Economic Foundations and Game TheoryPresentation OverviewEconomicsTrading AgentsEconomics of Trading AgentsResourcesTwo Types of AgentsConsumer PreferencesConsumer Preferences (2)Preferences Expressed as UtilityConsumer EndowmentsSimple Exchange EconomyPrice SystemsSolutionsSolution QualityEquilibriumClassic Agent BehaviorGeneral EquilibriumGeneral Equilibrium ExistenceProduction EconomiesFundamental TheoremsLimitations of G.E. ModelG.E. SummaryTatonnementMechanism DesignProtocolsTwo Sides of the Same CoinGame TheorySummaryA GameExamplePlay the GameNormal (Strategic) FormPareto EfficiencySlide 35Dominant StrategyDominant Strategy EquilibriumIterated Strict DominanceSlide 39Slide 40Slide 41Dominant Strategy EvaluationNo Dominant Strategy equilibrium.Nash EquilibriumSlide 45Slide 46StrategiesMixed-Strategy EquilibriumMixed Strategy equilibriumAssumptions So FarStage GamesExample: MatchsticksGame Tree for 4-MatchsticksSub-game AnalysisSlide 55Extensive FormInformationHidden-Move MatchsticksFlip-a-Coin MatchsticksAI: Minimax searchAdvanced Topics in Game TheorySlide 62Slide 63Game Theory UsesGame Theory Conclusions1Economic Foundations and Game TheoryPeter Wurman2Presentation Overview EconomicsEconomics of Trading AgentsEconomic modelingGeneral Equilibrium and its LimitationsMechanism designIntroduction to Game TheoryPareto Efficiency and Dominant strategyNash EquilibriumMixed StrategiesExtensive Form and Sub-game AnalysisAdvanced Topics in Game Theory3EconomicsStudy 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.4Trading AgentsAgent: software to which we ascribeBeliefs and knowledge;Rationality;Competence;Autonomy.Trading agent: software that participates in an electronic market andIs 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.5Economics of Trading AgentsWe will consider economics of trading agents as software entities.Elements of an Economic ModelResources;Agents;Market Infrastructure.6ResourcesResourcesLimited;Consumed (private) or shared (public).FormalizationN is the number of resources types;xi is an amount of resource i;x is a N-vector of quantities.7Two Types of AgentsConsumersDerive value from owning/consuming resources.ProducersHave technologies to transform resources;Goal is to make money (distributed to shareholders).Both have private information.8Consumer PreferencesPreferences (>, ≥)Total preorder over all bundles x in X x ≥ x’ or x’ ≥ x (completeness) x ≥ x’ and x’ ≥ x” implies x ≥ x” (transitivity)9Consumer Preferences (2)Often, we assume convexityFor all in [0,1], x ≥ x” and x’ ≥ x” and x ≠ x’ implies [x + (1-) x’] ≥ x”x1x2xx’x”10Preferences Expressed as UtilityGenerally, we express preferences as a utility function:uj(x) assigns a numeric value to all bundlesOften, we assume that utility is quasi-linear in one resource:uj(x) = vj(x) + m,where m is money11Consumer EndowmentsConsumers generally begin with some resources, denoted ej.Often, these endowments do not maximize the agent’s utility.Agents engage in economic activities.12Simple Exchange Economy• Suppose all participants are consumers• How do we determine resources to exchange?• What is a “good” allocation?Agent 1 Agent 2Agent 3Agent 1 Agent 2Agent 313Price SystemsAssociate a price pi with each resource iPrices 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 value14SolutionsAn allocation assigns quantities of each resource to each consumerFeasible allocations satisfyMaterial balance which requires that, for all i, xi,j = ei,j ;Other feasibility constraints.15Solution QualityPareto efficiencyThere is no other solution in whichone agent is strictly better off, andno agent is worse off.Global efficiency (when utility is quasilinear)Corresponds to maximizing j uj(xj);Unique.16EquilibriumGeneral DefinitionA state from which no agent wishes to deviate.Equilibrium concepts make assumptions aboutAgent knowledge;Agent behaviors.Equilibrium questionsDo equilibria exist? How many?Do they support efficient solutions?17Classic Agent BehaviorCompetitive assumptionAgents solve optimization problem:Find a bundle that maximizes agent’s utility,xi* = argmaxx uj(x);Subject to agent’s budget, piei,j ;Assuming prices are given.Agents truthfully state their demand (supply) zi = xi* - ei .18General EquilibriumDefinition: A price vector and allocation such thatAll agents are maximizing their utility with respect to the prices;No resource is over demanded.Also called Competitive or Walrasian equilibrium.19General Equilibrium ExistenceA competitive equilibrium exists in an exchange economy ifThere is a positive endowment of every good;Preferences are continuous, strongly convex, and strongly monotone.One sufficient condition for existence is gross substitutabilityRaising the price of one good will not decrease the demand of another.20Production EconomiesWe allow agents to transform resources from one type to another.Competitive Equilibrium exist ifProduction technologies have convex or constant returns to scale.21Fundamental TheoremsFirst Welfare TheoremAny competitive equilibrium is Pareto efficient.Second Welfare TheoremIf preferences and technologies are convex, any feasible Pareto solution is a Competitive equilibrium for some price vector and set of endowments.22Limitations of G.E. ModelWhen are the assumptions violated?When agents have market powerWhen prices are nonlinearWhen agent preferences have Externalities;nonconvexities (discreteness);Complementarities.23G.E. SummaryGeneral Equilibrium Theory providesSome conditions under which competitive equilibria exist and are unique.Justification for price systems.But...We
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