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UMBC CMSC 691 - Economic Foundations and Game Theory

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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 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 Theory3Economics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.4Trading 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.5Economics of Trading AgentsWe will consider economics of trading agents as software entities.Elements of an Economic ModelResources;Agents;Market Infrastructure.6Resources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.7Two 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.8Consumer 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)9Consumer Preferences (2)Often, we assume convexityFor all  in [0,1], x ≥ x” and x’ ≥ x” and x ≠ x’ implies [x + (1-) x’] ≥ x”x1x2xx’x”10Preferences 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 money11Consumer EndowmentsConsumers 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 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 value14SolutionsAn allocation assigns quantities of each resource to each consumerFeasible allocations satisfyMaterial balance which requires that, for all i,  xi,j =  ei,j ;Other feasibility constraints.15Solution 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 j uj(xj);Unique.16Equilibrium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?17Classic Agent BehaviorCompetitive assumptionAgents 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 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.19General 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.20Production 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.21Fundamental 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.22Limitations 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.23G.E. SummaryGeneral Equilibrium Theory providesSome conditions under which competitive equilibria exist and are unique.Justification for price systems.But...We


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UMBC CMSC 691 - Economic Foundations and Game Theory

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