: • uj(x) assigns a numeric value to all bundles • Often, we assume that utility is quasi-linear in one resource: • uj(x) = vj(x) + m,where m is money

11. Consumer Endowments • Consumers generally begin with some resources, denoted ej. • Often, these endowments do not maximize the agent’s utility. • Agents engage in economic activities.

12. Simple Exchange Economy • Suppose all participants are consumers Agent 1 Agent 2 Agent 1 Agent 2 Agent 3 Agent 3 • How do we determine resources to exchange? • What is a “good” allocation?

13. Price Systems • Associate a price pi with each resource i • Prices specify resource exchange rates: • One unit of i can be exchanged for pi/ph units of h. • Present a common scale on which to measure resource value. • Very compact representation of value

14. Solutions • An allocation assigns quantities of each resource to each consumer • Feasible allocations satisfy • Material balance which requires that, for all i, Sxi,j = Sei,j ; • Other feasibility constraints.

15. Solution Quality • Pareto efficiency • There is no other solution in which • one agent is strictly better off, and • no agent is worse off. • Global efficiency (when utility is quasilinear) • Corresponds to maximizing Sjuj(xj); • Unique.

16. Equilibrium • General Definition • A state from which no agent wishes to deviate. • Equilibrium concepts make assumptions about • Agent knowledge; • Agent behaviors. • Equilibrium questions • Do equilibria exist? • How many? • Do they support efficient solutions?

17. Classic Agent Behavior • Competitive assumption • Agents solve optimization problem: • Find a bundle that maximizes agent’s utility,xi* = argmaxxuj(x); • Subject to agent’s budget, Spiei,j ; • Assuming prices are given. • Agents truthfully state their demand (supply) • zi = xi* - ei .

18. General Equilibrium • Definition: A price vector and allocation such that • All agents are maximizing their utility with respect to the prices; • No resource is over demanded. • Also called Competitive or Walrasian equilibrium.

19. General Equilibrium Existence • A competitive equilibrium exists in an exchange economy if • There is a positive endowment of every good; • Preferences are continuous, strongly convex, and strongly monotone. • One sufficient condition for existence is gross substitutability • Raising the price of one good will not decrease the demand of another.

20. Production Economies • We allow agents to transform resources from one type to another. • Competitive Equilibrium exist if • Production technologies have convex or constant returns to scale.

21. Fundamental Theorems • First Welfare Theorem • Any competitive equilibrium is Pareto efficient. • Second Welfare Theorem • If preferences and technologies are convex, any feasible Pareto solution is a Competitive equilibrium for some price vector and set of endowments.

22. Limitations of G.E. Model • When are the assumptions violated? • When agents have market power • When prices are nonlinear • When agent preferences have • Externalities; • nonconvexities (discreteness); • Complementarities.

23. G.E. Summary • General Equilibrium Theory provides • Some conditions under which competitive equilibria exist and are unique. • Justification for price systems. • But... • We have said nothing about how to reach equilibrium

24. Tatonnement • Tatonnement is the iterative price adjustment scheme proposed by Leon Walras (1874) • Auctioneer announces prices; • Agents respond with demands; • Auctioneer adjusts price of most overdemanded resource. • Convergence of tatonnement iterative price adjustment guaranteed if gross substitutability holds.

25. Mechanism Design • General Definition • An allocation mechanism is a set of rules that define • Allowable agent actions; • Information that is revealed. • Examples • Tattonement; • Auctions; • Fixed pricing.

26. Protocols • A protocol is a combination of a mechanism and assumptions on the agents’ behavior; • Tatonnement & competitive assumption = Walrasian protocol. • Protocols allows us to analyze systems when • General Equilibrium conditions do not hold; • Competitive assumptions are violated; • Perfect rationality is intractable.

27. Two Sides of the Same Coin • Given assumptions about the agents, how do we design an allocation mechanism? • Given an allocation mechanism, how do we design an agent to participate in it?

28. Game Theory • Game theory is a general tool for • analyzing mechanisms • synthesizing strategies

29. Summary • The design of trading agents should be informed by economics. • General Equilibrium is the foundation of modern economic theory. • Competitive behavior is a simple form of competence. • But there is much more to the story…

30. A Game • Players • Actions • Payoffs • Information • Finite game: has finite number of players and finite number of decision alternatives for each player. • We will consider examples of two-person games. • Zero-sum game: the sum of players’ payoffs equals zero. • Two-person-zero-sum games: one player’s loss is the other player’s gain.

31. Example • Players: Red & Blue • Actions • Red: join or pass • Blue: join or pass • Payoffs Red’s payoffs Blue’s payoffs

32. Play the Game Red’s payoffs Blue’s payoffs

33. Normal (Strategic) Form Red’s payoffs Blue’s payoffs “Prisoners’ Dilemma”

34. Pareto Efficiency • Pareto Efficiency: • There is no other solution in which • An agent is strictly better off; • No agent is worse off.

35. Pareto Efficiency • Pareto Efficiency: • There is no other solution in which • An agent is strictly better off; • No agent is worse off.

36. Dominant Strategy • Dominant Strategy: • A strategy for which the payoffs are better regardless of the other player’s choice.

37. Dominant Strategy Equilibrium • Dominant Strategy: • A strategy for which the payoffs are better regardless of the other player’s choice; • Red plays join; • Blue plays join.

38. Iterated Strict Dominance • Repeatedly rule out strategies until only one remains

39. Iterated Strict Dominance • Repeatedly rule out strategies until only one remains Dominates

40. Iterated Strict Dominance • Repeatedly rule out strategies until only one remains Dominates

41. Iterated Strict Dominance • Repeatedly rule out strategies until only one remains

42. Dominant Strategy Evaluation • When they exist, they are conclusive (unique). • Often they don’t exist.

43. No Dominant Strategy equilibrium. • Dominant strategy equilibrium does not exist for pure strategies. • Zero-sum game. • A solution exists if the game is played repeatedly. “Matching pennies”

44. Nash Equilibrium • An outcome is a Nash equilibrium if each player’s strategy is an optimal response given the other players’ strategies. “Battle of the Sexes”

45. Nash Equilibrium • An outcome is a Nash equilibrium if each player’s strategy is an optimal response given the other players’ strategies. • If red plays B, blue should play B. • If blue plays B, red should play B.

46. Nash Equilibrium • An outcome is a Nash equilibrium if each player’s strategy is an optimal response given the other players’ strategies. • If red plays F, blue should play F. • If blue plays F, red should play F.

47. Strategies • Strategy space • Si = {si1, si2,…sin} • Pure strategy • A single action, sij • Mixed strategy • A probability distribution over pure strategies si = {(pi1, si1), (pi2, si2),…(pin, sin)} where Sjpij = 1 • Von Neumann’s Discovery: every two-person zero-sum game has a maximin solution, in pure or mixed strategies.

48. Mixed-Strategy Equilibrium • A mixed-strategy equilibrium • Red plays {(1/3, F)(2/3, B)} • Blue plays {(2/3,F)(1/3, B)} • E(ured) = 2/3, E(ublue) = 2/3 • No other combination of probabilities is a Nash equilibrium

49. Mixed Strategy equilibrium • Every finite strategic-form game has a mixed-strategy equilibrium (Nash, 1950). • No pure-strategy equilibrium. • Mixed-strategy equilibrium: • Red plays {(1/2, H)(1/2, T)}; • Blue plays {(1/2,H)(1/2, T)}. “Matching Pennies”

50. Assumptions So Far • Complete information: • Agents know each other’s strategy space and payoffs. • Common knowledge: • Moreover, each agent knows the other knows… • No communication • Single round